Finite Group Databasebeta

← #10684353

Subgroups — #10684353

Order 768.

366 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 81 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#76181C211yesmaximal, characteristic
2#69608C211yesmaximal, characteristic
3#68692C211yesmaximal, characteristic
4#68639C211yesmaximal, characteristic
5C3×#1271C211yesmaximal, characteristic
6#67480C211yesmaximal, characteristic
7#67477C211yesmaximal, characteristic
8#7807—31nomaximal, Sylow
9C3×#389C2211yescharacteristic
10C2×#818—21no—
11C2×#817—21no—
12C2×#816—21no—
13#4985—21no—
14#4980—21no—
15#4979—21no—
16#4976—21no—
17C2×#808—21no—
18#4935C2211yescharacteristic
19#4929C2211yescharacteristic
20#4928C2211yescharacteristic
21#4660—21no—
22#4653—21no—
23C3×#336C2211yescharacteristic
24C3×#336C2211yescharacteristic
25#4551—21no—
26#4529—21no—
27#4524C2211yescharacteristic
28#2805—31no—
29#2096—31no—
30#2084—31no—
31#2083—31no—
32#1272—31no—
33#1271—31no—
34#1271D311yescharacteristic
35C22×Dic6D411yescharacteristic
36C22×C3×C8—21no—
37C22×C3×C8—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4D411yescharacteristic
40C2×C3×C4⋊C4D411yescharacteristic
41C2×C3×C42C2311yescharacteristic
42C2×C4⋊Dic3—21no—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3D411yescharacteristic
46C2×C4⋊Dic3D411yescharacteristic
47C2×C4⋊Dic3D411yescharacteristic
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×C4×Dic3—21no—
52C22×C3⋊C8—21no—
53C3×#95—21no—
54C3⋊#95—21no—
55C3⋊#95—21no—
56C3⋊#95—21no—
57C3⋊#95—21no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62#818—42no—
63#817—42no—
64#816—42no—
65#808—42no—
66C2×#107—61no—
67C2×#106—61no—
68C2×#106—61no—
69C2×#103—61no—
70#400—61no—
71#400—61no—
72#395—61no—
73#391—31no—
74#389—31no—
75#389—61no—
76#389C2×D311yescharacteristic
77#387—61no—
78#387—61no—
79#384—31no—
80#340—61no—
81#340—61no—
82#336—31no—
83#336C2×D311yescharacteristic
84#336C2×D311yescharacteristic
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4C2×D411yescharacteristic
88C22×C3×C4C2×D411yescharacteristic
89C22×C3×C4C2×D411yescharacteristic
90C22×Dic3—21no—
91C22×Dic3—21no—
92C22×Dic3—21no—
93C22×Dic3—21no—
94C22×Dic3—41no—
95C2×Dic6—21no—
96C2×Dic6—21no—
97C2×Dic6—41no—
98C2×Dic6—41no—
99C2×C3×C8—22no—
100C2×C3×C8—22no—
101C2×C3×C8—41no—
102C2×C3×C8—41no—
103C2×C3×C8—41no—
104C2×C3×C8—41no—
105C3×C4⋊C4—21no—
106C3×C4⋊C4—21no—
107C3×C4⋊C4—41no—
108C3×C4⋊C4—41no—
109C3×C4⋊C4—41no—
110C3×C42—41no—
111C4⋊Dic3—21no—
112C4⋊Dic3—41no—
113C4⋊Dic3—41no—
114C4⋊Dic3—41no—
115C4⋊Dic3—41no—
116C4⋊Dic3—41no—
117C4⋊Dic3—41no—
118C4⋊Dic3—41no—
119C4⋊Dic3—41no—
120C4⋊Dic3SD1612yes—
121Dic3⋊C4—41no—
122Dic3⋊C4—41no—
123Dic3⋊C4—81no—
124Dic3⋊C4—81no—
125C4×Dic3—81no—
126C2×C3⋊C8—22no—
127C2×C3⋊C8—41no—
128C2×C3⋊C8—41no—
129C22×Q8—31no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—61no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—31no—
135C2×C4⋊C4—31no—
136C2×C4⋊C4—31no—
137C2×C4⋊C4—61no—
138C2×C4⋊C4—61no—
139C2×C4⋊C4—61no—
140C2×C4⋊C4—61no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4D1211yescharacteristic
144C2×C4⋊C4D1211yescharacteristic
145C2×C42—61no—
146C2×C42C22×D311yescharacteristic
147#107—122no—
148#106—122no—
149#106—122no—
150#103—122no—
151#95—21no—
152#95—61no—
153#95—61no—
154#95—61no—
155#95—61no—
156#95—61no—
157#95—61no—
158#95—61no—
159#95—61no—
160C23×C3#12011yescharacteristic
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—41no—
168C2×C3×C4—41no—
169C2×C3×C4—41no—
170C2×C3×C4—41no—
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×C3×C4C2×SD1611yescharacteristic
174C2×C3×C4#13511yescharacteristic
175C2×C3×C4#13511yescharacteristic
176C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
177C2×C3×C4#13711yescharacteristic
178C2×C3×C4#13711yescharacteristic
179C2×Dic3—22no—
180C2×Dic3—41no—
181C2×Dic3—41no—
182C2×Dic3—41no—
183C2×Dic3—41no—
184C2×Dic3—41no—
185C2×Dic3—41no—
186C2×Dic3—41no—
187C2×Dic3—41no—
188C2×Dic3—41no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C2×Dic3—81no—
192C2×Dic3—81no—
193C2×Dic3—81no—
194Dic6—42no—
195Dic6—81no—
196C3×C8—42no—
197C3×C8—42no—
198C3⋊C8—42no—
199C2×Q8—61no—
200C2×Q8—61no—
201C2×Q8—121no—
202C2×Q8—121no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—61no—
206C22×C4—61no—
207C22×C4—61no—
208C22×C4—61no—
209C22×C4—121no—
210C22×C4C2×D1211yescharacteristic
211C22×C4D3×D411yescharacteristic
212C22×C4D3×D411yescharacteristic
213C2×C8—22no—
214C2×C8—22no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—62no—
220C2×C8—121no—
221C2×C8—121no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—32no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—61no—
229C4⋊C4—121no—
230C4⋊C4—121no—
231C4⋊C4—121no—
232C4⋊C4—121no—
233C4⋊C4—121no—
234C4⋊C4—121no—
235C4⋊C4—121no—
236C4⋊C4—121no—
237C4⋊C4—121no—
238C4⋊C4—121no—
239C4⋊C4—241no—
240C4⋊C4—241no—
241C42—41no—
242C42—241no—
243C22×C3#44811yescharacteristic
244C22×C3#44911yescharacteristic
245C22×C3#45211yescharacteristic
246C22×C3#45311yescharacteristic
247C22×C3#45611yescharacteristic
248C22×C3#45611yescharacteristic
249C22×C3#45811yescharacteristic
250C3×C4—21no—
251C3×C4—21no—
252C3×C4—21no—
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—81no—
257C3×C4—81no—
258Dic3—42no—
259Dic3—81no—
260Dic3—81no—
261Dic3—81no—
262Dic3—161no—
263C23#88211yescharacteristic
264Q8—122no—
265Q8—241no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—62no—
279C2×C4—121no—
280C2×C4—121no—
281C2×C4—121no—
282C2×C4—121no—
283C2×C4—121no—
284C2×C4—121no—
285C2×C4—121no—
286C2×C4—121no—
287C2×C4—121no—
288C2×C4—121no—
289C2×C4—121no—
290C2×C4—241no—
291C2×C4—241no—
292C2×C4—241no—
293C2×C4#90411yescharacteristic
294C2×C4#90811yescharacteristic
295C2×C4#91211yescharacteristic
296C2×C4D3×SD1611yescharacteristic
297C2×C4#91511yescharacteristic
298C2×C4#91811yescharacteristic
299C8—42no—
300C8—42no—
301C8—122no—
302C2×C3#160911yescharacteristic
303C2×C3#161111yescharacteristic
304C2×C3#161811yescharacteristic
305C2×C3#162211yescharacteristic
306C2×C3#165011yescharacteristic
307C2×C3#165311yescharacteristic
308C2×C3#166011yescharacteristic
309C22#473411yescharacteristic
310C22#474211yescharacteristic
311C22#475211yescharacteristic
312C22#478311yescharacteristic
313C22#480611yescharacteristic
314C22#482211yescharacteristic
315C22#482611yescharacteristic
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—81no—
323C4—81no—
324C4—122no—
325C4—241no—
326C4—241no—
327C4—241no—
328C4—481no—
329C3#780711yescharacteristic, complemented, Sylow
330C2#6813411yescharacteristic
331C2#6820911yescharacteristic
332C2#6824411yescharacteristic
333C2#6829111yescharacteristic
334C2#6832911yescharacteristic
335C2#6837211yescharacteristic
336C2#6846311yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1271C2
socle, prime radicalC23×C3#120
hypercenter#1271D3
hypocenterC3#7807
LCSG ▷ C2×C3×C42 ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
329. C3#7807semidirect

Maximal subgroups

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

subgroupindex
#761812
#696082
#686922
#686392
C3×#12712
#674802
#674772
#78073