Finite Group Databasebeta

← #10691968

Subgroups — #10691968

Order 768.

464 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 111 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#82304C211yesmaximal, characteristic
2#82296C211yesmaximal, characteristic
3C3×#1804C211yesmaximal, characteristic
4#70429C211yesmaximal, characteristic, complemented
5#69546C211yesmaximal, characteristic, complemented
6#67707C211yesmaximal, characteristic, complemented
7#67631C211yesmaximal, characteristic, complemented
8#9307—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×C3×C4⋊C8C2211yescharacteristic
12C2×C3×#102—21no—
13C3×#382C2211yescharacteristic
14#5215—21no—
15#5207—22no—
16#5012—22no—
17#5012—22no—
18#5009—22no—
19#5008C412yescomplemented
20#5007C412yescomplemented
21#4964—21no—
22#4925C2211yescharacteristic
23#4925C2211yescharacteristic
24#4924C2211yescharacteristic
25C3×#340—21no—
26#4484—22no—
27#4452—22no—
28#3077—31no—
29#2975—31no—
30#1804D311yescharacteristic
31#1518—31no—
32#1481—31no—
33#1301—31no—
34#1286—31no—
35C22×C3×D4D411yescharacteristic
36C23×C3×C4—21no—
37C22×C3×C8—21no—
38C3×C4×D4—21no—
39C3×C4×D4—21no—
40C3×C4×D4C2×C412yes—
41C3×C4×D4D412yes—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4D411yescharacteristic
44C2×C3×C22⋊C4—21no—
45C2×C3×C42C2311yescharacteristic
46#926—21no—
47#924—21no—
48#924—21no—
49C2×#208—21no—
50C2×#208—24no—
51#870—21no—
52#869C2×C411yescharacteristic
53#869C2×C411yescharacteristic
54C4×Dic6—21no—
55C4×Dic6—21no—
56C3×C4⋊C8—21no—
57C3×C4⋊C8—21no—
58C3×#102—42no—
59C3×#95—21no—
60C3×#95—21no—
61#832—22no—
62#832—22no—
63#830—24no—
64#830—21no—
65#830D412yes—
66#808—22no—
67#807—22no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)C2×C412yes—
72C4×C3⋊C8—21no—
73C4×C3⋊C8D412yes—
74#533—31no—
75C2×C4×D4C2×D311yescharacteristic
76#478—62no—
77#476—32no—
78#474—32no—
79#436—62no—
80#436—62no—
81#431—31no—
82#431—31no—
83#428—61no—
84#423—31no—
85C2×C4⋊C8C2×D311yescharacteristic
86#418—61no—
87C2×#102—21no—
88#407—62no—
89#382C2×D311yescharacteristic
90#340—21no—
91#328—62no—
92Q8⋊C8—62no—
93C24×C3—21no—
94C2×C3×D4—21no—
95C2×C3×D4—21no—
96C2×C3×D4—21no—
97C2×C3×D4—21no—
98C2×C3×D4—21no—
99C2×C3×D4C22⋊C412yes—
100C22×C3×C4—21no—
101C22×C3×C4—22no—
102C22×C3×C4—22no—
103C22×C3×C4—41no—
104C22×C3×C4—82no—
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C2×Dic6—21no—
109C2×Dic6—21no—
110C2×C3×C8—41no—
111C2×C3×C8—41no—
112C2×C3×C8—41no—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—41no—
115C3×C4⋊C4—41no—
116C3×C4⋊C4C22⋊C412yes—
117C3×C22⋊C4—22no—
118C3×C22⋊C4—22no—
119C3×C42C22×C411yescharacteristic
120C3×C42C2×D411yescharacteristic
121C3×C42C2×D411yescharacteristic
122C3×C42#4111yescharacteristic
123#217—41no—
124C4⋊Dic3—21no—
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128Dic3⋊C4—41no—
129Dic3⋊C4—41no—
130C4×Dic3—41no—
131#208—41no—
132#208—41no—
133#208—84no—
134C2×C3⋊C8—22no—
135C2×C3⋊C8—22no—
136C2×C3⋊C8—22no—
137C22×D4#6711yescharacteristic
138C23×C4—21no—
139C2×M16—61no—
140C2×M16—64no—
141C22×C8—21no—
142C4⋊Q8—31no—
143C4⋊Q8—31no—
144#125—61no—
145C22⋊Q8—61no—
146C22⋊Q8—61no—
147C4×Q8—61no—
148C4×Q8—61no—
149C4×D4—21no—
150C4×D4—21no—
151C4×D4C4×D312yes—
152C4×D4D1212yes—
153#117—61no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4#6711yescharacteristic
156C2×C22⋊C4—21no—
157C2×C42C22×D311yescharacteristic
158#107—62no—
159#106—62no—
160C4⋊C8—21no—
161C4⋊C8—21no—
162C4⋊C8—32no—
163C4⋊C8—61no—
164C4⋊C8—61no—
165C4⋊C8—61no—
166#102—42no—
167#102—62no—
168#102—62no—
169C22⋊C8—32no—
170C22⋊C8—64no—
171C22⋊C8—61no—
172C4×C8—32no—
173C4×C8—61no—
174#95—21no—
175#95—21no—
176C23×C3—22no—
177C23×C3—41no—
178C23×C3—41no—
179C23×C3—81no—
180C23×C3—81no—
181C23×C3#12111yescharacteristic
182C3×D4—22no—
183C3×D4—22no—
184C3×D4—41no—
185C3×D4—41no—
186C2×C3×C4—22no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—44no—
190C2×C3×C4—82no—
191C2×C3×C4—81no—
192C2×C3×C4—81no—
193C2×C3×C4—81no—
194C2×C3×C4—81no—
195C2×C3×C4C2×C22⋊C411yescharacteristic
196C2×C3×C4C4×D411yescharacteristic
197C2×C3×C4C4×D411yescharacteristic
198C2×C3×C4#12011yescharacteristic
199C2×C3×C4#12111yescharacteristic
200C2×C3×C4#12311yescharacteristic
201C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
202C2×C3×C4#13711yescharacteristic
203C2×C3×C4#13711yescharacteristic
204C2×C3×C4#13711yescharacteristic
205C2×Dic3—41no—
206C2×Dic3—41no—
207Dic6—42no—
208Dic6—81no—
209C3×C8—81no—
210C3⋊C8—42no—
211C3⋊C8—42no—
212C3⋊C8—42no—
213C24—21no—
214C2×Q8—61no—
215C2×Q8—61no—
216C2×D4—21no—
217C2×D4—21no—
218C2×D4—21no—
219C2×D4—21no—
220C2×D4—21no—
221C2×D4(C2×D3)⋊C412yes—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—41no—
226C22×C4—82no—
227C22×C4D3×D411yescharacteristic
228C22×C4#23711yescharacteristic
229C22×C4C2×#6711yescharacteristic
230M16—121no—
231M16—121no—
232M16—244no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—62no—
237C2×C8—62no—
238C2×C8—62no—
239C4⋊C4—21no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—121no—
247C4⋊C4—121no—
248C4⋊C4(C2×D3)⋊C412yes—
249C22⋊C4—22no—
250C22⋊C4—22no—
251C22⋊C4—121no—
252C42—121no—
253C42C2×C4×D311yescharacteristic
254C42C2×D1211yescharacteristic
255C42D3×D411yescharacteristic
256C42#23711yescharacteristic
257C22×C3—21no—
258C22×C3—21no—
259C22×C3—42no—
260C22×C3—42no—
261C22×C3—42no—
262C22×C3—82no—
263C22×C3—82no—
264C22×C3#38611yescharacteristic
265C22×C3#46811yescharacteristic
266C22×C3#47011yescharacteristic
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—42no—
272C3×C4—161no—
273C3×C4#41711yescharacteristic
274C3×C4#41911yescharacteristic
275C3×C4#46811yescharacteristic
276C3×C4#47011yescharacteristic
277Dic3—81no—
278Dic3—81no—
279C23—22no—
280C23—41no—
281C23—41no—
282C23—81no—
283C23—81no—
284C23#93911yescharacteristic
285Q8—122no—
286Q8—241no—
287D4—22no—
288D4—22no—
289D4—41no—
290D4—41no—
291C2×C4—22no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—44no—
295C2×C4—82no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—121no—
301C2×C4—121no—
302C2×C4#67⋊C411yescharacteristic
303C2×C4#67⋊C411yescharacteristic
304C2×C4#88211yescharacteristic
305C2×C4#88611yescharacteristic
306C2×C4C2×(C2×D3)⋊C411yescharacteristic
307C2×C4#93211yescharacteristic
308C2×C4#93911yescharacteristic
309C2×C4#94211yescharacteristic
310C2×C4#95111yescharacteristic
311C2×C4#95111yescharacteristic
312C8—81no—
313C8—122no—
314C8—122no—
315C8—122no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3—82no—
319C2×C3#188111yescharacteristic
320C2×C3#189411yescharacteristic
321C2×C3#190411yescharacteristic
322C22—21no—
323C22—21no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—82no—
328C22—82no—
329C22#495811yescharacteristic
330C22#504211yescharacteristic
331C22#505211yescharacteristic
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—42no—
337C4—161no—
338C4—241no—
339C4—241no—
340C4#496611yescharacteristic
341C4#500411yescharacteristic
342C4#504211yescharacteristic
343C4#505211yescharacteristic
344C3#930711yescharacteristic, complemented, Sylow
345C2—21no—
346C2—21no—
347C2—82no—
348C2#6951611yescharacteristic
349C2#7029311yescharacteristic
350C2#7031411yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1804C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#1804D3
hypocenterC3#9307
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
4. #70429C2semidirect
5. #69546C2semidirect
6. #67707C2semidirect
7. #67631C2semidirect
19. #5008C4semidirect
20. #5007C4semidirect
344. C3#9307semidirect

Maximal subgroups

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

subgroupindex
#823042
#822962
C3×#18042
#704292
#695462
#677072
#676312
#93073