Finite Group Databasebeta

← #10770310

Subgroups — #10770310

Order 768.

928 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3559C211yesmaximal, characteristic
2#83317C211yesmaximal, characteristic, complemented
3#83154C211yesmaximal, characteristic, complemented
4#83044C211yesmaximal, characteristic, complemented
5#71889C214yesmaximal, complemented
6#71880C214yesmaximal, complemented
7#71812C211yesmaximal, characteristic, complemented
8#71754C211yesmaximal, characteristic, complemented
9#71737C211yesmaximal, characteristic, complemented
10#18431—31nomaximal, Sylow
11C3×#563—21no—
12C3×#562—21no—
13C3×#554—21no—
14C3×#553—21no—
15C3×#553—21no—
16C3×#547—22no—
17C3×#547—21no—
18C3×#544—21no—
19C3×#540—21no—
20C3×#539—22no—
21C3×#539C2212yes—
22C3×#537C2212yes—
23C3×#532C2211yescharacteristic
24C3×#529C2211yescharacteristic
25C2×C3×#125C2211yescharacteristic
26C2×#869C2211yescharacteristic
27#5093—21no—
28#5076—21no—
29#5069—21no—
30#5060—21no—
31#5050C2212yes—
32#5040C2214yes—
33#5040C2212yes—
34#5031—21no—
35#5019—22no—
36#5019—21no—
37#4981—22no—
38#4979C2214yes—
39#4977—21no—
40#4975—21no—
41#4932C2211yescharacteristic
42#4926C2211yescharacteristic
43#4925C2211yescharacteristic
44#4538C2212yes—
45#4537C2212yes—
46#4537C2214yes—
47#4528C2214yes—
48#3559D311yescharacteristic
49#3218—31no—
50#3206—31no—
51#2660—31no—
52#1604—34no—
53#1603—34no—
54#1505—31no—
55#1500—31no—
56#1470—31no—
57C2×C3×#41—21no—
58C2×C3×#41—22no—
59C22×Dic6C2312yes—
60C3×C4⋊Q8—22no—
61C3×C4⋊Q8—21no—
62C3×C4⋊Q8D411yescharacteristic
63C3×#127D411yescharacteristic
64C3×#126—22no—
65C3×#126—22no—
66C3×#126—21no—
67C3×#125—21no—
68C3×#125—22no—
69C3×#125—21no—
70C3×#125—21no—
71C3×#125—21no—
72C3×#125—21no—
73C3×#125D411yescharacteristic
74C3×#124—21no—
75C3×#124D411yescharacteristic
76C3×#123—22no—
77C3×#123—22no—
78C3×#123—22no—
79C3×#123—22no—
80C3×#123—22no—
81C3×#123—21no—
82C3×#123—22no—
83C3×C22⋊Q8—22no—
84C3×C22⋊Q8—22no—
85C3×C22⋊Q8—22no—
86C3×C22⋊Q8—22no—
87C3×C22⋊Q8—22no—
88C3×C22⋊Q8—22no—
89C3×C22⋊Q8C2311yescharacteristic
90C3×#121—22no—
91C3×#121—21no—
92C3×#121—21no—
93C3×#121—22no—
94C3×#121—22no—
95C3×#121—21no—
96C3×#121—22no—
97C3×#121C2311yescharacteristic
98C3×#121C2312yes—
99C3×C4×Q8D411yescharacteristic
100C3×C4×D4—22no—
101C3×C4×D4—21no—
102C3×C4×D4—21no—
103C3×C4×D4—21no—
104C3×C4×D4—22no—
105C3×C4×D4—21no—
106C3×C4×D4D411yescharacteristic
107C3×C4×D4D412yes—
108C3×#117—21no—
109C3×#117—21no—
110C2×C3×C4⋊C4—22no—
111C2×C3×C4⋊C4—21no—
112C2×C3×C4⋊C4D411yescharacteristic
113C2×C3×C4⋊C4D411yescharacteristic
114C2×C3×C4⋊C4D412yes—
115C2×C3×C4⋊C4C2312yes—
116C2×C3×C42C2311yescharacteristic
117C2×#216—22no—
118C2×#214—22no—
119C2×#214—24no—
120C2×C4⋊Dic3—21no—
121C2×C4⋊Dic3—21no—
122C2×Dic3⋊C4—24no—
123C2×C4×Dic3—21no—
124#869—21no—
125#869—21no—
126#869—21no—
127#869—21no—
128C3⋊#95—21no—
129C3⋊#95—24no—
130C3⋊#95C2312yes—
131#830C2312yes—
132#830C2312yes—
133#810—24no—
134#810—22no—
135#810—22no—
136C4⋊(C3⋊C8)—21no—
137C4⋊(C3⋊C8)—21no—
138#803—21no—
139C4×C3⋊C8—21no—
140#563—21no—
141#562—21no—
142#554—21no—
143#553—21no—
144#553—21no—
145#547—22no—
146#547—21no—
147#544—21no—
148#540—21no—
149#539—22no—
150#539C2×D312yes—
151#537C2×D312yes—
152#532C2×D311yescharacteristic
153C2×C4⋊Q8—31no—
154#529C2×D311yescharacteristic
155C2×#125C2×D311yescharacteristic
156#497—61no—
157#492—61no—
158#491—61no—
159#487—61no—
160#462—61no—
161#461—62no—
162#461—61no—
163#451—32no—
164#448—34no—
165#448—32no—
166#432—31no—
167#431—31no—
168#398—62no—
169#395—34no—
170#384—61no—
171#382—61no—
172#376—31no—
173#328—32no—
174#327—34no—
175#327—32no—
176#324—34no—
177C3×#41—82no—
178C3×#41—84no—
179C2×C3×Q8—22no—
180C2×C3×Q8—21no—
181C2×C3×Q8C2×D411yescharacteristic
182C2×C3×D4—21no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4C2×D411yescharacteristic
189C2×C3×D4C2×D412yes—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—21no—
194C22×C3×C4—22no—
195C22×C3×C4C2×D412yes—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2411yescharacteristic
200C22×Dic3—22no—
201C22×Dic3—22no—
202C2×Dic6—22no—
203C2×Dic6—22no—
204C2×Dic6—82no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4C2×D412yes—
223C3×C4⋊C4C2×D411yescharacteristic
224C3×C4⋊C4C2×D411yescharacteristic
225C3×C22⋊C4—22no—
226C3×C22⋊C4—22no—
227C3×C22⋊C4—22no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—22no—
230C3×C22⋊C4—21no—
231C3×C22⋊C4—21no—
232C3×C22⋊C4—21no—
233C3×C22⋊C4—21no—
234C3×C22⋊C4—22no—
235C3×C22⋊C4—21no—
236C3×C22⋊C4—22no—
237C3×C42—22no—
238C3×C42—21no—
239C3×C42C2×D411yescharacteristic
240C3×C42C2×D411yescharacteristic
241C3×C42C2×D411yescharacteristic
242C3×C42C2×D411yescharacteristic
243#216—82no—
244#214—84no—
245#214—82no—
246C4⋊Dic3—41no—
247C4⋊Dic3—41no—
248C4⋊Dic3—41no—
249C4⋊Dic3—41no—
250Dic3⋊C4—84no—
251C4×Dic3—24no—
252C2×C3⋊C8—22no—
253C2×C3⋊C8—22no—
254C2×#41—21no—
255C2×#41—22no—
256C22×Q8—32no—
257C2×Q16—62no—
258C2×SD16—64no—
259C2×SD16—62no—
260C4⋊Q8—21no—
261C4⋊Q8—22no—
262C4⋊Q8—61no—
263C4⋊Q8—61no—
264C4⋊Q8—61no—
265C4⋊Q8—61no—
266C4⋊Q8#6711yescharacteristic
267#127#6711yescharacteristic
268#126—22no—
269#126—22no—
270#126—21no—
271#125—21no—
272#125—22no—
273#125—21no—
274#125—21no—
275#125—21no—
276#125—21no—
277#125#6711yescharacteristic
278#124—21no—
279#124#6711yescharacteristic
280#123—22no—
281#123—22no—
282#123—22no—
283#123—22no—
284#123—22no—
285#123—21no—
286#123—22no—
287C22⋊Q8—22no—
288C22⋊Q8—22no—
289C22⋊Q8—22no—
290C22⋊Q8—22no—
291C22⋊Q8—22no—
292C22⋊Q8—22no—
293C22⋊Q8C22×D311yescharacteristic
294#121—21no—
295#121—22no—
296#121—21no—
297#121—22no—
298#121—22no—
299#121—21no—
300#121—22no—
301#121C22×D311yescharacteristic
302#121C22×D312yes—
303C4×Q8#6711yescharacteristic
304C4×D4—22no—
305C4×D4—21no—
306C4×D4—21no—
307C4×D4—21no—
308C4×D4—22no—
309C4×D4—21no—
310C4×D4#6711yescharacteristic
311C4×D4#6712yes—
312#117—21no—
313#117—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4—64no—
319C2×C4⋊C4#6711yescharacteristic
320C2×C4⋊C4#6712yes—
321C2×C4⋊C4#6711yescharacteristic
322C2×C4⋊C4C22×D312yes—
323C2×C42—61no—
324C2×C42C22×D311yescharacteristic
325C4⋊C8—61no—
326C4⋊C8—61no—
327#103—64no—
328#103—62no—
329#103—62no—
330C22⋊C8—32no—
331C22⋊C8—32no—
332#97—61no—
333C4×C8—61no—
334#95—32no—
335#95—61no—
336#95—64no—
337C23×C3—21no—
338C23×C3—22no—
339C23×C3C22×D411yescharacteristic
340C3×Q8—41no—
341C3×Q8—81no—
342C3×Q8—82no—
343C3×D4—42no—
344C3×D4—41no—
345C3×D4—44no—
346C3×D4—42no—
347C3×D4—42no—
348C3×D4—81no—
349C3×D4—82no—
350C3×D4—82no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—21no—
358C2×C3×C4—22no—
359C2×C3×C4—24no—
360C2×C3×C4—41no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—82no—
364C2×C3×C4—82no—
365C2×C3×C4—82no—
366C2×C3×C4—82no—
367C2×C3×C4#12011yescharacteristic
368C2×C3×C4#12011yescharacteristic
369C2×C3×C4#12011yescharacteristic
370C2×C3×C4#12011yescharacteristic
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C22×D411yescharacteristic
373C2×Dic3—24no—
374C2×Dic3—42no—
375C2×Dic3—42no—
376C2×Dic3—44no—
377C2×Dic3—42no—
378Dic6—44no—
379Dic6—42no—
380Dic6—42no—
381C3⋊C8—42no—
382C3⋊C8—42no—
383#41—82no—
384#41—84no—
385C2×Q8—22no—
386C2×Q8—21no—
387C2×Q8—62no—
388C2×Q8—62no—
389C2×Q8—242no—
390C2×Q8C2×#6711yescharacteristic
391C2×D4—21no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4C2×#6711yescharacteristic
398C2×D4C2×#6712yes—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—62no—
405C22×C4—62no—
406C22×C4C2×#6712yes—
407C22×C4C2×#6711yescharacteristic
408C22×C4C2×#6711yescharacteristic
409C22×C4C2×#6712yes—
410C22×C4C23×D311yescharacteristic
411Q16—242no—
412SD16—244no—
413SD16—242no—
414C2×C8—62no—
415C2×C8—62no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—22no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—121no—
434C4⋊C4—121no—
435C4⋊C4—121no—
436C4⋊C4—121no—
437C4⋊C4—244no—
438C4⋊C4C2×#6712yes—
439C4⋊C4C2×#6711yescharacteristic
440C4⋊C4C2×#6711yescharacteristic
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—21no—
447C22⋊C4—21no—
448C22⋊C4—21no—
449C22⋊C4—21no—
450C22⋊C4—22no—
451C22⋊C4—21no—
452C22⋊C4—22no—
453C42—22no—
454C42—21no—
455C42—64no—
456C42C2×#6711yescharacteristic
457C42C2×#6711yescharacteristic
458C42C2×#6711yescharacteristic
459C42C2×#6711yescharacteristic
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—42no—
463C22×C3—41no—
464C22×C3—41no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—41no—
468C22×C3C2×#12011yescharacteristic
469C22×C3#57712yes—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—42no—
476C3×C4—42no—
477C3×C4—41no—
478C3×C4—41no—
479C3×C4—42no—
480C3×C4—41no—
481C3×C4#45612yes—
482Dic3—44no—
483Dic3—82no—
484C23—21no—
485C23—22no—
486C23C22×#6711yescharacteristic
487Q8—41no—
488Q8—81no—
489Q8—82no—
490Q8—124no—
491Q8—122no—
492Q8—122no—
493D4—42no—
494D4—41no—
495D4—44no—
496D4—42no—
497D4—42no—
498D4—81no—
499D4—82no—
500D4—82no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—64no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—122no—
519C2×C4—122no—
520C2×C4—124no—
521C2×C4—122no—
522C2×C4#95311yescharacteristic
523C2×C4#95311yescharacteristic
524C2×C4#95311yescharacteristic
525C2×C4#95311yescharacteristic
526C2×C4C22×#6711yescharacteristic
527C2×C4C22×#6711yescharacteristic
528C8—122no—
529C8—122no—
530C2×C3—22no—
531C2×C3—81no—
532C2×C3—82no—
533C2×C3#300911yescharacteristic
534C2×C3C2×#45611yescharacteristic
535C2×C3#302711yescharacteristic
536C22—21no—
537C22—21no—
538C22—42no—
539C22—41no—
540C22—41no—
541C22—42no—
542C22—42no—
543C22—41no—
544C22#583612yes—
545C22C2×#95311yescharacteristic
546C4—21no—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—42no—
552C4—42no—
553C4—41no—
554C4—41no—
555C4—42no—
556C4—41no—
557C4—124no—
558C4—242no—
559C4#524612yes—
560C3#1843111yescharacteristic, complemented, Sylow
561C2—22no—
562C2—81no—
563C2—82no—
564C2#8410511yescharacteristic
565C2C2×#524611yescharacteristic
566C2#8412611yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3559C2
socleC22×C3C2×#120
hypercenter#3559D3
hypocenterC3#18431
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3559

Complements

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

NH ≅ G/Ntype
2. #83317C2semidirect
3. #83154C2semidirect
4. #83044C2semidirect
5. #71889C2semidirect
6. #71880C2semidirect
7. #71812C2semidirect
8. #71754C2semidirect
9. #71737C2semidirect
560. C3#18431semidirect

Maximal subgroups

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

subgroupindex
C3×#35592
#833172
#831542
#830442
#718892
#718802
#718122
#717542
#717372
#184313