Finite Group Databasebeta

← #10692112

Subgroups — #10692112

Order 768.

492 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C3×#3110C211yesmaximal, characteristic
2#82325C211yesmaximal, characteristic
3#71841C211yesmaximal, characteristic, complemented
4#71836C211yesmaximal, characteristic, complemented
5#71599C211yesmaximal, characteristic
6#67750C211yesmaximal, characteristic, complemented
7#67748C211yesmaximal, characteristic, complemented
8#9327—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C3×#484—21no—
12C3×#474C412yescomplemented
13C3×#461C412yescomplemented
14C3×#450—21no—
15C3×#440—22no—
16C3×#432C2211yescharacteristic
17#5221—21no—
18#5212—21no—
19#5211—21no—
20#4984—21no—
21#4951—22no—
22C4×C4⋊Dic3C2211yescharacteristic
23#4932C2211yescharacteristic
24#4926C2211yescharacteristic
25C4×#208C2211yescharacteristic
26#4538—21no—
27#4538—21no—
28D4⋊(C3⋊C8)—22no—
29#4532—22no—
30#3110D311yescharacteristic
31#2319—31no—
32#1986—31no—
33#1506—31no—
34#1490—31no—
35#1300—31no—
36#1285—31no—
37C22×C3×D4D411yescharacteristic
38C23×C3×C4—21no—
39C2×C3×SD16—22no—
40C3×C4⋊Q8C2×C411yescharacteristic
41C3×C4⋊Q8C2×C411yescharacteristic
42C3×#125—21no—
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C3×C4×D4C2×C412yes—
50C3×C4×D4D412yes—
51C3×#117—21no—
52C2×C3×C4⋊C4D411yescharacteristic
53C2×C3×C22⋊C4—21no—
54C2×C3×C42C2311yescharacteristic
55C2×#217—22no—
56C2×#217—22no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—22no—
59C2×C4×Dic3—21no—
60C2×C4×Dic3D412yes—
61C2×#208—21no—
62C3×#107—22no—
63C3×C4⋊C8C2×C412yes—
64C3×#103—22no—
65C3×#102—22no—
66C3×C22⋊C8—21no—
67C3×#97—21no—
68C3⋊#95—22no—
69C3⋊#95—21no—
70C3⋊#95—21no—
71C3⋊#95D411yescharacteristic
72C3⋊#95D411yescharacteristic
73#830—21no—
74C4⋊(C3⋊C8)—21no—
75#803—21no—
76#803—21no—
77C4×C3⋊C8—21no—
78#533C2×D311yescharacteristic
79C2×C4×D4C2×D311yescharacteristic
80#484—21no—
81#474Dic312yescomplemented
82#461Dic312yescomplemented
83#450—21no—
84#440—22no—
85#432—31no—
86#432C2×D311yescharacteristic
87C4×M16—31no—
88#390—61no—
89#389—61no—
90#388—62no—
91#381—61no—
92#380—61no—
93C4×C4⋊C4—31no—
94#376—31no—
95#339—62no—
96#328—61no—
97#328—61no—
98D4⋊C8—62no—
99C24×C3—21no—
100C2×C3×Q8—21no—
101C2×C3×Q8—21no—
102C2×C3×D4—21no—
103C2×C3×D4—42no—
104C2×C3×D4C22⋊C412yes—
105C22×C3×C4—21no—
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4—82no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×Dic3—21no—
113C22×Dic3—22no—
114C22×Dic3—21no—
115C3×SD16—44no—
116C2×C3×C8—22no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4C22⋊C412yes—
125C3×C22⋊C4—22no—
126C3×C22⋊C4—22no—
127C3×C22⋊C4—41no—
128C3×C42—41no—
129C3×C42C22×C411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42#4111yescharacteristic
132C3×C42#4111yescharacteristic
133#217—82no—
134#217—82no—
135C4⋊Dic3—42no—
136C4⋊Dic3—82no—
137C4×Dic3—42no—
138C4×Dic3—42no—
139#208—22no—
140#208—41no—
141C2×C3⋊C8—41no—
142C2×C3⋊C8—41no—
143C22×D4#6711yescharacteristic
144C23×C4—21no—
145C2×SD16—22no—
146C2×M16—61no—
147C4⋊Q8C2×Dic311yescharacteristic
148C4⋊Q8C2×Dic311yescharacteristic
149#125—21no—
150C22⋊Q8—21no—
151C22⋊Q8—21no—
152C4×Q8—21no—
153C4×Q8—21no—
154C4×D4—21no—
155C4×D4—21no—
156C4×D4C2×Dic312yes—
157C4×D4#6712yes—
158#117—21no—
159C2×C4⋊C4—61no—
160C2×C4⋊C4—62no—
161C2×C4⋊C4#6711yescharacteristic
162C2×C22⋊C4—21no—
163C2×C22⋊C4—62no—
164C2×C22⋊C4—62no—
165C2×C42—32no—
166C2×C42—61no—
167C2×C42C22×D311yescharacteristic
168#107—22no—
169C4⋊C8—61no—
170C4⋊C8C2×Dic312yes—
171#103—22no—
172#102—22no—
173C22⋊C8—21no—
174C22⋊C8—61no—
175#97—21no—
176#97—61no—
177#97—61no—
178C4×C8—61no—
179#95—31no—
180#95—31no—
181#95—62no—
182#95—61no—
183#95—61no—
184C23×C3—22no—
185C23×C3—42no—
186C23×C3—42no—
187C23×C3—41no—
188C23×C3—41no—
189C23×C3#12011yescharacteristic
190C3×Q8—42no—
191C3×Q8—81no—
192C3×D4—22no—
193C3×D4—22no—
194C3×D4—42no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—22no—
198C2×C3×C4—22no—
199C2×C3×C4—41no—
200C2×C3×C4—42no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—82no—
204C2×C3×C4—82no—
205C2×C3×C4C2×C22⋊C411yescharacteristic
206C2×C3×C4C4×D411yescharacteristic
207C2×C3×C4C4×D411yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12311yescharacteristic
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—42no—
214C2×Dic3—41no—
215C2×Dic3—42no—
216C2×Dic3—41no—
217C2×Dic3—42no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C3×C8—42no—
221C3⋊C8—42no—
222C3⋊C8—81no—
223C24—21no—
224C2×Q8—21no—
225C2×Q8—21no—
226C2×D4—21no—
227C2×D4—42no—
228C2×D4#21712yes—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4—62no—
235C22×C4—82no—
236C22×C4D3×D411yescharacteristic
237C22×C4D3×D411yescharacteristic
238C22×C4C2×#6711yescharacteristic
239SD16—44no—
240M16—62no—
241M16—121no—
242C2×C8—22no—
243C2×C8—121no—
244C2×C8—121no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—122no—
253C4⋊C4—242no—
254C4⋊C4#21712yes—
255C22⋊C4—22no—
256C22⋊C4—22no—
257C22⋊C4—41no—
258C22⋊C4—242no—
259C22⋊C4—242no—
260C42—41no—
261C42—122no—
262C42—122no—
263C42#23711yescharacteristic
264C42#23711yescharacteristic
265C42C22×Dic311yescharacteristic
266C42C2×#6711yescharacteristic
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—42no—
270C22×C3—42no—
271C22×C3—42no—
272C22×C3—82no—
273C22×C3—82no—
274C22×C3#38611yescharacteristic
275C22×C3#45511yescharacteristic
276C22×C3#45611yescharacteristic
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—42no—
283C3×C4—81no—
284C3×C4—81no—
285C3×C4#42112yes—
286Dic3—81no—
287Dic3—82no—
288Dic3—81no—
289C23—22no—
290C23—42no—
291C23—42no—
292C23—41no—
293C23—41no—
294C23#93711yescharacteristic
295Q8—42no—
296Q8—81no—
297D4—22no—
298D4—22no—
299D4—42no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—122no—
314C2×C4—122no—
315C2×C4—121no—
316C2×C4—122no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4D4×Dic311yescharacteristic
320C2×C4D4×Dic311yescharacteristic
321C2×C4#93511yescharacteristic
322C2×C4#93911yescharacteristic
323C2×C4#93911yescharacteristic
324C2×C4C2×#21711yescharacteristic
325C8—42no—
326C8—122no—
327C8—241no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—82no—
331C2×C3#188211yescharacteristic
332C2×C3#189711yescharacteristic
333C2×C3#190411yescharacteristic
334C22—21no—
335C22—21no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22—82no—
340C22—82no—
341C22#520211yescharacteristic
342C22#520511yescharacteristic
343C22#522011yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—42no—
350C4—81no—
351C4—81no—
352C4—241no—
353C4—242no—
354C4—241no—
355C4#523612yes—
356C3#932711yescharacteristic, complemented, Sylow
357C2—21no—
358C2—21no—
359C2—82no—
360C2#7171211yescharacteristic
361C2#7208911yescharacteristic
362C2#7210011yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3110C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#3110D3
hypocenterC3#9327
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
3. #71841C2semidirect
4. #71836C2semidirect
6. #67750C2semidirect
7. #67748C2semidirect
12. C3×#474C4semidirect
13. C3×#461C4semidirect
81. #474Dic3semidirect
82. #461Dic3semidirect
356. C3#9327semidirect

Maximal subgroups

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

subgroupindex
C3×#31102
#823252
#718412
#718362
#715992
#677502
#677482
#93273