Finite Group Databasebeta

← #10694743

Subgroups — #10694743

Order 768.

392 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C2×#5056C211yesmaximal, characteristic
2C3×#1469C211yesmaximal, characteristic
3#71718C211yesmaximal, characteristic
4#70812C211yesmaximal, characteristic, complemented
5#70794C211yesmaximal, characteristic, complemented
6#70787C211yesmaximal, characteristic, complemented
7#70739C211yesmaximal, characteristic, complemented
8#9789—31nomaximal, Sylow
9C2×C3×#124C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C2×#835C411yescharacteristic
14C2×#835C411yescharacteristic
15C2×#832C411yescharacteristic
16C2×#832C411yescharacteristic
17#5056—21no—
18#5056—21no—
19#5056—21no—
20#5056—21no—
21#4926C2211yescharacteristic
22C4×#208C2211yescharacteristic
23C2×C4×C3⋊C8C2211yescharacteristic
24C3×#323—22no—
25#4540—22no—
26#4480—21no—
27#4479—21no—
28#4469—22no—
29#4465—22no—
30#4454—21no—
31C8×C3⋊C8—21no—
32C2×#488—31no—
33#1568—31no—
34#1528—31no—
35#1527—31no—
36#1470—31no—
37#1469D311yescharacteristic
38C8×M16—31no—
39C22×C3×Q8C2×C411yescharacteristic
40C22×C3×D4C2×C411yescharacteristic
41C3×#124—21no—
42C3×#124—21no—
43C3×#124—21no—
44C3×#124—21no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C42C2311yescharacteristic
48C2×C4⋊Dic3C2×C411yescharacteristic
49C2×C4⋊Dic3C2×C411yescharacteristic
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—21no—
52C2×#208—21no—
53C22×C3⋊C8C2×C411yescharacteristic
54C22×C3⋊C8C2×C411yescharacteristic
55#870—21no—
56#870—21no—
57#870—21no—
58#870—21no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×C22⋊C8—21no—
62C3×C22⋊C8D412yes—
63C3×#97—21no—
64C3×C4×C8—21no—
65#835—21no—
66#835—21no—
67#835—21no—
68#835—21no—
69#832—21no—
70#832—21no—
71#832—21no—
72#832—21no—
73#830—21no—
74C4⋊(C3⋊C8)—21no—
75#803—21no—
76#803D412yes—
77C4×C3⋊C8—21no—
78C4×C3⋊C8D411yescharacteristic
79C4×C3⋊C8D411yescharacteristic
80C4×C3⋊C8D411yescharacteristic
81C4×C3⋊C8D411yescharacteristic
82C2×#125—31no—
83C2×#124C2×D311yescharacteristic
84#488—61no—
85#488—61no—
86#488—61no—
87#488—61no—
88#432—31no—
89#432C2×D311yescharacteristic
90#431C2×D311yescharacteristic
91C2×#103—31no—
92C2×#103—31no—
93C2×#102—31no—
94C2×#102—31no—
95C4×M16—31no—
96C2×C4×C8—31no—
97#335—61no—
98#334—61no—
99#330—62no—
100#329—62no—
101#324—62no—
102#323—22no—
103#322—61no—
104C82—61no—
105C24×C3—21no—
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—41no—
109C2×C3×Q8—41no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C22×C3×C4—21no—
115C22×C3×C4C22×C411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4#4111yescharacteristic
118C22×Dic3—21no—
119C22×Dic3—21no—
120C2×C3×C8—22no—
121C2×C3×C8—41no—
122C3×C22⋊C4—41no—
123C3×C22⋊C4—41no—
124C3×C22⋊C4—41no—
125C3×C22⋊C4—41no—
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42#4111yescharacteristic
130C4⋊Dic3—21no—
131C4⋊Dic3—21no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134Dic3⋊C4—41no—
135Dic3⋊C4—41no—
136Dic3⋊C4—41no—
137Dic3⋊C4—41no—
138#208—24no—
139C2×C3⋊C8—22no—
140C2×C3⋊C8—41no—
141C2×C3⋊C8—41no—
142C2×C3⋊C8—41no—
143C2×C3⋊C8C22⋊C411yescharacteristic
144C2×C3⋊C8C22⋊C411yescharacteristic
145C2×C3⋊C8C22⋊C411yescharacteristic
146C2×C3⋊C8C22⋊C411yescharacteristic
147C22×Q8C4×D311yescharacteristic
148C22×D4C4×D311yescharacteristic
149C2×M16—61no—
150C22×C8—31no—
151C22×C8—31no—
152#125—61no—
153#125—61no—
154#125—61no—
155#125—61no—
156#124—21no—
157#124—21no—
158#124—21no—
159#124—21no—
160C2×C4⋊C4—31no—
161C2×C4⋊C4—31no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4—61no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C42C22×D311yescharacteristic
167C4⋊C8—21no—
168C4⋊C8—21no—
169C4⋊C8—61no—
170#103—61no—
171#103—61no—
172#103—61no—
173#103—61no—
174#102—61no—
175#102—61no—
176#102—61no—
177#102—61no—
178C22⋊C8—21no—
179C22⋊C8—61no—
180C22⋊C8#6712yes—
181#97—21no—
182#97—32no—
183#97—61no—
184C4×C8—21no—
185C4×C8—31no—
186C4×C8—31no—
187C4×C8—31no—
188C4×C8—31no—
189C4×C8—61no—
190C23×C3—41no—
191C23×C3—42no—
192C23×C3—81no—
193C23×C3—81no—
194C23×C3C4×D411yescharacteristic
195C3×Q8—41no—
196C3×Q8—41no—
197C3×Q8—81no—
198C3×D4—41no—
199C3×D4—41no—
200C3×D4—81no—
201C2×C3×C4—22no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—81no—
207C2×C3×C4C2×C22⋊C411yescharacteristic
208C2×C3×C4C4×D411yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12411yescharacteristic
212C2×C3×C4#12711yescharacteristic
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—42no—
216C2×Dic3—81no—
217C3×C8—42no—
218C3×C8—81no—
219C3⋊C8—24no—
220C3⋊C8—21no—
221C3⋊C8—21no—
222C3⋊C8—21no—
223C3⋊C8—21no—
224C3⋊C8—81no—
225C24—21no—
226C2×Q8—21no—
227C2×Q8—21no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×D4—21no—
231C2×D4—21no—
232C2×D4—41no—
233C2×D4—41no—
234C22×C4—21no—
235C22×C4—61no—
236C22×C4—61no—
237C22×C4C2×C4×D311yescharacteristic
238C22×C4#23511yescharacteristic
239C22×C4C2×#6711yescharacteristic
240M16—64no—
241C2×C8—22no—
242C2×C8—31no—
243C2×C8—31no—
244C2×C8—31no—
245C2×C8—31no—
246C2×C8—41no—
247C2×C8—62no—
248C2×C8—121no—
249C2×C8—121no—
250C2×C8—121no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—121no—
256C4⋊C4—121no—
257C4⋊C4—121no—
258C4⋊C4—121no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C42D3×D411yescharacteristic
264C42D3×D411yescharacteristic
265C42D3×D411yescharacteristic
266C42#23711yescharacteristic
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—81no—
270C22×C3—81no—
271C22×C3—81no—
272C22×C3—162no—
273C22×C3#39011yescharacteristic
274C22×C3#44312yes—
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—81no—
281C3×C4#41311yescharacteristic
282C3×C4#41311yescharacteristic
283Dic3—81no—
284Dic3—81no—
285C23—41no—
286C23—42no—
287C23—81no—
288C23—81no—
289C23C4×#6711yescharacteristic
290Q8—41no—
291Q8—41no—
292Q8—81no—
293D4—41no—
294D4—41no—
295D4—81no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—81no—
302C2×C4—122no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—241no—
306C2×C4D3×C22⋊C411yescharacteristic
307C2×C4#67⋊C411yescharacteristic
308C2×C4#88411yescharacteristic
309C2×C4#88511yescharacteristic
310C2×C4#93911yescharacteristic
311C2×C4#94011yescharacteristic
312C8—42no—
313C8—64no—
314C8—61no—
315C8—61no—
316C8—61no—
317C8—61no—
318C8—81no—
319C8—241no—
320C2×C3—22no—
321C2×C3—161no—
322C2×C3#196711yescharacteristic
323C2×C3#197411yescharacteristic
324C2×C3#197611yescharacteristic
325C22—21no—
326C22—21no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—162no—
331C22#495711yescharacteristic
332C22#514412yes—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—81no—
339C4—241no—
340C4—241no—
341C4#474511yescharacteristic
342C4#474511yescharacteristic
343C3#978911yescharacteristic, complemented, Sylow
344C2—22no—
345C2—161no—
346C2#6999911yescharacteristic
347C2#7004211yescharacteristic
348C2#7006511yescharacteristic
subgroupG/H
centerC22#4957
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1469C2
socleC22×C3#390
prime radicalC22×C3×D4C2×C4
cosocleC2×C3×C42C23
hypercenter#1469D3
hypocenterC3#9789
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1469

Complements

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

NH ≅ G/Ntype
4. #70812C2semidirect
5. #70794C2semidirect
6. #70787C2semidirect
7. #70739C2semidirect
343. C3#9789semidirect

Maximal subgroups

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

subgroupindex
C2×#50562
C3×#14692
#717182
#708122
#707942
#707872
#707392
#97893