Finite Group Databasebeta

← #10691152

Subgroups — #10691152

Order 768.

530 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82823C211yesmaximal, characteristic
2#78634C211yesmaximal, characteristic
3C3×#1508C211yesmaximal, characteristic, complemented
4#71688C211yesmaximal, characteristic, complemented
5C4×#806C211yesmaximal, characteristic
6#67682C211yesmaximal, characteristic, complemented
7#67616C211yesmaximal, characteristic, complemented
8#9184—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×C4×#67C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C3×C4×C22⋊C4C2211yescharacteristic
13#5231C411yescharacteristic, complemented
14#5230C411yescharacteristic, complemented
15#5211C2211yescharacteristic
16C4×#217C2211yescharacteristic
17#5207C2211yescharacteristic
18#5128—22no—
19#4992C411yescharacteristic, complemented
20#4989—21no—
21#4983C411yescharacteristic, complemented
22#4978—21no—
23#4959—21no—
24C2×#806—21no—
25#4952—21no—
26#4729—22no—
27#4726—21no—
28#4718—21no—
29C3×#328—22no—
30#4538—22no—
31#4527—21no—
32#2975—31no—
33#2638—31no—
34C4×#99—31no—
35#1508—31no—
36#1508D311yescharacteristic, complemented
37#1284—31no—
38#1284—31no—
39C22×C3×Q8C2×C411yescharacteristic
40C23×C3×C4C2311yescharacteristic
41C22×#67D411yescharacteristic
42C22×C4×D3D411yescharacteristic
43C2×C3×M16—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4C2×C411yescharacteristic
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C42—21no—
53C2×#217—21no—
54C2×#217—21no—
55C2×#217D411yescharacteristic
56C4×#67—21no—
57C4×#67—21no—
58C4×#67—21no—
59C4×#67—21no—
60C2×(C2×D3)⋊C4—21no—
61C2×(C2×D3)⋊C4—21no—
62C2×(C2×D3)⋊C4C2×C411yescharacteristic
63C2×Dic3⋊C4—21no—
64C2×Dic3⋊C4—21no—
65C2×Dic3⋊C4C2×C411yescharacteristic
66C2×C4×Dic3—21no—
67C2×C4×Dic3D411yescharacteristic
68C2×#208—21no—
69C2×ΓL(1,𝔽25)—22no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8D412yes—
72C3×#95—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95C2×C411yescharacteristic
80C3⋊#95C2×C411yescharacteristic
81#830—21no—
82#830D412yes—
83(C2×D3)⋊C8—22no—
84Dic3⋊C8—22no—
85C8×Dic3—22no—
86#806—24no—
87C2×C22⋊Q8C2×D311yescharacteristic
88C2×C4×D4—31no—
89#436—62no—
90#436—62no—
91#431—61no—
92C2×#99—61no—
93#407—31no—
94#407—61no—
95#407C2×D311yescharacteristic
96#398—31no—
97#397—31no—
98#397—61no—
99#395—31no—
100#394—61no—
101#393—31no—
102#387—61no—
103#385—61no—
104#381—31no—
105C4×C22⋊C4—31no—
106C4×C22⋊C4C2×D311yescharacteristic
107#342—61no—
108#328—22no—
109#328—62no—
110C24×C3C2×D411yescharacteristic
111C23×D3C22⋊C411yescharacteristic
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8—81no—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—41no—
118C22×C3×C4—81no—
119C22×C3×C4C22×C411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4#4111yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C2×#67—21no—
126C2×#67—21no—
127C2×#67—22no—
128C2×#67—41no—
129C22×Dic3—21no—
130C22×Dic3—21no—
131C22×Dic3—41no—
132C22×Dic3C22⋊C411yescharacteristic
133C22×Dic3C22⋊C411yescharacteristic
134C22×Dic3C22⋊C411yescharacteristic
135C2×C4×D3—21no—
136C2×C4×D3—21no—
137C2×C4×D3—81no—
138C3×M16—81no—
139C2×C3×C8—22no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C22⋊C4—22no—
145C3×C22⋊C4—41no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C42—81no—
149#217—21no—
150#217—21no—
151#217—22no—
152#217—41no—
153#217—81no—
154(C2×D3)⋊C4—21no—
155(C2×D3)⋊C4—21no—
156(C2×D3)⋊C4—42no—
157(C2×D3)⋊C4—81no—
158Dic3⋊C4—21no—
159Dic3⋊C4—21no—
160Dic3⋊C4—42no—
161Dic3⋊C4—81no—
162C4×Dic3—21no—
163C4×Dic3—21no—
164C4×Dic3—81no—
165#208—81no—
166C2×C3⋊C8—22no—
167ΓL(1,𝔽25)—82no—
168C22×Q8C4×D311yescharacteristic
169C22×D4—31no—
170C23×C4—31no—
171C23×C4C22×D311yescharacteristic
172C2×M16—21no—
173C2×M16—62no—
174C2×M16—61no—
175C22⋊Q8—21no—
176C22⋊Q8—21no—
177C22⋊Q8—21no—
178C22⋊Q8—21no—
179C4×D4—61no—
180C4×D4—61no—
181C4×D4—61no—
182C4×D4—61no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4C4×D311yescharacteristic
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—31no—
191C2×C22⋊C4—31no—
192C2×C22⋊C4—61no—
193C2×C22⋊C4—61no—
194C2×C22⋊C4—61no—
195C2×C22⋊C4—61no—
196C2×C42—21no—
197C2×C42—31no—
198C2×C42—61no—
199C4⋊C8—62no—
200#99—64no—
201C22⋊C8—21no—
202C22⋊C8—32no—
203C22⋊C8—62no—
204C22⋊C8—61no—
205C22⋊C8#6712yes—
206C4×C8—62no—
207#95—21no—
208#95—31no—
209#95—31no—
210#95—61no—
211#95—61no—
212#95—61no—
213#95—61no—
214#95—61no—
215#95—61no—
216C23×C3—21no—
217C23×C3—21no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3C2×C22⋊C411yescharacteristic
221C23×C3#12111yescharacteristic
222C23×C3#12111yescharacteristic
223C22×D3—21no—
224C22×D3—21no—
225C22×D3—21no—
226C22×D3—41no—
227C22×D3—41no—
228C3×Q8—81no—
229C3×Q8—81no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—42no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—81no—
244C2×C3×C4—81no—
245C2×C3×C4C4×D411yescharacteristic
246C2×C3×C4C4×D411yescharacteristic
247C2×C3×C4#12011yescharacteristic
248C2×C3×C4#12311yescharacteristic
249#67—42no—
250#67—42no—
251C2×Dic3—21no—
252C2×Dic3—21no—
253C2×Dic3—21no—
254C2×Dic3—21no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258C2×Dic3—41no—
259C2×Dic3—42no—
260C2×Dic3—41no—
261C2×Dic3—41no—
262C2×Dic3—41no—
263C2×Dic3—41no—
264C2×Dic3—81no—
265C2×Dic3—81no—
266C2×Dic3#10412yes—
267C4×D3—42no—
268C4×D3—81no—
269C3×C8—42no—
270C3⋊C8—42no—
271C24—31no—
272C24D3×D411yescharacteristic
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—81no—
276C2×D4—62no—
277C2×D4—61no—
278C2×D4—61no—
279C2×D4—121no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—31no—
283C22×C4—31no—
284C22×C4—31no—
285C22×C4—41no—
286C22×C4—61no—
287C22×C4—61no—
288C22×C4—61no—
289C22×C4—61no—
290C22×C4—81no—
291C22×C4—121no—
292C22×C4—241no—
293C22×C4C2×C4×D311yescharacteristic
294C22×C4#23511yescharacteristic
295C22×C4D3×D411yescharacteristic
296C22×C4D3×D411yescharacteristic
297C22×C4#23711yescharacteristic
298C22×C4C2×#6711yescharacteristic
299M16—81no—
300M16—242no—
301M16—241no—
302C2×C8—22no—
303C2×C8—62no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—61no—
309C4⋊C4—61no—
310C4⋊C4—122no—
311C4⋊C4—241no—
312C22⋊C4—22no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—61no—
317C22⋊C4—61no—
318C22⋊C4—61no—
319C22⋊C4—61no—
320C22⋊C4—62no—
321C22⋊C4—122no—
322C22⋊C4—121no—
323C22⋊C4—241no—
324C22⋊C4—241no—
325C42—61no—
326C42—61no—
327C42—81no—
328C42—241no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—81no—
338C22×C3—81no—
339C22×C3#38611yescharacteristic
340C22×C3C2×#10411yescharacteristic
341C22×C3#42111yescharacteristic
342C2×D3—42no—
343C2×D3—41no—
344C2×D3—41no—
345C2×D3—41no—
346C2×D3—81no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—82no—
352C3×C4—81no—
353C3×C4—81no—
354Dic3—22no—
355Dic3—41no—
356Dic3—41no—
357Dic3—81no—
358Dic3—81no—
359Dic3—82no—
360C23—21no—
361C23—21no—
362C23—41no—
363C23—41no—
364C23—61no—
365C23—61no—
366C23—61no—
367C23—121no—
368C23—121no—
369C23D3×C22⋊C411yescharacteristic
370C23#88411yescharacteristic
371C23#93811yescharacteristic
372Q8—81no—
373Q8—81no—
374D4—122no—
375D4—122no—
376C2×C4—32no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—61no—
391C2×C4—61no—
392C2×C4—61no—
393C2×C4—61no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—122no—
397C2×C4—121no—
398C2×C4—121no—
399C2×C4—122no—
400C2×C4—121no—
401C2×C4—121no—
402C2×C4—121no—
403C2×C4—121no—
404C2×C4—121no—
405C2×C4—121no—
406C2×C4—241no—
407C2×C4—241no—
408C2×C4—241no—
409C2×C4#67⋊C411yescharacteristic
410C2×C4#88311yescharacteristic
411C2×C4C4×#6711yescharacteristic
412C2×C4#93711yescharacteristic
413C8—42no—
414C8—122no—
415C2×C3—21no—
416C2×C3—21no—
417C2×C3—41no—
418C2×C3—41no—
419C2×C3#187111yescharacteristic
420C2×C3#190911yescharacteristic
421C2×C3#191311yescharacteristic
422D3—81no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—81no—
432C22—81no—
433C22—122no—
434C22—121no—
435C22—121no—
436C22—121no—
437C22—241no—
438C22D3×#10411yescharacteristic
439C22#482111yescharacteristic
440C22#495611yescharacteristic
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—62no—
446C4—82no—
447C4—81no—
448C4—81no—
449C4—121no—
450C4—121no—
451C4—241no—
452C4—241no—
453C4—242no—
454C3#918411yescharacteristic, complemented, Sylow
455C2—21no—
456C2—21no—
457C2—41no—
458C2—41no—
459C2—241no—
460C2#7003411yescharacteristic
461C2#7035011yescharacteristic
462C2#7038111yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1508C2
socleC22×C3#386
prime radicalC22×#67D4
cosocleC23×C3×C4C23
hypercenter#1508D3
hypocenterC3#9184
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1508

Complements

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

NH ≅ G/Ntype
3. C3×#1508C2semidirect
4. #71688C2semidirect
6. #67682C2semidirect
7. #67616C2semidirect
13. #5231C4semidirect
14. #5230C4semidirect
19. #4992C4semidirect
21. #4983C4semidirect
36. #1508D3semidirect
454. C3#9184semidirect

Maximal subgroups

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

subgroupindex
#828232
#786342
C3×#15082
#716882
C4×#8062
#676822
#676162
#91843