Finite Group Databasebeta

← #10955433

Subgroups — #10955433

Order 768.

2314 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#237C211yesmaximal, characteristic
2C3×#2319C211yesmaximal, characteristic, complemented
3#82325C211yesmaximal, characteristic, complemented
4#76303C212yesmaximal, complemented
5#76301C212yesmaximal
6#75982C212yesmaximal, complemented
7#75979C212yesmaximal, complemented
8#75476C214yesmaximal, complemented
9#75473C214yesmaximal, complemented
10#74055C212yesmaximal, complemented
11#74051C212yesmaximal, complemented
12#73870C212yesmaximal, complemented
13#73863C212yesmaximal, complemented
14#73752C211yesmaximal, characteristic, complemented
15#73616C211yesmaximal, characteristic, complemented
16#73413C211yesmaximal, characteristic, complemented
17#73383C211yesmaximal, characteristic, complemented
18#34197—31nomaximal, Sylow
19C22×#237C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22C2×C4×#67C2212yes—
23C22×C4×Dic3C2212yes—
24C4×#237C4116yescomplemented
25C2×#892C2211yescharacteristic
26C2×#887C2212yes—
27C2×#887C2211yescharacteristic
28C2×#67⋊C4C2212yes—
29C2×#877C2212yes—
30C2×C42×D3C2211yescharacteristic
31C2×C4×Dic6C2212yes—
32C2×C4×Dic6C2211yescharacteristic
33C3×#390C2212yes—
34C3×#389C2212yes—
35C3×#388C2214yes—
36C3×#381C2212yes—
37C3×#380C2212yes—
38C3×C4×C4⋊C4C2211yescharacteristic
39C3×#376C2211yescharacteristic
40#5221C2212yes—
41#5212C2212yes—
42#5211C2212yes—
43#4990C2214yes—
44#4987C2212yes—
45#4984C2212yes—
46#4977C2212yes—
47#4977C2214yes—
48#4976C2214yes—
49Dic3×C4⋊C4C2212yes—
50Dic3×C4⋊C4C2214yes—
51#4957C2214yes—
52#4957C2214yes—
53#4951C2214yes—
54#4945C2214yes—
55#4944C2214yes—
56#4943C2214yes—
57#4942C2214yes—
58Dic3×C22⋊C4C2214yes—
59Dic3×C22⋊C4C2214yes—
60C4×(C2×D3)⋊C4C2212yes—
61C4×(C2×D3)⋊C4C2212yes—
62C4×C4⋊Dic3C2212yes—
63C4×C4⋊Dic3C2211yescharacteristic
64#4932C2212yes—
65#4932C2211yescharacteristic
66C4×Dic3⋊C4C2212yes—
67C4×Dic3⋊C4C2214yes—
68#4670C2214yes—
69#4667C2211yescharacteristic
70#4667C2211yescharacteristic
71#4664C2212yes—
72#4664C2212yes—
73#4651C2212yes—
74#4651C2211yescharacteristic
75#4651C2211yescharacteristic
76#4650C2214yes—
77#4650C2212yes—
78#4649C2212yes—
79#4649C2212yes—
80#4649C2214yes—
81#4648C2214yes—
82#4648C2212yes—
83#4648C2212yes—
84#4647C2214yes—
85C2×C4×#41—31no—
86#2357—32no—
87#2357—34no—
88#2351—32no—
89#2346—32no—
90#2346—34no—
91#2341—32no—
92#2319—32no—
93#2319—32no—
94#2319—31no—
95#2319D311yescharacteristic, complemented
96#2283—32no—
97#2283—32no—
98#2283—31no—
99#2283—31no—
100#2276—31no—
101#2276—31no—
102C22×C3×D4C2311yescharacteristic
103C23×C3×C4C2312yes—
104C22×#67C2312yes—
105C23×Dic3C2312yes—
106C23×Dic3C2312yes—
107C2×#237—28no—
108C2×#237C2×C418yes—
109C22×C4×D3C2311yescharacteristic
110C22×C4×D3C2311yescharacteristic
111C22×C4×D3C2311yescharacteristic
112C22×Dic6C2311yescharacteristic
113C3×C4×D4C2×C418yes—
114C2×C3×C4⋊C4C2312yes—
115C2×C3×C4⋊C4C2314yes—
116C2×C3×C4⋊C4C2311yescharacteristic
117C2×C3×C22⋊C4C2314yes—
118C2×C3×C22⋊C4C2314yes—
119C2×C3×C22⋊C4C2312yes—
120C2×C3×C42C2312yes—
121C2×C3×C42C2312yes—
122C2×C3×C42C2311yescharacteristic
123C2×#217C2314yes—
124C2×#217C2314yes—
125C2×#217C2312yes—
126D4×Dic3C2×C418yes—
127C4×#67C2×C4116yes—
128C2×(C2×D3)⋊C4C2312yes—
129C2×(C2×D3)⋊C4C2312yes—
130C2×(C2×D3)⋊C4C2314yes—
131C2×(C2×D3)⋊C4C2312yes—
132C2×C4⋊Dic3C2312yes—
133C2×C4⋊Dic3C2314yes—
134C2×C4⋊Dic3C2311yescharacteristic
135C2×Dic3⋊C4C2314yes—
136C2×Dic3⋊C4C2314yes—
137C2×Dic3⋊C4C2312yes—
138C2×Dic3⋊C4C2312yes—
139C2×Dic3⋊C4C2312yes—
140C2×Dic3⋊C4C2312yes—
141C2×C4×Dic3C2×C4116yes—
142C2×C4×Dic3C2314yes—
143C2×C4×Dic3C2314yes—
144C2×C4×Dic3C2312yes—
145C2×C4×Dic3C2314yes—
146C2×C4×Dic3C2314yes—
147C2×C4×Dic3C2312yes—
148C2×C4×Dic3C2312yes—
149C2×C4×Dic3C2312yes—
150C2×C4×Dic3C2311yescharacteristic
151C2×C4×Dic3C2312yes—
152C2×C4×Dic3C2311yescharacteristic
153C2×C4×Dic3C2311yescharacteristic
154#892C2×C418yes—
155#887—28no—
156#887C2×C418yes—
157#67⋊C4C2×C4116yes—
158#877C2×C4116yes—
159C42×D3C2×C418yes—
160C4×Dic6—28no—
161C4×Dic6C2×C418yes—
162C3×#95C2314yes—
163C3×#95C2311yescharacteristic
164C3×#95C2312yes—
165C3×#95C2312yes—
166C3×#95C2311yescharacteristic
167C3⋊#95C2314yes—
168C3⋊#95C2312yes—
169C3⋊#95C2311yescharacteristic
170C3⋊#95C2312yes—
171C3⋊#95C2314yes—
172C3⋊#95C2314yes—
173C3⋊#95C2314yes—
174C3⋊#95C2314yes—
175C3⋊#95C2311yescharacteristic
176C3⋊#95C2312yes—
177C3⋊#95C2312yes—
178C3⋊#95C2314yes—
179C3⋊#95C2312yes—
180C3⋊#95C2312yes—
181C22×#41—31no—
182C4×#41—316no—
183C2×C4×Q8—31no—
184C2×C4×Q8—32no—
185C2×C4×Q8—31no—
186C2×C4×Q8—32no—
187C2×C4×D4—31no—
188C2×C4×D4—32no—
189C2×C4×D4—32no—
190C2×C4×D4C2×D311yescharacteristic
191C2×#117—31no—
192C2×#117—32no—
193C22×C42—32no—
194C22×C42—31no—
195#390—32no—
196#390—34no—
197#390—34no—
198#390C2×D312yes—
199#389—34no—
200#389—32no—
201#389—34no—
202#389C2×D312yes—
203#388—34no—
204#388—34no—
205#388—34no—
206#388—34no—
207#388—34no—
208#388C2×D314yes—
209#383—31no—
210#383—32no—
211#383—31no—
212#382—34no—
213#382—32no—
214#382—32no—
215#382—34no—
216#381—32no—
217#381—34no—
218#381—32no—
219#381—32no—
220#381C2×D312yes—
221#380—32no—
222#380—32no—
223#380—34no—
224#380—31no—
225#380—31no—
226#380C2×D312yes—
227C4×C4⋊C4—32no—
228C4×C4⋊C4—34no—
229C4×C4⋊C4—31no—
230C4×C4⋊C4—34no—
231C4×C4⋊C4—32no—
232C4×C4⋊C4—32no—
233C4×C4⋊C4—32no—
234C4×C4⋊C4—32no—
235C4×C4⋊C4—34no—
236C4×C4⋊C4C2×D311yescharacteristic
237C4×C22⋊C4—34no—
238C4×C22⋊C4—34no—
239C4×C22⋊C4—32no—
240C4×C22⋊C4—32no—
241#376—31no—
242#376—32no—
243#376—32no—
244#376—32no—
245#376—34no—
246#376C2×D311yescharacteristic
247C24×C3C2412yes—
248C23×D3C2411yescharacteristic
249C2×C3×D4—24no—
250C2×C3×D4C22×C414yes—
251C22×C3×C4—44no—
252C22×C3×C4C22×C418yes—
253C22×C3×C4C2412yes—
254C22×C3×C4C2414yes—
255C22×C3×C4C2412yes—
256C22×C3×C4C2411yescharacteristic
257C22×C3×C4C2411yescharacteristic
258C22×C3×C4C2412yes—
259C22×C3×C4C2411yescharacteristic
260C2×#67—28no—
261C2×#67C22×C418yes—
262C22×Dic3—28no—
263C22×Dic3—42no—
264C22×Dic3—42no—
265C22×Dic3C22×C418yes—
266C22×Dic3C22×C418yes—
267C22×Dic3C2412yes—
268C22×Dic3C2412yes—
269C22×Dic3C2414yes—
270C22×Dic3C2411yescharacteristic
271C22×Dic3C2411yescharacteristic
272C22×Dic3C2412yes—
273C22×Dic3C2411yescharacteristic
274C22×Dic3C2412yes—
275#237—232no—
276C2×C4×D3—24no—
277C2×C4×D3—42no—
278C2×C4×D3—41no—
279C2×C4×D3—41no—
280C2×C4×D3C22×C414yes—
281C2×C4×D3C22×C414yes—
282C2×C4×D3C22×C414yes—
283C2×Dic6C22×C414yes—
284C2×Dic6#4118yes—
285C3×C4⋊C4—24no—
286C3×C4⋊C4—44no—
287C3×C4⋊C4C22×C414yes—
288C3×C22⋊C4—44no—
289C3×C22⋊C4—44no—
290C3×C22⋊C4C22×C418yes—
291C3×C42—24no—
292C3×C42—42no—
293C3×C42C22×C414yes—
294#217—44no—
295#217—44no—
296#217C22×C418yes—
297(C2×D3)⋊C4—42no—
298(C2×D3)⋊C4—44no—
299(C2×D3)⋊C4—42no—
300(C2×D3)⋊C4C22×C418yes—
301C4⋊Dic3—24no—
302C4⋊Dic3—44no—
303C4⋊Dic3C22×C414yes—
304Dic3⋊C4—28no—
305Dic3⋊C4—28no—
306Dic3⋊C4—42no—
307Dic3⋊C4—42no—
308Dic3⋊C4C22×C418yes—
309Dic3⋊C4C22×C418yes—
310C4×Dic3—24no—
311C4×Dic3—28no—
312C4×Dic3—28no—
313C4×Dic3—44no—
314C4×Dic3—44no—
315C4×Dic3—42no—
316C4×Dic3C22×C418yes—
317C4×Dic3C22×C414yes—
318C4×Dic3C22×C414yes—
319C4×Dic3C22×C414yes—
320C4×Dic3C22×C418yes—
321C4×Dic3C22×C418yes—
322C2×#41—38no—
323C2×#41—68no—
324C22×Q8—31no—
325C22×D4—32no—
326C22×D4C22×D311yescharacteristic
327C23×C4—32no—
328C23×C4—32no—
329C23×C4—31no—
330C23×C4—31no—
331C23×C4—31no—
332C23×C4C22×D312yes—
333C4×Q8—38no—
334C4×Q8—38no—
335C4×Q8—68no—
336C4×Q8—68no—
337C4×D4—38no—
338C4×D4—316no—
339C4×D4—316no—
340C4×D4C4×D318yes—
341#117—38no—
342#117—316no—
343C2×C4⋊C4—32no—
344C2×C4⋊C4—31no—
345C2×C4⋊C4—32no—
346C2×C4⋊C4—34no—
347C2×C4⋊C4—34no—
348C2×C4⋊C4—32no—
349C2×C4⋊C4—32no—
350C2×C4⋊C4—32no—
351C2×C4⋊C4—34no—
352C2×C4⋊C4C22×D312yes—
353C2×C4⋊C4C22×D314yes—
354C2×C4⋊C4C22×D311yescharacteristic
355C2×C22⋊C4—32no—
356C2×C22⋊C4—32no—
357C2×C22⋊C4—34no—
358C2×C22⋊C4—34no—
359C2×C22⋊C4—32no—
360C2×C22⋊C4—34no—
361C2×C22⋊C4—32no—
362C2×C22⋊C4C22×D314yes—
363C2×C22⋊C4C22×D314yes—
364C2×C22⋊C4C22×D312yes—
365C2×C42—316no—
366C2×C42—32no—
367C2×C42—31no—
368C2×C42—32no—
369C2×C42—32no—
370C2×C42—31no—
371C2×C42—31no—
372C2×C42—38no—
373C2×C42—34no—
374C2×C42—34no—
375C2×C42—32no—
376C2×C42—34no—
377C2×C42—34no—
378C2×C42—32no—
379C2×C42C22×D312yes—
380C2×C42C22×D312yes—
381C2×C42C22×D311yescharacteristic
382#95—31no—
383#95—34no—
384#95—32no—
385#95—32no—
386#95—31no—
387#95—34no—
388#95—34no—
389#95—32no—
390#95—34no—
391#95—34no—
392#95—32no—
393#95—34no—
394#95—32no—
395#95—32no—
396#95C22×D311yescharacteristic
397#95C22×D312yes—
398#95C22×D314yes—
399#95C22×D311yescharacteristic
400#95C22×D312yes—
401C23×C3—24no—
402C23×C3—22no—
403C23×C3—24no—
404C23×C3—22no—
405C23×C3C23×C414yes—
406C23×C3C2511yescharacteristic
407C22×D3—22no—
408C22×D3—22no—
409C22×D3—21no—
410C22×D3—21no—
411C22×D3C23×C412yes—
412C3×D4—28no—
413C2×C3×C4—21no—
414C2×C3×C4—24no—
415C2×C3×C4—21no—
416C2×C3×C4—22no—
417C2×C3×C4—22no—
418C2×C3×C4—24no—
419C2×C3×C4—22no—
420C2×C3×C4—22no—
421C2×C3×C4—24no—
422C2×C3×C4—22no—
423C2×C3×C4—22no—
424C2×C3×C4—22no—
425C2×C3×C4—24no—
426C2×C3×C4—48no—
427C2×C3×C4C23×C412yes—
428C2×C3×C4C23×C414yes—
429C2×C3×C4C23×C412yes—
430C2×C3×C4C23×C412yes—
431C2×C3×C4C2×#4114yes—
432#67—216no—
433C2×Dic3—22no—
434C2×Dic3—22no—
435C2×Dic3—22no—
436C2×Dic3—22no—
437C2×Dic3—21no—
438C2×Dic3—21no—
439C2×Dic3—216no—
440C2×Dic3—22no—
441C2×Dic3—22no—
442C2×Dic3—24no—
443C2×Dic3—22no—
444C2×Dic3—24no—
445C2×Dic3—22no—
446C2×Dic3—22no—
447C2×Dic3—24no—
448C2×Dic3—22no—
449C2×Dic3—48no—
450C2×Dic3C23×C412yes—
451C2×Dic3C23×C412yes—
452C2×Dic3C23×C412yes—
453C2×Dic3C23×C414yes—
454C2×Dic3C23×C414yes—
455C2×Dic3C2×#4118yes—
456C4×D3—28no—
457C4×D3—44no—
458C4×D3—44no—
459Dic6#117116yes—
460C24—31no—
461C24C23×D312yes—
462#41—632no—
463C2×Q8—34no—
464C2×Q8—38no—
465C2×D4—24no—
466C2×D4—38no—
467C2×D4—68no—
468C2×D4C2×C4×D314yes—
469C22×C4—32no—
470C22×C4—38no—
471C22×C4—31no—
472C22×C4—31no—
473C22×C4—32no—
474C22×C4—31no—
475C22×C4—38no—
476C22×C4—34no—
477C22×C4—34no—
478C22×C4—34no—
479C22×C4—32no—
480C22×C4—32no—
481C22×C4—34no—
482C22×C4—44no—
483C22×C4—68no—
484C22×C4—64no—
485C22×C4—121no—
486C22×C4—122no—
487C22×C4—121no—
488C22×C4—122no—
489C22×C4—122no—
490C22×C4C2×C4×D318yes—
491C22×C4C23×D312yes—
492C22×C4C23×D314yes—
493C22×C4C23×D312yes—
494C22×C4C23×D311yescharacteristic
495C22×C4C23×D311yescharacteristic
496C22×C4C23×D312yes—
497C22×C4C23×D311yescharacteristic
498C4⋊C4—24no—
499C4⋊C4—38no—
500C4⋊C4—38no—
501C4⋊C4—34no—
502C4⋊C4—44no—
503C4⋊C4—68no—
504C4⋊C4—68no—
505C4⋊C4—64no—
506C4⋊C4—122no—
507C4⋊C4—122no—
508C4⋊C4—124no—
509C4⋊C4C2×C4×D314yes—
510C22⋊C4—38no—
511C22⋊C4—38no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—122no—
515C22⋊C4—124no—
516C22⋊C4—122no—
517C22⋊C4—124no—
518C22⋊C4—124no—
519C22⋊C4C2×C4×D318yes—
520C42—24no—
521C42—34no—
522C42—38no—
523C42—34no—
524C42—38no—
525C42—38no—
526C42—34no—
527C42—42no—
528C42—68no—
529C42—68no—
530C42—64no—
531C42—122no—
532C42—124no—
533C42—124no—
534C42C2×C4×D314yes—
535C22×C3—24no—
536C22×C3—24no—
537C22×C3—24no—
538C22×C3—44no—
539C22×C3—44no—
540C22×C3C24×C411yescharacteristic
541C22×C3C22×#4112yes—
542C22×C3#58512yes—
543C22×C3#58511yescharacteristic
544C22×C3#58511yescharacteristic
545C2×D3—22no—
546C2×D3—22no—
547C2×D3—22no—
548C2×D3—42no—
549C2×D3—42no—
550C3×C4—22no—
551C3×C4—22no—
552C3×C4—24no—
553C3×C4—42no—
554C3×C4—42no—
555C3×C4—44no—
556C3×C4C2×#11714yes—
557Dic3—22no—
558Dic3—22no—
559Dic3—22no—
560Dic3—24no—
561Dic3—42no—
562Dic3—44no—
563Dic3—42no—
564Dic3C2×#11718yes—
565C23—24no—
566C23—22no—
567C23—24no—
568C23—22no—
569C23—32no—
570C23—62no—
571C23—62no—
572C23—61no—
573C23—61no—
574C23C22×C4×D314yes—
575C23C24×D311yescharacteristic
576Q8—316no—
577D4—28no—
578D4—616no—
579C2×C4—21no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—34no—
593C2×C4—38no—
594C2×C4—32no—
595C2×C4—32no—
596C2×C4—32no—
597C2×C4—34no—
598C2×C4—48no—
599C2×C4—616no—
600C2×C4—62no—
601C2×C4—62no—
602C2×C4—62no—
603C2×C4—61no—
604C2×C4—62no—
605C2×C4—61no—
606C2×C4—68no—
607C2×C4—62no—
608C2×C4—64no—
609C2×C4—62no—
610C2×C4—62no—
611C2×C4—64no—
612C2×C4—62no—
613C2×C4—62no—
614C2×C4—64no—
615C2×C4—62no—
616C2×C4—124no—
617C2×C4—124no—
618C2×C4—128no—
619C2×C4C22×C4×D312yes—
620C2×C4C22×C4×D314yes—
621C2×C4C22×C4×D312yes—
622C2×C4C22×C4×D312yes—
623C2×C4D3×#4114yes—
624C2×C3—44no—
625C2×C3C22×#11711yescharacteristic
626C2×C3#342811yescharacteristic
627C2×C3#342811yescharacteristic
628C2×C3#352412yes—
629C2×C3#352412yes—
630D3—42no—
631C22—24no—
632C22—24no—
633C22—24no—
634C22—44no—
635C22—44no—
636C22—62no—
637C22—62no—
638C22—62no—
639C22—122no—
640C22—122no—
641C22C23×C4×D311yescharacteristic
642C22C2×D3×#4112yes—
643C22#596612yes—
644C22#596611yescharacteristic
645C22#596811yescharacteristic
646C4—22no—
647C4—24no—
648C4—22no—
649C4—38no—
650C4—42no—
651C4—42no—
652C4—44no—
653C4—62no—
654C4—62no—
655C4—64no—
656C4—62no—
657C4—122no—
658C4—124no—
659C4—122no—
660C4D3×#11714yes—
661C3#3419711yescharacteristic, complemented, Sylow
662C2—44no—
663C2—122no—
664C2C2×D3×#11711yescharacteristic
665C2#8585611yescharacteristic
666C2#8588511yescharacteristic
667C2#8647112yes—
668C2#8652812yes—
subgroupG/H
center, FrattiniC23C24×D3
commutatorC22×C3C24×C4
FittingC3×#2319C2
socle, cosocleC23×C3C25
prime radicalC22×#237C22
hypercenter#2319D3
hypocenterC3#34197
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2319

Complements

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

NH ≅ G/Ntype
2. C3×#2319C2semidirect
3. #82325C2semidirect
4. #76303C2semidirect
6. #75982C2semidirect
7. #75979C2semidirect
8. #75476C2semidirect
9. #75473C2semidirect
10. #74055C2semidirect
11. #74051C2semidirect
12. #73870C2semidirect
13. #73863C2semidirect
14. #73752C2semidirect
15. #73616C2semidirect
16. #73413C2semidirect
17. #73383C2semidirect
24. C4×#237C4semidirect
95. #2319D3semidirect
661. C3#34197semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#2372
C3×#23192
#823252
#763032
#763012
#759822
#759792
#754762
#754732
#740552
#740512
#738702
#738632
#737522
#736162
#734132
#733832
#341973