Finite Group Databasebeta

← #10800175

Subgroups — #10800175

Order 768.

1186 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5528C211yesmaximal, characteristic
2C2×#4707C211yesmaximal, characteristic, complemented
3C2×#4685C211yesmaximal, characteristic, complemented
4C3×#1750C211yesmaximal, characteristic, complemented
5#71582C211yesmaximal, characteristic, complemented
6#69958C212yesmaximal, complemented
7#69916C212yesmaximal, complemented
8#69830C211yesmaximal, characteristic, complemented
9#69753C211yesmaximal, characteristic, complemented
10#68669C212yesmaximal, complemented
11#68619C212yesmaximal, complemented
12#21378—31nomaximal, Sylow
13C2×C3×#117C2211yescharacteristic
14C2×C4×#67C2211yescharacteristic
15C2×C4×#67C2211yescharacteristic
16C22×ΓL(1,𝔽25)C2212yes—
17#5528—24no—
18#5528—24no—
19C2×#898C411yescharacteristic
20C2×#896C411yescharacteristic
21C2×#894C411yescharacteristic
22C2×#889C411yescharacteristic
23C2×#885C411yescharacteristic
24C2×#884C411yescharacteristic
25C2×#883C411yescharacteristic
26C2×#879C411yescharacteristic
27C2×C4×D12C2211yescharacteristic
28C2×#872C2211yescharacteristic
29C2×C42×D3C2211yescharacteristic
30C2×C4×Dic6C2211yescharacteristic
31C2×C3×C22⋊C8C2212yes—
32C2×C3×#97C2211yescharacteristic
33C2×C3×C4×C8C2211yescharacteristic
34C2×#830C2212yes—
35C2×(C2×D3)⋊C8C2212yes—
36C2×(C2×D3)⋊C8C2212yes—
37C2×(C2×D3)⋊C8C2212yes—
38C2×Dic3⋊C8C2212yes—
39C2×C8×Dic3C2212yes—
40C2×#803C2211yescharacteristic
41C2×C4×C3⋊C8C2211yescharacteristic
42#4707—24no—
43#4685—24no—
44C3×#336C2212yes—
45#4550C2212yes—
46#4550C2212yes—
47#4524C2212yes—
48C2×#529—31no—
49C2×#431—31no—
50C2×C4×M16—31no—
51#1984—32no—
52#1954—32no—
53#1870—32no—
54#1870—32no—
55#1750—31no—
56#1750—31no—
57#1750—31no—
58#1750D311yescharacteristic, complemented
59C23×C3×C4C2311yescharacteristic
60C22×#67C2×C411yescharacteristic
61C22×#67C2×C411yescharacteristic
62C22×D12C2×C411yescharacteristic
63C22×C4×D3C2×C411yescharacteristic
64C22×C4×D3C2×C411yescharacteristic
65C22×C4×D3C2311yescharacteristic
66C22×C4×D3C2311yescharacteristic
67C22×Dic6C2×C411yescharacteristic
68C22×C3×C8C2312yes—
69C22×C3×C8C2312yes—
70C3×#117—22no—
71C3×#117—22no—
72C2×C3×C4⋊C4C2×C411yescharacteristic
73C2×C3×C4⋊C4C2×C411yescharacteristic
74C2×C3×C22⋊C4C2×C411yescharacteristic
75C2×C3×C22⋊C4C2×C411yescharacteristic
76C2×C3×C42C2311yescharacteristic
77C2×C3×C42C2311yescharacteristic
78C2×#217C2×C411yescharacteristic
79C2×#217C2×C411yescharacteristic
80C4×#67—22no—
81C4×#67—22no—
82C4×#67—22no—
83C4×#67—22no—
84C2×(C2×D3)⋊C4C2×C411yescharacteristic
85C2×(C2×D3)⋊C4C2×C411yescharacteristic
86C2×(C2×D3)⋊C4C2×C411yescharacteristic
87C2×(C2×D3)⋊C4C2×C411yescharacteristic
88C2×(C2×D3)⋊C4C2×C411yescharacteristic
89C2×(C2×D3)⋊C4C2×C411yescharacteristic
90C2×C4⋊Dic3C2×C411yescharacteristic
91C2×C4⋊Dic3C2×C411yescharacteristic
92C2×Dic3⋊C4C2×C411yescharacteristic
93C2×Dic3⋊C4C2×C411yescharacteristic
94C2×Dic3⋊C4C2×C411yescharacteristic
95C2×Dic3⋊C4C2×C411yescharacteristic
96C2×Dic3⋊C4C2×C411yescharacteristic
97C2×Dic3⋊C4C2×C411yescharacteristic
98C2×C4×Dic3C2×C411yescharacteristic
99C2×C4×Dic3C2×C411yescharacteristic
100C2×C4×Dic3C2311yescharacteristic
101C2×C4×Dic3C2311yescharacteristic
102C22×C3⋊C8C2312yes—
103C22×C3⋊C8C2312yes—
104C2×ΓL(1,𝔽25)—24no—
105C2×ΓL(1,𝔽25)—24no—
106C2×ΓL(1,𝔽25)—24no—
107#898—24no—
108#896—24no—
109#894—24no—
110#889—24no—
111#885—24no—
112#884—24no—
113#883—24no—
114#879—24no—
115C4×D12—22no—
116C4×D12—22no—
117#872—22no—
118#872—22no—
119C42×D3D414yes—
120C42×D3D414yes—
121C4×Dic6—22no—
122C4×Dic6—22no—
123C3×C22⋊C8—24no—
124C3×#97—22no—
125C3×C4×C8—22no—
126#830—24no—
127(C2×D3)⋊C8—24no—
128(C2×D3)⋊C8—24no—
129(C2×D3)⋊C8—24no—
130Dic3⋊C8—24no—
131C8×Dic3—24no—
132#803—22no—
133C4×C3⋊C8—22no—
134C22×M16—32no—
135#529—64no—
136#529—64no—
137C2×#126—31no—
138C2×#126—31no—
139C2×#125—31no—
140C2×#124—31no—
141C2×#123—31no—
142C2×#123—31no—
143C2×C22⋊Q8—31no—
144C2×#121—31no—
145C2×C4×Q8—31no—
146C2×C4×D4—31no—
147C2×C4×D4—31no—
148C2×C4×D4—31no—
149C2×#117—31no—
150C2×#117C2×D311yescharacteristic
151C22×C42—31no—
152#431—64no—
153C2×C4⋊C8—32no—
154C2×C22⋊C8—32no—
155C2×C22⋊C8—32no—
156C2×C22⋊C8—32no—
157C2×C22⋊C8—32no—
158C2×C22⋊C8C2×D312yes—
159C4×M16—64no—
160C2×#97—31no—
161C2×#97C2×D311yescharacteristic
162C2×C4×C8—32no—
163C2×C4×C8—31no—
164C2×C4×C8C2×D311yescharacteristic
165#336—32no—
166#336—32no—
167#336—32no—
168#336C2×D312yes—
169C24×C3C22×C411yescharacteristic
170C23×D3C22×C411yescharacteristic
171C23×D3C22×C411yescharacteristic
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4C22×C411yescharacteristic
177C22×C3×C4C22×C411yescharacteristic
178C22×C3×C4C22×C411yescharacteristic
179C22×C3×C4C22×C411yescharacteristic
180C22×C3×C4C22×C411yescharacteristic
181C22×C3×C4C2411yescharacteristic
182C2×#67—22no—
183C2×#67—22no—
184C2×#67—24no—
185C2×#67—42no—
186C22×Dic3C22×C411yescharacteristic
187C22×Dic3C22×C411yescharacteristic
188C22×Dic3C22×C411yescharacteristic
189C22×Dic3C22×C411yescharacteristic
190C22×Dic3C22×C411yescharacteristic
191C22×Dic3C22×C411yescharacteristic
192C2×D12—22no—
193C2×D12—42no—
194C2×C4×D3—22no—
195C2×C4×D3—22no—
196C2×C4×D3—24no—
197C2×C4×D3—21no—
198C2×C4×D3—21no—
199C2×C4×D3—22no—
200C2×C4×D3—22no—
201C2×C4×D3—42no—
202C2×C4×D3C22⋊C414yes—
203C2×C4×D3C22⋊C414yes—
204C2×C4×D3C2×D412yes—
205C2×C4×D3C2×D412yes—
206C2×Dic6—22no—
207C2×Dic6—24no—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C2×C3×C8—22no—
213C2×C3×C8#4114yes—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—22no—
216C3×C22⋊C4—22no—
217C3×C22⋊C4—22no—
218C3×C42—21no—
219C3×C42—21no—
220C3×C42C2×D412yes—
221C3×C42C2×D412yes—
222#217—22no—
223#217—22no—
224(C2×D3)⋊C4—22no—
225(C2×D3)⋊C4—22no—
226(C2×D3)⋊C4—22no—
227(C2×D3)⋊C4—22no—
228(C2×D3)⋊C4—22no—
229(C2×D3)⋊C4—22no—
230C4⋊Dic3—22no—
231C4⋊Dic3—22no—
232Dic3⋊C4—22no—
233Dic3⋊C4—22no—
234Dic3⋊C4—22no—
235Dic3⋊C4—22no—
236Dic3⋊C4—22no—
237Dic3⋊C4—22no—
238C4×Dic3—21no—
239C4×Dic3—21no—
240C4×Dic3C22⋊C414yes—
241C4×Dic3C22⋊C414yes—
242C4×Dic3C2×D412yes—
243C4×Dic3C2×D412yes—
244C2×C3⋊C8—22no—
245C2×C3⋊C8—22no—
246C2×C3⋊C8—22no—
247C2×C3⋊C8—22no—
248C2×C3⋊C8—22no—
249C2×C3⋊C8#4114yes—
250ΓL(1,𝔽25)—216no—
251C22×Q8—31no—
252C22×D4—31no—
253C22×D4—31no—
254C22×D4—31no—
255C23×C4—31no—
256C23×C4—31no—
257C23×C4—31no—
258C23×C4—31no—
259C23×C4C22×D311yescharacteristic
260C2×M16—64no—
261C2×M16—64no—
262C2×M16—64no—
263C22×C8—32no—
264C22×C8—32no—
265C22×C8C22×D312yes—
266C22×C8C22×D312yes—
267#126—64no—
268#126—64no—
269#125—64no—
270#124—64no—
271#123—64no—
272#123—64no—
273C22⋊Q8—64no—
274#121—64no—
275C4×Q8—62no—
276C4×Q8—62no—
277C4×D4—62no—
278C4×D4—62no—
279C4×D4—62no—
280C4×D4—62no—
281C4×D4—62no—
282C4×D4—62no—
283#117—22no—
284#117—22no—
285#117—62no—
286#117—62no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4C4×D311yescharacteristic
296C2×C4⋊C4C4×D311yescharacteristic
297C2×C22⋊C4—31no—
298C2×C22⋊C4—31no—
299C2×C22⋊C4—31no—
300C2×C22⋊C4—31no—
301C2×C22⋊C4—31no—
302C2×C22⋊C4—31no—
303C2×C22⋊C4—31no—
304C2×C22⋊C4—31no—
305C2×C22⋊C4C4×D311yescharacteristic
306C2×C22⋊C4C4×D311yescharacteristic
307C2×C42—31no—
308C2×C42—31no—
309C2×C42—31no—
310C2×C42—34no—
311C2×C42—34no—
312C2×C42—31no—
313C2×C42C22×D311yescharacteristic
314C2×C42C22×D311yescharacteristic
315C4⋊C8—64no—
316C22⋊C8—24no—
317C22⋊C8—64no—
318C22⋊C8—64no—
319C22⋊C8—64no—
320C22⋊C8—64no—
321#97—22no—
322#97—62no—
323C4×C8—22no—
324C4×C8—64no—
325C4×C8—62no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—21no—
329C23×C3—22no—
330C23×C3—22no—
331C23×C3C23×C411yescharacteristic
332C22×D3—21no—
333C22×D3—21no—
334C22×D3—22no—
335C22×D3—22no—
336C22×D3—21no—
337C22×D3—21no—
338C22×D3—21no—
339C22×D3—22no—
340C22×D3—22no—
341C22×D3C2×C22⋊C412yes—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4C2×C22⋊C412yes—
354C2×C3×C4C2×C22⋊C412yes—
355C2×C3×C4C22×D411yescharacteristic
356C2×C3×C4C22×D411yescharacteristic
357C2×C3×C4C2×#4112yes—
358C2×C3×C4C2×#4112yes—
359#67—44no—
360#67—44no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—22no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—22no—
370C2×Dic3—22no—
371C2×Dic3C2×C22⋊C412yes—
372C2×Dic3C2×C22⋊C412yes—
373C2×Dic3C2×C22⋊C412yes—
374C2×Dic3#13114yes—
375D12—44no—
376C4×D3—24no—
377C4×D3—24no—
378C4×D3—28no—
379C4×D3—44no—
380C4×D3—42no—
381C4×D3—42no—
382Dic6—44no—
383C3×C8—24no—
384C3×C8—42no—
385C3⋊C8—24no—
386C3⋊C8—42no—
387C24—31no—
388C24—31no—
389C24C2×C4×D311yescharacteristic
390C2×Q8—62no—
391C2×Q8—64no—
392C2×D4—62no—
393C2×D4—62no—
394C2×D4—64no—
395C2×D4—62no—
396C2×D4—122no—
397C2×D4—122no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—31no—
403C22×C4—31no—
404C22×C4—31no—
405C22×C4—34no—
406C22×C4—32no—
407C22×C4—32no—
408C22×C4—34no—
409C22×C4—31no—
410C22×C4—31no—
411C22×C4—31no—
412C22×C4—64no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—62no—
419C22×C4—122no—
420C22×C4C2×C4×D311yescharacteristic
421C22×C4C2×C4×D311yescharacteristic
422C22×C4C2×C4×D311yescharacteristic
423C22×C4C2×C4×D311yescharacteristic
424C22×C4C2×C4×D311yescharacteristic
425C22×C4C23×D311yescharacteristic
426M16—616no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—34no—
433C2×C8—62no—
434C2×C8—62no—
435C2×C8—62no—
436C2×C8—62no—
437C2×C8—62no—
438C2×C8#23514yes—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—62no—
442C4⋊C4—62no—
443C4⋊C4—62no—
444C4⋊C4—62no—
445C4⋊C4—62no—
446C4⋊C4—62no—
447C4⋊C4—62no—
448C4⋊C4—62no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—62no—
452C22⋊C4—62no—
453C22⋊C4—62no—
454C22⋊C4—62no—
455C22⋊C4—62no—
456C22⋊C4—62no—
457C22⋊C4—62no—
458C22⋊C4—62no—
459C42—21no—
460C42—21no—
461C42—34no—
462C42—34no—
463C42—32no—
464C42—32no—
465C42—61no—
466C42—61no—
467C42D3×D412yes—
468C42D3×D412yes—
469C22×C3—41no—
470C22×C3—41no—
471C22×C3—42no—
472C22×C3—42no—
473C22×C3—41no—
474C22×C3C22×C22⋊C411yescharacteristic
475C22×C3C4×#4111yescharacteristic
476C22×C3C4×#4111yescharacteristic
477C22×C3C2×#13112yes—
478C22×C3#56812yes—
479C2×D3—22no—
480C2×D3—22no—
481C2×D3—22no—
482C2×D3—42no—
483C2×D3—42no—
484C2×D3—41no—
485C2×D3—41no—
486C2×D3—42no—
487C2×D3—42no—
488C2×D3—41no—
489C3×C4—22no—
490C3×C4—22no—
491C3×C4—41no—
492C3×C4—41no—
493C3×C4—41no—
494C3×C4#52512yes—
495C3×C4#53212yes—
496Dic3—22no—
497Dic3—22no—
498Dic3—41no—
499Dic3—41no—
500Dic3—41no—
501Dic3#40814yes—
502C23—21no—
503C23—21no—
504C23—22no—
505C23—22no—
506C23—21no—
507C23—32no—
508C23—61no—
509C23—62no—
510C23—62no—
511C23—61no—
512C23—61no—
513C23—61no—
514C23—62no—
515C23—62no—
516C23—61no—
517C23C22×C4×D311yescharacteristic
518Q8—124no—
519D4—124no—
520D4—124no—
521D4—124no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—32no—
532C2×C4—34no—
533C2×C4—32no—
534C2×C4—32no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—68no—
538C2×C4—61no—
539C2×C4—62no—
540C2×C4—61no—
541C2×C4—64no—
542C2×C4—64no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—62no—
548C2×C4—62no—
549C2×C4—61no—
550C2×C4—124no—
551C2×C4—122no—
552C2×C4—122no—
553C2×C4D3×C22⋊C412yes—
554C2×C4D3×C22⋊C412yes—
555C2×C4C2×#23512yes—
556C2×C4C2×D3×D411yescharacteristic
557C2×C4C2×D3×D411yescharacteristic
558C2×C4D3×#4112yes—
559C8—24no—
560C8—42no—
561C8—64no—
562C8—122no—
563C2×C3—81no—
564C2×C3#229711yescharacteristic
565C2×C3C2×#40811yescharacteristic
566C2×C3#287911yescharacteristic
567C2×C3#296912yes—
568C2×C3#296912yes—
569D3—42no—
570D3—81no—
571C22—42no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—42no—
576C22—62no—
577C22—62no—
578C22—62no—
579C22—122no—
580C22—122no—
581C22—122no—
582C22—121no—
583C22—121no—
584C22—121no—
585C22—122no—
586C22C4×#23511yescharacteristic
587C22C2×D3×C22⋊C411yescharacteristic
588C22#552211yescharacteristic
589C22D3×#13112yes—
590C22#575012yes—
591C4—22no—
592C4—22no—
593C4—34no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—62no—
598C4—62no—
599C4—121no—
600C4—121no—
601C4—121no—
602C4#554712yes—
603C4#554812yes—
604C3#2137811yescharacteristic, complemented, Sylow
605C2—81no—
606C2—122no—
607C2—241no—
608C2#7356311yescharacteristic
609C2#7855912yes—
610C2D3×#40811yescharacteristic
611C2#7866211yescharacteristic
612C2#7936912yes—
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC3×#1750C2
prime radicalC2×#884C4
cosocleC22×C3×C4C24
hypercenter#1750D3
hypocenterC3#21378
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ #1750

Complements

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

NH ≅ G/Ntype
2. C2×#4707C2semidirect
3. C2×#4685C2semidirect
4. C3×#1750C2semidirect
5. #71582C2semidirect
6. #69958C2semidirect
7. #69916C2semidirect
8. #69830C2semidirect
9. #69753C2semidirect
10. #68669C2semidirect
11. #68619C2semidirect
58. #1750D3semidirect
604. C3#21378semidirect

Maximal subgroups

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

subgroupindex
C2×#55282
C2×#47072
C2×#46852
C3×#17502
#715822
#699582
#699162
#698302
#697532
#686692
#686192
#213783