Finite Group Databasebeta

← #10691975

Subgroups — #10691975

Order 768.

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

normal subgroups only

This group has 416 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#82436C211yesmaximal, characteristic
2#82431C211yesmaximal, characteristic
3C3×#1805C211yesmaximal, characteristic
4#70429C211yesmaximal, characteristic
5#69546C211yesmaximal, characteristic
6#67707C211yesmaximal, characteristic
7#67631C211yesmaximal, characteristic
8#9308—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×C3×C4⋊C8C2211yescharacteristic
12C2×C3×#103—21no—
13C3×#382C2211yescharacteristic
14#5225—21no—
15#5024—22no—
16#5024—22no—
17#5022—22no—
18#5021C412yescomplemented
19#5020C412yescomplemented
20#4964—21no—
21#4926—22no—
22#4925C2211yescharacteristic
23#4925C2211yescharacteristic
24#4924C2211yescharacteristic
25C3×#340—21no—
26#4484—22no—
27#4452—22no—
28#3078—31no—
29#2983—31no—
30#1805D311yescharacteristic
31#1518—31no—
32#1481—31no—
33#1301—31no—
34#1286—31no—
35C22×C3×Q8D411yescharacteristic
36C22×C3×C8—21no—
37C3×C4×Q8—21no—
38C3×C4×Q8—21no—
39C3×C4×Q8C2×C412yes—
40C3×C4×Q8D412yes—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4D411yescharacteristic
44C2×C3×C42—21no—
45C2×C3×C42C2311yescharacteristic
46#926—21no—
47#924—21no—
48#924—21no—
49C2×#208—21no—
50#870—21no—
51#869C2×C411yescharacteristic
52#869C2×C411yescharacteristic
53C4×Dic6—21no—
54C4×Dic6—21no—
55C3×C4⋊C8—21no—
56C3×C4⋊C8—21no—
57C3×#103—42no—
58C3×#95—21no—
59C3×#95—21no—
60#835—22no—
61#835—22no—
62#830—21no—
63#830D412yes—
64#808—22no—
65#807—22no—
66C4⋊(C3⋊C8)—24no—
67C4⋊(C3⋊C8)—21no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)C2×C412yes—
71#803—24no—
72C4×C3⋊C8—21no—
73C4×C3⋊C8D412yes—
74#533—31no—
75C2×C4×Q8C2×D311yescharacteristic
76#479—62no—
77#477—32no—
78#475—32no—
79#446—62no—
80#446—62no—
81#432—62no—
82#431—31no—
83#431—31no—
84#428—61no—
85#423—31no—
86C2×C4⋊C8C2×D311yescharacteristic
87#419—61no—
88C2×#103—21no—
89#382C2×D311yescharacteristic
90#340—21no—
91#328—62no—
92Q8⋊C8—62no—
93C2×C3×Q8—21no—
94C2×C3×Q8—21no—
95C2×C3×Q8—21no—
96C2×C3×Q8—21no—
97C2×C3×Q8—21no—
98C2×C3×Q8C22⋊C412yes—
99C22×C3×C4—21no—
100C22×C3×C4—21no—
101C22×C3×C4—41no—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4#4111yescharacteristic
105C2×Dic6—21no—
106C2×Dic6—21no—
107C2×C3×C8—41no—
108C2×C3×C8—41no—
109C2×C3×C8—41no—
110C3×C4⋊C4—22no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—22no—
113C3×C4⋊C4—41no—
114C3×C4⋊C4—41no—
115C3×C4⋊C4C22⋊C412yes—
116C3×C42—22no—
117C3×C42—22no—
118C3×C42C22×C411yescharacteristic
119C3×C42C2×D411yescharacteristic
120C3×C42C2×D411yescharacteristic
121C3×C42#4111yescharacteristic
122#217—41no—
123C4⋊Dic3—21no—
124C4⋊Dic3—21no—
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127Dic3⋊C4—41no—
128Dic3⋊C4—41no—
129C4×Dic3—41no—
130#208—41no—
131#208—41no—
132C2×C3⋊C8—22no—
133C2×C3⋊C8—22no—
134C2×C3⋊C8—22no—
135C22×Q8#6711yescharacteristic
136C2×M16—61no—
137C22×C8—21no—
138C4⋊Q8—31no—
139C4⋊Q8—31no—
140#125—61no—
141C22⋊Q8—61no—
142C22⋊Q8—61no—
143C4×Q8—21no—
144C4×Q8—21no—
145C4×Q8—61no—
146C4×Q8—61no—
147C4×Q8C4×D312yes—
148C4×Q8D1212yes—
149#117—61no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4#6711yescharacteristic
153C2×C42—21no—
154C2×C42C22×D311yescharacteristic
155#107—62no—
156#106—62no—
157C4⋊C8—21no—
158C4⋊C8—21no—
159C4⋊C8—32no—
160C4⋊C8—64no—
161C4⋊C8—61no—
162C4⋊C8—61no—
163C4⋊C8—61no—
164#103—42no—
165#103—62no—
166#103—62no—
167C22⋊C8—32no—
168C22⋊C8—61no—
169#97—64no—
170C4×C8—32no—
171C4×C8—61no—
172#95—21no—
173#95—21no—
174C23×C3#12111yescharacteristic
175C3×Q8—22no—
176C3×Q8—22no—
177C3×Q8—41no—
178C3×Q8—41no—
179C2×C3×C4—22no—
180C2×C3×C4—22no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—81no—
186C2×C3×C4—81no—
187C2×C3×C4—81no—
188C2×C3×C4—81no—
189C2×C3×C4C2×C22⋊C411yescharacteristic
190C2×C3×C4C4×D411yescharacteristic
191C2×C3×C4C4×D411yescharacteristic
192C2×C3×C4#12011yescharacteristic
193C2×C3×C4#12111yescharacteristic
194C2×C3×C4#12311yescharacteristic
195C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
196C2×C3×C4#13711yescharacteristic
197C2×C3×C4#13711yescharacteristic
198C2×C3×C4#13711yescharacteristic
199C2×Dic3—41no—
200C2×Dic3—41no—
201Dic6—42no—
202Dic6—81no—
203C3×C8—81no—
204C3⋊C8—42no—
205C3⋊C8—42no—
206C3⋊C8—42no—
207C2×Q8—21no—
208C2×Q8—21no—
209C2×Q8—21no—
210C2×Q8—21no—
211C2×Q8—21no—
212C2×Q8—61no—
213C2×Q8—61no—
214C2×Q8(C2×D3)⋊C412yes—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—41no—
218C22×C4D3×D411yescharacteristic
219C22×C4#23711yescharacteristic
220C22×C4C2×#6711yescharacteristic
221M16—121no—
222M16—121no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—62no—
227C2×C8—62no—
228C2×C8—62no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—21no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4—121no—
239C4⋊C4—121no—
240C4⋊C4(C2×D3)⋊C412yes—
241C22⋊C4—121no—
242C42—22no—
243C42—22no—
244C42—121no—
245C42C2×C4×D311yescharacteristic
246C42C2×D1211yescharacteristic
247C42D3×D411yescharacteristic
248C42#23711yescharacteristic
249C22×C3—21no—
250C22×C3—21no—
251C22×C3#38611yescharacteristic
252C22×C3#46811yescharacteristic
253C22×C3#47011yescharacteristic
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—21no—
258C3×C4—42no—
259C3×C4—42no—
260C3×C4—161no—
261C3×C4#41711yescharacteristic
262C3×C4#41911yescharacteristic
263C3×C4#46811yescharacteristic
264C3×C4#47011yescharacteristic
265Dic3—81no—
266Dic3—81no—
267C23#93911yescharacteristic
268Q8—22no—
269Q8—22no—
270Q8—41no—
271Q8—41no—
272Q8—122no—
273Q8—241no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—121no—
285C2×C4—121no—
286C2×C4#67⋊C411yescharacteristic
287C2×C4#67⋊C411yescharacteristic
288C2×C4#88211yescharacteristic
289C2×C4#88611yescharacteristic
290C2×C4C2×(C2×D3)⋊C411yescharacteristic
291C2×C4#93211yescharacteristic
292C2×C4#93911yescharacteristic
293C2×C4#94211yescharacteristic
294C2×C4#95111yescharacteristic
295C2×C4#95111yescharacteristic
296C8—81no—
297C8—122no—
298C8—122no—
299C8—122no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3#188111yescharacteristic
303C2×C3#189411yescharacteristic
304C2×C3#190511yescharacteristic
305C22—21no—
306C22—21no—
307C22#495811yescharacteristic
308C22#504211yescharacteristic
309C22#505211yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—42no—
315C4—42no—
316C4—161no—
317C4—241no—
318C4—241no—
319C4#496611yescharacteristic
320C4#500411yescharacteristic
321C4#504211yescharacteristic
322C4#505211yescharacteristic
323C3#930811yescharacteristic, complemented, Sylow
324C2—21no—
325C2—21no—
326C2#6951611yescharacteristic
327C2#7029311yescharacteristic
328C2#7033411yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1805C2
socleC22×C3#386
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenter#1805D3
hypocenterC3#9308
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
18. #5021C4semidirect
19. #5020C4semidirect
323. C3#9308semidirect

Maximal subgroups

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

subgroupindex
#824362
#824312
C3×#18052
#704292
#695462
#677072
#676312
#93083