Finite Group Databasebeta

← #10764140

Subgroups — #10764140

Order 768.

924 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3511C211yesmaximal, characteristic
2#82523C211yesmaximal, characteristic, complemented
3#82522C211yesmaximal, characteristic, complemented
4#71833C212yesmaximal, complemented
5#71828C212yesmaximal, complemented
6#71810C211yesmaximal, characteristic, complemented
7#71808C212yesmaximal, complemented
8#71806C212yesmaximal, complemented
9#71805C211yesmaximal, characteristic, complemented
10#71775C211yesmaximal, characteristic, complemented
11#71773C211yesmaximal, characteristic, complemented
12#17781—31nomaximal, Sylow
13C3×#556—21no—
14C3×#555—21no—
15C3×#554—21no—
16C3×#553—22no—
17C3×#548—22no—
18C3×#547—21no—
19C3×#542C414yes—
20C3×#537C412yes—
21C3×#533C2212yes—
22C3×#532C2211yescharacteristic
23C3×#532C2212yes—
24C2×C3×C4⋊Q8C2211yescharacteristic
25C3×#520C412yes—
26C3×C4×#41C2211yescharacteristic
27#5024—21no—
28Q8×C3⋊C8—21no—
29#5012—21no—
30#5012—22no—
31#5011—22no—
32D4×C3⋊C8—21no—
33#4926C2212yes—
34#4925C2211yescharacteristic
35#4925C2211yescharacteristic
36#4925C2212yes—
37#4924C2211yescharacteristic
38#4924C2211yescharacteristic
39#4924C2212yes—
40C2×C4⋊(C3⋊C8)C2212yes—
41#4547C2212yes—
42#4547C2212yes—
43#4547C2212yes—
44#4547C2212yes—
45#4546C2211yescharacteristic
46#4546C2211yescharacteristic
47#4546C2212yes—
48#4544C2211yescharacteristic
49#4544C2211yescharacteristic
50#4544C2212yes—
51#3511D311yescharacteristic
52#2991—31no—
53#2990—31no—
54#1560—31no—
55#1559—32no—
56#1553—32no—
57#1549—31no—
58#1547—32no—
59#1546—31no—
60#1543—31no—
61#1542—32no—
62C2×C3×#41—22no—
63C2×C3×#41C2×C412yes—
64C2×C3×#41C2×C411yescharacteristic
65C22×C3×Q8—21no—
66C3×C4⋊Q8—21no—
67C3×C4⋊Q8—21no—
68C3×C4⋊Q8C2×C412yes—
69C3×C4⋊Q8C2311yescharacteristic
70C3×C4⋊Q8C2312yes—
71C3×#127C2311yescharacteristic
72C3×#126—24no—
73C3×#125—22no—
74C3×#124—24no—
75C3×#124—22no—
76C3×#124C2×C412yes—
77C3×#123—24no—
78C3×C22⋊Q8—22no—
79C3×C22⋊Q8—22no—
80C3×C22⋊Q8—24no—
81C3×C22⋊Q8C2×C414yes—
82C3×#121—22no—
83C3×#121—24no—
84C3×#121—21no—
85C3×#121—21no—
86C3×#121—22no—
87C3×#121C2×C414yes—
88C3×C4×Q8—22no—
89C3×C4×Q8C2×C414yes—
90C3×C4×Q8D411yescharacteristic
91C3×C4×Q8D411yescharacteristic
92C3×C4×D4—24no—
93C3×C4×D4—21no—
94C3×C4×D4—22no—
95C3×C4×D4—21no—
96C3×C4×D4C2×C414yes—
97C3×C4×D4D411yescharacteristic
98C3×C4×D4D412yes—
99C3×C4×D4D411yescharacteristic
100C3×C4×D4D412yes—
101C3×#117—22no—
102C3×#117—22no—
103C3×#117C2×C411yescharacteristic
104C3×#117C2×C412yes—
105C2×C3×C4⋊C4—21no—
106C2×C3×C4⋊C4—22no—
107C2×C3×C4⋊C4C2×C412yes—
108C2×C3×C42C2312yes—
109C2×C3×C42C2311yescharacteristic
110C2×#208—21no—
111C2×#208—21no—
112C2×#208—22no—
113C22×C3⋊C8—22no—
114#830—21no—
115#830—21no—
116#830—22no—
117#830—22no—
118C4⋊(C3⋊C8)—22no—
119C4⋊(C3⋊C8)—21no—
120C4⋊(C3⋊C8)—21no—
121C4⋊(C3⋊C8)—22no—
122C4⋊(C3⋊C8)C2311yescharacteristic
123C4⋊(C3⋊C8)C2312yes—
124C4⋊(C3⋊C8)C2312yes—
125C4⋊(C3⋊C8)C2311yescharacteristic
126C4⋊(C3⋊C8)C2311yescharacteristic
127C4⋊(C3⋊C8)C2311yescharacteristic
128#803—22no—
129C4×C3⋊C8—21no—
130C4×C3⋊C8—21no—
131C4×C3⋊C8—22no—
132#556—21no—
133#555—21no—
134#554—21no—
135#553—22no—
136#548—22no—
137#547—21no—
138#542Dic314yes—
139#537Dic312yes—
140#533C2×D312yes—
141#532C2×D311yescharacteristic
142#532C2×D312yes—
143C2×C4⋊Q8C2×D311yescharacteristic
144#520Dic312yes—
145C4×#41C2×D311yescharacteristic
146#446—61no—
147C8×Q8—61no—
148#436—61no—
149#436—62no—
150#435—62no—
151C8×D4—61no—
152#432—32no—
153#431—32no—
154#431—31no—
155#431—31no—
156#423—31no—
157#423—32no—
158#423—31no—
159C2×C4⋊C8—32no—
160#333—31no—
161#333—32no—
162#333—31no—
163#332—32no—
164#332—32no—
165#332—32no—
166#332—32no—
167#331—31no—
168#331—31no—
169#331—32no—
170C3×#41—42no—
171C3×#41—44no—
172C3×#41—82no—
173C3×#41—82no—
174C2×C3×Q8—22no—
175C2×C3×Q8—22no—
176C2×C3×Q8—81no—
177C2×C3×Q8—81no—
178C2×C3×Q8C22⋊C411yescharacteristic
179C2×C3×Q8C22×C412yes—
180C2×C3×D4—22no—
181C2×C3×D4—24no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4C22⋊C411yescharacteristic
185C2×C3×D4C22⋊C412yes—
186C2×C3×D4C22×C412yes—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—24no—
190C22×C3×C4C22×C411yescharacteristic
191C22×C3×C4C22×C412yes—
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D412yes—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—24no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4C22⋊C411yescharacteristic
204C3×C4⋊C4C22⋊C411yescharacteristic
205C3×C4⋊C4C22⋊C412yes—
206C3×C4⋊C4C22⋊C412yes—
207C3×C4⋊C4C22×C414yes—
208C3×C22⋊C4—24no—
209C3×C22⋊C4—24no—
210C3×C22⋊C4—24no—
211C3×C22⋊C4—22no—
212C3×C22⋊C4—22no—
213C3×C22⋊C4C22⋊C412yes—
214C3×C22⋊C4C22⋊C412yes—
215C3×C22⋊C4C22⋊C412yes—
216C3×C42—24no—
217C3×C42C22×C412yes—
218C3×C42C22×C411yescharacteristic
219C3×C42C2×D411yescharacteristic
220C3×C42C2×D411yescharacteristic
221C3×C42C2×D412yes—
222C3×C42C2×D412yes—
223C3×C42C2411yescharacteristic
224#208—81no—
225#208—82no—
226#208—81no—
227C2×C3⋊C8—21no—
228C2×C3⋊C8—21no—
229C2×C3⋊C8—21no—
230C2×C3⋊C8—21no—
231C2×C3⋊C8—22no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8—82no—
234C2×#41—22no—
235C2×#41C2×Dic312yes—
236C2×#41C2×Dic311yescharacteristic
237C22×Q8—21no—
238C2×M16—61no—
239C2×M16—61no—
240C2×M16—62no—
241C22×C8—62no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8C2×Dic312yes—
245C4⋊Q8C22×D311yescharacteristic
246C4⋊Q8C22×D312yes—
247#127C22×D311yescharacteristic
248#126—24no—
249#125—22no—
250#124—24no—
251#124—22no—
252#124C2×Dic312yes—
253#123—24no—
254C22⋊Q8—22no—
255C22⋊Q8—22no—
256C22⋊Q8—24no—
257C22⋊Q8C2×Dic314yes—
258#121—22no—
259#121—24no—
260#121—21no—
261#121—21no—
262#121—22no—
263#121C2×Dic314yes—
264C4×Q8—22no—
265C4×Q8C2×Dic314yes—
266C4×Q8#6711yescharacteristic
267C4×Q8#6711yescharacteristic
268C4×D4—24no—
269C4×D4—21no—
270C4×D4—22no—
271C4×D4—21no—
272C4×D4C2×Dic314yes—
273C4×D4#6711yescharacteristic
274C4×D4#6712yes—
275C4×D4#6712yes—
276C4×D4#6711yescharacteristic
277#117—22no—
278#117—22no—
279#117C2×Dic311yescharacteristic
280#117C2×Dic312yes—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4C2×Dic312yes—
284C2×C42C22×D311yescharacteristic
285C2×C42C22×D312yes—
286C4⋊C8—31no—
287C4⋊C8—32no—
288C4⋊C8—31no—
289C4⋊C8—32no—
290C4⋊C8—31no—
291C4⋊C8—31no—
292C4⋊C8—62no—
293C4⋊C8—61no—
294C4⋊C8—61no—
295C4⋊C8—62no—
296C22⋊C8—61no—
297C22⋊C8—61no—
298C22⋊C8—62no—
299C22⋊C8—62no—
300#97—62no—
301C4×C8—61no—
302C4×C8—61no—
303C4×C8—62no—
304C23×C3—22no—
305C23×C3C2×C22⋊C412yes—
306C23×C3C2×C22⋊C411yescharacteristic
307C3×Q8—22no—
308C3×Q8—42no—
309C3×Q8—42no—
310C3×Q8—42no—
311C3×Q8—82no—
312C3×D4—22no—
313C3×D4—42no—
314C3×D4—42no—
315C3×D4—44no—
316C3×D4—41no—
317C3×D4—41no—
318C3×D4—41no—
319C3×D4—41no—
320C3×D4—84no—
321C2×C3×C4—21no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—21no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—24no—
332C2×C3×C4—42no—
333C2×C3×C4—44no—
334C2×C3×C4—84no—
335C2×C3×C4—82no—
336C2×C3×C4C2×C22⋊C412yes—
337C2×C3×C4C2×C22⋊C411yescharacteristic
338C2×C3×C4C2×C22⋊C411yescharacteristic
339C2×C3×C4C2×C22⋊C412yes—
340C2×C3×C4C2×C22⋊C412yes—
341C2×C3×C4C2×C22⋊C411yescharacteristic
342C2×C3×C4C23×C411yescharacteristic
343C2×C3×C4C22×D411yescharacteristic
344C2×C3×C4C22×D411yescharacteristic
345C3⋊C8—42no—
346C3⋊C8—41no—
347C3⋊C8—41no—
348C3⋊C8—41no—
349C3⋊C8—42no—
350C3⋊C8—41no—
351#41—42no—
352#41—44no—
353#41—82no—
354#41—82no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—81no—
358C2×Q8—81no—
359C2×Q8#21711yescharacteristic
360C2×Q8C22×Dic312yes—
361C2×D4—22no—
362C2×D4—24no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4#21711yescharacteristic
366C2×D4#21712yes—
367C2×D4C22×Dic312yes—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4C22×Dic311yescharacteristic
372C22×C4C22×Dic312yes—
373C22×C4C2×#6711yescharacteristic
374C22×C4C2×#6712yes—
375C22×C4C2×#6711yescharacteristic
376C22×C4C2×#6712yes—
377M16—241no—
378M16—242no—
379M16—241no—
380C2×C8—61no—
381C2×C8—61no—
382C2×C8—61no—
383C2×C8—61no—
384C2×C8—62no—
385C2×C8—62no—
386C2×C8—242no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—22no—
390C4⋊C4—24no—
391C4⋊C4—22no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4#21711yescharacteristic
395C4⋊C4#21711yescharacteristic
396C4⋊C4#21712yes—
397C4⋊C4#21712yes—
398C4⋊C4C22×Dic314yes—
399C22⋊C4—24no—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—22no—
403C22⋊C4—22no—
404C22⋊C4#21712yes—
405C22⋊C4#21712yes—
406C22⋊C4#21712yes—
407C42—24no—
408C42C22×Dic312yes—
409C42C22×Dic311yescharacteristic
410C42C2×#6711yescharacteristic
411C42C2×#6711yescharacteristic
412C42C2×#6712yes—
413C42C2×#6712yes—
414C42C23×D311yescharacteristic
415C22×C3—21no—
416C22×C3—22no—
417C22×C3—21no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—21no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3C22×C22⋊C411yescharacteristic
425C3×C4—21no—
426C3×C4—22no—
427C3×C4—22no—
428C3×C4—21no—
429C3×C4—22no—
430C3×C4—21no—
431C3×C4—42no—
432C3×C4—44no—
433C3×C4#41312yes—
434C3×C4#57611yescharacteristic
435C3×C4#57711yescharacteristic
436C3×C4#57711yescharacteristic
437C3×C4#57811yescharacteristic
438C23—22no—
439C23C2×#21712yes—
440C23C2×#21711yescharacteristic
441Q8—22no—
442Q8—42no—
443Q8—42no—
444Q8—42no—
445Q8—82no—
446D4—22no—
447D4—42no—
448D4—42no—
449D4—44no—
450D4—41no—
451D4—41no—
452D4—41no—
453D4—41no—
454D4—84no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—84no—
469C2×C4—82no—
470C2×C4C2×#21712yes—
471C2×C4C2×#21711yescharacteristic
472C2×C4C2×#21711yescharacteristic
473C2×C4C2×#21712yes—
474C2×C4C2×#21712yes—
475C2×C4C2×#21711yescharacteristic
476C2×C4C23×Dic311yescharacteristic
477C2×C4C22×#6711yescharacteristic
478C2×C4C22×#6711yescharacteristic
479C8—122no—
480C8—121no—
481C8—121no—
482C8—121no—
483C8—122no—
484C8—121no—
485C2×C3—41no—
486C2×C3—42no—
487C2×C3—82no—
488C2×C3C2×#41311yescharacteristic
489C2×C3#289711yescharacteristic
490C2×C3#289811yescharacteristic
491C22—21no—
492C22—22no—
493C22—21no—
494C22—22no—
495C22—22no—
496C22—21no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22C22×#21711yescharacteristic
501C4—21no—
502C4—22no—
503C4—22no—
504C4—21no—
505C4—22no—
506C4—21no—
507C4—42no—
508C4—44no—
509C4#523712yes—
510C4#583511yescharacteristic
511C4#583611yescharacteristic
512C4#583711yescharacteristic
513C4#583811yescharacteristic
514C3#1778111yescharacteristic, complemented, Sylow
515C2—41no—
516C2—42no—
517C2—82no—
518C2C2×#523711yescharacteristic
519C2#8409911yescharacteristic
520C2#8411111yescharacteristic
subgroupG/H
centerC22C22×#217
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
Fitting, prime radicalC3×#3511C2
socleC22×C3C22×C22⋊C4
cosocleC3×C42C24
hypercenter#3511D3
hypocenterC3#17781
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×#41 ◁ #3511

Complements

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

NH ≅ G/Ntype
2. #82523C2semidirect
3. #82522C2semidirect
4. #71833C2semidirect
5. #71828C2semidirect
6. #71810C2semidirect
7. #71808C2semidirect
8. #71806C2semidirect
9. #71805C2semidirect
10. #71775C2semidirect
11. #71773C2semidirect
514. C3#17781semidirect

Maximal subgroups

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

subgroupindex
C3×#35112
#825232
#825222
#718332
#718282
#718102
#718082
#718062
#718052
#717752
#717732
#177813