Finite Group Databasebeta

← #10683603

Subgroups — #10683603

Order 768.

384 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 101 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#76192C211yesmaximal, characteristic
2#68607C211yesmaximal, characteristic
3#68602C211yesmaximal, characteristic
4C3×#1271C211yesmaximal, characteristic
5#67638C211yesmaximal, characteristic
6#67480C211yesmaximal, characteristic
7#67480C211yesmaximal, characteristic
8#7673—31nomaximal, Sylow
9C3×#389C2211yescharacteristic
10C2×Dic3⋊C8—21no—
11C2×C8×Dic3—21no—
12#4984C411yescharacteristic
13#4980—21no—
14#4978—21no—
15#4973C411yescharacteristic
16#4973C411yescharacteristic
17#4972C411yescharacteristic
18C42×Dic3C2211yescharacteristic
19#4929C2211yescharacteristic
20#4654—21no—
21#4651—21no—
22C3×#336C2211yescharacteristic
23C3×#336C2211yescharacteristic
24#4550—21no—
25#4550—21no—
26#4524C2211yescharacteristic
27#4524C2211yescharacteristic
28#2863—31no—
29#1839—31no—
30C4×C4⋊C8—31no—
31#1272—31no—
32#1272—31no—
33#1271—31no—
34#1271D311yescharacteristic
35C22×Dic6C2×C411yescharacteristic
36C22×C3×C8—21no—
37C22×C3×C8—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4C2×C411yescharacteristic
40C2×C3×C4⋊C4C2×C411yescharacteristic
41C2×C3×C42C2311yescharacteristic
42C2×C4⋊Dic3—21no—
43C2×C4⋊Dic3C2×C411yescharacteristic
44C2×Dic3⋊C4—21no—
45C2×C4×Dic3C2×C411yescharacteristic
46C2×C4×Dic3C2×C411yescharacteristic
47C2×C4×Dic3D411yescharacteristic
48C2×C4×Dic3D411yescharacteristic
49C2×C4×Dic3D411yescharacteristic
50C2×C4×Dic3D411yescharacteristic
51C22×C3⋊C8—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—
62Dic3⋊C8—24no—
63C8×Dic3—42no—
64C2×C4⋊C8—61no—
65C2×C4×C8—61no—
66#391—31no—
67#391—31no—
68#391—31no—
69#389—31no—
70#389—31no—
71#389C2×D311yescharacteristic
72#387—61no—
73#387—61no—
74#387—61no—
75#383—61no—
76C43—31no—
77#336—31no—
78#336—31no—
79#336—61no—
80#336—61no—
81#336C2×D311yescharacteristic
82#336C2×D311yescharacteristic
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4C22×C411yescharacteristic
86C22×C3×C4C2×D411yescharacteristic
87C22×C3×C4C2×D411yescharacteristic
88C22×Dic3—21no—
89C22×Dic3—21no—
90C22×Dic3C22⋊C411yescharacteristic
91C22×Dic3C22⋊C411yescharacteristic
92C22×Dic3C22⋊C411yescharacteristic
93C22×Dic3C22⋊C411yescharacteristic
94C2×Dic6—21no—
95C2×Dic6—21no—
96C2×Dic6—41no—
97C2×Dic6—41no—
98C2×C3×C8—22no—
99C2×C3×C8—41no—
100C2×C3×C8—41no—
101C2×C3×C8—41no—
102C2×C3×C8—41no—
103C2×C3×C8—41no—
104C3×C4⋊C4—21no—
105C3×C4⋊C4—21no—
106C3×C4⋊C4—41no—
107C3×C4⋊C4—41no—
108C3×C4⋊C4—41no—
109C3×C42—41no—
110C4⋊Dic3—42no—
111C4⋊Dic3—41no—
112Dic3⋊C4—42no—
113C4×Dic3—21no—
114C4×Dic3—21no—
115C4×Dic3—21no—
116C4×Dic3—21no—
117C4×Dic3—22no—
118C4×Dic3—41no—
119C4×Dic3—41no—
120C4×Dic3SD1612yes—
121C4×Dic3Q1612yes—
122C2×C3⋊C8—22no—
123C2×C3⋊C8—41no—
124C2×C3⋊C8—41no—
125C2×C3⋊C8—41no—
126C2×C3⋊C8—41no—
127C2×C3⋊C8—41no—
128C22×Q8—31no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—61no—
132C22×C8—61no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—31no—
135C2×C4⋊C4—61no—
136C2×C4⋊C4—61no—
137C2×C4⋊C4C4×D311yescharacteristic
138C2×C4⋊C4C4×D311yescharacteristic
139C2×C42—31no—
140C2×C42—31no—
141C2×C42—31no—
142C2×C42—31no—
143C2×C42—31no—
144C2×C42—31no—
145C2×C42C22×D311yescharacteristic
146C4⋊C8—64no—
147C4×C8—122no—
148#95—21no—
149#95—61no—
150#95—61no—
151#95—61no—
152#95—61no—
153#95—61no—
154#95—61no—
155#95—61no—
156#95—61no—
157C23×C3C2×C22⋊C411yescharacteristic
158C2×C3×C4—21no—
159C2×C3×C4—21no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—41no—
165C2×C3×C4—41no—
166C2×C3×C4—41no—
167C2×C3×C4—41no—
168C2×C3×C4—41no—
169C2×C3×C4—41no—
170C2×C3×C4#13111yescharacteristic
171C2×C3×C4#13111yescharacteristic
172C2×C3×C4C2×SD1611yescharacteristic
173C2×C3×C4C2×Q1611yescharacteristic
174C2×C3×C4#13511yescharacteristic
175C2×C3×C4#13511yescharacteristic
176C2×Dic3—21no—
177C2×Dic3—21no—
178C2×Dic3—21no—
179C2×Dic3—21no—
180C2×Dic3—21no—
181C2×Dic3—21no—
182C2×Dic3—21no—
183C2×Dic3—21no—
184C2×Dic3—21no—
185C2×Dic3—41no—
186C2×Dic3—41no—
187C2×Dic3—41no—
188C2×Dic3—41no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C2×Dic3#10312yes—
192C2×Dic3#10312yes—
193C2×Dic3#10412yes—
194Dic6—81no—
195Dic6—81no—
196C3×C8—42no—
197C3×C8—81no—
198C3⋊C8—42no—
199C3⋊C8—81no—
200C2×Q8—61no—
201C2×Q8—61no—
202C2×Q8—121no—
203C2×Q8—121no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—31no—
207C22×C4—31no—
208C22×C4—31no—
209C22×C4—31no—
210C22×C4—61no—
211C22×C4—61no—
212C22×C4C2×C4×D311yescharacteristic
213C22×C4D3×D411yescharacteristic
214C22×C4D3×D411yescharacteristic
215C2×C8—22no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—62no—
222C2×C8—121no—
223C2×C8—121no—
224C2×C8—121no—
225C2×C8—121no—
226C2×C8—121no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—121no—
233C4⋊C4—122no—
234C4⋊C4—122no—
235C42—32no—
236C42—32no—
237C42—41no—
238C42—62no—
239C42—61no—
240C42—61no—
241C42—61no—
242C42—61no—
243C42—121no—
244C42—121no—
245C22×C3#40811yescharacteristic
246C22×C3#41011yescharacteristic
247C22×C3#41311yescharacteristic
248C22×C3C2×#10311yescharacteristic
249C22×C3#41611yescharacteristic
250C22×C3C2×#10411yescharacteristic
251C22×C3#42111yescharacteristic
252C3×C4—21no—
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—21no—
258C3×C4—81no—
259C3×C4—81no—
260Dic3—22no—
261Dic3—22no—
262Dic3—22no—
263Dic3—41no—
264Dic3—81no—
265Dic3—81no—
266C23D3×C22⋊C411yescharacteristic
267Q8—241no—
268Q8—241no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—32no—
276C2×C4—32no—
277C2×C4—32no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—61no—
285C2×C4—61no—
286C2×C4—61no—
287C2×C4—61no—
288C2×C4—61no—
289C2×C4—61no—
290C2×C4—61no—
291C2×C4—61no—
292C2×C4—61no—
293C2×C4—121no—
294C2×C4—121no—
295C2×C4—121no—
296C2×C4—121no—
297C2×C4—121no—
298C2×C4—121no—
299C2×C4#90111yescharacteristic
300C2×C4#90711yescharacteristic
301C2×C4#91211yescharacteristic
302C2×C4D3×SD1611yescharacteristic
303C2×C4#91611yescharacteristic
304C2×C4D3×Q1611yescharacteristic
305C8—42no—
306C8—81no—
307C8—122no—
308C8—241no—
309C2×C3#147311yescharacteristic
310C2×C3#148111yescharacteristic
311C2×C3#149411yescharacteristic
312C2×C3#150311yescharacteristic
313C2×C3#152111yescharacteristic
314C2×C3#152911yescharacteristic
315C2×C3#154211yescharacteristic
316C22#472711yescharacteristic
317C22#474011yescharacteristic
318C22#475111yescharacteristic
319C22#477211yescharacteristic
320C22D3×#10311yescharacteristic
321C22D3×#10411yescharacteristic
322C22#482111yescharacteristic
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—62no—
330C4—62no—
331C4—62no—
332C4—81no—
333C4—81no—
334C4—121no—
335C4—241no—
336C4—241no—
337C3#767311yescharacteristic, complemented, Sylow
338C2#6812911yescharacteristic
339C2#6817711yescharacteristic
340C2#6823211yescharacteristic
341C2#6828311yescharacteristic
342C2#6830911yescharacteristic
343C2#6836411yescharacteristic
344C2#6841811yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1271C2
socle, prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1271D3
hypocenterC3#7673
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
337. C3#7673semidirect

Maximal subgroups

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

subgroupindex
#761922
#686072
#686022
C3×#12712
#676382
#674802
#76733