Finite Group Databasebeta

← #10823633

Subgroups — #10823633

Order 768.

1074 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3461C211yesmaximal, characteristic
2C4×#951C212yesmaximal, complemented
3C4×#944C211yesmaximal, characteristic, complemented
4#71905C211yesmaximal, characteristic, complemented
5#71900C211yesmaximal, characteristic, complemented
6#71881C211yesmaximal, characteristic
7#71876C211yesmaximal, characteristic
8#71868C212yesmaximal, complemented
9#71863C212yesmaximal, complemented
10C4×#837C212yesmaximal, complemented
11C4×#836C211yesmaximal, characteristic, complemented
12#23590—31nomaximal, Sylow
13C3×#544C2211yescharacteristic
14C3×#542—22no—
15C3×#541—22no—
16C3×#539—22no—
17C3×#537C2211yescharacteristic
18C3×#536C2212yes—
19C3×#533—22no—
20C3×#529—22no—
21C3×#529—22no—
22C3×C4×#41C2212yes—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×#951C2212yes—
25C2×#944C2211yescharacteristic
26C2×C4×Dic6C2211yescharacteristic
27C2×#837C2212yes—
28C2×#837C2212yes—
29#5232C2212yes—
30#5229C2211yescharacteristic
31C2×#836C2212yes—
32#5030—22no—
33C4×#216—22no—
34#5018—22no—
35C4×#214—22no—
36#5004C2212yes—
37#4995C2212yes—
38#4976—22no—
39#4975C2211yescharacteristic
40#4973C2211yescharacteristic
41C4×Dic3⋊C4—22no—
42C42×Dic3C2211yescharacteristic
43C4×#208C2212yes—
44#4532C2212yes—
45#4532C2212yes—
46#4530C2211yescharacteristic
47#4530C2211yescharacteristic
48#3461D311yescharacteristic
49C4×#137—32no—
50C4×C4⋊Q8—31no—
51#2060—31no—
52#2039—31no—
53#2038—32no—
54#2028—31no—
55#2007—31no—
56#2005—32no—
57C4×#104—32no—
58C4×#101—31no—
59C2×C3×#41C2312yes—
60C22×C3×Q8C2311yescharacteristic
61C22×Dic6C2311yescharacteristic
62C3×C4⋊Q8—22no—
63C3×C4⋊Q8—22no—
64C3×#126—24no—
65C3×#126—24no—
66C3×#125—22no—
67C3×#125—24no—
68C3×#125—22no—
69C3×#124—22no—
70C3×#124—22no—
71C3×#123—24no—
72C3×#123—24no—
73C3×C22⋊Q8—24no—
74C3×C22⋊Q8—22no—
75C3×C22⋊Q8—22no—
76C3×C22⋊Q8—22no—
77C3×C22⋊Q8—22no—
78C3×#121—22no—
79C3×#121—22no—
80C3×C4×Q8—22no—
81C3×C4×Q8—22no—
82C3×C4×Q8—22no—
83C3×C4×Q8D412yes—
84C3×C4×Q8D412yes—
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×C4×D4D412yes—
89C3×C4×D4D412yes—
90C3×#117—24no—
91C3×#117—22no—
92C3×#117C2312yes—
93C2×C3×C4⋊C4D412yes—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C42—22no—
96C2×C3×C42D412yes—
97C2×C3×C42C2311yescharacteristic
98#951—24no—
99#951—24no—
100#944—22no—
101#944—22no—
102C2×#216—22no—
103C2×#214—22no—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—22no—
106C2×Dic3⋊C4—22no—
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3C2311yescharacteristic
112C2×C4×Dic3C2311yescharacteristic
113C2×#208C2312yes—
114C2×#208C2312yes—
115C4×Dic6—22no—
116C4×Dic6—22no—
117#837—24no—
118#837—24no—
119#836—24no—
120#835—22no—
121#832—22no—
122C3⋊#95—22no—
123C3⋊#95—22no—
124#810—22no—
125#810—22no—
126#808—22no—
127#807—22no—
128#803—22no—
129C4×C3⋊C8—22no—
130C2×#137—32no—
131#544C2×D311yescharacteristic
132#542—22no—
133#541—22no—
134#539—22no—
135#537C2×D311yescharacteristic
136#536C2×D312yes—
137#533—22no—
138C2×C4⋊Q8—31no—
139#529—22no—
140#529—22no—
141C4×#41C2×D312yes—
142C2×C4×Q8—31no—
143C2×C4×Q8C2×D311yescharacteristic
144#441—62no—
145#440—62no—
146C4×Q16—62no—
147C4×SD16—62no—
148#428—32no—
149C2×#104—32no—
150C2×#104—32no—
151#419—32no—
152#417—32no—
153C2×#101—32no—
154C4×M16—32no—
155#391—31no—
156#391—31no—
157#389—62no—
158#384—31no—
159C4×C4⋊C4—62no—
160C43—31no—
161#343—31no—
162#343—31no—
163#339—32no—
164#339—32no—
165C3×#41—22no—
166C3×#41—22no—
167C3×#41—22no—
168C3×#41—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8C2×D412yes—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4C2×D412yes—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2411yescharacteristic
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C2×Dic6—22no—
191C2×Dic6—22no—
192C2×Dic6—21no—
193C2×Dic6#4112yes—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—24no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4C2×D412yes—
204C3×C4⋊C4C2×D412yes—
205C3×C22⋊C4—24no—
206C3×C22⋊C4—24no—
207C3×C22⋊C4—24no—
208C3×C22⋊C4—22no—
209C3×C22⋊C4—22no—
210C3×C22⋊C4—22no—
211C3×C42—22no—
212C3×C42—22no—
213C3×C42—22no—
214C3×C42—22no—
215C3×C42—22no—
216C3×C42C2×D412yes—
217C3×C42C2×D412yes—
218#216—28no—
219#214—28no—
220C4⋊Dic3—22no—
221Dic3⋊C4—28no—
222Dic3⋊C4—44no—
223Dic3⋊C4—42no—
224C4×Dic3—24no—
225C4×Dic3—21no—
226C4×Dic3—22no—
227C4×Dic3—42no—
228C4×Dic3—41no—
229C4×Dic3—41no—
230C4×Dic3#4112yes—
231#208—22no—
232#208—22no—
233#208—22no—
234#208#4114yes—
235C2×C3⋊C8—22no—
236C2×C3⋊C8—22no—
237C2×#41C22×D312yes—
238C22×Q8—31no—
239C22×Q8C22×D311yescharacteristic
240#137—64no—
241#137—64no—
242C2×Q16—62no—
243C2×SD16—62no—
244C2×M16—32no—
245C2×M16—32no—
246C4⋊Q8—22no—
247C4⋊Q8—22no—
248C4⋊Q8—62no—
249C4⋊Q8—62no—
250#126—24no—
251#126—24no—
252#125—22no—
253#125—24no—
254#125—22no—
255#124—22no—
256#124—22no—
257#123—24no—
258#123—24no—
259C22⋊Q8—24no—
260C22⋊Q8—22no—
261C22⋊Q8—22no—
262C22⋊Q8—22no—
263C22⋊Q8—22no—
264#121—22no—
265#121—22no—
266C4×Q8—22no—
267C4×Q8—22no—
268C4×Q8—22no—
269C4×Q8—62no—
270C4×Q8—62no—
271C4×Q8#6712yes—
272C4×Q8#6712yes—
273C4×D4—22no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4#6712yes—
277C4×D4#6712yes—
278#117—24no—
279#117—22no—
280#117C22×D312yes—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—62no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—62no—
285C2×C4⋊C4#6712yes—
286C2×C4⋊C4C22×D311yescharacteristic
287C2×C42—22no—
288C2×C42—31no—
289C2×C42—31no—
290C2×C42—61no—
291C2×C42—61no—
292C2×C42—61no—
293C2×C42#6712yes—
294C2×C42C22×D311yescharacteristic
295#107—62no—
296#106—62no—
297#104—64no—
298#104—64no—
299#103—62no—
300#103—62no—
301#103—62no—
302#102—62no—
303#101—64no—
304#97—62no—
305C4×C8—62no—
306#95—62no—
307#95—62no—
308C23×C3—22no—
309C23×C3C22×D411yescharacteristic
310C3×Q8—22no—
311C3×Q8—22no—
312C3×Q8—42no—
313C3×Q8—42no—
314C3×Q8—82no—
315C3×D4—22no—
316C3×D4—22no—
317C3×D4—42no—
318C3×D4—42no—
319C3×D4—82no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—41no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—44no—
332C2×C3×C4—42no—
333C2×C3×C4—82no—
334C2×C3×C4#12011yescharacteristic
335C2×C3×C4#12012yes—
336C2×C3×C4#12011yescharacteristic
337C2×C3×C4C22×D412yes—
338C2×C3×C4C2×#4111yescharacteristic
339C2×C3×C4C2×#4112yes—
340C2×C3×C4C2×#4111yescharacteristic
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—22no—
345C2×Dic3—22no—
346C2×Dic3—41no—
347C2×Dic3—41no—
348C2×Dic3—41no—
349C2×Dic3—41no—
350C2×Dic3—42no—
351C2×Dic3—41no—
352Dic6—24no—
353Dic6—42no—
354C3⋊C8—24no—
355C3⋊C8—42no—
356#41—22no—
357#41—22no—
358#41—22no—
359#41—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—22no—
365C2×Q8—32no—
366C2×Q8—61no—
367C2×Q8—62no—
368C2×Q8—62no—
369C2×Q8C2×#6712yes—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4C2×#6712yes—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—61no—
377C22×C4—61no—
378C22×C4—61no—
379C22×C4—61no—
380C22×C4C2×#6712yes—
381C22×C4C2×#6711yescharacteristic
382C22×C4C2×#6712yes—
383C22×C4C2×#6711yescharacteristic
384C22×C4C23×D311yescharacteristic
385Q16—68no—
386SD16—68no—
387M16—34no—
388M16—62no—
389M16—62no—
390M16—62no—
391C2×C8—62no—
392C2×C8—62no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—68no—
403C4⋊C4—62no—
404C4⋊C4—124no—
405C4⋊C4—122no—
406C4⋊C4C2×#6712yes—
407C4⋊C4C2×#6712yes—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—32no—
420C42—64no—
421C42—61no—
422C42—62no—
423C42—122no—
424C42—121no—
425C42—121no—
426C42C2×#6712yes—
427C42C2×#6712yes—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—42no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3C2×#12011yescharacteristic
434C22×C3#52511yescharacteristic
435C22×C3#53211yescharacteristic
436C3×C4—21no—
437C3×C4—22no—
438C3×C4—42no—
439C3×C4—42no—
440C3×C4—42no—
441C3×C4—42no—
442C3×C4—42no—
443C3×C4#45612yes—
444C3×C4#52512yes—
445C3×C4#53212yes—
446Dic3—24no—
447Dic3—42no—
448Dic3—42no—
449Dic3—81no—
450C23—22no—
451C23C22×#6711yescharacteristic
452Q8—22no—
453Q8—22no—
454Q8—42no—
455Q8—42no—
456Q8—64no—
457Q8—82no—
458Q8—122no—
459D4—22no—
460D4—22no—
461D4—42no—
462D4—42no—
463D4—82no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—41no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—62no—
478C2×C4—62no—
479C2×C4—62no—
480C2×C4—62no—
481C2×C4—62no—
482C2×C4—82no—
483C2×C4—121no—
484C2×C4—121no—
485C2×C4—121no—
486C2×C4—121no—
487C2×C4—122no—
488C2×C4—121no—
489C2×C4#95311yescharacteristic
490C2×C4#95312yes—
491C2×C4#95311yescharacteristic
492C2×C4D3×#4111yescharacteristic
493C2×C4D3×#4112yes—
494C2×C4D3×#4111yescharacteristic
495C2×C4C22×#6712yes—
496C8—64no—
497C8—122no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—82no—
501C2×C3#238811yescharacteristic
502C2×C3C2×#45611yescharacteristic
503C2×C3#301811yescharacteristic
504C22—21no—
505C22—21no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22#582811yescharacteristic
510C22#583211yescharacteristic
511C22C2×#95311yescharacteristic
512C4—21no—
513C4—22no—
514C4—42no—
515C4—42no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—64no—
520C4—122no—
521C4—122no—
522C4—241no—
523C4#524612yes—
524C4#582812yes—
525C4#583212yes—
526C3#2359011yescharacteristic, complemented, Sylow
527C2—21no—
528C2—21no—
529C2—82no—
530C2#8409511yescharacteristic
531C2C2×#524611yescharacteristic
532C2#8412111yescharacteristic
subgroupG/H
centerC2×C4#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3461C2
socleC22×C3C2×#120
prime radicalC3×#537C22
hypercenter#3461D3
hypocenterC3#23590
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3461

Complements

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

NH ≅ G/Ntype
2. C4×#951C2semidirect
3. C4×#944C2semidirect
4. #71905C2semidirect
5. #71900C2semidirect
8. #71868C2semidirect
9. #71863C2semidirect
10. C4×#837C2semidirect
11. C4×#836C2semidirect
526. C3#23590semidirect

Maximal subgroups

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

subgroupindex
C3×#34612
C4×#9512
C4×#9442
#719052
#719002
#718812
#718762
#718682
#718632
C4×#8372
C4×#8362
#235903