Finite Group Databasebeta

← #10693280

Subgroups — #10693280

Order 768.

428 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 89 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#82315C211yesmaximal, characteristic
2#79266C211yesmaximal, characteristic, complemented
3C3×#1486C211yesmaximal, characteristic
4#71603C211yesmaximal, characteristic
5#70510C211yesmaximal, characteristic, complemented
6#67747C211yesmaximal, characteristic, complemented
7#67716C211yesmaximal, characteristic, complemented
8#9530—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13#5218C412yes—
14#5019—21no—
15#5018—22no—
16#5007—21no—
17#4970C412yes—
18#4926C2211yescharacteristic
19#4926C2211yescharacteristic
20#4926C2211yescharacteristic
21#4828—22no—
22#4720—22no—
23#4679—21no—
24C8×Dic6—21no—
25C3×D4⋊C8—21no—
26C3×D4⋊C8—21no—
27#4525—22no—
28#4482—21no—
29#4481—21no—
30#3110—31no—
31#2985—31no—
32#1559—31no—
33#1486D311yescharacteristic
34#1466—31no—
35#1307—31no—
36#1287—31no—
37C22×C3×D4C2×C411yescharacteristic
38C23×C3×C4—21no—
39C2×C3×M16—21no—
40C3×C4×D4—21no—
41C3×C4×D4—21no—
42C3×C4×D4—21no—
43C3×C4×D4—21no—
44C2×C3×C4⋊C4C2×C411yescharacteristic
45C2×C3×C22⋊C4—21no—
46C2×C3×C42C2311yescharacteristic
47C2×#214—22no—
48#926—21no—
49#924—21no—
50#924C2×C412yes—
51#870—21no—
52#869D411yescharacteristic
53#869D411yescharacteristic
54C4×Dic6—21no—
55C4×Dic6D412yes—
56C3×C4⋊C8—21no—
57C3×C4⋊C8—21no—
58C3×C4⋊C8—21no—
59C3×C22⋊C8—21no—
60C3×C4×C8D411yescharacteristic
61C3×C4×C8D411yescharacteristic
62#832—22no—
63#830—21no—
64#830C2×C412yes—
65#830D412yes—
66#815—22no—
67Dic3⋊C8—22no—
68Dic3⋊C8—22no—
69C8×Dic3—22no—
70#810—22no—
71#807—22no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75#803—21no—
76#803—21no—
77#803—21no—
78#533—31no—
79C2×C4×D4C2×D311yescharacteristic
80#484—32no—
81#474—61no—
82#461—61no—
83#450—32no—
84#446—62no—
85#446—61no—
86C8×Q8—61no—
87#440—62no—
88#433—62no—
89#432—31no—
90#432—31no—
91#432—31no—
92#431C2×D311yescharacteristic
93#423C2×D311yescharacteristic
94#333—61no—
95#332—61no—
96D4⋊C8—21no—
97D4⋊C8—21no—
98#323—62no—
99C24×C3—21no—
100C2×C3×D4—21no—
101C2×C3×D4—21no—
102C2×C3×D4—42no—
103C22×C3×C4—21no—
104C22×C3×C4—41no—
105C22×C3×C4—41no—
106C22×C3×C4—81no—
107C22×C3×C4—81no—
108C22×C3×C4C22×C411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4#4111yescharacteristic
111C2×Dic6—21no—
112C2×Dic6C22⋊C412yes—
113C3×M16—81no—
114C2×C3×C8—21no—
115C2×C3×C8—21no—
116C2×C3×C8—41no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C22⋊C4—41no—
120C3×C22⋊C4—41no—
121C3×C42C2×D411yescharacteristic
122C3×C42C2×D411yescharacteristic
123C3×C42C2×D411yescharacteristic
124C3×C42#4111yescharacteristic
125#217—22no—
126#214—44no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130C4⋊Dic3C22⋊C412yes—
131Dic3⋊C4—22no—
132Dic3⋊C4—22no—
133C4×Dic3—22no—
134C2×C3⋊C8—22no—
135C2×C3⋊C8—22no—
136C2×C3⋊C8—41no—
137C22×D4C4×D311yescharacteristic
138C23×C4—21no—
139C2×SD16—62no—
140C2×M16—21no—
141C4⋊Q8—31no—
142C4⋊Q8—31no—
143#125—61no—
144C22⋊Q8—32no—
145C22⋊Q8—61no—
146C4×Q8—32no—
147C4×Q8—61no—
148C4×D4—21no—
149C4×D4—21no—
150C4×D4—21no—
151C4×D4—21no—
152#117—61no—
153C2×C4⋊C4C4×D311yescharacteristic
154C2×C22⋊C4—21no—
155C2×C42C22×D311yescharacteristic
156#107—62no—
157C4⋊C8—21no—
158C4⋊C8—21no—
159C4⋊C8—21no—
160C4⋊C8—62no—
161C4⋊C8—62no—
162C4⋊C8—61no—
163C4⋊C8—61no—
164C4⋊C8—61no—
165#103—62no—
166#102—62no—
167C22⋊C8—21no—
168C22⋊C8—32no—
169C22⋊C8—32no—
170C22⋊C8—61no—
171#97—62no—
172#97—61no—
173#97—61no—
174#97—61no—
175C4×C8—62no—
176C4×C8#6711yescharacteristic
177C4×C8#6711yescharacteristic
178C23×C3—22no—
179C23×C3—41no—
180C23×C3—41no—
181C23×C3—81no—
182C23×C3—81no—
183C23×C3C4×D411yescharacteristic
184C3×D4—22no—
185C3×D4—22no—
186C3×D4—41no—
187C3×D4—41no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—82no—
194C2×C3×C4—81no—
195C2×C3×C4—81no—
196C2×C3×C4—81no—
197C2×C3×C4C2×C22⋊C411yescharacteristic
198C2×C3×C4C4×D411yescharacteristic
199C2×C3×C4#12011yescharacteristic
200C2×C3×C4#12111yescharacteristic
201C2×C3×C4#12111yescharacteristic
202C2×C3×C4#12311yescharacteristic
203C2×Dic3—22no—
204C2×Dic3—22no—
205Dic6—41no—
206Dic6—41no—
207Dic6—42no—
208C3×C8—41no—
209C3×C8—41no—
210C3×C8—81no—
211C3⋊C8—42no—
212C3⋊C8—42no—
213C3⋊C8—81no—
214C24—21no—
215C2×Q8—32no—
216C2×Q8—61no—
217C2×D4—21no—
218C2×D4—21no—
219C2×D4—42no—
220C22×C4—21no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4C2×C4×D311yescharacteristic
226C22×C4D3×D411yescharacteristic
227C22×C4#23711yescharacteristic
228SD16—124no—
229M16—81no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—41no—
233C2×C8—62no—
234C2×C8—62no—
235C2×C8—121no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—32no—
239C4⋊C4—62no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—62no—
243C4⋊C4—61no—
244C22⋊C4—41no—
245C22⋊C4—41no—
246C22⋊C4—62no—
247C42—62no—
248C42#23511yescharacteristic
249C42D3×D411yescharacteristic
250C42D3×D411yescharacteristic
251C42C2×#6711yescharacteristic
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—42no—
256C22×C3—82no—
257C22×C3—82no—
258C22×C3—81no—
259C22×C3—81no—
260C22×C3#38611yescharacteristic
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—81no—
267C3×C4#47111yescharacteristic
268C3×C4#47211yescharacteristic
269Dic3—42no—
270Dic3—42no—
271C23—22no—
272C23—41no—
273C23—41no—
274C23—81no—
275C23—81no—
276C23#67⋊C411yescharacteristic
277Q8—122no—
278Q8—121no—
279Q8—121no—
280D4—22no—
281D4—22no—
282D4—41no—
283D4—41no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—62no—
290C2×C4—62no—
291C2×C4—82no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4D3×C22⋊C411yescharacteristic
296C2×C4#88311yescharacteristic
297C2×C4#88411yescharacteristic
298C2×C4C4×#6711yescharacteristic
299C2×C4#93711yescharacteristic
300C2×C4#93811yescharacteristic
301C8—41no—
302C8—41no—
303C8—81no—
304C8—122no—
305C8—122no—
306C8—241no—
307C2×C3—41no—
308C2×C3—82no—
309C2×C3#189111yescharacteristic
310C2×C3#190511yescharacteristic
311C2×C3#190711yescharacteristic
312C22—21no—
313C22—21no—
314C22—21no—
315C22—42no—
316C22—82no—
317C22—82no—
318C22—81no—
319C22—81no—
320C22#495611yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—81no—
327C4—122no—
328C4—122no—
329C4#516011yescharacteristic
330C4#517711yescharacteristic
331C3#953011yescharacteristic, complemented, Sylow
332C2—41no—
333C2—82no—
334C2#6999211yescharacteristic
335C2#7021811yescharacteristic
336C2#7041011yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1486C2
socleC22×C3#386
prime radicalC22×C3×D4C2×C4
cosocleC2×C3×C42C23
hypercenter#1486D3
hypocenterC3#9530
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1486

Complements

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

NH ≅ G/Ntype
2. #79266C2semidirect
5. #70510C2semidirect
6. #67747C2semidirect
7. #67716C2semidirect
331. C3#9530semidirect

Maximal subgroups

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

subgroupindex
#823152
#792662
C3×#14862
#716032
#705102
#677472
#677162
#95303