Finite Group Databasebeta

← C67×#3348 #22557779

Subgroups — C67×#3348 #22557779

Order 8576.

348 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#555×C67C211yesmaximal, characteristic
2#552×C67C211yesmaximal, characteristic, complemented
3#501×C67C211yesmaximal, characteristic, complemented
4#485×C67C211yesmaximal, characteristic, complemented
5#485×C67C211yesmaximal, characteristic, complemented
6#482×C67C211yesmaximal, characteristic, complemented
7#481×C67C211yesmaximal, characteristic, complemented
8#478×C67C211yesmaximal, characteristic, complemented
9#475×C67C211yesmaximal, characteristic, complemented
10#468×C67C211yesmaximal, characteristic, complemented
11#465×C67C211yesmaximal, characteristic, complemented
12#440×C67C211yesmaximal, characteristic, complemented
13#435×C67C211yesmaximal, characteristic, complemented
14#428×C67C211yesmaximal, characteristic, complemented
15#427×C67C211yesmaximal, characteristic, complemented
16C2×SD16×C67C2211yescharacteristic
17C2×M16×C67C2211yescharacteristic
18C22×C8×C67C2211yescharacteristic
19C4⋊Q8×C67C2211yescharacteristic
20#126×C67—21no—
21#125×C67—21no—
22#125×C67C2211yescharacteristic
23#124×C67—21no—
24#124×C67—21no—
25#123×C67—21no—
26C22⋊Q8×C67C2211yescharacteristic
27C22⋊Q8×C67C2211yescharacteristic
28#121×C67—21no—
29#121×C67C2211yescharacteristic
30#121×C67C2211yescharacteristic
31C4×Q8×C67C2211yescharacteristic
32C4×D4×C67C2211yescharacteristic
33C4×D4×C67C2211yescharacteristic
34#117×C67—21no—
35#117×C67—21no—
36#117×C67C2211yescharacteristic
37#117×C67C2211yescharacteristic
38#117×C67C2211yescharacteristic
39C2×C4⋊C4×C67C2211yescharacteristic
40#107×C67C2211yescharacteristic
41#107×C67C2211yescharacteristic
42#107×C67C2211yescharacteristic
43#107×C67C2211yescharacteristic
44#106×C67C2211yescharacteristic
45#106×C67C2211yescharacteristic
46#106×C67C2211yescharacteristic
47#106×C67C2211yescharacteristic
48#106×C67C2211yescharacteristic
49C4⋊C8×C67C2211yescharacteristic
50#103×C67C2211yescharacteristic
51#103×C67C2211yescharacteristic
52#103×C67C2211yescharacteristic
53#102×C67C2211yescharacteristic
54#102×C67C2211yescharacteristic
55#102×C67C2211yescharacteristic
56C22⋊C8×C67C2211yescharacteristic
57C22⋊C8×C67C2211yescharacteristic
58#97×C67C2211yescharacteristic
59C2×Q8×C67—21no—
60C2×Q8×C67C2311yescharacteristic
61C2×D4×C67—21no—
62C2×D4×C67—21no—
63C2×D4×C67D411yescharacteristic
64C2×D4×C67C2311yescharacteristic
65C22×C4×C67—21no—
66C22×C4×C67—21no—
67C22×C4×C67C2311yescharacteristic
68C22×C4×C67C2311yescharacteristic
69SD16×C67—22no—
70M16×C67—22no—
71C2×C8×C67—22no—
72C2×C8×C67C2311yescharacteristic
73C2×C8×C67C2311yescharacteristic
74C2×C8×C67C2311yescharacteristic
75C2×C8×C67C2311yescharacteristic
76C4⋊C4×C67—21no—
77C4⋊C4×C67—21no—
78C4⋊C4×C67—21no—
79C4⋊C4×C67—21no—
80C4⋊C4×C67—21no—
81C4⋊C4×C67—21no—
82C4⋊C4×C67—21no—
83C4⋊C4×C67D411yescharacteristic
84C4⋊C4×C67C2311yescharacteristic
85C4⋊C4×C67C2311yescharacteristic
86C4⋊C4×C67C2311yescharacteristic
87C4⋊C4×C67C2311yescharacteristic
88C4⋊C4×C67C2311yescharacteristic
89C4⋊C4×C67C2311yescharacteristic
90C22⋊C4×C67—21no—
91C22⋊C4×C67—21no—
92C22⋊C4×C67—21no—
93C22⋊C4×C67—21no—
94C22⋊C4×C67—21no—
95C22⋊C4×C67—21no—
96C22⋊C4×C67—21no—
97C22⋊C4×C67—21no—
98C22⋊C4×C67D411yescharacteristic
99C22⋊C4×C67D411yescharacteristic
100C42×C67—21no—
101C42×C67—21no—
102C42×C67—21no—
103C42×C67—21no—
104C42×C67C2311yescharacteristic
105C23×C67—21no—
106C23×C67C2×D411yescharacteristic
107C23×C67C2×D411yescharacteristic
108Q8×C67—21no—
109Q8×C67—21no—
110Q8×C67—81no—
111D4×C67—21no—
112D4×C67—21no—
113D4×C67—41no—
114D4×C67—81no—
115D4×C67—81no—
116C2×C4×C67—21no—
117C2×C4×C67—21no—
118C2×C4×C67—21no—
119C2×C4×C67—21no—
120C2×C4×C67—21no—
121C2×C4×C67—21no—
122C2×C4×C67—21no—
123C2×C4×C67—21no—
124C2×C4×C67—21no—
125C2×C4×C67—21no—
126C2×C4×C67—21no—
127C2×C4×C67—42no—
128C2×C4×C67—42no—
129C2×C4×C67C2×D411yescharacteristic
130C2×C4×C67C2×D411yescharacteristic
131C2×C4×C67C2×D411yescharacteristic
132C2×C4×C67C2×D411yescharacteristic
133C2×C4×C67C2411yescharacteristic
134C8×C67—21no—
135C8×C67—21no—
136C8×C67—21no—
137C8×C67#4112yes—
138C22×C67—21no—
139C22×C67—21no—
140C22×C67—21no—
141C22×C67—21no—
142C22×C67—21no—
143C22×C67—21no—
144C22×C67—41no—
145C22×C67—41no—
146C22×C67—41no—
147C22×C67C22×D411yescharacteristic
148C4×C67—21no—
149C4×C67—21no—
150C4×C67—21no—
151C4×C67—21no—
152C4×C67—41no—
153C4×C67—41no—
154C4×C67—41no—
155C4×C67—41no—
156C4×C67—41no—
157C4×C67—41no—
158C4×C67—41no—
159C4×C67C2×#4111yescharacteristic
160C4×C672−1+411yescharacteristic
161C2×C67—41no—
162C2×C67—41no—
163C2×C67—81no—
164C2×C67#54711yescharacteristic
165C2×C67#57511yescharacteristic
166C2×C67#57711yescharacteristic
167#3348C6711yesmaximal, characteristic, direct product factor, Sylow
168C67#334811yescharacteristic, direct product factor, Sylow
169#555C2×C6711yescharacteristic
170#552C2×C6711yescharacteristic, complemented
171#501C2×C6711yescharacteristic, complemented
172#485C2×C6711yescharacteristic, complemented
173#485C2×C6711yescharacteristic, complemented
174#482C2×C6711yescharacteristic, complemented
175#481C2×C6711yescharacteristic, complemented
176#478C2×C6711yescharacteristic, complemented
177#475C2×C6711yescharacteristic, complemented
178#468C2×C6711yescharacteristic, complemented
179#465C2×C6711yescharacteristic, complemented
180#440C2×C6711yescharacteristic, complemented
181#435C2×C6711yescharacteristic, complemented
182#428C2×C6711yescharacteristic, complemented
183#427C2×C6711yescharacteristic, complemented
184C2×SD16C22×C6711yescharacteristic
185C2×M16C22×C6711yescharacteristic
186C22×C8C22×C6711yescharacteristic
187C4⋊Q8C22×C6711yescharacteristic
188#126—21no—
189#125—21no—
190#125C22×C6711yescharacteristic
191#124—21no—
192#124—21no—
193#123—21no—
194C22⋊Q8C22×C6711yescharacteristic
195C22⋊Q8C22×C6711yescharacteristic
196#121—21no—
197#121C22×C6711yescharacteristic
198#121C22×C6711yescharacteristic
199C4×Q8C22×C6711yescharacteristic
200C4×D4C22×C6711yescharacteristic
201C4×D4C22×C6711yescharacteristic
202#117—21no—
203#117—21no—
204#117C22×C6711yescharacteristic
205#117C22×C6711yescharacteristic
206#117C22×C6711yescharacteristic
207C2×C4⋊C4C22×C6711yescharacteristic
208#107C22×C6711yescharacteristic
209#107C22×C6711yescharacteristic
210#107C22×C6711yescharacteristic
211#107C22×C6711yescharacteristic
212#106C22×C6711yescharacteristic
213#106C22×C6711yescharacteristic
214#106C22×C6711yescharacteristic
215#106C22×C6711yescharacteristic
216#106C22×C6711yescharacteristic
217C4⋊C8C22×C6711yescharacteristic
218#103C22×C6711yescharacteristic
219#103C22×C6711yescharacteristic
220#103C22×C6711yescharacteristic
221#102C22×C6711yescharacteristic
222#102C22×C6711yescharacteristic
223#102C22×C6711yescharacteristic
224C22⋊C8C22×C6711yescharacteristic
225C22⋊C8C22×C6711yescharacteristic
226#97C22×C6711yescharacteristic
227C2×Q8—21no—
228C2×Q8C23×C6711yescharacteristic
229C2×D4—21no—
230C2×D4—21no—
231C2×D4D4×C6711yescharacteristic
232C2×D4C23×C6711yescharacteristic
233C22×C4—21no—
234C22×C4—21no—
235C22×C4C23×C6711yescharacteristic
236C22×C4C23×C6711yescharacteristic
237SD16—22no—
238M16—22no—
239C2×C8—22no—
240C2×C8C23×C6711yescharacteristic
241C2×C8C23×C6711yescharacteristic
242C2×C8C23×C6711yescharacteristic
243C2×C8C23×C6711yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4D4×C6711yescharacteristic
252C4⋊C4C23×C6711yescharacteristic
253C4⋊C4C23×C6711yescharacteristic
254C4⋊C4C23×C6711yescharacteristic
255C4⋊C4C23×C6711yescharacteristic
256C4⋊C4C23×C6711yescharacteristic
257C4⋊C4C23×C6711yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4D4×C6711yescharacteristic
267C22⋊C4D4×C6711yescharacteristic
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42C23×C6711yescharacteristic
273C23—21no—
274C23C2×D4×C6711yescharacteristic
275C23C2×D4×C6711yescharacteristic
276Q8—21no—
277Q8—21no—
278Q8—81no—
279D4—21no—
280D4—21no—
281D4—41no—
282D4—81no—
283D4—81no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4C2×D4×C6711yescharacteristic
298C2×C4C2×D4×C6711yescharacteristic
299C2×C4C2×D4×C6711yescharacteristic
300C2×C4C2×D4×C6711yescharacteristic
301C2×C4C24×C6711yescharacteristic
302C8—21no—
303C8—21no—
304C8—21no—
305C8#41×C6712yes—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22C22×D4×C6711yescharacteristic
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4C2×#41×C6711yescharacteristic
328C42−1+4×C6711yescharacteristic
329C2—41no—
330C2—41no—
331C2—81no—
332C2#547×C6711yescharacteristic
333C2#575×C6711yescharacteristic
334C2#577×C6711yescharacteristic
subgroupG/H
center, socleC22×C67C22×D4
commutator, Frattini, cosocleC2×C4C24×C67
prime radical#555×C67C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C4×D4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. #552×C67C2semidirect
3. #501×C67C2semidirect
4. #485×C67C2semidirect
5. #485×C67C2semidirect
6. #482×C67C2semidirect
7. #481×C67C2semidirect
8. #478×C67C2semidirect
9. #475×C67C2semidirect
10. #468×C67C2semidirect
11. #465×C67C2semidirect
12. #440×C67C2semidirect
13. #435×C67C2semidirect
14. #428×C67C2semidirect
15. #427×C67C2semidirect
167. #3348C67direct (N × H)
168. C67#3348direct (N × H)
170. #552C2×C67semidirect
171. #501C2×C67semidirect
172. #485C2×C67semidirect
173. #485C2×C67semidirect
174. #482C2×C67semidirect
175. #481C2×C67semidirect
176. #478C2×C67semidirect
177. #475C2×C67semidirect
178. #468C2×C67semidirect
179. #465C2×C67semidirect
180. #440C2×C67semidirect
181. #435C2×C67semidirect
182. #428C2×C67semidirect
183. #427C2×C67semidirect

Maximal subgroups

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

subgroupindex
#555×C672
#552×C672
#501×C672
#485×C672
#482×C672
#481×C672
#478×C672
#475×C672
#468×C672
#465×C672
#440×C672
#435×C672
#428×C672
#427×C672
#334867