Finite Group Databasebeta

← #10704607

Subgroups — #10704607

Order 768.

446 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 87 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#82226C211yesmaximal, characteristic
2C3×#1464C211yesmaximal, characteristic
3#71572C211yesmaximal, characteristic
4#67447C212yesmaximal
5#67446C212yesmaximal
6#11518—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C2×#926C411yescharacteristic
9C22×C4×Dic3C411yescharacteristic
10C3×#432—21no—
11#4972—21no—
12#4972—21no—
13#4946C411yescharacteristic
14#4946C411yescharacteristic
15#4946C2211yescharacteristic
16#4946C2211yescharacteristic
17C4×Dic3⋊C4—21no—
18C4×Dic3⋊C4—21no—
19#4926—21no—
20C3×#324—21no—
21C3×#323C2212yes—
22#4528—21no—
23#4525C2212yes—
24#2296—31no—
25#1464—31no—
26#1464D311yescharacteristic
27#1326—32no—
28#1326—32no—
29C23×C3×C4C2×C411yescharacteristic
30C23×Dic3C2×C411yescharacteristic
31C23×Dic3C2×C411yescharacteristic
32C3×#117—41no—
33C3×#117—41no—
34C2×C3×C4⋊C4D411yescharacteristic
35C2×C3×C4⋊C4D411yescharacteristic
36C2×C3×C22⋊C4C2×C411yescharacteristic
37C2×C3×C22⋊C4C2311yescharacteristic
38C2×C3×C42D411yescharacteristic
39C2×C3×C42D411yescharacteristic
40C2×#217C2×C411yescharacteristic
41C2×#217C2×C411yescharacteristic
42#926—22no—
43#926—22no—
44C2×C4⋊Dic3—21no—
45C2×Dic3⋊C4—21no—
46C2×Dic3⋊C4—21no—
47C2×Dic3⋊C4—21no—
48C2×Dic3⋊C4—21no—
49C2×C4×Dic3—21no—
50C2×C4×Dic3—21no—
51C2×C4×Dic3—21no—
52C2×C4×Dic3—41no—
53C2×C4×Dic3—41no—
54C3×C4⋊C8—41no—
55C3×C22⋊C8—22no—
56C3×#97—41no—
57C3⋊#95—21no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61#830—22no—
62C4⋊(C3⋊C8)—41no—
63#803—41no—
64C2×#117—31no—
65C2×#117C2×D311yescharacteristic
66C22×C42—31no—
67#432—21no—
68#432—61no—
69#389—61no—
70#389—61no—
71#385—31no—
72#385—31no—
73#385—31no—
74#385—31no—
75C4×C4⋊C4—61no—
76C4×C4⋊C4—61no—
77#324—21no—
78#324—61no—
79#323—32no—
80#323C2×D312yes—
81C24×C3C22×C411yescharacteristic
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—22no—
87C22×C3×C4C22⋊C411yescharacteristic
88C22×C3×C4C22⋊C411yescharacteristic
89C22×C3×C4C22⋊C411yescharacteristic
90C22×C3×C4C22⋊C411yescharacteristic
91C22×C3×C4C2×D411yescharacteristic
92C22×C3×C4C2×D411yescharacteristic
93C22×Dic3—21no—
94C22×Dic3—21no—
95C22×Dic3—21no—
96C22×Dic3—21no—
97C22×Dic3—42no—
98C22×Dic3—42no—
99C22×Dic3—41no—
100C22×Dic3—41no—
101C2×C3×C8—42no—
102C3×C4⋊C4—41no—
103C3×C4⋊C4—41no—
104C3×C22⋊C4—41no—
105C3×C22⋊C4—41no—
106C3×C42—21no—
107C3×C42—21no—
108C3×C42—21no—
109C3×C42—21no—
110#217—21no—
111#217—21no—
112#217M1612yes—
113#217M1612yes—
114C4⋊Dic3—24no—
115Dic3⋊C4—42no—
116Dic3⋊C4—42no—
117Dic3⋊C4—42no—
118Dic3⋊C4—42no—
119C4×Dic3—24no—
120C4×Dic3—42no—
121C4×Dic3—41no—
122C4×Dic3—41no—
123C2×C3⋊C8—42no—
124C23×C4—31no—
125C23×C4—31no—
126C23×C4C4×D311yescharacteristic
127#117—41no—
128#117—41no—
129#117—62no—
130#117—62no—
131C2×C4⋊C4—61no—
132C2×C4⋊C4—61no—
133C2×C4⋊C4—61no—
134C2×C4⋊C4—61no—
135C2×C4⋊C4—61no—
136C2×C4⋊C4D1211yescharacteristic
137C2×C4⋊C4#6711yescharacteristic
138C2×C22⋊C4—31no—
139C2×C22⋊C4—31no—
140C2×C22⋊C4C4×D311yescharacteristic
141C2×C22⋊C4C22×D311yescharacteristic
142C2×C42—61no—
143C2×C42—61no—
144C2×C42—61no—
145C2×C42—121no—
146C2×C42—121no—
147C2×C42D1211yescharacteristic
148C2×C42#6711yescharacteristic
149C4⋊C8—41no—
150C4⋊C8—121no—
151C22⋊C8—22no—
152C22⋊C8—62no—
153#97—41no—
154#97—121no—
155#95—61no—
156#95—61no—
157#95—61no—
158#95—61no—
159C23×C3—21no—
160C23×C3—21no—
161C23×C3—41no—
162C23×C3—41no—
163C23×C3C2×C22⋊C411yescharacteristic
164C23×C3C2×M1611yescharacteristic
165C23×C3C2×M1611yescharacteristic
166C2×C3×C4—22no—
167C2×C3×C4—22no—
168C2×C3×C4—22no—
169C2×C3×C4—22no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4#9911yescharacteristic
183C2×C3×C4#9911yescharacteristic
184C2×C3×C4#10112yes—
185C2×Dic3—22no—
186C2×Dic3—22no—
187C2×Dic3—22no—
188C2×Dic3—22no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C2×Dic3—41no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C2×Dic3—42no—
196C2×Dic3—42no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—42no—
200C2×Dic3—42no—
201C3×C8—82no—
202C3⋊C8—82no—
203C24C2×C4×D311yescharacteristic
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—61no—
210C22×C4—61no—
211C22×C4—61no—
212C22×C4—61no—
213C22×C4—122no—
214C22×C4—122no—
215C22×C4—121no—
216C22×C4—121no—
217C22×C4(C2×D3)⋊C411yescharacteristic
218C22×C4(C2×D3)⋊C411yescharacteristic
219C22×C4(C2×D3)⋊C411yescharacteristic
220C22×C4(C2×D3)⋊C411yescharacteristic
221C22×C4C2×D1211yescharacteristic
222C22×C4C2×#6711yescharacteristic
223C2×C8—42no—
224C2×C8—122no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—64no—
228C4⋊C4—122no—
229C4⋊C4—122no—
230C4⋊C4—122no—
231C4⋊C4—122no—
232C22⋊C4—32no—
233C22⋊C4—32no—
234C22⋊C4—41no—
235C22⋊C4—41no—
236C22⋊C4—61no—
237C22⋊C4—61no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—64no—
243C42—122no—
244C42—121no—
245C42—121no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—41no—
251C22×C3—41no—
252C22×C3—42no—
253C22×C3—41no—
254C22×C3—41no—
255C22×C3#40711yescharacteristic
256C22×C3C2×#9911yescharacteristic
257C22×C3C2×#10111yescharacteristic
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—22no—
261C3×C4—41no—
262C3×C4—41no—
263C3×C4—41no—
264C3×C4—41no—
265Dic3—42no—
266Dic3—42no—
267Dic3—42no—
268Dic3—42no—
269C23—21no—
270C23—21no—
271C23—41no—
272C23—41no—
273C23D3×M1611yescharacteristic
274C23D3×M1611yescharacteristic
275C23C2×(C2×D3)⋊C411yescharacteristic
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—62no—
293C2×C4—62no—
294C2×C4—62no—
295C2×C4—62no—
296C2×C4—121no—
297C2×C4—121no—
298C2×C4—121no—
299C2×C4—121no—
300C2×C4—121no—
301C2×C4—121no—
302C2×C4—122no—
303C2×C4—122no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—122no—
307C2×C4—122no—
308C2×C4#80611yescharacteristic
309C2×C4#80611yescharacteristic
310C2×C4#82412yes—
311C8—82no—
312C8—242no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—41no—
316C2×C3—41no—
317C2×C3#146311yescharacteristic
318C2×C3#212011yescharacteristic
319C2×C3#212011yescharacteristic
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—42no—
327C22—41no—
328C22—41no—
329C22C2×#80611yescharacteristic
330C22#512611yescharacteristic
331C22C2×#82411yescharacteristic
332C4—21no—
333C4—21no—
334C4—22no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—122no—
340C4—122no—
341C4—122no—
342C4—122no—
343C3#1151811yescharacteristic, complemented, Sylow
344C2—21no—
345C2—21no—
346C2—41no—
347C2—41no—
348C2#6997111yescharacteristic
349C2#7005611yescharacteristic
350C2#7014511yescharacteristic
subgroupG/H
centerC22C2×#806
commutator, prime radicalC24×C3C22×C4
FrattiniC2×C22⋊C4C22×D3
FittingC3×#1464C2
socleC22×C3C2×#99
cosocleC2×C3×C22⋊C4C23
hypercenter#1464D3
hypocenterC3#11518
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
343. C3#11518semidirect

Maximal subgroups

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

subgroupindex
#822262
C3×#14642
#715722
#674472
#674462
#115183