Finite Group Databasebeta

← #10704589

Subgroups — #10704589

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#82224C211yesmaximal, characteristic
2C3×#1464C211yesmaximal, characteristic
3#71572C211yesmaximal, characteristic
4#67453C212yesmaximal, complemented
5#67452C212yesmaximal, complemented
6#11513—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C2×#926C411yescharacteristic
9C22×C4×Dic3C411yescharacteristic
10C3×#432—21no—
11#4977C411yescharacteristic
12#4977C2211yescharacteristic
13#4975C411yescharacteristic
14#4975C2211yescharacteristic
15#4942—21no—
16#4942—21no—
17C4×Dic3⋊C4—21no—
18C4×Dic3⋊C4—21no—
19#4926—21no—
20C3×#324C2212yes—
21C3×#323—21no—
22#4528C2212yes—
23#4525—21no—
24#2285—31no—
25#1464—31no—
26#1464D311yescharacteristic
27#1325—32no—
28#1324—32no—
29C23×C3×C4C2×C411yescharacteristic
30C23×Dic3—21no—
31C3×#117—41no—
32C3×#117—41no—
33C2×C3×C4⋊C4C2×C411yescharacteristic
34C2×C3×C4⋊C4C2311yescharacteristic
35C2×C3×C22⋊C4D411yescharacteristic
36C2×C3×C22⋊C4D411yescharacteristic
37C2×C3×C42D411yescharacteristic
38C2×C3×C42D411yescharacteristic
39C2×#217—21no—
40#926—41no—
41#926—41no—
42C2×C4⋊Dic3C2×C411yescharacteristic
43C2×C4⋊Dic3C2×C411yescharacteristic
44C2×Dic3⋊C4—21no—
45C2×Dic3⋊C4—21no—
46C2×Dic3⋊C4—21no—
47C2×Dic3⋊C4—21no—
48C2×C4×Dic3—21no—
49C2×C4×Dic3—22no—
50C2×C4×Dic3—22no—
51C2×C4×Dic3—21no—
52C2×C4×Dic3C2×C411yescharacteristic
53C2×C4×Dic3C2×C411yescharacteristic
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#384—31no—
70#384—31no—
71#382—31no—
72#382—31no—
73#380—61no—
74#380—61no—
75C4×C4⋊C4—61no—
76C4×C4⋊C4—61no—
77#324—32no—
78#324C2×D312yes—
79#323—21no—
80#323—61no—
81C24×C3C22⋊C411yescharacteristic
82C22×C3×C4—22no—
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
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—24no—
95C22×Dic3—21no—
96C22×Dic3—21no—
97C22×Dic3—21no—
98C22×Dic3—42no—
99C22×Dic3—42no—
100C2×C3×C8—42no—
101C3×C4⋊C4—41no—
102C3×C4⋊C4—41no—
103C3×C22⋊C4—41no—
104C3×C22⋊C4—41no—
105C3×C42—21no—
106C3×C42—21no—
107C3×C42—21no—
108C3×C42—21no—
109#217—42no—
110C4⋊Dic3—41no—
111C4⋊Dic3—41no—
112Dic3⋊C4—42no—
113Dic3⋊C4—42no—
114Dic3⋊C4—42no—
115Dic3⋊C4—42no—
116C4×Dic3—24no—
117C4×Dic3—21no—
118C4×Dic3—21no—
119C4×Dic3—41no—
120C4×Dic3—41no—
121C4×Dic3M1612yes—
122C4×Dic3M1612yes—
123C2×C3⋊C8—42no—
124C23×C4—61no—
125C23×C4C4×D311yescharacteristic
126#117—41no—
127#117—41no—
128#117—121no—
129#117—121no—
130C2×C4⋊C4—31no—
131C2×C4⋊C4—31no—
132C2×C4⋊C4—61no—
133C2×C4⋊C4—61no—
134C2×C4⋊C4—61no—
135C2×C4⋊C4—61no—
136C2×C4⋊C4C4×D311yescharacteristic
137C2×C4⋊C4C22×D311yescharacteristic
138C2×C22⋊C4—61no—
139C2×C22⋊C4D1211yescharacteristic
140C2×C22⋊C4#6711yescharacteristic
141C2×C42—31no—
142C2×C42—31no—
143C2×C42—62no—
144C2×C42—62no—
145C2×C42—61no—
146C2×C42—61no—
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×C3#10112yes—
164C23×C3C2×C22⋊C411yescharacteristic
165C2×C3×C4—22no—
166C2×C3×C4—22no—
167C2×C3×C4—22no—
168C2×C3×C4—22no—
169C2×C3×C4—21no—
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—41no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4#9911yescharacteristic
182C2×C3×C4#9911yescharacteristic
183C2×C3×C4C2×M1611yescharacteristic
184C2×C3×C4C2×M1611yescharacteristic
185C2×Dic3—22no—
186C2×Dic3—22no—
187C2×Dic3—22no—
188C2×Dic3—22no—
189C2×Dic3—48no—
190C2×Dic3—41no—
191C2×Dic3—41no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C3×C8—82no—
199C3⋊C8—82no—
200C24(C2×D3)⋊C411yescharacteristic
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—61no—
207C22×C4—64no—
208C22×C4—61no—
209C22×C4—61no—
210C22×C4—61no—
211C22×C4—122no—
212C22×C4—122no—
213C22×C4(C2×D3)⋊C411yescharacteristic
214C22×C4(C2×D3)⋊C411yescharacteristic
215C22×C4(C2×D3)⋊C411yescharacteristic
216C22×C4C2×C4×D311yescharacteristic
217C22×C4C2×D1211yescharacteristic
218C22×C4C2×#6711yescharacteristic
219C2×C8—42no—
220C2×C8—122no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—121no—
224C4⋊C4—121no—
225C4⋊C4—122no—
226C4⋊C4—122no—
227C4⋊C4—122no—
228C4⋊C4—122no—
229C22⋊C4—41no—
230C22⋊C4—41no—
231C22⋊C4—122no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—32no—
237C42—32no—
238C42—64no—
239C42—61no—
240C42—61no—
241C42—121no—
242C42—121no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—22no—
246C22×C3—42no—
247C22×C3—42no—
248C22×C3—41no—
249C22×C3—41no—
250C22×C3#40711yescharacteristic
251C22×C3C2×#9911yescharacteristic
252C22×C3C2×#10111yescharacteristic
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—41no—
258C3×C4—41no—
259C3×C4—41no—
260C3×C4—41no—
261Dic3—24no—
262Dic3—42no—
263Dic3—81no—
264Dic3—81no—
265C23—21no—
266C23—21no—
267C23—41no—
268C23—41no—
269C23#82412yes—
270C23C2×(C2×D3)⋊C411yescharacteristic
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—62no—
288C2×C4—62no—
289C2×C4—62no—
290C2×C4—62no—
291C2×C4—128no—
292C2×C4—121no—
293C2×C4—121no—
294C2×C4—121no—
295C2×C4—121no—
296C2×C4—121no—
297C2×C4—121no—
298C2×C4—121no—
299C2×C4—121no—
300C2×C4#80611yescharacteristic
301C2×C4#80611yescharacteristic
302C2×C4D3×M1611yescharacteristic
303C2×C4D3×M1611yescharacteristic
304C8—82no—
305C8—242no—
306C2×C3—21no—
307C2×C3—21no—
308C2×C3—42no—
309C2×C3#146311yescharacteristic
310C2×C3#212611yescharacteristic
311C2×C3#213211yescharacteristic
312C22—21no—
313C22—21no—
314C22—22no—
315C22—42no—
316C22—42no—
317C22—41no—
318C22—41no—
319C22C2×#80611yescharacteristic
320C22#512611yescharacteristic
321C22C2×#82411yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—64no—
331C4—122no—
332C4—241no—
333C4—241no—
334C3#1151311yescharacteristic, complemented, Sylow
335C2—21no—
336C2—21no—
337C2—42no—
338C2#6997111yescharacteristic
339C2#7006011yescharacteristic
340C2#7014811yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1464C2
socleC22×C3C2×#99
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1464D3
hypocenterC3#11513
LCSG ▷ C22×C3×C4 ▷ 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
4. #67453C2semidirect
5. #67452C2semidirect
334. C3#11513semidirect

Maximal subgroups

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

subgroupindex
#822242
C3×#14642
#715722
#674532
#674522
#115133