Finite Group Databasebeta

← #10705371

Subgroups — #10705371

Order 768.

568 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 107 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#82286C211yesmaximal, characteristic
2C3×#1740C211yesmaximal, characteristic, complemented
3#71571C211yesmaximal, characteristic, complemented
4#67489C214yesmaximal, complemented
5#11693—31nomaximal, Sylow
6C2×C3×#117—21no—
7C2×C3×#117C2211yescharacteristic
8C2×#930—21no—
9C2×#874C411yescharacteristic, complemented
10#4991—21no—
11#4991C412yescomplemented
12#4960—21no—
13#4960C2212yes—
14#4939—21no—
15#4936C411yescharacteristic, complemented
16#4925—21no—
17C3×#342—21no—
18C3×#342C2212yes—
19#4528—21no—
20#4525C2212yes—
21#2619—31no—
22#1740D311yescharacteristic, complemented
23#1458—31no—
24#1343—34no—
25C23×C3×C4D411yescharacteristic
26C22×#67—22no—
27C22×D12—21no—
28C22×D12—21no—
29C22×D12C2×C412yes—
30C3×#117—24no—
31C3×#117—41no—
32C3×#117—41no—
33C3×#117—42no—
34C2×C3×C4⋊C4—41no—
35C2×C3×C4⋊C4C2×C411yescharacteristic
36C2×C3×C4⋊C4D411yescharacteristic
37C2×C3×C22⋊C4—22no—
38C2×C3×C22⋊C4D411yescharacteristic
39C2×C3×C22⋊C4C2311yescharacteristic
40C2×C3×C42—41no—
41C2×C3×C42C2×C411yescharacteristic
42C2×C3×C42D411yescharacteristic
43#930—42no—
44#930—42no—
45C2×(C2×D3)⋊C4—22no—
46C2×(C2×D3)⋊C4—22no—
47C2×(C2×D3)⋊C4—22no—
48C2×C4⋊Dic3—21no—
49#874—41no—
50#874—41no—
51C3⋊#95—21no—
52C3⋊#95C2×C412yes—
53#830—22no—
54C4⋊(C3⋊C8)—41no—
55C4×C3⋊C8—41no—
56C2×#127—31no—
57C2×#121—61no—
58C2×#117—21no—
59C2×#117C2×D311yescharacteristic
60#431—61no—
61#394—32no—
62#394—61no—
63#392—32no—
64#392—61no—
65#390—61no—
66#382—31no—
67#342—21no—
68#342C2×D312yes—
69#324—61no—
70#323—32no—
71C24×C3C2×D411yescharacteristic
72C23×D3—22no—
73C22×C3×C4—21no—
74C22×C3×C4—21no—
75C22×C3×C4—22no—
76C22×C3×C4—42no—
77C22×C3×C4C22⋊C411yescharacteristic
78C22×C3×C4C22⋊C411yescharacteristic
79C22×C3×C4C22⋊C411yescharacteristic
80C22×C3×C4C22⋊C411yescharacteristic
81C22×C3×C4D812yes—
82C22×C3×C4SD1612yes—
83C22×C3×C4C22×C411yescharacteristic
84C22×C3×C4C2×D411yescharacteristic
85C2×#67—42no—
86C2×#67—42no—
87C2×#67—82no—
88C2×#67—82no—
89C22×Dic3—22no—
90C2×D12—42no—
91C2×D12—42no—
92C2×D12—41no—
93C2×D12—41no—
94C2×D12—42no—
95C2×D12—42no—
96C2×D12—82no—
97C2×D12—82no—
98C3×C4⋊C4—41no—
99C3×C4⋊C4—41no—
100C3×C4⋊C4—44no—
101C3×C22⋊C4—24no—
102C3×C22⋊C4—41no—
103C3×C22⋊C4—41no—
104C3×C22⋊C4—42no—
105C3×C42—21no—
106C3×C42—21no—
107C3×C42—21no—
108C3×C42—21no—
109C3×C42—44no—
110(C2×D3)⋊C4—44no—
111(C2×D3)⋊C4—82no—
112(C2×D3)⋊C4—82no—
113C4⋊Dic3—42no—
114C2×C3⋊C8—42no—
115C22×D4—32no—
116C22×D4—62no—
117C22×D4—61no—
118C22×D4—61no—
119C23×C4#6711yescharacteristic
120#127—121no—
121#127—121no—
122#121—122no—
123#121—122no—
124#117—24no—
125#117—41no—
126#117—41no—
127#117—42no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—61no—
130C2×C4⋊C4C4×D311yescharacteristic
131C2×C4⋊C4D1211yescharacteristic
132C2×C22⋊C4—22no—
133C2×C22⋊C4—62no—
134C2×C22⋊C4—62no—
135C2×C22⋊C4—62no—
136C2×C22⋊C4#6711yescharacteristic
137C2×C22⋊C4C22×D311yescharacteristic
138C2×C42—41no—
139C2×C42C4×D311yescharacteristic
140C2×C42D1211yescharacteristic
141C4⋊C8—121no—
142C22⋊C8—62no—
143C4×C8—121no—
144#95—32no—
145#95—61no—
146C23×C3—21no—
147C23×C3—21no—
148C23×C3—41no—
149C23×C3—41no—
150C23×C3C2×C22⋊C411yescharacteristic
151C23×C3C2×D811yescharacteristic
152C23×C3C2×SD1611yescharacteristic
153C22×D3—42no—
154C22×D3—42no—
155C22×D3—42no—
156C22×D3—42no—
157C22×D3—42no—
158C22×D3—42no—
159C22×D3—42no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4—22no—
163C2×C3×C4—22no—
164C2×C3×C4—22no—
165C2×C3×C4—22no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—41no—
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4—44no—
175C2×C3×C4—82no—
176C2×C3×C4—82no—
177C2×C3×C4#9911yescharacteristic
178C2×C3×C4#9911yescharacteristic
179C2×C3×C4#10212yes—
180C2×C3×C4#10212yes—
181C2×C3×C4#10412yes—
182#67—84no—
183#67—84no—
184C2×Dic3—42no—
185C2×Dic3—42no—
186C2×Dic3—42no—
187D12—84no—
188D12—84no—
189D12—82no—
190D12—82no—
191C3⋊C8—82no—
192C24—62no—
193C24C2×#6711yescharacteristic
194C2×D4—122no—
195C2×D4—122no—
196C2×D4—122no—
197C2×D4—122no—
198C2×D4—122no—
199C2×D4—121no—
200C2×D4—121no—
201C2×D4—122no—
202C2×D4—242no—
203C2×D4—242no—
204C2×D4—242no—
205C2×D4—242no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—42no—
210C22×C4—62no—
211C22×C4(C2×D3)⋊C411yescharacteristic
212C22×C4(C2×D3)⋊C411yescharacteristic
213C22×C4(C2×D3)⋊C411yescharacteristic
214C22×C4(C2×D3)⋊C411yescharacteristic
215C22×C4#21312yes—
216C22×C4#21512yes—
217C22×C4C2×C4×D311yescharacteristic
218C22×C4C2×D1211yescharacteristic
219C2×C8—122no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—44no—
223C4⋊C4—122no—
224C22⋊C4—24no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—42no—
228C22⋊C4—124no—
229C22⋊C4—242no—
230C22⋊C4—242no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—44no—
236C22×C3—21no—
237C22×C3—21no—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3—41no—
241C22×C3—41no—
242C22×C3—41no—
243C22×C3—41no—
244C22×C3—81no—
245C22×C3C2×#9911yescharacteristic
246C22×C3C2×#10211yescharacteristic
247C22×C3C2×#10411yescharacteristic
248C2×D3—82no—
249C2×D3—82no—
250C2×D3—82no—
251C2×D3—82no—
252C2×D3—82no—
253C2×D3—82no—
254C2×D3—82no—
255C3×C4—22no—
256C3×C4—41no—
257C3×C4—41no—
258C3×C4—41no—
259C3×C4—41no—
260C3×C4—84no—
261C3×C4#32712yes—
262C3×C4#32712yes—
263Dic3—82no—
264C23—21no—
265C23—21no—
266C23—41no—
267C23—41no—
268C23—122no—
269C23—122no—
270C23—122no—
271C23—122no—
272C23—122no—
273C23—122no—
274C23—122no—
275C23C2×(C2×D3)⋊C411yescharacteristic
276C23C2×#21311yescharacteristic
277C23C2×#21511yescharacteristic
278D4—244no—
279D4—244no—
280D4—244no—
281D4—244no—
282D4—242no—
283D4—242no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—44no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—122no—
302C2×C4—122no—
303C2×C4—122no—
304C2×C4#80512yes—
305C2×C4#80611yescharacteristic
306C2×C4#80611yescharacteristic
307C2×C4#80912yes—
308C2×C4#80912yes—
309C8—242no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—41no—
313C2×C3—41no—
314C2×C3C2×#32711yescharacteristic
315C2×C3#212211yescharacteristic
316C2×C3#212811yescharacteristic
317D3—162no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22—242no—
328C22—242no—
329C22—242no—
330C22—242no—
331C22—242no—
332C22—242no—
333C22—242no—
334C22C2×#80511yescharacteristic
335C22C2×#80611yescharacteristic
336C22C2×#80911yescharacteristic
337C4—22no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—84no—
343C4—242no—
344C4#445112yes—
345C4#445112yes—
346C3#1169311yescharacteristic, complemented, Sylow
347C2—21no—
348C2—21no—
349C2—41no—
350C2—41no—
351C2—482no—
352C2C2×#445111yescharacteristic
353C2#7005711yescharacteristic
354C2#7006111yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C22⋊C4C22×D3
FittingC3×#1740C2
socleC22×C3C2×#99
prime radical#82286C2
cosocleC2×C3×C22⋊C4C23
hypercenter#1740D3
hypocenterC3#11693
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1740

Complements

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

NH ≅ G/Ntype
2. C3×#1740C2semidirect
3. #71571C2semidirect
4. #67489C2semidirect
9. C2×#874C4semidirect
11. #4991C4semidirect
15. #4936C4semidirect
22. #1740D3semidirect
346. C3#11693semidirect

Maximal subgroups

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

subgroupindex
#822862
C3×#17402
#715712
#674892
#116933