Finite Group Databasebeta

← #10767971

Subgroups — #10767971

Order 768.

1106 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86280C211yesmaximal, characteristic, complemented
2C3×#3110C211yesmaximal, characteristic, complemented
3#82347C211yesmaximal, characteristic
4#82315C211yesmaximal, characteristic, complemented
5#71836C211yesmaximal, characteristic, complemented
6#71176C214yesmaximal, complemented
7#70976C214yesmaximal, complemented
8#70836C211yesmaximal, characteristic, complemented
9#70827C211yesmaximal, characteristic, complemented
10#18180—31nomaximal, Sylow
11C3×#533C2211yescharacteristic
12C2×C3×C4×D4C2211yescharacteristic
13C2×C4×#67—22no—
14#5730—21no—
15#5727—21no—
16#5690—22no—
17#5634C2214yes—
18D3×C22⋊Q8—22no—
19#5579—22no—
20#5579—21no—
21Q8×D12—21no—
22#5521—21no—
23#5519—21no—
24C2×C4×D12C2211yescharacteristic
25#5470C2211yescharacteristic
26C3×#484C2212yes—
27C3×#474—21no—
28C3×#461—21no—
29C3×#450C2212yes—
30C3×#440—22no—
31C3×#432C2211yescharacteristic
32#5223C2212yes—
33#5218C2212yes—
34#5019—21no—
35#5018—22no—
36#5007—21no—
37#4993C2212yes—
38#4970C2212yes—
39#4960—21no—
40#4949—21no—
41#4932C2211yescharacteristic
42#4926C2211yescharacteristic
43#4538C2214yes—
44#4468C2214yes—
45#4467C2214yes—
46#3501—31no—
47#3110—31no—
48#3110D311yescharacteristic, complemented
49#2397—31no—
50#1614—34no—
51#1596—34no—
52#1506—31no—
53#1506—31no—
54#1466—31no—
55C22×C3×D4C2311yescharacteristic
56C23×C3×C4—21no—
57C22×#67—22no—
58C2×D3×Q8—22no—
59C22×D12D411yescharacteristic
60C22×C4×D3D412yes—
61C2×C3×SD16—22no—
62C3×C4⋊Q8D411yescharacteristic, complemented
63C3×C4⋊Q8D411yescharacteristic, complemented
64C3×#125—21no—
65C3×C22⋊Q8—21no—
66C3×C22⋊Q8C2312yes—
67C3×C4×Q8—21no—
68C3×C4×Q8D412yescomplemented
69C3×C4×D4—21no—
70C3×C4×D4—21no—
71C3×C4×D4—21no—
72C3×C4×D4—21no—
73C3×#117—21no—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C22⋊C4—21no—
76C2×C3×C42C2311yescharacteristic
77C2×#217—22no—
78#946—22no—
79#946—24no—
80C2×#214—22no—
81C4×#67—48no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4C2312yes—
84#926—21no—
85C2×C4⋊Dic3D411yescharacteristic
86#924—21no—
87#924C2312yes—
88C2×Dic3⋊C4—22no—
89C2×C4×Dic3—21no—
90#897—22no—
91#897—24no—
92#896—24no—
93#896—24no—
94#896—24no—
95#893—22no—
96#893—22no—
97D3×C4⋊C4—24no—
98D3×C4⋊C4—22no—
99#890—22no—
100#888—22no—
101D3×C22⋊C4—24no—
102#878—24no—
103C4×D12—22no—
104C4×D12—22no—
105C4×D12—21no—
106C4×D12—21no—
107C4×D12—21no—
108C4×D12—21no—
109#870—22no—
110#870—21no—
111#869—22no—
112#869D411yescharacteristic, complemented
113#869D411yescharacteristic, complemented
114C4×Dic6—21no—
115C4×Dic6D412yescomplemented
116C3×#107—22no—
117C3×C4⋊C8—21no—
118C3×#103—22no—
119C3×#102—22no—
120C3×C22⋊C8C2312yes—
121C3×#97—21no—
122#832—22no—
123C3⋊#95—21no—
124C3⋊#95C2312yes—
125#830C2312yes—
126#810—22no—
127#807—22no—
128C4⋊(C3⋊C8)—21no—
129#803—21no—
130#554—61no—
131#554—62no—
132#554—61no—
133#554—62no—
134D4×Q8—61no—
135D4×Q8—61no—
136#544—61no—
137#543—34no—
138#533—31no—
139#533C2×D311yescharacteristic
140C2×C22⋊Q8—62no—
141#518—61no—
142C2×C4×D4—31no—
143C2×C4×D4—62no—
144C2×C4×D4C2×D311yescharacteristic
145#484—32no—
146#484C2×D312yes—
147#474—21no—
148#474—61no—
149#461—21no—
150#461—61no—
151#450—32no—
152#450C2×D312yes—
153#440—22no—
154#440—62no—
155#432—31no—
156#432C2×D311yescharacteristic
157#399—32no—
158#399—61no—
159#392—32no—
160#392—61no—
161#376—31no—
162#328—34no—
163#328—34no—
164#323—34no—
165C24×C3—21no—
166C23×D3C2×D412yes—
167C2×C3×Q8—21no—
168C2×C3×Q8C2×D412yes—
169C2×C3×D4—21no—
170C2×C3×D4—21no—
171C2×C3×D4—42no—
172C22×C3×C4—21no—
173C22×C3×C4—44no—
174C22×C3×C4—41no—
175C22×C3×C4—41no—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2411yescharacteristic
179C2×#67—44no—
180C2×#67—44no—
181C2×#67—44no—
182C22×Dic3—22no—
183C22×Dic3C2×D412yes—
184D3×Q8—84no—
185C2×D12—24no—
186C2×D12—21no—
187C2×D12—21no—
188C2×C4×D3—22no—
189C2×C4×D3—24no—
190C2×C4×D3—24no—
191C2×C4×D3—22no—
192C2×C4×D3—44no—
193C2×Dic6—21no—
194C2×Dic6—24no—
195C2×Dic6C2×D412yes—
196C3×SD16—44no—
197C2×C3×C8—22no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4C2×D412yes—
206C3×C22⋊C4—22no—
207C3×C22⋊C4—41no—
208C3×C22⋊C4—41no—
209C3×C42—22no—
210C3×C42C2×D411yescharacteristic
211C3×C42C2×D411yescharacteristic
212C3×C42C2×D411yescharacteristic
213C3×C42C2×D411yescharacteristic
214#217—22no—
215#217—44no—
216#214—44no—
217(C2×D3)⋊C4—24no—
218(C2×D3)⋊C4—24no—
219(C2×D3)⋊C4—22no—
220(C2×D3)⋊C4—22no—
221(C2×D3)⋊C4—44no—
222C4⋊Dic3—24no—
223C4⋊Dic3—21no—
224C4⋊Dic3—21no—
225C4⋊Dic3—21no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3C2×D412yes—
229Dic3⋊C4—24no—
230Dic3⋊C4—24no—
231Dic3⋊C4—22no—
232Dic3⋊C4—22no—
233Dic3⋊C4—44no—
234C4×Dic3—24no—
235C4×Dic3—22no—
236C2×C3⋊C8—22no—
237C22×Q8—62no—
238C22×D4—31no—
239C22×D4—62no—
240C22×D4C22×D311yescharacteristic
241C23×C4—21no—
242C23×C4—32no—
243C2×SD16—22no—
244C2×SD16—62no—
245C4⋊Q8—31no—
246C4⋊Q8—31no—
247C4⋊Q8—62no—
248C4⋊Q8—62no—
249C4⋊Q8#6711yescharacteristic, complemented
250C4⋊Q8#6711yescharacteristic, complemented
251#125—21no—
252#125—62no—
253#125—62no—
254#125—61no—
255C22⋊Q8—21no—
256C22⋊Q8—32no—
257C22⋊Q8—64no—
258C22⋊Q8—64no—
259C22⋊Q8—64no—
260C22⋊Q8—62no—
261C22⋊Q8—64no—
262C22⋊Q8—62no—
263C22⋊Q8—64no—
264C22⋊Q8—61no—
265C22⋊Q8—64no—
266C22⋊Q8C22×D312yes—
267C4×Q8—21no—
268C4×Q8—32no—
269C4×Q8—61no—
270C4×Q8#6712yescomplemented
271C4×D4—21no—
272C4×D4—21no—
273C4×D4—21no—
274C4×D4—21no—
275C4×D4—62no—
276C4×D4—62no—
277C4×D4—62no—
278C4×D4—62no—
279C4×D4—61no—
280C4×D4—61no—
281C4×D4—61no—
282C4×D4—61no—
283C4×D4—128no—
284#117—21no—
285#117—61no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—62no—
288C2×C4⋊C4—64no—
289C2×C4⋊C4—62no—
290C2×C4⋊C4C22×D311yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—32no—
293C2×C22⋊C4—64no—
294C2×C22⋊C4—62no—
295C2×C22⋊C4—62no—
296C2×C42—61no—
297C2×C42C22×D311yescharacteristic
298#107—22no—
299#107—62no—
300C4⋊C8—21no—
301C4⋊C8—61no—
302#103—22no—
303#103—62no—
304#102—22no—
305#102—62no—
306C22⋊C8—32no—
307C22⋊C8C22×D312yes—
308#97—21no—
309#97—61no—
310#95—32no—
311#95—61no—
312C23×C3—22no—
313C23×C3—42no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3—42no—
317C23×C3C22×D411yescharacteristic
318C22×D3—22no—
319C22×D3—22no—
320C22×D3—42no—
321C22×D3—42no—
322C22×D3#13714yes—
323C3×Q8—42no—
324C3×Q8—42no—
325C3×D4—22no—
326C3×D4—22no—
327C3×D4—41no—
328C3×D4—41no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—41no—
334C2×C3×C4—42no—
335C2×C3×C4—41no—
336C2×C3×C4—82no—
337C2×C3×C4—82no—
338C2×C3×C4#12011yescharacteristic
339C2×C3×C4#12011yescharacteristic
340C2×C3×C4#12011yescharacteristic
341C2×C3×C4#12011yescharacteristic
342C2×C3×C4C22×D411yescharacteristic
343C2×C3×C4C22×D411yescharacteristic
344#67—416no—
345C2×Dic3—24no—
346C2×Dic3—22no—
347C2×Dic3—22no—
348C2×Dic3—22no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352D12—24no—
353D12—24no—
354C4×D3—44no—
355C4×D3—82no—
356C4×D3—84no—
357C4×D3—84no—
358Dic6—42no—
359Dic6—42no—
360Dic6—84no—
361C3×C8—42no—
362C3⋊C8—42no—
363C24—21no—
364C24—32no—
365C2×Q8—21no—
366C2×Q8—32no—
367C2×Q8—64no—
368C2×Q8—61no—
369C2×Q8—244no—
370C2×Q8C2×#6712yes—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—42no—
374C2×D4—64no—
375C2×D4—61no—
376C2×D4—61no—
377C2×D4—124no—
378C2×D4—124no—
379C2×D4—124no—
380C22×C4—21no—
381C22×C4—32no—
382C22×C4—44no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—64no—
386C22×C4—64no—
387C22×C4—62no—
388C22×C4—62no—
389C22×C4—62no—
390C22×C4—124no—
391C22×C4D3×D411yescharacteristic
392C22×C4D3×D411yescharacteristic
393C22×C4C23×D311yescharacteristic
394SD16—44no—
395SD16—124no—
396C2×C8—22no—
397C2×C8—62no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—22no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—32no—
406C4⋊C4—64no—
407C4⋊C4—64no—
408C4⋊C4—64no—
409C4⋊C4—61no—
410C4⋊C4—61no—
411C4⋊C4—62no—
412C4⋊C4—61no—
413C4⋊C4—61no—
414C4⋊C4—61no—
415C4⋊C4—62no—
416C4⋊C4—124no—
417C4⋊C4C2×#6712yes—
418C22⋊C4—22no—
419C22⋊C4—41no—
420C22⋊C4—41no—
421C22⋊C4—64no—
422C22⋊C4—64no—
423C22⋊C4—62no—
424C22⋊C4—62no—
425C22⋊C4—62no—
426C22⋊C4—124no—
427C22⋊C4—124no—
428C42—22no—
429C42—64no—
430C42—62no—
431C42D3×D411yescharacteristic
432C42D3×D411yescharacteristic
433C42C2×#6711yescharacteristic
434C42C2×#6711yescharacteristic
435C22×C3—21no—
436C22×C3—21no—
437C22×C3—44no—
438C22×C3—42no—
439C22×C3—82no—
440C22×C3—81no—
441C22×C3—81no—
442C22×C3C2×#12011yescharacteristic
443C22×C3C2×#13712yes—
444C2×D3—24no—
445C2×D3—42no—
446C2×D3—42no—
447C2×D3—44no—
448C2×D3—82no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—21no—
453C3×C4—21no—
454C3×C4—42no—
455C3×C4—42no—
456C3×C4—81no—
457C3×C4#45312yes—
458Dic3—44no—
459Dic3—42no—
460Dic3—42no—
461Dic3—42no—
462C23—22no—
463C23—34no—
464C23—42no—
465C23—41no—
466C23—42no—
467C23—41no—
468C23—62no—
469C23—62no—
470C23—122no—
471C23—122no—
472C23C2×D3×D411yescharacteristic
473Q8—42no—
474Q8—42no—
475Q8—122no—
476Q8—122no—
477Q8—244no—
478D4—22no—
479D4—22no—
480D4—41no—
481D4—41no—
482D4—64no—
483D4—64no—
484D4—1216no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—64no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—62no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—124no—
501C2×C4—122no—
502C2×C4—242no—
503C2×C4—244no—
504C2×C4—244no—
505C2×C4#93711yescharacteristic
506C2×C4#93711yescharacteristic
507C2×C4#93711yescharacteristic
508C2×C4#93711yescharacteristic
509C2×C4C2×D3×D411yescharacteristic
510C2×C4C22×#6711yescharacteristic
511C8—42no—
512C8—122no—
513C2×C3—22no—
514C2×C3—82no—
515C2×C3#300411yescharacteristic
516C2×C3C2×#45311yescharacteristic
517C2×C3#302511yescharacteristic
518D3—44no—
519C22—21no—
520C22—21no—
521C22—44no—
522C22—42no—
523C22—64no—
524C22—82no—
525C22—81no—
526C22—81no—
527C22—124no—
528C22—122no—
529C22—122no—
530C22—242no—
531C22D3×#13712yes—
532C22C2×#93711yescharacteristic
533C4—21no—
534C4—21no—
535C4—21no—
536C4—21no—
537C4—21no—
538C4—42no—
539C4—42no—
540C4—81no—
541C4—124no—
542C4—122no—
543C4—122no—
544C4—122no—
545C4#519812yes—
546C3#1818011yescharacteristic, complemented, Sylow
547C2—22no—
548C2—82no—
549C2—124no—
550C2#8293611yescharacteristic
551C2C2×#519811yescharacteristic
552C2#8397911yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3110C2
socleC22×C3C2×#120
prime radical#82347C2
hypercenter#3110D3
hypocenterC3#18180
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
1. #86280C2semidirect
2. C3×#3110C2semidirect
4. #82315C2semidirect
5. #71836C2semidirect
6. #71176C2semidirect
7. #70976C2semidirect
8. #70836C2semidirect
9. #70827C2semidirect
48. #3110D3semidirect
62. C3×C4⋊Q8D4semidirect
63. C3×C4⋊Q8D4semidirect
68. C3×C4×Q8D4semidirect
112. #869D4semidirect
113. #869D4semidirect
115. C4×Dic6D4semidirect
249. C4⋊Q8#67semidirect
250. C4⋊Q8#67semidirect
270. C4×Q8#67semidirect
546. C3#18180semidirect

Maximal subgroups

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

subgroupindex
#862802
C3×#31102
#823472
#823152
#718362
#711762
#709762
#708362
#708272
#181803