Finite Group Databasebeta

← #10690370

Subgroups — #10690370

Order 768.

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

normal subgroups only

This group has 424 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
1#82447C211yesmaximal, characteristic
2#79045C211yesmaximal, characteristic
3C3×#1487C211yesmaximal, characteristic
4#71635C211yesmaximal, characteristic
5#70776C211yesmaximal, characteristic
6#67741C211yesmaximal, characteristic
7#67714C211yesmaximal, characteristic
8#9048—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13#5228C412yes—
14#5122—21no—
15#5119—21no—
16#5031—21no—
17C4×#216—22no—
18#5021—21no—
19#4970C412yes—
20#4925C2211yescharacteristic
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22C2×#803C2211yescharacteristic
23#4828—22no—
24#4828—22no—
25C3×Q8⋊C8—22no—
26#4524—21no—
27#4524—21no—
28#4482—21no—
29#4481—21no—
30#3104—31no—
31#2961—31no—
32#1985—31no—
33#1542—31no—
34#1487D311yescharacteristic
35#1301—31no—
36#1281—31no—
37C22×C3×Q8C2×C411yescharacteristic
38C2×C3×M16—21no—
39C3×C4×Q8—21no—
40C3×C4×Q8—21no—
41C3×C4×Q8—21no—
42C3×C4×Q8—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4C2×C411yescharacteristic
45C2×C3×C42—21no—
46C2×C3×C42C2311yescharacteristic
47C2×#216—22no—
48#926—21no—
49#924—21no—
50#924C2×C412yes—
51C22×C3⋊C8—21no—
52C22×C3⋊C8D411yescharacteristic
53C22×C3⋊C8D411yescharacteristic
54#870—21no—
55#869D411yescharacteristic
56#869D411yescharacteristic
57C4×Dic6—21no—
58C4×Dic6D412yes—
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×C4⋊C8D412yes—
62C3×C22⋊C8—21no—
63C3×C4×C8—21no—
64#835—22no—
65#830C2×C412yes—
66#815—22no—
67Dic3⋊C8—24no—
68C8×Dic3—22no—
69#810—22no—
70#807—22no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74#803—21no—
75#803—21no—
76C4×C3⋊C8—21no—
77#533—31no—
78C2×C4×Q8C2×D311yescharacteristic
79#484—32no—
80#477—61no—
81#462—61no—
82#451—32no—
83#446—62no—
84#446—62no—
85C4×Q16—62no—
86#431—31no—
87#431C2×D311yescharacteristic
88#424—61no—
89#423C2×D311yescharacteristic
90C2×C4⋊C8—31no—
91#405—61no—
92C2×#97—31no—
93#336—61no—
94#336—61no—
95#333—61no—
96#332—61no—
97Q8⋊C8—22no—
98C2×C3×Q8—21no—
99C2×C3×Q8—21no—
100C2×C3×Q8—42no—
101C22×C3×C4—21no—
102C22×C3×C4—21no—
103C22×C3×C4C22×C411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4#4111yescharacteristic
106C2×Dic6—21no—
107C2×Dic6C22⋊C412yes—
108C3×M16—41no—
109C3×M16—41no—
110C2×C3×C8—22no—
111C2×C3×C8—41no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—41no—
115C3×C4⋊C4—41no—
116C3×C42—41no—
117C3×C42—41no—
118C3×C42C2×D411yescharacteristic
119C3×C42C2×D411yescharacteristic
120C3×C42C2×D411yescharacteristic
121C3×C42#4111yescharacteristic
122#217—22no—
123#216—44no—
124C4⋊Dic3—21no—
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127C4⋊Dic3C22⋊C412yes—
128Dic3⋊C4—22no—
129Dic3⋊C4—22no—
130C4×Dic3—22no—
131C2×C3⋊C8—21no—
132C2×C3⋊C8—21no—
133C2×C3⋊C8—22no—
134C2×C3⋊C8—21no—
135C2×C3⋊C8—21no—
136C2×C3⋊C8—21no—
137C2×C3⋊C8—41no—
138C2×C3⋊C8—41no—
139C2×C3⋊C8—41no—
140C2×C3⋊C8Q1612yescomplemented
141C22×Q8C4×D311yescharacteristic
142C2×Q16—62no—
143C2×M16—21no—
144C22×C8—31no—
145C22×C8—31no—
146C22×C8—61no—
147C4⋊Q8—31no—
148C4⋊Q8—31no—
149#125—61no—
150C22⋊Q8—32no—
151C22⋊Q8—61no—
152C4×Q8—21no—
153C4×Q8—21no—
154C4×Q8—21no—
155C4×Q8—21no—
156C4×Q8—32no—
157C4×Q8—61no—
158#117—61no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4C4×D311yescharacteristic
161C2×C42—21no—
162C2×C42C22×D311yescharacteristic
163#107—62no—
164C4⋊C8—21no—
165C4⋊C8—21no—
166C4⋊C8—64no—
167C4⋊C8—61no—
168C4⋊C8—61no—
169C4⋊C8—61no—
170C4⋊C8#6712yes—
171#103—62no—
172#103—62no—
173C22⋊C8—21no—
174C22⋊C8—32no—
175#97—62no—
176#97—61no—
177#97—61no—
178C4×C8—21no—
179C4×C8—62no—
180C4×C8—61no—
181C23×C3C4×D411yescharacteristic
182C3×Q8—22no—
183C3×Q8—22no—
184C3×Q8—41no—
185C3×Q8—41no—
186C2×C3×C4—22no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—42no—
192C2×C3×C4—41no—
193C2×C3×C4C2×C22⋊C411yescharacteristic
194C2×C3×C4C4×D411yescharacteristic
195C2×C3×C4#12011yescharacteristic
196C2×C3×C4#12111yescharacteristic
197C2×C3×C4#12111yescharacteristic
198C2×C3×C4#12311yescharacteristic
199C2×C3×C4C2×Q1611yescharacteristic
200C2×C3×C4#13511yescharacteristic
201C2×C3×C4#13711yescharacteristic
202C2×C3×C4#13711yescharacteristic
203C2×Dic3—22no—
204C2×Dic3—22no—
205Dic6—42no—
206Dic6—81no—
207C3×C8—42no—
208C3×C8—81no—
209C3⋊C8—22no—
210C3⋊C8—42no—
211C3⋊C8—41no—
212C3⋊C8—81no—
213C2×Q8—21no—
214C2×Q8—21no—
215C2×Q8—32no—
216C2×Q8—42no—
217C2×Q8—61no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4C2×C4×D311yescharacteristic
221C22×C4D3×D411yescharacteristic
222C22×C4#23711yescharacteristic
223Q16—124no—
224M16—41no—
225M16—41no—
226C2×C8—22no—
227C2×C8—32no—
228C2×C8—41no—
229C2×C8—61no—
230C2×C8—61no—
231C2×C8—62no—
232C2×C8—61no—
233C2×C8—61no—
234C2×C8—61no—
235C2×C8—121no—
236C2×C8—121no—
237C2×C8—121no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—32no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—62no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—62no—
247C4⋊C4—61no—
248C22⋊C4—62no—
249C42—41no—
250C42—41no—
251C42—62no—
252C42#23511yescharacteristic
253C42D3×D411yescharacteristic
254C42D3×D411yescharacteristic
255C42C2×#6711yescharacteristic
256C22×C3—21no—
257C22×C3—21no—
258C22×C3#38611yescharacteristic
259C22×C3C4×Q1611yescharacteristic
260C22×C3#44111yescharacteristic
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—42no—
266C3×C4—81no—
267C3×C4#45111yescharacteristic
268C3×C4#45211yescharacteristic
269C3×C4#46211yescharacteristic
270C3×C4#46411yescharacteristic
271Dic3—42no—
272Dic3—42no—
273C23#67⋊C411yescharacteristic
274Q8—22no—
275Q8—22no—
276Q8—41no—
277Q8—41no—
278Q8—122no—
279Q8—241no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—62no—
288C2×C4—62no—
289C2×C4D3×C22⋊C411yescharacteristic
290C2×C4#88311yescharacteristic
291C2×C4#88411yescharacteristic
292C2×C4#91511yescharacteristic
293C2×C4#91611yescharacteristic
294C2×C4D3×Q1611yescharacteristic
295C2×C4#91811yescharacteristic
296C2×C4C4×#6711yescharacteristic
297C2×C4#93711yescharacteristic
298C2×C4#93811yescharacteristic
299C8—42no—
300C8—62no—
301C8—81no—
302C8—122no—
303C8—121no—
304C8—241no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3#187811yescharacteristic
308C2×C3#188111yescharacteristic
309C2×C3#189111yescharacteristic
310C22—21no—
311C22—21no—
312C22#478911yescharacteristic
313C22#479011yescharacteristic
314C22#495611yescharacteristic
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—42no—
320C4—81no—
321C4—122no—
322C4—122no—
323C4#479511yescharacteristic
324C4#479811yescharacteristic
325C4#517311yescharacteristic
326C4#518611yescharacteristic
327C3#904811yescharacteristic, complemented, Sylow
328C2—21no—
329C2—21no—
330C2#6999211yescharacteristic
331C2#7028111yescharacteristic
332C2#7041611yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1487C2
socleC22×C3#386
prime radicalC23×C3C4×D4
cosocleC2×C3×C42C23
hypercenter#1487D3
hypocenterC3#9048
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
140. C2×C3⋊C8Q16semidirect
327. C3#9048semidirect

Maximal subgroups

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

subgroupindex
#824472
#790452
C3×#14872
#716352
#707762
#677412
#677142
#90483