Finite Group Databasebeta

← #10705802

Subgroups — #10705802

Order 768.

460 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 73 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#76799C211yesmaximal, characteristic
2#68952C211yesmaximal, characteristic
3#68121C211yesmaximal, characteristic
4C3×#1351C211yesmaximal, characteristic, complemented
5#67646C211yesmaximal, characteristic, complemented
6#67490C211yesmaximal, characteristic, complemented
7#67449C211yesmaximal, characteristic, complemented
8#11787—31nomaximal, Sylow
9C2×#944C411yescharacteristic, complemented
10C2×#67⋊C4C2211yescharacteristic
11C3×#393C2211yescharacteristic
12#5231C411yescharacteristic, complemented
13#4957—21no—
14C2×#806—21no—
15#4947C2211yescharacteristic
16#4942—21no—
17Dic3×C22⋊C4—21no—
18#4726—21no—
19#4673C411yescharacteristic, complemented
20#4667—21no—
21#4650—21no—
22#4649C411yescharacteristic, complemented
23C3×#342C2211yescharacteristic
24C3×#323C2211yescharacteristic
25#4526C2211yescharacteristic
26#4525C2211yescharacteristic
27#4469—21no—
28#2563—31no—
29C4×#99—31no—
30#1465—31no—
31#1351—31no—
32#1351—31no—
33#1351—31no—
34#1351D311yescharacteristic, complemented
35C22×C3×Q8C2×C411yescharacteristic
36C22×#67D411yescharacteristic
37C23×Dic3D411yescharacteristic
38C22×C4×D3D411yescharacteristic
39C22×Dic6C2×C411yescharacteristic
40C2×C3×C22⋊C4—21no—
41C2×C3×C22⋊C4—21no—
42C2×C3×C22⋊C4C2311yescharacteristic
43C2×#217—21no—
44C2×#217—21no—
45#944—41no—
46#944—41no—
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—21no—
49C2×(C2×D3)⋊C4C2×C411yescharacteristic
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4D411yescharacteristic
54C2×C4×Dic3—21no—
55C2×C4×Dic3—41no—
56C2×C4×Dic3C2×C411yescharacteristic
57C2×ΓL(1,𝔽25)—41no—
58#67⋊C4—41no—
59#67⋊C4—41no—
60C3×C22⋊C8—21no—
61C3×#95—21no—
62C3×#95C2×C411yescharacteristic
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—41no—
67C3⋊#95C2×C411yescharacteristic
68#830—21no—
69(C2×D3)⋊C8—41no—
70#806—24no—
71C2×C4⋊Q8—31no—
72C2×C4×D4—31no—
73C2×#99—61no—
74#407—61no—
75#393—31no—
76#393—31no—
77#393—31no—
78#393C2×D311yescharacteristic
79#390—61no—
80#382—61no—
81#380—61no—
82#380—61no—
83C4×C4⋊C4—31no—
84C4×C22⋊C4—61no—
85#342—31no—
86#342C2×D311yescharacteristic
87#324—61no—
88#323—31no—
89#323C2×D311yescharacteristic
90C24×C3C2×D411yescharacteristic
91C23×D3C22⋊C411yescharacteristic
92C2×C3×Q8—41no—
93C2×C3×Q8—81no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—41no—
97C22×C3×C4C22×C411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C2×#67—21no—
100C2×#67—21no—
101C2×#67—22no—
102C2×#67—41no—
103C22×Dic3—21no—
104C22×Dic3—21no—
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—42no—
108C22×Dic3—41no—
109C22×Dic3C22⋊C411yescharacteristic
110C22×Dic3C22⋊C411yescharacteristic
111C22×Dic3C22⋊C411yescharacteristic
112C2×C4×D3—21no—
113C2×C4×D3—21no—
114C2×C4×D3—42no—
115C2×Dic6—41no—
116C2×Dic6—81no—
117C2×C3×C8—41no—
118C3×C22⋊C4—22no—
119C3×C22⋊C4—41no—
120C3×C22⋊C4—41no—
121C3×C22⋊C4—81no—
122#217—22no—
123#217—41no—
124#217—81no—
125(C2×D3)⋊C4—41no—
126(C2×D3)⋊C4—81no—
127(C2×D3)⋊C4—81no—
128Dic3⋊C4—41no—
129Dic3⋊C4—42no—
130Dic3⋊C4—42no—
131Dic3⋊C4—41no—
132Dic3⋊C4—41no—
133C4×Dic3—21no—
134C4×Dic3—21no—
135C4×Dic3—42no—
136C4×Dic3—82no—
137C2×C3⋊C8—41no—
138ΓL(1,𝔽25)—82no—
139C22×Q8—31no—
140C22×Q8C4×D311yescharacteristic
141C22×D4—31no—
142C23×C4—31no—
143C23×C4—31no—
144C2×M16—121no—
145C4⋊Q8—121no—
146C4⋊Q8—121no—
147C4×D4—121no—
148C4×D4—121no—
149C2×C4⋊C4—31no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4—61no—
152C2×C4⋊C4—61no—
153C2×C22⋊C4—21no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—31no—
156C2×C22⋊C4—61no—
157C2×C22⋊C4—61no—
158C2×C22⋊C4—61no—
159C2×C22⋊C4—61no—
160C2×C22⋊C4C22×D311yescharacteristic
161C2×C42—31no—
162C2×C42—61no—
163C2×C42—121no—
164#99—64no—
165C22⋊C8—21no—
166C22⋊C8—61no—
167C22⋊C8—121no—
168#95—21no—
169#95—31no—
170#95—61no—
171#95—61no—
172#95—61no—
173#95—121no—
174#95C4×D311yescharacteristic
175C23×C3—21no—
176C23×C3—21no—
177C23×C3—41no—
178C23×C3—41no—
179C23×C3C2×C22⋊C411yescharacteristic
180C23×C3#13711yescharacteristic
181C23×C3#13711yescharacteristic
182C22×D3—21no—
183C22×D3—21no—
184C22×D3—21no—
185C22×D3—41no—
186C22×D3—41no—
187C3×Q8—82no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—42no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—81no—
200C2×C3×C4—81no—
201#67—42no—
202#67—42no—
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3—22no—
206C2×Dic3—21no—
207C2×Dic3—42no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—42no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—81no—
217C2×Dic3—81no—
218C2×Dic3—81no—
219C2×Dic3#9911yescharacteristic
220C2×Dic3#9911yescharacteristic
221C2×Dic3#10412yes—
222C4×D3—42no—
223C4×D3—81no—
224Dic6—82no—
225C3×C8—81no—
226C3⋊C8—81no—
227C24—31no—
228C24D3×D411yescharacteristic
229C2×Q8—41no—
230C2×Q8—81no—
231C2×Q8—121no—
232C2×Q8—241no—
233C2×D4—62no—
234C2×D4—61no—
235C2×D4—61no—
236C2×D4—121no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—31no—
240C22×C4—31no—
241C22×C4—31no—
242C22×C4—41no—
243C22×C4—61no—
244C22×C4—61no—
245C22×C4—61no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4—61no—
249C22×C4—122no—
250C22×C4—122no—
251C22×C4—121no—
252C22×C4C2×C4×D311yescharacteristic
253C22×C4D3×D411yescharacteristic
254M16—242no—
255C2×C8—41no—
256C2×C8—121no—
257C4⋊C4—121no—
258C4⋊C4—121no—
259C4⋊C4—121no—
260C4⋊C4—122no—
261C4⋊C4—122no—
262C22⋊C4—22no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—62no—
266C22⋊C4—81no—
267C22⋊C4—121no—
268C22⋊C4—121no—
269C22⋊C4—241no—
270C22⋊C4—241no—
271C22⋊C4—241no—
272C42—61no—
273C42—61no—
274C42—122no—
275C42—242no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3—81no—
285C22×C3C2×#9911yescharacteristic
286C22×C3#41911yescharacteristic
287C22×C3C2×#10411yescharacteristic
288C2×D3—42no—
289C2×D3—41no—
290C2×D3—41no—
291C2×D3—41no—
292C2×D3—81no—
293C3×C4—41no—
294C3×C4—41no—
295C3×C4—81no—
296C3×C4—82no—
297C3×C4—81no—
298Dic3—21no—
299Dic3—22no—
300Dic3—41no—
301Dic3—81no—
302Dic3—82no—
303Dic3—81no—
304Dic3#35412yes—
305C23—21no—
306C23—21no—
307C23—41no—
308C23—41no—
309C23—61no—
310C23—61no—
311C23—61no—
312C23—121no—
313C23—121no—
314C23D3×C22⋊C411yescharacteristic
315C23#91511yescharacteristic
316C23#91811yescharacteristic
317Q8—82no—
318Q8—242no—
319D4—122no—
320D4—122no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—31no—
324C2×C4—31no—
325C2×C4—32no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—62no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—122no—
342C2×C4—122no—
343C2×C4—121no—
344C2×C4—122no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4—241no—
352C2×C4—241no—
353C2×C4—241no—
354C2×C4—241no—
355C8—81no—
356C8—241no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—41no—
360C2×C3—41no—
361C2×C3#150811yescharacteristic
362C2×C3#212211yescharacteristic
363C2×C3C2×#35411yescharacteristic
364D3—81no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—81no—
374C22—122no—
375C22—121no—
376C22—121no—
377C22—121no—
378C22—241no—
379C22D3×#9911yescharacteristic
380C22#480211yescharacteristic
381C22D3×#10411yescharacteristic
382C4—32no—
383C4—41no—
384C4—41no—
385C4—61no—
386C4—62no—
387C4—81no—
388C4—82no—
389C4—81no—
390C4—121no—
391C4—241no—
392C4—242no—
393C4—241no—
394C3#1178711yescharacteristic, complemented, Sylow
395C2—21no—
396C2—21no—
397C2—41no—
398C2—41no—
399C2—241no—
400C2#6833511yescharacteristic
401C2#6913711yescharacteristic
402C2D3×#35411yescharacteristic
subgroupG/H
centerC22D3×#99
commutatorC22×C3×C4C22×C4
FrattiniC2×C22⋊C4C22×D3
FittingC3×#1351C2
socleC22×C3C2×#99
prime radicalC22×#67D4
cosocleC2×C3×C22⋊C4C23
hypercenter#1351D3
hypocenterC3#11787
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1351

Complements

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

NH ≅ G/Ntype
4. C3×#1351C2semidirect
5. #67646C2semidirect
6. #67490C2semidirect
7. #67449C2semidirect
9. C2×#944C4semidirect
12. #5231C4semidirect
19. #4673C4semidirect
22. #4649C4semidirect
34. #1351D3semidirect
394. C3#11787semidirect

Maximal subgroups

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

subgroupindex
#767992
#689522
#681212
C3×#13512
#676462
#674902
#674492
#117873