Finite Group Databasebeta

← #10705397

Subgroups — #10705397

Order 768.

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

normal subgroups only

This group has 552 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×#1742C211yesmaximal, characteristic, complemented
3#71571C211yesmaximal, characteristic, complemented
4#67493C214yesmaximal, complemented
5#11699—31nomaximal, Sylow
6C22×C3×M16—21no—
7C2×C3×#117C2211yescharacteristic
8C2×#930—21no—
9C2×#874C411yescharacteristic
10#4991—21no—
11#4991C2212yes—
12#4960—21no—
13#4960C412yes—
14#4939—21no—
15#4936C411yescharacteristic
16#4925—21no—
17C3×#343—21no—
18C3×#343C2212yes—
19#4528C2212yes—
20#4525—21no—
21#2619—31no—
22#1742D311yescharacteristic, complemented
23#1458—31no—
24#1345—34no—
25C23×C3×C4D411yescharacteristic
26C22×#67—22no—
27C22×D12—21no—
28C22×D12—21no—
29C22×D12C2×C412yes—
30C2×C3×M16—24no—
31C2×C3×M16—22no—
32C2×C3×M16—41no—
33C2×C3×M16—42no—
34C22×C3×C8—41no—
35C3×#117—41no—
36C3×#117—41no—
37C2×C3×C4⋊C4D411yescharacteristic
38C2×C3×C4⋊C4C2311yescharacteristic
39C2×C3×C22⋊C4C2×C411yescharacteristic
40C2×C3×C22⋊C4D411yescharacteristic
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—
56C22×M16—21no—
57C2×#127—31no—
58C2×#121—61no—
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#343—21no—
68#343C2×D312yes—
69#324—32no—
70#323—61no—
71C24×C3C22⋊C411yescharacteristic
72C23×D3—22no—
73C22×C3×C4—21no—
74C22×C3×C4—21no—
75C22×C3×C4—22no—
76C22×C3×C4C22⋊C411yescharacteristic
77C22×C3×C4C22⋊C411yescharacteristic
78C22×C3×C4C22⋊C411yescharacteristic
79C22×C3×C4D812yes—
80C22×C3×C4SD1612yes—
81C22×C3×C4C22×C411yescharacteristic
82C22×C3×C4C2×D411yescharacteristic
83C22×C3×C4C2×D411yescharacteristic
84C2×#67—44no—
85C2×#67—82no—
86C2×#67—82no—
87C22×Dic3—22no—
88C2×D12—42no—
89C2×D12—42no—
90C2×D12—41no—
91C2×D12—41no—
92C2×D12—42no—
93C2×D12—42no—
94C2×D12—82no—
95C2×D12—82no—
96C3×M16—24no—
97C3×M16—44no—
98C3×M16—42no—
99C2×C3×C8—42no—
100C2×C3×C8—44no—
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(C2×D3)⋊C4—42no—
110(C2×D3)⋊C4—42no—
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
120C2×M16—24no—
121C2×M16—22no—
122C2×M16—41no—
123C2×M16—42no—
124C22×C8—41no—
125#127—121no—
126#127—121no—
127#121—122no—
128#121—122no—
129#117—41no—
130#117—41no—
131C2×C4⋊C4—61no—
132C2×C4⋊C4#6711yescharacteristic
133C2×C4⋊C4C22×D311yescharacteristic
134C2×C22⋊C4—62no—
135C2×C22⋊C4—62no—
136C2×C22⋊C4—62no—
137C2×C22⋊C4C4×D311yescharacteristic
138C2×C22⋊C4D1211yescharacteristic
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×C3#10212yes—
151C23×C3C2×C22⋊C411yescharacteristic
152C22×D3—42no—
153C22×D3—42no—
154C22×D3—42no—
155C22×D3—42no—
156C22×D3—42no—
157C22×D3—42no—
158C22×D3—42no—
159C2×C3×C4—21no—
160C2×C3×C4—21no—
161C2×C3×C4—22no—
162C2×C3×C4—22no—
163C2×C3×C4—22no—
164C2×C3×C4—22no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—41no—
170C2×C3×C4—41no—
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×C3×C4#9911yescharacteristic
174C2×C3×C4#9911yescharacteristic
175C2×C3×C4#10212yes—
176C2×C3×C4#10412yes—
177C2×C3×C4C2×D811yescharacteristic
178C2×C3×C4C2×SD1611yescharacteristic
179#67—88no—
180C2×Dic3—42no—
181C2×Dic3—42no—
182C2×Dic3—42no—
183D12—84no—
184D12—84no—
185D12—82no—
186D12—82no—
187C3×C8—44no—
188C3⋊C8—82no—
189C24—62no—
190C24(C2×D3)⋊C411yescharacteristic
191C2×D4—122no—
192C2×D4—122no—
193C2×D4—124no—
194C2×D4—122no—
195C2×D4—121no—
196C2×D4—121no—
197C2×D4—122no—
198C2×D4—242no—
199C2×D4—242no—
200C2×D4—242no—
201C2×D4—242no—
202C22×C4—22no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—62no—
206C22×C4(C2×D3)⋊C411yescharacteristic
207C22×C4(C2×D3)⋊C411yescharacteristic
208C22×C4(C2×D3)⋊C411yescharacteristic
209C22×C4#21312yes—
210C22×C4#21512yes—
211C22×C4C2×C4×D311yescharacteristic
212C22×C4C2×D1211yescharacteristic
213C22×C4C2×#6711yescharacteristic
214M16—24no—
215M16—44no—
216M16—42no—
217C2×C8—42no—
218C2×C8—44no—
219C2×C8—122no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—122no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—122no—
226C22⋊C4—122no—
227C22⋊C4—242no—
228C22⋊C4—242no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—21no—
233C22×C3—21no—
234C22×C3—21no—
235C22×C3—22no—
236C22×C3—42no—
237C22×C3—41no—
238C22×C3—41no—
239C22×C3—81no—
240C22×C3C2×#9911yescharacteristic
241C22×C3C2×#10211yescharacteristic
242C22×C3C2×#10411yescharacteristic
243C2×D3—82no—
244C2×D3—82no—
245C2×D3—82no—
246C2×D3—82no—
247C2×D3—82no—
248C2×D3—82no—
249C2×D3—82no—
250C3×C4—21no—
251C3×C4—21no—
252C3×C4—41no—
253C3×C4—41no—
254C3×C4—41no—
255C3×C4—41no—
256C3×C4#32712yes—
257C3×C4#32712yes—
258Dic3—82no—
259C23—21no—
260C23—21no—
261C23—41no—
262C23—41no—
263C23—122no—
264C23—122no—
265C23—122no—
266C23—122no—
267C23—122no—
268C23—122no—
269C23—122no—
270C23#80912yes—
271C23C2×(C2×D3)⋊C411yescharacteristic
272D4—244no—
273D4—244no—
274D4—248no—
275D4—242no—
276D4—242no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—122no—
292C2×C4—122no—
293C2×C4—122no—
294C2×C4#80512yes—
295C2×C4#80611yescharacteristic
296C2×C4#80611yescharacteristic
297C2×C4#80912yes—
298C2×C4C2×#21311yescharacteristic
299C2×C4C2×#21511yescharacteristic
300C8—44no—
301C8—242no—
302C2×C3—21no—
303C2×C3—21no—
304C2×C3—42no—
305C2×C3C2×#32711yescharacteristic
306C2×C3#212211yescharacteristic
307C2×C3#213411yescharacteristic
308D3—162no—
309C22—21no—
310C22—21no—
311C22—22no—
312C22—42no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22—242no—
317C22—242no—
318C22—242no—
319C22—242no—
320C22—242no—
321C22—242no—
322C22—242no—
323C22C2×#80511yescharacteristic
324C22C2×#80611yescharacteristic
325C22C2×#80911yescharacteristic
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—242no—
333C4#445112yes—
334C4#445112yes—
335C3#1169911yescharacteristic, complemented, Sylow
336C2—21no—
337C2—21no—
338C2—42no—
339C2—482no—
340C2C2×#445111yescharacteristic
341C2#7014611yescharacteristic
342C2#7014911yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1742C2
socleC22×C3C2×#99
prime radical#82286C2
cosocleC2×C3×C4⋊C4C23
hypercenter#1742D3
hypocenterC3#11699
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1742

Complements

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

NH ≅ G/Ntype
2. C3×#1742C2semidirect
3. #71571C2semidirect
4. #67493C2semidirect
22. #1742D3semidirect
335. C3#11699semidirect

Maximal subgroups

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

subgroupindex
#822862
C3×#17422
#715712
#674932
#116993