Finite Group Databasebeta

← #10707604

Subgroups — #10707604

Order 768.

458 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 57 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#77815C211yesmaximal, characteristic, complemented
2C3×#1610C211yesmaximal, characteristic, complemented
3#71878C211yesmaximal, characteristic
4#71877C211yesmaximal, characteristic, complemented
5#67650C211yesmaximal, characteristic, complemented
6#67456C211yesmaximal, characteristic, complemented
7#67452C211yesmaximal, characteristic, complemented
8#12154—31nomaximal, Sylow
9C3×#537C2211yescharacteristic
10C2×#892—21no—
11C3×#480—21no—
12C3×#398C2211yescharacteristic
13C2×#837—21no—
14#5232—21no—
15#5036—21no—
16#4994—21no—
17#4991C2211yescharacteristic
18#4990—21no—
19#4977C2211yescharacteristic
20#4668—21no—
21#4665—21no—
22#4655—21no—
23C3×#327—21no—
24C3×#324C2211yescharacteristic
25#4537—21no—
26#4530C2211yescharacteristic
27#4528C2211yescharacteristic
28#2609—31no—
29#2040—31no—
30#1610D311yescharacteristic, complemented
31#1598—31no—
32#1346—31no—
33#1345—31no—
34#1324—31no—
35C2×C3×#41—21no—
36C22×D12D411yescharacteristic
37C22×C4×D3—21no—
38C3×C22⋊Q8—41no—
39C3×#121—21no—
40C3×#121—21no—
41C3×#121—41no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4D411yescharacteristic
44C2×C3×C4⋊C4C2311yescharacteristic
45C2×#213—41no—
46C2×(C2×D3)⋊C4—21no—
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—21no—
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3D411yescharacteristic
53C2×Dic3⋊C4—21no—
54C2×C4×Dic3—21no—
55C2×C4×Dic3D411yescharacteristic
56C2×#208—21no—
57#892—42no—
58#892—42no—
59C3×#107—41no—
60C3×#102—41no—
61C3×C22⋊C8—21no—
62C3×#95—21no—
63C3×#95D411yescharacteristic
64#837—42no—
65#835—41no—
66#832—41no—
67C3⋊#95—21no—
68C3⋊#95—21no—
69C3⋊#95—21no—
70C3⋊#95D411yescharacteristic
71#830—21no—
72#809—41no—
73#537C2×D311yescharacteristic
74C2×#117—61no—
75#480—21no—
76#447—61no—
77C2×#104—61no—
78#417—61no—
79#398C2×D311yescharacteristic
80#396—61no—
81#394—31no—
82#388—61no—
83#387—61no—
84#386—61no—
85#385—61no—
86#382—31no—
87#343—31no—
88#327—21no—
89#327—61no—
90#324—31no—
91#324C2×D311yescharacteristic
92C23×D3—21no—
93C3×#41—41no—
94C3×#41—41no—
95C3×#41—41no—
96C3×#41—41no—
97C2×C3×Q8—21no—
98C2×C3×D4—21no—
99C2×C3×D4—21no—
100C2×C3×D4—21no—
101C2×C3×D4—41no—
102C22×C3×C4—21no—
103C22×C3×C4—21no—
104C22×C3×C4—41no—
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×Dic3—21no—
109C22×Dic3—21no—
110C22×Dic3—21no—
111C2×D12—41no—
112C2×D12—42no—
113C2×C4×D3—42no—
114C2×C4×D3—42no—
115C2×C4×D3—42no—
116C2×C3×C8—41no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C22⋊C4—41no—
124C3×C22⋊C4—41no—
125#213—82no—
126(C2×D3)⋊C4—42no—
127(C2×D3)⋊C4—42no—
128(C2×D3)⋊C4—42no—
129(C2×D3)⋊C4—41no—
130(C2×D3)⋊C4—41no—
131(C2×D3)⋊C4—81no—
132C4⋊Dic3—42no—
133C4⋊Dic3—41no—
134Dic3⋊C4—42no—
135C4×Dic3—21no—
136C4×Dic3—42no—
137C4×Dic3D812yescomplemented
138#208—41no—
139#208—41no—
140C2×C3⋊C8—41no—
141C2×C3⋊C8—41no—
142C2×#41—21no—
143C22×D4—31no—
144C23×C4—61no—
145C2×D8—121no—
146C2×M16—61no—
147C22⋊Q8—41no—
148#121—21no—
149#121—21no—
150#121—41no—
151#117—122no—
152#117—122no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—31no—
155C2×C4⋊C4—61no—
156C2×C4⋊C4—61no—
157C2×C4⋊C4#6711yescharacteristic
158C2×C4⋊C4C22×D311yescharacteristic
159C2×C22⋊C4—61no—
160C2×C22⋊C4—61no—
161C2×C22⋊C4—61no—
162C2×C22⋊C4—61no—
163C2×C22⋊C4—61no—
164C2×C42—31no—
165C2×C42—61no—
166#107—41no—
167#104—122no—
168#103—121no—
169#102—41no—
170#102—121no—
171#102—121no—
172C22⋊C8—21no—
173C22⋊C8—61no—
174#95—21no—
175#95—31no—
176#95—61no—
177#95—61no—
178#95—61no—
179#95#6711yescharacteristic
180C23×C3—41no—
181C23×C3#12011yescharacteristic
182C22×D3—22no—
183C22×D3—41no—
184C22×D3—41no—
185C22×D3—41no—
186C22×D3—41no—
187C22×D3—41no—
188C22×D3—41no—
189C3×Q8—41no—
190C3×Q8—41no—
191C3×D4—41no—
192C3×D4—41no—
193C3×D4—81no—
194C3×D4—81no—
195C3×D4—81no—
196C3×D4—81no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—22no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—81no—
211C2×C3×C4—81no—
212C2×C3×C4C2×D811yescharacteristic
213C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
214C2×Dic3—22no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223D12—42no—
224D12—81no—
225C4×D3—48no—
226C3×C8—81no—
227C3⋊C8—81no—
228C3⋊C8—81no—
229C24—61no—
230#41—41no—
231#41—41no—
232#41—41no—
233#41—41no—
234C2×Q8—21no—
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4—41no—
239C2×D4—121no—
240C2×D4—122no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—41no—
244C22×C4—61no—
245C22×C4—61no—
246C22×C4—61no—
247C22×C4—122no—
248C22×C4—122no—
249C22×C4—122no—
250C22×C4D3×D411yescharacteristic
251C22×C4D3×D411yescharacteristic
252C22×C4C2×#6711yescharacteristic
253D8—242no—
254M16—121no—
255M16—121no—
256C2×C8—41no—
257C2×C8—121no—
258C2×C8—121no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—121no—
266C4⋊C4—122no—
267C4⋊C4—122no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—122no—
271C22⋊C4—122no—
272C22⋊C4—122no—
273C22⋊C4—121no—
274C22⋊C4—121no—
275C22⋊C4—241no—
276C42—32no—
277C42—61no—
278C42—122no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—81no—
282C22×C3—81no—
283C22×C3—81no—
284C22×C3#44711yescharacteristic
285C22×C3#45411yescharacteristic
286C22×C3B(4,𝔽2)11yescharacteristic
287C2×D3—42no—
288C2×D3—42no—
289C2×D3—42no—
290C2×D3—81no—
291C2×D3—81no—
292C2×D3—81no—
293C2×D3—81no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—42no—
297C3×C4—41no—
298C3×C4—41no—
299C3×C4—81no—
300C3×C4—81no—
301Dic3—42no—
302Dic3—81no—
303Dic3—81no—
304C23—41no—
305C23—62no—
306C23—121no—
307C23—121no—
308C23—121no—
309C23—121no—
310C23—121no—
311C23—121no—
312C23#93711yescharacteristic
313Q8—41no—
314Q8—41no—
315D4—41no—
316D4—41no—
317D4—81no—
318D4—81no—
319D4—81no—
320D4—81no—
321D4—122no—
322D4—241no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—62no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—128no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4—121no—
348C2×C4D3×D811yescharacteristic
349C2×C4#91111yescharacteristic
350C8—81no—
351C8—241no—
352C8—241no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—161no—
356C2×C3#159811yescharacteristic
357C2×C3#219711yescharacteristic
358C2×C3#220411yescharacteristic
359D3—82no—
360C22—21no—
361C22—21no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—122no—
366C22—122no—
367C22—122no—
368C22—241no—
369C22—241no—
370C22—241no—
371C22—241no—
372C22#503211yescharacteristic
373C22#515611yescharacteristic
374C22#520111yescharacteristic
375C4—21no—
376C4—21no—
377C4—42no—
378C4—41no—
379C4—41no—
380C4—81no—
381C4—81no—
382C4—122no—
383C4—241no—
384C4—241no—
385C3#1215411yescharacteristic, complemented, Sylow
386C2—21no—
387C2—21no—
388C2—161no—
389C2—242no—
390C2#7097111yescharacteristic
391C2#7099411yescharacteristic
392C2#7100011yescharacteristic
subgroupG/H
centerC22#5032
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1610C2
socleC22×C3B(4,𝔽2)
prime radical#71878C2
hypercenter#1610D3
hypocenterC3#12154
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1610

Complements

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

NH ≅ G/Ntype
1. #77815C2semidirect
2. C3×#1610C2semidirect
4. #71877C2semidirect
5. #67650C2semidirect
6. #67456C2semidirect
7. #67452C2semidirect
30. #1610D3semidirect
137. C4×Dic3D8semidirect
385. C3#12154semidirect

Maximal subgroups

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

subgroupindex
#778152
C3×#16102
#718782
#718772
#676502
#674562
#674522
#121543