Finite Group Databasebeta

← #10955434

Subgroups — #10955434

Order 768.

2314 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
1C2×C4×#237C211yesmaximal, characteristic
2C3×#2357C211yesmaximal, characteristic, complemented
3#82338C211yesmaximal, characteristic, complemented
4#76301C212yesmaximal
5#76295C212yesmaximal, complemented
6#75473C214yesmaximal, complemented
7#75458C214yesmaximal, complemented
8#74157C211yesmaximal, characteristic, complemented
9#74147C211yesmaximal, characteristic, complemented
10#74061C212yesmaximal, complemented
11#74051C212yesmaximal, complemented
12#73752C211yesmaximal, characteristic, complemented
13#73637C211yesmaximal, characteristic, complemented
14#73633C212yesmaximal, complemented
15#73616C212yesmaximal, complemented
16#73463C212yesmaximal, complemented
17#73437C212yesmaximal, 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×#388C2214yes—
35C3×#382C2211yescharacteristic
36C3×#381C2212yes—
37C3×C4×C4⋊C4C2211yescharacteristic
38C3×C4×C4⋊C4C2212yes—
39C3×C4×C22⋊C4C2212yes—
40#5221C2212yes—
41#5211C2212yes—
42C4×#217C2212yes—
43#4987C2212yes—
44#4987C2212yes—
45#4977C2212yes—
46#4976C2214yes—
47Dic3×C4⋊C4C2212yes—
48Dic3×C4⋊C4C2212yes—
49#4972C2212yes—
50#4957C2214yes—
51#4957C2214yes—
52#4951C2214yes—
53#4945C2214yes—
54#4944C2214yes—
55#4943C2214yes—
56#4942C2214yes—
57#4942C2214yes—
58Dic3×C22⋊C4C2214yes—
59#4940C2214yes—
60C4×(C2×D3)⋊C4C2212yes—
61#4936C2211yescharacteristic
62C4×C4⋊Dic3C2211yescharacteristic
63#4932C2212yes—
64#4932C2214yes—
65C4×Dic3⋊C4C2214yes—
66C4×Dic3⋊C4C2214yes—
67#4928C2212yes—
68#4669C2214yes—
69#4666C2212yes—
70#4664C2211yescharacteristic
71#4664C2211yescharacteristic
72#4664C2212yes—
73#4651C2211yescharacteristic
74#4651C2211yescharacteristic
75#4651C2212yes—
76#4650C2214yes—
77#4650C2212yes—
78#4650C2212yes—
79#4649C2212yes—
80#4649C2212yes—
81#4649C2214yes—
82#4648C2214yes—
83#4648C2212yes—
84#4647C2214yes—
85C2×C4×#41—31no—
86#2357—32no—
87#2357—32no—
88#2357—31no—
89#2357D311yescharacteristic, complemented
90#2351—31no—
91#2351—31no—
92#2346—34no—
93#2346—32no—
94#2341—32no—
95#2319—32no—
96#2319—34no—
97#2283—31no—
98#2283—32no—
99#2283—31no—
100#2283—32no—
101#2276—32no—
102C22×C3×D4C2311yescharacteristic
103C23×C3×C4C2312yes—
104C22×#67C2312yes—
105C23×Dic3C2312yes—
106C23×Dic3C2312yes—
107C2×#237—24no—
108C2×#237—24no—
109C2×#237C2×C418yes—
110C22×C4×D3C2311yescharacteristic
111C22×C4×D3C2311yescharacteristic
112C22×C4×D3C2311yescharacteristic
113C22×Dic6C2311yescharacteristic
114C3×C4×D4C2×C418yes—
115C2×C3×C4⋊C4C2312yes—
116C2×C3×C4⋊C4C2312yes—
117C2×C3×C4⋊C4C2311yescharacteristic
118C2×C3×C22⋊C4C2314yes—
119C2×C3×C22⋊C4C2314yes—
120C2×C3×C22⋊C4C2312yes—
121C2×C3×C42C2312yes—
122C2×C3×C42C2314yes—
123C2×C3×C42C2311yescharacteristic
124C2×#217C2314yes—
125C2×#217C2314yes—
126C2×#217C2312yes—
127D4×Dic3C2×C418yes—
128C4×#67C2×C4116yes—
129C2×(C2×D3)⋊C4C2314yes—
130C2×(C2×D3)⋊C4C2312yes—
131C2×(C2×D3)⋊C4C2312yes—
132C2×(C2×D3)⋊C4C2312yes—
133C2×C4⋊Dic3C2312yes—
134C2×C4⋊Dic3C2312yes—
135C2×C4⋊Dic3C2311yescharacteristic
136C2×Dic3⋊C4C2314yes—
137C2×Dic3⋊C4C2314yes—
138C2×Dic3⋊C4C2314yes—
139C2×Dic3⋊C4C2314yes—
140C2×Dic3⋊C4C2312yes—
141C2×Dic3⋊C4C2312yes—
142C2×C4×Dic3C2×C4116yes—
143C2×C4×Dic3C2314yes—
144C2×C4×Dic3C2314yes—
145C2×C4×Dic3C2312yes—
146C2×C4×Dic3C2314yes—
147C2×C4×Dic3C2314yes—
148C2×C4×Dic3C2312yes—
149C2×C4×Dic3C2311yescharacteristic
150C2×C4×Dic3C2312yes—
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×#95C2312yes—
163C3×#95C2314yes—
164C3×#95C2311yescharacteristic
165C3×#95C2312yes—
166C3×#95C2311yescharacteristic
167C3⋊#95C2314yes—
168C3⋊#95C2312yes—
169C3⋊#95C2314yes—
170C3⋊#95C2314yes—
171C3⋊#95C2314yes—
172C3⋊#95C2312yes—
173C3⋊#95C2311yescharacteristic
174C3⋊#95C2312yes—
175C3⋊#95C2312yes—
176C3⋊#95C2311yescharacteristic
177C3⋊#95C2312yes—
178C3⋊#95C2314yes—
179C3⋊#95C2312yes—
180C3⋊#95C2312yes—
181C3⋊#95C2312yes—
182C22×#41—31no—
183C4×#41—316no—
184C2×C4×Q8—31no—
185C2×C4×Q8—32no—
186C2×C4×Q8—31no—
187C2×C4×Q8—32no—
188C2×C4×D4—31no—
189C2×C4×D4—32no—
190C2×C4×D4—32no—
191C2×C4×D4C2×D311yescharacteristic
192C2×#117—31no—
193C2×#117—32no—
194C22×C42—32no—
195C22×C42—31no—
196#390—32no—
197#390—34no—
198#390—34no—
199#390C2×D312yes—
200#389—32no—
201#389—34no—
202#389—32no—
203#389—34no—
204#388—34no—
205#388—34no—
206#388—34no—
207#388—34no—
208#388—34no—
209#388C2×D314yes—
210#383—31no—
211#383—32no—
212#383—31no—
213#382—32no—
214#382—31no—
215#382—34no—
216#382—32no—
217#382—32no—
218#382C2×D311yescharacteristic
219#381—32no—
220#381—34no—
221#381—32no—
222#381—31no—
223#381—31no—
224#381C2×D312yes—
225#380—32no—
226#380—32no—
227#380—34no—
228#380—34no—
229C4×C4⋊C4—32no—
230C4×C4⋊C4—32no—
231C4×C4⋊C4—34no—
232C4×C4⋊C4—34no—
233C4×C4⋊C4—31no—
234C4×C4⋊C4—34no—
235C4×C4⋊C4—32no—
236C4×C4⋊C4—32no—
237C4×C4⋊C4C2×D312yes—
238C4×C4⋊C4C2×D311yescharacteristic
239C4×C22⋊C4—32no—
240C4×C22⋊C4—34no—
241C4×C22⋊C4—32no—
242C4×C22⋊C4—32no—
243C4×C22⋊C4C2×D312yes—
244#376—32no—
245#376—34no—
246#376—34no—
247#376—32no—
248C24×C3C2412yes—
249C23×D3C2411yescharacteristic
250C2×C3×D4—22no—
251C2×C3×D4—22no—
252C2×C3×D4C22×C414yes—
253C22×C3×C4—44no—
254C22×C3×C4C22×C418yes—
255C22×C3×C4C2412yes—
256C22×C3×C4C2412yes—
257C22×C3×C4C2414yes—
258C22×C3×C4C2411yescharacteristic
259C22×C3×C4C2412yes—
260C22×C3×C4C2411yescharacteristic
261C22×C3×C4C2411yescharacteristic
262C2×#67—24no—
263C2×#67—24no—
264C2×#67C22×C418yes—
265C22×Dic3—24no—
266C22×Dic3—24no—
267C22×Dic3—44no—
268C22×Dic3C22×C418yes—
269C22×Dic3C22×C418yes—
270C22×Dic3C2412yes—
271C22×Dic3C2412yes—
272C22×Dic3C2414yes—
273C22×Dic3C2411yescharacteristic
274C22×Dic3C2411yescharacteristic
275C22×Dic3C2411yescharacteristic
276C22×Dic3C2412yes—
277C22×Dic3C2412yes—
278#237—232no—
279C2×C4×D3—22no—
280C2×C4×D3—22no—
281C2×C4×D3—42no—
282C2×C4×D3—42no—
283C2×C4×D3C22×C414yes—
284C2×C4×D3C22×C414yes—
285C2×C4×D3C22×C414yes—
286C2×Dic6C22×C414yes—
287C2×Dic6#4114yes—
288C2×Dic6#4114yes—
289C3×C4⋊C4—24no—
290C3×C4⋊C4—42no—
291C3×C4⋊C4C22×C414yes—
292C3×C22⋊C4—44no—
293C3×C22⋊C4—44no—
294C3×C22⋊C4C22×C418yes—
295C3×C42—24no—
296C3×C42—44no—
297C3×C42C22×C414yes—
298#217—44no—
299#217—44no—
300#217C22×C418yes—
301(C2×D3)⋊C4—42no—
302(C2×D3)⋊C4—44no—
303(C2×D3)⋊C4—42no—
304(C2×D3)⋊C4C22×C418yes—
305C4⋊Dic3—24no—
306C4⋊Dic3—42no—
307C4⋊Dic3C22×C414yes—
308Dic3⋊C4—28no—
309Dic3⋊C4—28no—
310Dic3⋊C4—44no—
311Dic3⋊C4—44no—
312Dic3⋊C4C22×C418yes—
313Dic3⋊C4C22×C418yes—
314C4×Dic3—24no—
315C4×Dic3—28no—
316C4×Dic3—28no—
317C4×Dic3—44no—
318C4×Dic3—44no—
319C4×Dic3C22×C418yes—
320C4×Dic3C22×C414yes—
321C4×Dic3C22×C414yes—
322C4×Dic3C22×C414yes—
323C4×Dic3C22×C418yes—
324C4×Dic3C22×C418yes—
325C2×#41—38no—
326C2×#41—64no—
327C2×#41—64no—
328C22×Q8—31no—
329C22×D4—32no—
330C22×D4C22×D311yescharacteristic
331C23×C4—32no—
332C23×C4—32no—
333C23×C4—31no—
334C23×C4—31no—
335C23×C4—31no—
336C23×C4C22×D312yes—
337C4×Q8—38no—
338C4×Q8—38no—
339C4×Q8—68no—
340C4×Q8—68no—
341C4×D4—38no—
342C4×D4—316no—
343C4×D4—316no—
344C4×D4C4×D318yes—
345#117—38no—
346#117—316no—
347C2×C4⋊C4—32no—
348C2×C4⋊C4—32no—
349C2×C4⋊C4—31no—
350C2×C4⋊C4—34no—
351C2×C4⋊C4—34no—
352C2×C4⋊C4—32no—
353C2×C4⋊C4—34no—
354C2×C4⋊C4—34no—
355C2×C4⋊C4—32no—
356C2×C4⋊C4C22×D312yes—
357C2×C4⋊C4C22×D312yes—
358C2×C4⋊C4C22×D311yescharacteristic
359C2×C22⋊C4—32no—
360C2×C22⋊C4—32no—
361C2×C22⋊C4—34no—
362C2×C22⋊C4—34no—
363C2×C22⋊C4—32no—
364C2×C22⋊C4—32no—
365C2×C22⋊C4—34no—
366C2×C22⋊C4C22×D314yes—
367C2×C22⋊C4C22×D314yes—
368C2×C22⋊C4C22×D312yes—
369C2×C42—316no—
370C2×C42—31no—
371C2×C42—32no—
372C2×C42—32no—
373C2×C42—31no—
374C2×C42—32no—
375C2×C42—31no—
376C2×C42—38no—
377C2×C42—34no—
378C2×C42—34no—
379C2×C42—32no—
380C2×C42—34no—
381C2×C42—34no—
382C2×C42C22×D312yes—
383C2×C42C22×D314yes—
384C2×C42C22×D311yescharacteristic
385#95—32no—
386#95—32no—
387#95—34no—
388#95—32no—
389#95—34no—
390#95—32no—
391#95—32no—
392#95—34no—
393#95—34no—
394#95—34no—
395#95—32no—
396#95—32no—
397#95—32no—
398#95—31no—
399#95—31no—
400#95C22×D314yes—
401#95C22×D312yes—
402#95C22×D312yes—
403#95C22×D311yescharacteristic
404#95C22×D311yescharacteristic
405C23×C3—22no—
406C23×C3—22no—
407C23×C3—24no—
408C23×C3—24no—
409C23×C3C23×C414yes—
410C23×C3C2511yescharacteristic
411C22×D3—21no—
412C22×D3—21no—
413C22×D3—22no—
414C22×D3—22no—
415C22×D3C23×C412yes—
416C3×D4—28no—
417C2×C3×C4—24no—
418C2×C3×C4—22no—
419C2×C3×C4—22no—
420C2×C3×C4—22no—
421C2×C3×C4—24no—
422C2×C3×C4—22no—
423C2×C3×C4—24no—
424C2×C3×C4—22no—
425C2×C3×C4—24no—
426C2×C3×C4—22no—
427C2×C3×C4—22no—
428C2×C3×C4—22no—
429C2×C3×C4—48no—
430C2×C3×C4C23×C412yes—
431C2×C3×C4C23×C414yes—
432C2×C3×C4C23×C412yes—
433C2×C3×C4C23×C412yes—
434C2×C3×C4C2×#4112yes—
435C2×C3×C4C2×#4112yes—
436#67—216no—
437C2×Dic3—21no—
438C2×Dic3—21no—
439C2×Dic3—22no—
440C2×Dic3—24no—
441C2×Dic3—22no—
442C2×Dic3—216no—
443C2×Dic3—22no—
444C2×Dic3—22no—
445C2×Dic3—24no—
446C2×Dic3—22no—
447C2×Dic3—24no—
448C2×Dic3—22no—
449C2×Dic3—24no—
450C2×Dic3—22no—
451C2×Dic3—22no—
452C2×Dic3—48no—
453C2×Dic3C23×C412yes—
454C2×Dic3C23×C412yes—
455C2×Dic3C23×C412yes—
456C2×Dic3C23×C414yes—
457C2×Dic3C23×C414yes—
458C2×Dic3C2×#4114yes—
459C2×Dic3C2×#4114yes—
460C4×D3—28no—
461C4×D3—44no—
462C4×D3—44no—
463Dic6#117116yes—
464C24—31no—
465C24C23×D312yes—
466#41—632no—
467C2×Q8—34no—
468C2×Q8—34no—
469C2×Q8—34no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—38no—
473C2×D4—64no—
474C2×D4—64no—
475C2×D4C2×C4×D314yes—
476C22×C4—32no—
477C22×C4—38no—
478C22×C4—31no—
479C22×C4—31no—
480C22×C4—31no—
481C22×C4—32no—
482C22×C4—38no—
483C22×C4—34no—
484C22×C4—34no—
485C22×C4—34no—
486C22×C4—32no—
487C22×C4—32no—
488C22×C4—34no—
489C22×C4—44no—
490C22×C4—64no—
491C22×C4—64no—
492C22×C4—62no—
493C22×C4—62no—
494C22×C4—122no—
495C22×C4—122no—
496C22×C4—124no—
497C22×C4C2×C4×D318yes—
498C22×C4C23×D312yes—
499C22×C4C23×D312yes—
500C22×C4C23×D314yes—
501C22×C4C23×D311yescharacteristic
502C22×C4C23×D312yes—
503C22×C4C23×D311yescharacteristic
504C22×C4C23×D311yescharacteristic
505C4⋊C4—24no—
506C4⋊C4—38no—
507C4⋊C4—38no—
508C4⋊C4—34no—
509C4⋊C4—42no—
510C4⋊C4—68no—
511C4⋊C4—68no—
512C4⋊C4—64no—
513C4⋊C4—124no—
514C4⋊C4—124no—
515C4⋊C4—122no—
516C4⋊C4C2×C4×D314yes—
517C22⋊C4—38no—
518C22⋊C4—38no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—124no—
522C22⋊C4—122no—
523C22⋊C4—122no—
524C22⋊C4—124no—
525C22⋊C4—124no—
526C22⋊C4C2×C4×D318yes—
527C42—24no—
528C42—34no—
529C42—38no—
530C42—34no—
531C42—38no—
532C42—38no—
533C42—34no—
534C42—44no—
535C42—68no—
536C42—68no—
537C42—64no—
538C42—124no—
539C42—124no—
540C42C2×C4×D314yes—
541C22×C3—24no—
542C22×C3—24no—
543C22×C3—24no—
544C22×C3—44no—
545C22×C3—44no—
546C22×C3C24×C411yescharacteristic
547C22×C3C22×#4111yescharacteristic
548C22×C3C22×#4111yescharacteristic
549C22×C3#58512yes—
550C22×C3#58512yes—
551C2×D3—22no—
552C2×D3—22no—
553C2×D3—22no—
554C2×D3—42no—
555C2×D3—42no—
556C3×C4—22no—
557C3×C4—22no—
558C3×C4—24no—
559C3×C4—44no—
560C3×C4—42no—
561C3×C4—42no—
562C3×C4C2×#11714yes—
563Dic3—22no—
564Dic3—22no—
565Dic3—22no—
566Dic3—24no—
567Dic3—42no—
568Dic3—42no—
569Dic3—44no—
570Dic3C2×#11718yes—
571C23—22no—
572C23—22no—
573C23—24no—
574C23—24no—
575C23—32no—
576C23—61no—
577C23—61no—
578C23—62no—
579C23—62no—
580C23C22×C4×D314yes—
581C23C24×D311yescharacteristic
582Q8—316no—
583D4—28no—
584D4—616no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—34no—
598C2×C4—34no—
599C2×C4—34no—
600C2×C4—32no—
601C2×C4—32no—
602C2×C4—34no—
603C2×C4—32no—
604C2×C4—48no—
605C2×C4—616no—
606C2×C4—61no—
607C2×C4—61no—
608C2×C4—64no—
609C2×C4—62no—
610C2×C4—62no—
611C2×C4—68no—
612C2×C4—62no—
613C2×C4—62no—
614C2×C4—64no—
615C2×C4—62no—
616C2×C4—64no—
617C2×C4—62no—
618C2×C4—62no—
619C2×C4—64no—
620C2×C4—62no—
621C2×C4—124no—
622C2×C4—124no—
623C2×C4—128no—
624C2×C4C22×C4×D312yes—
625C2×C4C22×C4×D314yes—
626C2×C4C22×C4×D312yes—
627C2×C4C22×C4×D312yes—
628C2×C4D3×#4112yes—
629C2×C4D3×#4112yes—
630C2×C3—44no—
631C2×C3C22×#11711yescharacteristic
632C2×C3#342811yescharacteristic
633C2×C3#342811yescharacteristic
634C2×C3#352412yes—
635C2×C3#352412yes—
636D3—42no—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—44no—
641C22—44no—
642C22—62no—
643C22—62no—
644C22—62no—
645C22—122no—
646C22—122no—
647C22C23×C4×D311yescharacteristic
648C22C2×D3×#4111yescharacteristic
649C22C2×D3×#4111yescharacteristic
650C22#596412yes—
651C22#596612yes—
652C4—22no—
653C4—22no—
654C4—24no—
655C4—38no—
656C4—44no—
657C4—42no—
658C4—42no—
659C4—62no—
660C4—64no—
661C4—62no—
662C4—62no—
663C4—122no—
664C4—122no—
665C4—124no—
666C4D3×#11714yes—
667C3#3419711yescharacteristic, complemented, Sylow
668C2—44no—
669C2—122no—
670C2C2×D3×#11711yescharacteristic
671C2#8583411yescharacteristic
672C2#8585611yescharacteristic
673C2#8689112yes—
674C2#8705612yes—
subgroupG/H
center, FrattiniC23C24×D3
commutatorC22×C3C24×C4
FittingC3×#2357C2
socle, cosocleC23×C3C25
prime radicalC22×#237C22
hypercenter#2357D3
hypocenterC3#34197
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2357

Complements

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

NH ≅ G/Ntype
2. C3×#2357C2semidirect
3. #82338C2semidirect
5. #76295C2semidirect
6. #75473C2semidirect
7. #75458C2semidirect
8. #74157C2semidirect
9. #74147C2semidirect
10. #74061C2semidirect
11. #74051C2semidirect
12. #73752C2semidirect
13. #73637C2semidirect
14. #73633C2semidirect
15. #73616C2semidirect
16. #73463C2semidirect
17. #73437C2semidirect
24. C4×#237C4semidirect
89. #2357D3semidirect
667. 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×#23572
#823382
#763012
#762952
#754732
#754582
#741572
#741472
#740612
#740512
#737522
#736372
#736332
#736162
#734632
#734372
#341973