Finite Group Databasebeta

← C67×#3345 #22557776

Subgroups — C67×#3345 #22557776

Order 8576.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#553×C67C211yesmaximal, characteristic, complemented
2#547×C67C211yesmaximal, characteristic, complemented
3#501×C67C211yesmaximal, characteristic, complemented
4#482×C67C211yesmaximal, characteristic, complemented
5#481×C67C211yesmaximal, characteristic, complemented
6#481×C67C211yesmaximal, characteristic, complemented
7#480×C67C211yesmaximal, characteristic, complemented
8#478×C67C211yesmaximal, characteristic, complemented
9#476×C67C211yesmaximal, characteristic, complemented
10#469×C67C211yesmaximal, characteristic, complemented
11#466×C67C211yesmaximal, characteristic, complemented
12#442×C67C211yesmaximal, characteristic, complemented
13#435×C67C211yesmaximal, characteristic, complemented
14#428×C67C211yesmaximal, characteristic, complemented
15C2×#106×C67C211yesmaximal, characteristic, complemented
16C2×#41×C67—21no—
17C2×D8×C67C2211yescharacteristic
18C2×M16×C67C2211yescharacteristic
19C22×C8×C67C2211yescharacteristic
20C4⋊Q8×C67C2211yescharacteristic
21#126×C67—21no—
22#125×C67C2211yescharacteristic
23#123×C67—21no—
24#123×C67—21no—
25#123×C67—21no—
26C22⋊Q8×C67—21no—
27#121×C67—21no—
28#121×C67C2211yescharacteristic
29#121×C67C2211yescharacteristic
30#121×C67C2211yescharacteristic
31#121×C67C2211yescharacteristic
32C4×D4×C67—21no—
33C4×D4×C67C2211yescharacteristic
34C4×D4×C67C2211yescharacteristic
35C4×D4×C67C2211yescharacteristic
36#117×C67C2211yescharacteristic
37C2×C4⋊C4×C67C2211yescharacteristic
38C2×C4⋊C4×C67C2211yescharacteristic
39C2×C4⋊C4×C67C2211yescharacteristic
40#107×C67C2211yescharacteristic
41#107×C67C2211yescharacteristic
42#107×C67C2211yescharacteristic
43#106×C67C2211yescharacteristic
44#106×C67C2211yescharacteristic
45#106×C67C2211yescharacteristic
46#106×C67C2211yescharacteristic
47#106×C67C2211yescharacteristic
48#106×C67C2211yescharacteristic
49C4⋊C8×C67C2211yescharacteristic
50#102×C67C2211yescharacteristic
51#102×C67C2211yescharacteristic
52#102×C67C2211yescharacteristic
53#102×C67C2211yescharacteristic
54#102×C67C2211yescharacteristic
55#102×C67C2211yescharacteristic
56C22⋊C8×C67C2211yescharacteristic
57C22⋊C8×C67C2211yescharacteristic
58#97×C67C2211yescharacteristic
59#41×C67—44no—
60C2×Q8×C67—21no—
61C2×D4×C67—21no—
62C2×D4×C67—21no—
63C2×D4×C67—21no—
64C2×D4×C67—21no—
65C2×D4×C67D411yescharacteristic
66C2×D4×C67C2311yescharacteristic
67C2×D4×C67C2311yescharacteristic
68C22×C4×C67—21no—
69C22×C4×C67—21no—
70C22×C4×C67—21no—
71C22×C4×C67—21no—
72C22×C4×C67—21no—
73C22×C4×C67C2311yescharacteristic
74C22×C4×C67C2311yescharacteristic
75D8×C67—22no—
76M16×C67—22no—
77C2×C8×C67—22no—
78C2×C8×C67C2311yescharacteristic
79C2×C8×C67C2311yescharacteristic
80C2×C8×C67C2311yescharacteristic
81C2×C8×C67C2311yescharacteristic
82C4⋊C4×C67—21no—
83C4⋊C4×C67—21no—
84C4⋊C4×C67—21no—
85C4⋊C4×C67—21no—
86C4⋊C4×C67—21no—
87C4⋊C4×C67—21no—
88C4⋊C4×C67D411yescharacteristic
89C4⋊C4×C67C2311yescharacteristic
90C4⋊C4×C67C2311yescharacteristic
91C4⋊C4×C67C2311yescharacteristic
92C4⋊C4×C67C2311yescharacteristic
93C4⋊C4×C67C2311yescharacteristic
94C4⋊C4×C67C2311yescharacteristic
95C22⋊C4×C67—21no—
96C22⋊C4×C67—21no—
97C22⋊C4×C67—21no—
98C22⋊C4×C67—21no—
99C22⋊C4×C67—21no—
100C22⋊C4×C67—21no—
101C22⋊C4×C67—21no—
102C22⋊C4×C67D411yescharacteristic
103C22⋊C4×C67D411yescharacteristic
104C42×C67—21no—
105C42×C67C2311yescharacteristic
106C23×C67—21no—
107C23×C67—21no—
108C23×C67C2×D411yescharacteristic
109C23×C67C2×D411yescharacteristic
110Q8×C67—42no—
111D4×C67—22no—
112D4×C67—21no—
113D4×C67—21no—
114D4×C67—21no—
115D4×C67—21no—
116D4×C67—42no—
117D4×C67—42no—
118D4×C67—42no—
119D4×C67—81no—
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—21no—
128C2×C4×C67—21no—
129C2×C4×C67—21no—
130C2×C4×C67—42no—
131C2×C4×C67—42no—
132C2×C4×C67—42no—
133C2×C4×C67—81no—
134C2×C4×C67—81no—
135C2×C4×C67C2×D411yescharacteristic
136C2×C4×C67C2×D411yescharacteristic
137C2×C4×C67C2×D411yescharacteristic
138C2×C4×C67C2×D411yescharacteristic
139C2×C4×C67C2411yescharacteristic
140C8×C67—21no—
141C8×C67—21no—
142C8×C67—21no—
143C8×C67#4112yescomplemented
144C22×C67—21no—
145C22×C67—21no—
146C22×C67—21no—
147C22×C67—21no—
148C22×C67—21no—
149C22×C67—41no—
150C22×C67—41no—
151C22×C67—41no—
152C22×C67—41no—
153C22×C67—41no—
154C22×C67—41no—
155C22×C67AGL(1,ℤ/8)12yes—
156C22×C67C22×D411yescharacteristic
157C4×C67—21no—
158C4×C67—21no—
159C4×C67—21no—
160C4×C67—21no—
161C4×C67—41no—
162C4×C67—41no—
163C4×C67—41no—
164C4×C67—41no—
165C4×C67—41no—
166C4×C67—41no—
167C4×C67C2×#4111yescharacteristic
168C4×C672−1+411yescharacteristic
169C2×C67—22no—
170C2×C67—41no—
171C2×C67—81no—
172C2×C67—81no—
173C2×C67#54711yescharacteristic
174C2×C67C2×AGL(1,ℤ/8)11yescharacteristic
175C2×C67#57711yescharacteristic
176#3345C6711yesmaximal, characteristic, direct product factor, Sylow
177C67#334511yescharacteristic, direct product factor, Sylow
178#553C2×C6711yescharacteristic, complemented
179#547C2×C6711yescharacteristic, complemented
180#501C2×C6711yescharacteristic, complemented
181#482C2×C6711yescharacteristic, complemented
182#481C2×C6711yescharacteristic, complemented
183#481C2×C6711yescharacteristic, complemented
184#480C2×C6711yescharacteristic, complemented
185#478C2×C6711yescharacteristic, complemented
186#476C2×C6711yescharacteristic, complemented
187#469C2×C6711yescharacteristic, complemented
188#466C2×C6711yescharacteristic, complemented
189#442C2×C6711yescharacteristic, complemented
190#435C2×C6711yescharacteristic, complemented
191#428C2×C6711yescharacteristic, complemented
192C2×#106C2×C6711yescharacteristic, complemented
193C2×#41—21no—
194C2×D8C22×C6711yescharacteristic
195C2×M16C22×C6711yescharacteristic
196C22×C8C22×C6711yescharacteristic
197C4⋊Q8C22×C6711yescharacteristic
198#126—21no—
199#125C22×C6711yescharacteristic
200#123—21no—
201#123—21no—
202#123—21no—
203C22⋊Q8—21no—
204#121—21no—
205#121C22×C6711yescharacteristic
206#121C22×C6711yescharacteristic
207#121C22×C6711yescharacteristic
208#121C22×C6711yescharacteristic
209C4×D4—21no—
210C4×D4C22×C6711yescharacteristic
211C4×D4C22×C6711yescharacteristic
212C4×D4C22×C6711yescharacteristic
213#117C22×C6711yescharacteristic
214C2×C4⋊C4C22×C6711yescharacteristic
215C2×C4⋊C4C22×C6711yescharacteristic
216C2×C4⋊C4C22×C6711yescharacteristic
217#107C22×C6711yescharacteristic
218#107C22×C6711yescharacteristic
219#107C22×C6711yescharacteristic
220#106C22×C6711yescharacteristic
221#106C22×C6711yescharacteristic
222#106C22×C6711yescharacteristic
223#106C22×C6711yescharacteristic
224#106C22×C6711yescharacteristic
225#106C22×C6711yescharacteristic
226C4⋊C8C22×C6711yescharacteristic
227#102C22×C6711yescharacteristic
228#102C22×C6711yescharacteristic
229#102C22×C6711yescharacteristic
230#102C22×C6711yescharacteristic
231#102C22×C6711yescharacteristic
232#102C22×C6711yescharacteristic
233C22⋊C8C22×C6711yescharacteristic
234C22⋊C8C22×C6711yescharacteristic
235#97C22×C6711yescharacteristic
236#41—44no—
237C2×Q8—21no—
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4D4×C6711yescharacteristic
243C2×D4C23×C6711yescharacteristic
244C2×D4C23×C6711yescharacteristic
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4C23×C6711yescharacteristic
251C22×C4C23×C6711yescharacteristic
252D8—22no—
253M16—22no—
254C2×C8—22no—
255C2×C8C23×C6711yescharacteristic
256C2×C8C23×C6711yescharacteristic
257C2×C8C23×C6711yescharacteristic
258C2×C8C23×C6711yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4D4×C6711yescharacteristic
266C4⋊C4C23×C6711yescharacteristic
267C4⋊C4C23×C6711yescharacteristic
268C4⋊C4C23×C6711yescharacteristic
269C4⋊C4C23×C6711yescharacteristic
270C4⋊C4C23×C6711yescharacteristic
271C4⋊C4C23×C6711yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4D4×C6711yescharacteristic
280C22⋊C4D4×C6711yescharacteristic
281C42—21no—
282C42C23×C6711yescharacteristic
283C23—21no—
284C23—21no—
285C23C2×D4×C6711yescharacteristic
286C23C2×D4×C6711yescharacteristic
287Q8—42no—
288D4—22no—
289D4—21no—
290D4—21no—
291D4—21no—
292D4—21no—
293D4—42no—
294D4—42no—
295D4—42no—
296D4—81no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4C2×D4×C6711yescharacteristic
313C2×C4C2×D4×C6711yescharacteristic
314C2×C4C2×D4×C6711yescharacteristic
315C2×C4C2×D4×C6711yescharacteristic
316C2×C4C24×C6711yescharacteristic
317C8—21no—
318C8—21no—
319C8—21no—
320C8#41×C6712yescomplemented
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22AGL(1,ℤ/8)×C6712yes—
333C22C22×D4×C6711yescharacteristic
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4C2×#41×C6711yescharacteristic
345C42−1+4×C6711yescharacteristic
346C2—22no—
347C2—41no—
348C2—81no—
349C2—81no—
350C2#547×C6711yescharacteristic
351C2C2×AGL(1,ℤ/8)×C6711yescharacteristic
352C2#577×C6711yescharacteristic
subgroupG/H
center, socleC22×C67C22×D4
commutator, Frattini, cosocleC2×C4C24×C67
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
1. #553×C67C2semidirect
2. #547×C67C2semidirect
3. #501×C67C2semidirect
4. #482×C67C2semidirect
5. #481×C67C2semidirect
6. #481×C67C2semidirect
7. #480×C67C2semidirect
8. #478×C67C2semidirect
9. #476×C67C2semidirect
10. #469×C67C2semidirect
11. #466×C67C2semidirect
12. #442×C67C2semidirect
13. #435×C67C2semidirect
14. #428×C67C2semidirect
15. C2×#106×C67C2semidirect
143. C8×C67#41semidirect
176. #3345C67direct (N × H)
177. C67#3345direct (N × H)
178. #553C2×C67semidirect
179. #547C2×C67semidirect
180. #501C2×C67semidirect
181. #482C2×C67semidirect
182. #481C2×C67semidirect
183. #481C2×C67semidirect
184. #480C2×C67semidirect
185. #478C2×C67semidirect
186. #476C2×C67semidirect
187. #469C2×C67semidirect
188. #466C2×C67semidirect
189. #442C2×C67semidirect
190. #435C2×C67semidirect
191. #428C2×C67semidirect
192. C2×#106C2×C67semidirect
320. C8#41×C67semidirect

Maximal subgroups

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

subgroupindex
#553×C672
#547×C672
#501×C672
#482×C672
#481×C672
#480×C672
#478×C672
#476×C672
#469×C672
#466×C672
#442×C672
#435×C672
#428×C672
C2×#106×C672
#334567