Finite Group Databasebeta

← #10683677

Subgroups — #10683677

Order 768.

410 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 103 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#68603C211yesmaximal, characteristic
3#68602C211yesmaximal, characteristic
4C3×#1271C211yesmaximal, characteristic
5#67638C211yesmaximal, characteristic
6#67480C211yesmaximal, characteristic
7#67480C211yesmaximal, characteristic
8#7689—31nomaximal, Sylow
9C3×#389C2211yescharacteristic
10C2×#815—21no—
11C2×Dic3⋊C8—21no—
12C2×Dic3⋊C8—21no—
13C2×C8×Dic3—21no—
14#4984C411yescharacteristic
15#4980—21no—
16#4978—21no—
17#4973C411yescharacteristic
18#4973C411yescharacteristic
19#4972C411yescharacteristic
20C42×Dic3C2211yescharacteristic
21#4929C2211yescharacteristic
22#4654—21no—
23#4651—21no—
24C3×#336C2211yescharacteristic
25C3×#336C2211yescharacteristic
26#4524C2211yescharacteristic
27#4524C2211yescharacteristic
28#2863—31no—
29#1767—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—
62#815—42no—
63Dic3⋊C8—24no—
64Dic3⋊C8—24no—
65C8×Dic3—42no—
66C2×C4⋊C8—61no—
67C2×C4⋊C8—61no—
68C2×#97—61no—
69C2×C4×C8—61no—
70#391—31no—
71#391—31no—
72#391—31no—
73#389—31no—
74#389—31no—
75#389C2×D311yescharacteristic
76#387—61no—
77#387—61no—
78#387—61no—
79#383—61no—
80C43—31no—
81#336—31no—
82#336—31no—
83#336C2×D311yescharacteristic
84#336C2×D311yescharacteristic
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4C22×C411yescharacteristic
88C22×C3×C4C2×D411yescharacteristic
89C22×C3×C4C2×D411yescharacteristic
90C22×Dic3—21no—
91C22×Dic3—21no—
92C22×Dic3C22⋊C411yescharacteristic
93C22×Dic3C22⋊C411yescharacteristic
94C22×Dic3C22⋊C411yescharacteristic
95C22×Dic3C22⋊C411yescharacteristic
96C2×Dic6—21no—
97C2×Dic6—21no—
98C2×Dic6—41no—
99C2×Dic6—41no—
100C2×C3×C8—22no—
101C2×C3×C8—22no—
102C2×C3×C8—41no—
103C2×C3×C8—41no—
104C2×C3×C8—41no—
105C2×C3×C8—41no—
106C3×C4⋊C4—21no—
107C3×C4⋊C4—21no—
108C3×C4⋊C4—41no—
109C3×C4⋊C4—41no—
110C3×C4⋊C4—41no—
111C3×C42—41no—
112C4⋊Dic3—42no—
113C4⋊Dic3—41no—
114Dic3⋊C4—42no—
115C4×Dic3—21no—
116C4×Dic3—21no—
117C4×Dic3—21no—
118C4×Dic3—21no—
119C4×Dic3—22no—
120C4×Dic3—22no—
121C4×Dic3—41no—
122C4×Dic3SD1612yes—
123C4×Dic3Q1612yes—
124C2×C3⋊C8—22no—
125C2×C3⋊C8—22no—
126C2×C3⋊C8—41no—
127C2×C3⋊C8—41no—
128C2×C3⋊C8—41no—
129C2×C3⋊C8—41no—
130C22×Q8—31no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—61no—
134C22×C8—61no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—31no—
137C2×C4⋊C4—61no—
138C2×C4⋊C4—61no—
139C2×C4⋊C4C4×D311yescharacteristic
140C2×C4⋊C4C4×D311yescharacteristic
141C2×C42—31no—
142C2×C42—31no—
143C2×C42—31no—
144C2×C42—31no—
145C2×C42—31no—
146C2×C42—31no—
147C2×C42C22×D311yescharacteristic
148C4⋊C8—64no—
149C4⋊C8—64no—
150#97—122no—
151C4×C8—122no—
152#95—21no—
153#95—61no—
154#95—61no—
155#95—61no—
156#95—61no—
157#95—61no—
158#95—61no—
159#95—61no—
160#95—61no—
161C23×C3C2×C22⋊C411yescharacteristic
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—41no—
169C2×C3×C4—41no—
170C2×C3×C4—41no—
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4#13111yescharacteristic
175C2×C3×C4#13111yescharacteristic
176C2×C3×C4C2×SD1611yescharacteristic
177C2×C3×C4C2×Q1611yescharacteristic
178C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
179C2×C3×C4#13711yescharacteristic
180C2×Dic3—21no—
181C2×Dic3—21no—
182C2×Dic3—21no—
183C2×Dic3—21no—
184C2×Dic3—21no—
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3—21no—
188C2×Dic3—41no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C2×Dic3—41no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3#10312yes—
195C2×Dic3#10312yes—
196C2×Dic3#10412yes—
197C2×Dic3#10412yes—
198Dic6—81no—
199Dic6—81no—
200C3×C8—42no—
201C3×C8—42no—
202C3⋊C8—42no—
203C3⋊C8—42no—
204C2×Q8—61no—
205C2×Q8—61no—
206C2×Q8—121no—
207C2×Q8—121no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—31no—
211C22×C4—31no—
212C22×C4—31no—
213C22×C4—31no—
214C22×C4—61no—
215C22×C4—61no—
216C22×C4C2×C4×D311yescharacteristic
217C22×C4D3×D411yescharacteristic
218C22×C4D3×D411yescharacteristic
219C2×C8—22no—
220C2×C8—22no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—62no—
226C2×C8—62no—
227C2×C8—121no—
228C2×C8—121no—
229C2×C8—121no—
230C2×C8—121no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—121no—
237C4⋊C4—122no—
238C4⋊C4—122no—
239C42—32no—
240C42—32no—
241C42—41no—
242C42—62no—
243C42—62no—
244C42—61no—
245C42—61no—
246C42—61no—
247C42—61no—
248C42—121no—
249C22×C3#40811yescharacteristic
250C22×C3#41011yescharacteristic
251C22×C3#41311yescharacteristic
252C22×C3C2×#10311yescharacteristic
253C22×C3#41711yescharacteristic
254C22×C3C2×#10411yescharacteristic
255C22×C3C2×#10411yescharacteristic
256C3×C4—21no—
257C3×C4—21no—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—81no—
263C3×C4—81no—
264Dic3—22no—
265Dic3—22no—
266Dic3—22no—
267Dic3—22no—
268Dic3—81no—
269Dic3—81no—
270C23D3×C22⋊C411yescharacteristic
271Q8—241no—
272Q8—241no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—32no—
280C2×C4—32no—
281C2×C4—32no—
282C2×C4—32no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—61no—
290C2×C4—61no—
291C2×C4—61no—
292C2×C4—61no—
293C2×C4—61no—
294C2×C4—61no—
295C2×C4—61no—
296C2×C4—61no—
297C2×C4—121no—
298C2×C4—121no—
299C2×C4—121no—
300C2×C4—121no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4#90111yescharacteristic
304C2×C4#90711yescharacteristic
305C2×C4#91111yescharacteristic
306C2×C4D3×SD1611yescharacteristic
307C2×C4#91511yescharacteristic
308C2×C4D3×Q1611yescharacteristic
309C8—42no—
310C8—42no—
311C8—122no—
312C8—122no—
313C2×C3#147311yescharacteristic
314C2×C3#148311yescharacteristic
315C2×C3#149311yescharacteristic
316C2×C3#150311yescharacteristic
317C2×C3#150911yescharacteristic
318C2×C3#152411yescharacteristic
319C2×C3#155011yescharacteristic
320C22#472711yescharacteristic
321C22#474011yescharacteristic
322C22#475111yescharacteristic
323C22#477111yescharacteristic
324C22D3×#10311yescharacteristic
325C22D3×#10411yescharacteristic
326C22D3×#10411yescharacteristic
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—62no—
334C4—62no—
335C4—62no—
336C4—62no—
337C4—81no—
338C4—81no—
339C4—241no—
340C4—241no—
341C3#768911yescharacteristic, complemented, Sylow
342C2#6812911yescharacteristic
343C2#6817011yescharacteristic
344C2#6824011yescharacteristic
345C2#6827911yescharacteristic
346C2#6830911yescharacteristic
347C2#6836111yescharacteristic
348C2#6842311yescharacteristic
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#7689
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
341. C3#7689semidirect

Maximal subgroups

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

subgroupindex
#761922
#686032
#686022
C3×#12712
#676382
#674802
#76893