Finite Group Databasebeta

← #11660523

Subgroups — #11660523

Order 768.

1806 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85680C211yesmaximal, characteristic
2C3×#2920C211yesmaximal, characteristic, complemented
3C2×#4925C211yesmaximal, characteristic, complemented
4#79341C214yesmaximal, complemented
5#79338C212yesmaximal, complemented
6#79316C212yesmaximal, complemented
7#79283C214yesmaximal, complemented
8#79277C212yesmaximal, complemented
9#79276C212yesmaximal, complemented
10#79053C214yesmaximal, complemented
11#78635C214yesmaximal, complemented
12#78523C214yesmaximal, complemented
13#60279—31nomaximal, Sylow
14C22×C3×C42C2211yescharacteristic
15#5791C412yes—
16#5791C412yes—
17#5791C2212yes—
18#5479C412yes—
19#5479C412yes—
20#5479C412yes—
21#5479C412yes—
22#5479C2212yes—
23#5479C2212yes—
24#5477C411yescharacteristic
25#5477C411yescharacteristic
26#5477C2211yescharacteristic
27C4×#235C2212yes—
28C4×#235C2212yes—
29C4×#235C2212yes—
30#5470C411yescharacteristic
31#5470C411yescharacteristic
32#5470C2211yescharacteristic
33C3×#432C2212yes—
34C3×#432C2212yes—
35C3×#431C2212yes—
36C3×#431C2212yes—
37C3×#423C2212yes—
38C3×#407C2212yes—
39C3×C4×M16C2212yes—
40C2×#830C2212yes—
41#5128C2218yes—
42#5127C2218yes—
43#4925C2212yes—
44#4925C2212yes—
45#4925C2212yes—
46#4925C2212yes—
47C2×C4⋊(C3⋊C8)C2212yes—
48C2×C4×C3⋊C8C2211yescharacteristic
49C2×C4×C3⋊C8C2211yescharacteristic
50#4840C2218yescomplemented
51#4839C2218yescomplemented
52#4838C2214yescomplemented
53#4834C2214yescomplemented
54#4831C2214yescomplemented
55#4830C2214yescomplemented
56#4730C2218yes—
57#4727C2218yes—
58#67⋊C8C2218yes—
59#4720C2218yes—
60#4706C2218yescomplemented
61#4701C2218yescomplemented
62C4×ΓL(1,𝔽25)C2214yescomplemented
63#4685C2214yescomplemented
64#4685C2214yescomplemented
65C4×D3×C8C2214yescomplemented
66#3444—31no—
67#2993—34no—
68#2992—34no—
69#2985—32no—
70#2981—32no—
71#2976—34no—
72#2965—32no—
73#2965—34no—
74C8×#41—32no—
75#2922—34no—
76#2920D311yescharacteristic, complemented
77C2×#431—31no—
78C23×C3×C4C2×C411yescharacteristic
79C23×C3×C4C2×C411yescharacteristic
80C23×C3×C4C2311yescharacteristic
81C2×#235C2×C412yes—
82C2×#235C2×C412yes—
83C2×#235C2312yes—
84C2×C3×M16C2314yes—
85C2×C3×C42C2×C411yescharacteristic
86C2×C3×C42C2×C411yescharacteristic
87C2×C3×C42C2×C411yescharacteristic
88C2×C3×C42C2×C411yescharacteristic
89C2×C3×C42C2×C411yescharacteristic
90C2×C3×C42C2×C411yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C2×C3×C42C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97#953C2×C414yes—
98#930C2×C412yes—
99#930C2×C412yes—
100#930C2×C412yes—
101#930C2×C412yes—
102#930C2×C412yes—
103#930C2×C412yes—
104#929C2×C414yes—
105#929C2×C414yes—
106#929C2×C414yes—
107C4×#67C2×C414yes—
108C4×#67C2×C414yes—
109C4×#67C2×C414yes—
110C4×#67C2×C414yes—
111C4×#67C2314yes—
112C4×#67C2314yes—
113#926C2×C412yes—
114#926C2×C412yes—
115#926C2312yes—
116#924C2×C412yes—
117#924C2×C412yes—
118#924C2×C412yes—
119#924C2×C412yes—
120#924C2×C412yes—
121#924C2×C412yes—
122C22×C3⋊C8C2312yes—
123C22×C3⋊C8C2312yes—
124C2×ΓL(1,𝔽25)C2318yes—
125C2×D3×C8C2318yes—
126#876C2×C414yes—
127#876C2×C414yes—
128#875C2×C412yes—
129#875C2×C412yes—
130#875C2×C412yes—
131#874C2×C411yescharacteristic
132C4×D12C2×C412yes—
133C4×D12C2×C412yes—
134C4×D12C2×C412yes—
135C4×D12C2×C412yes—
136C4×D12C2312yes—
137C4×D12C2312yes—
138#872C2×C414yes—
139#872C2×C414yes—
140#872C2314yes—
141C42×D3C2314yes—
142#870C2×C412yes—
143#870C2×C412yes—
144#870C2×C412yes—
145#869C2×C411yescharacteristic
146#869C2×C411yescharacteristic
147#869C2×C411yescharacteristic
148C4×Dic6C2×C412yes—
149C4×Dic6C2×C412yes—
150C4×Dic6C2×C412yes—
151C4×Dic6C2×C412yes—
152C4×Dic6C2312yes—
153C4×Dic6C2312yes—
154C3×C4⋊C8C2314yes—
155C3×C4⋊C8C2314yes—
156C3×C22⋊C8C2314yes—
157C3×C22⋊C8C2314yes—
158C3×#97C2314yes—
159C3×C4×C8C2314yes—
160#830C2314yes—
161#830C2314yes—
162(C2×D3)⋊C8C2318yes—
163(C2×D3)⋊C8C2318yes—
164#815C2318yes—
165Dic3⋊C8C2318yes—
166Dic3⋊C8C2318yes—
167C8×Dic3C2318yes—
168C4⋊(C3⋊C8)C2314yes—
169C4⋊(C3⋊C8)C2314yes—
170C4×C3⋊C8C2312yes—
171C4×C3⋊C8C2312yes—
172C4×C3⋊C8C2312yes—
173C4×C3⋊C8C2312yes—
174#533—31no—
175#533—31no—
176#533—31no—
177#532—31no—
178#532—31no—
179#532—31no—
180#529—32no—
181#529—32no—
182#529—32no—
183#529—32no—
184#529—32no—
185#529—32no—
186#525—32no—
187#525—32no—
188#525—32no—
189C4×#41—32no—
190C4×#41—32no—
191C4×#41—32no—
192C22×C42C2×D311yescharacteristic
193#446—34no—
194#446—38no—
195C8×Q8—34no—
196#436—38no—
197#436—34no—
198#435—38no—
199#435—38no—
200#435—38no—
201C8×D4—34no—
202C8×D4—38no—
203#433—38no—
204#433—38no—
205#432C2×D312yes—
206#432C2×D312yes—
207#431—32no—
208#431—32no—
209#431—32no—
210#431—32no—
211#431—34no—
212#431—34no—
213#431C2×D312yes—
214#431C2×D312yes—
215#424—38no—
216#423C2×D312yes—
217C2×C4⋊C8—32no—
218#408—38no—
219#407C2×D312yes—
220C2×C22⋊C8—32no—
221C4×M16—34no—
222C4×M16C2×D312yes—
223C2×C4×C8—31no—
224C2×C4×C8—31no—
225C2×C4×C8—34no—
226C24×C3C22×C411yescharacteristic
227C22×C3×C4—24no—
228C22×C3×C4—42no—
229C22×C3×C4—42no—
230C22×C3×C4C22×C411yescharacteristic
231C22×C3×C4C22×C411yescharacteristic
232C22×C3×C4C22×C411yescharacteristic
233C22×C3×C4C22×C411yescharacteristic
234C22×C3×C4C22×C411yescharacteristic
235C22×C3×C4C22×C411yescharacteristic
236C22×C3×C4C22×C411yescharacteristic
237C22×C3×C4C22×C411yescharacteristic
238C22×C3×C4C22×C411yescharacteristic
239C22×C3×C4C22×C411yescharacteristic
240C22×C3×C4C22×C411yescharacteristic
241C22×C3×C4C22×C411yescharacteristic
242C22×C3×C4C22×C411yescharacteristic
243C22×C3×C4C22×C411yescharacteristic
244C22×C3×C4C22×C411yescharacteristic
245C22×C3×C4C2411yescharacteristic
246C22×C3×C4C2411yescharacteristic
247C22×C3×C4C2411yescharacteristic
248C2×#67C22×C414yes—
249C2×#67C22×C414yes—
250C2×#67C22×C414yes—
251#235—28no—
252#235—44no—
253#235—44no—
254C2×D12C22×C412yes—
255C2×D12C22×C412yes—
256C2×D12C22×C412yes—
257C2×C4×D3C22×C414yes—
258C2×C4×D3C22×C414yes—
259C2×C4×D3C2414yes—
260C2×Dic6C22×C412yes—
261C2×Dic6C22×C412yes—
262C2×Dic6C22×C412yes—
263C3×M16—44no—
264C2×C3×C8C2418yes—
265C3×C42C22×C411yescharacteristic
266C3×C42C22×C411yescharacteristic
267C3×C42C22×C411yescharacteristic
268C3×C42C22×C411yescharacteristic
269C3×C42C22×C411yescharacteristic
270C3×C42C22×C411yescharacteristic
271C3×C42C22×C411yescharacteristic
272C3×C42C22×C411yescharacteristic
273C3×C42C22×C411yescharacteristic
274C3×C42C22×C411yescharacteristic
275C3×C42C22×C411yescharacteristic
276C3×C42C22×C411yescharacteristic
277C3×C42C2411yescharacteristic
278C3×C42C2411yescharacteristic
279C3×C42C2411yescharacteristic
280C3×C42C2411yescharacteristic
281#217C22×C414yes—
282#217C22×C414yes—
283#217C22×C414yes—
284(C2×D3)⋊C4C22×C414yes—
285(C2×D3)⋊C4C22×C414yes—
286(C2×D3)⋊C4C22×C414yes—
287(C2×D3)⋊C4C22×C414yes—
288(C2×D3)⋊C4C22×C414yes—
289(C2×D3)⋊C4C22×C414yes—
290C4⋊Dic3C22×C412yes—
291C4⋊Dic3C22×C412yes—
292C4⋊Dic3C22×C412yes—
293C4⋊Dic3C22×C412yes—
294C4⋊Dic3C22×C412yes—
295C4⋊Dic3C22×C412yes—
296Dic3⋊C4C22×C414yes—
297Dic3⋊C4C22×C414yes—
298Dic3⋊C4C22×C414yes—
299Dic3⋊C4C22×C414yes—
300Dic3⋊C4C22×C414yes—
301Dic3⋊C4C22×C414yes—
302C4×Dic3C22×C414yes—
303C4×Dic3C22×C414yes—
304C4×Dic3C2414yes—
305C2×C3⋊C8—42no—
306C2×C3⋊C8—42no—
307C2×C3⋊C8C2414yes—
308C2×C3⋊C8C2414yes—
309ΓL(1,𝔽25)—48no—
310D3×C8—48no—
311C2×#41—32no—
312C2×#41—32no—
313C2×#41—32no—
314C23×C4C4×D311yescharacteristic
315C23×C4C4×D311yescharacteristic
316C23×C4C22×D311yescharacteristic
317C2×M16—38no—
318C2×M16C22×D314yes—
319C22×C8—32no—
320C22×C8—32no—
321C22×C8—38no—
322C4⋊Q8—31no—
323C4⋊Q8—31no—
324C4⋊Q8—31no—
325#127—31no—
326#126—34no—
327#126—34no—
328#125—32no—
329#125—32no—
330#125—32no—
331#124—32no—
332#124—32no—
333#124—32no—
334#123—34no—
335#123—34no—
336#123—34no—
337C22⋊Q8—32no—
338C22⋊Q8—32no—
339C22⋊Q8—32no—
340C22⋊Q8—32no—
341C22⋊Q8—32no—
342C22⋊Q8—32no—
343#121—32no—
344#121—32no—
345#121—32no—
346#121—32no—
347#121—32no—
348#121—32no—
349#120—34no—
350C4×Q8—32no—
351C4×Q8—32no—
352C4×Q8—32no—
353C4×Q8—32no—
354C4×Q8—32no—
355C4×Q8—32no—
356C4×D4—34no—
357C4×D4—34no—
358C4×D4—34no—
359C4×D4—34no—
360C4×D4—34no—
361C4×D4—34no—
362C4×D4—32no—
363C4×D4—32no—
364C4×D4—32no—
365C4×D4—32no—
366C4×D4—32no—
367C4×D4—32no—
368#117—32no—
369#117—32no—
370#117—32no—
371#117—34no—
372#117—34no—
373#117—34no—
374C2×C42—34no—
375C2×C42C4×D311yescharacteristic
376C2×C42C4×D311yescharacteristic
377C2×C42C4×D311yescharacteristic
378C2×C42C4×D311yescharacteristic
379C2×C42C4×D311yescharacteristic
380C2×C42C4×D311yescharacteristic
381C2×C42C22×D311yescharacteristic
382C2×C42C22×D311yescharacteristic
383C2×C42C22×D311yescharacteristic
384C2×C42C22×D311yescharacteristic
385C2×C42C22×D311yescharacteristic
386C2×C42C22×D311yescharacteristic
387C4⋊C8—38no—
388C4⋊C8—38no—
389C4⋊C8—34no—
390C4⋊C8—34no—
391C4⋊C8C22×D314yes—
392C4⋊C8C22×D314yes—
393C22⋊C8—34no—
394C22⋊C8—34no—
395C22⋊C8—38no—
396C22⋊C8—38no—
397C22⋊C8C22×D314yes—
398C22⋊C8C22×D314yes—
399#97—38no—
400#97C22×D314yes—
401C4×C8—38no—
402C4×C8—32no—
403C4×C8—32no—
404C4×C8—32no—
405C4×C8—32no—
406C4×C8C22×D314yes—
407C23×C3—22no—
408C23×C3—41no—
409C23×C3—41no—
410C23×C3C23×C411yescharacteristic
411C23×C3C23×C411yescharacteristic
412C23×C3C23×C411yescharacteristic
413C22×D3C23×C414yes—
414C2×C3×C4—22no—
415C2×C3×C4—22no—
416C2×C3×C4—22no—
417C2×C3×C4—22no—
418C2×C3×C4—22no—
419C2×C3×C4—22no—
420C2×C3×C4—22no—
421C2×C3×C4—22no—
422C2×C3×C4—22no—
423C2×C3×C4—22no—
424C2×C3×C4—22no—
425C2×C3×C4—41no—
426C2×C3×C4—41no—
427C2×C3×C4—41no—
428C2×C3×C4—41no—
429C2×C3×C4—41no—
430C2×C3×C4—41no—
431C2×C3×C4#13114yes—
432C2×C3×C4C23×C411yescharacteristic
433C2×C3×C4C23×C411yescharacteristic
434C2×C3×C4C23×C411yescharacteristic
435C2×C3×C4C23×C411yescharacteristic
436C2×C3×C4C23×C411yescharacteristic
437C2×C3×C4C23×C411yescharacteristic
438C2×C3×C4C23×C411yescharacteristic
439C2×C3×C4C23×C411yescharacteristic
440C2×C3×C4C23×C411yescharacteristic
441C2×C3×C4C23×C411yescharacteristic
442C2×C3×C4C23×C411yescharacteristic
443C2×C3×C4C2511yescharacteristic
444#67—28no—
445#67—44no—
446#67—44no—
447C2×Dic3C23×C414yes—
448C2×Dic3C23×C414yes—
449C2×Dic3C23×C414yes—
450D12—24no—
451D12—42no—
452D12—42no—
453C4×D3—28no—
454C4×D3—28no—
455C4×D3—28no—
456Dic6—24no—
457Dic6—42no—
458Dic6—42no—
459C3×C8—28no—
460C3⋊C8—24no—
461C3⋊C8—24no—
462C24C2×C4×D311yescharacteristic
463#41—68no—
464#41—124no—
465#41—124no—
466C2×Q8—32no—
467C2×Q8—32no—
468C2×Q8—32no—
469C2×D4—34no—
470C2×D4—34no—
471C2×D4—34no—
472C2×D4—32no—
473C2×D4—32no—
474C2×D4—32no—
475C22×C4—24no—
476C22×C4—34no—
477C22×C4—34no—
478C22×C4—34no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4C2×C4×D311yescharacteristic
482C22×C4C2×C4×D311yescharacteristic
483C22×C4C2×C4×D311yescharacteristic
484C22×C4C2×C4×D311yescharacteristic
485C22×C4C2×C4×D311yescharacteristic
486C22×C4C2×C4×D311yescharacteristic
487C22×C4C2×C4×D311yescharacteristic
488C22×C4C2×C4×D311yescharacteristic
489C22×C4C2×C4×D311yescharacteristic
490C22×C4C2×C4×D311yescharacteristic
491C22×C4C2×C4×D311yescharacteristic
492C22×C4C2×C4×D311yescharacteristic
493C22×C4C2×C4×D311yescharacteristic
494C22×C4C2×C4×D311yescharacteristic
495C22×C4C2×C4×D311yescharacteristic
496C22×C4C23×D311yescharacteristic
497C22×C4C23×D311yescharacteristic
498C22×C4C23×D311yescharacteristic
499M16—44no—
500M16—128no—
501C2×C8—34no—
502C2×C8—34no—
503C2×C8—128no—
504C2×C8—122no—
505C2×C8—122no—
506C2×C8C23×D318yes—
507C4⋊C4—32no—
508C4⋊C4—32no—
509C4⋊C4—32no—
510C4⋊C4—32no—
511C4⋊C4—32no—
512C4⋊C4—32no—
513C4⋊C4—34no—
514C4⋊C4—34no—
515C4⋊C4—34no—
516C4⋊C4—34no—
517C4⋊C4—34no—
518C4⋊C4—34no—
519C22⋊C4—34no—
520C22⋊C4—34no—
521C22⋊C4—34no—
522C22⋊C4—34no—
523C22⋊C4—34no—
524C22⋊C4—34no—
525C22⋊C4—34no—
526C22⋊C4—34no—
527C22⋊C4—34no—
528C42—34no—
529C42—34no—
530C42—34no—
531C42C2×C4×D311yescharacteristic
532C42C2×C4×D311yescharacteristic
533C42C2×C4×D311yescharacteristic
534C42C2×C4×D311yescharacteristic
535C42C2×C4×D311yescharacteristic
536C42C2×C4×D311yescharacteristic
537C42C2×C4×D311yescharacteristic
538C42C2×C4×D311yescharacteristic
539C42C2×C4×D311yescharacteristic
540C42C2×C4×D311yescharacteristic
541C42C2×C4×D311yescharacteristic
542C42C2×C4×D311yescharacteristic
543C42C23×D311yescharacteristic
544C42C23×D311yescharacteristic
545C42C23×D311yescharacteristic
546C42C23×D311yescharacteristic
547C22×C3—21no—
548C22×C3—21no—
549C22×C3—21no—
550C22×C3—21no—
551C22×C3—21no—
552C22×C3—21no—
553C22×C3—21no—
554C22×C3—21no—
555C22×C3—44no—
556C22×C3C2×#13112yes—
557C22×C3C24×C411yescharacteristic
558C2×D3—24no—
559C2×D3—24no—
560C2×D3—24no—
561C3×C4—21no—
562C3×C4—21no—
563C3×C4—21no—
564C3×C4—21no—
565C3×C4—21no—
566C3×C4—21no—
567C3×C4—21no—
568C3×C4—21no—
569C3×C4C2×#13112yes—
570C3×C4C2×#13112yes—
571C3×C4#56812yes—
572C3×C4#58512yes—
573Dic3—24no—
574Dic3—24no—
575Dic3—24no—
576C23—22no—
577C23—34no—
578C23—41no—
579C23—41no—
580C23C22×C4×D311yescharacteristic
581C23C22×C4×D311yescharacteristic
582C23C22×C4×D311yescharacteristic
583Q8—64no—
584Q8—122no—
585Q8—122no—
586D4—68no—
587D4—64no—
588D4—122no—
589D4—122no—
590D4—124no—
591D4—124no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—34no—
604C2×C4—34no—
605C2×C4—34no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—68no—
613C2×C4—68no—
614C2×C4—68no—
615C2×C4#90714yes—
616C2×C4C22×C4×D311yescharacteristic
617C2×C4C22×C4×D311yescharacteristic
618C2×C4C22×C4×D311yescharacteristic
619C2×C4C22×C4×D311yescharacteristic
620C2×C4C22×C4×D311yescharacteristic
621C2×C4C22×C4×D311yescharacteristic
622C2×C4C22×C4×D311yescharacteristic
623C2×C4C22×C4×D311yescharacteristic
624C2×C4C22×C4×D311yescharacteristic
625C2×C4C22×C4×D311yescharacteristic
626C2×C4C22×C4×D311yescharacteristic
627C2×C4C24×D311yescharacteristic
628C8—28no—
629C8—64no—
630C8—64no—
631C2×C3—22no—
632C2×C3—41no—
633C2×C3—41no—
634C2×C3#342811yescharacteristic
635C2×C3C22×#13111yescharacteristic
636C2×C3C2×#56811yescharacteristic
637D3—44no—
638C22—21no—
639C22—21no—
640C22—21no—
641C22—21no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—44no—
647C22—64no—
648C22—64no—
649C22—64no—
650C22C2×#90712yes—
651C22C23×C4×D311yescharacteristic
652C4—21no—
653C4—21no—
654C4—21no—
655C4—21no—
656C4—21no—
657C4—21no—
658C4—21no—
659C4—21no—
660C4—64no—
661C4—64no—
662C4—64no—
663C4C2×#90712yes—
664C4C2×#90712yes—
665C4#574512yes—
666C4#596412yes—
667C3#6027911yescharacteristic, complemented, Sylow
668C2—22no—
669C2—41no—
670C2—41no—
671C2—124no—
672C2#8573111yescharacteristic
673C2C22×#90711yescharacteristic
674C2C2×#574511yescharacteristic
subgroupG/H
centerC42C23×D3
commutator, socleC22×C3C24×C4
FrattiniC2×C4C24×D3
FittingC3×#2920C2
prime radical#85680C2
cosocleC2×C3×C4C25
hypercenter#2920D3
hypocenterC3#60279
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C42 ◁ #2920

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C3×#2920C2semidirect
3. C2×#4925C2semidirect
4. #79341C2semidirect
5. #79338C2semidirect
6. #79316C2semidirect
7. #79283C2semidirect
8. #79277C2semidirect
9. #79276C2semidirect
10. #79053C2semidirect
11. #78635C2semidirect
12. #78523C2semidirect
50. #4840C22semidirect
51. #4839C22semidirect
52. #4838C22semidirect
53. #4834C22semidirect
54. #4831C22semidirect
55. #4830C22semidirect
60. #4706C22semidirect
61. #4701C22semidirect
62. C4×ΓL(1,𝔽25)C22semidirect
63. #4685C22semidirect
64. #4685C22semidirect
65. C4×D3×C8C22semidirect
76. #2920D3semidirect
667. C3#60279semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#856802
C3×#29202
C2×#49252
#793412
#793382
#793162
#792832
#792772
#792762
#790532
#786352
#785232
#602793