Finite Group Databasebeta

← C67×#3308 #22557739

Subgroups — C67×#3308 #22557739

Order 8576.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#548×C67C211yesmaximal, characteristic, complemented
2#547×C67C211yesmaximal, characteristic, complemented
3#489×C67C211yesmaximal, characteristic, complemented
4#484×C67C211yesmaximal, characteristic, complemented
5#480×C67C211yesmaximal, characteristic, complemented
6#477×C67C211yesmaximal, characteristic, complemented
7#468×C67C211yesmaximal, characteristic, complemented
8#466×C67C211yesmaximal, characteristic, complemented
9#463×C67C211yesmaximal, characteristic, complemented
10#449×C67C211yesmaximal, characteristic, complemented
11#448×C67C211yesmaximal, characteristic, complemented
12#440×C67C211yesmaximal, characteristic, complemented
13#435×C67C211yesmaximal, characteristic, complemented
14#417×C67C211yesmaximal, characteristic, complemented
15C2×#103×C67C211yesmaximal, characteristic, complemented
16C2×#41×C67—21no—
17C2×#41×C67C2211yescharacteristic
18C22×Q8×C67C2211yescharacteristic
19C2×SD16×C67C2211yescharacteristic
20C2×SD16×C67C2211yescharacteristic
21C2×D8×C67C2211yescharacteristic
22C2×M16×C67C2211yescharacteristic
23C22×C8×C67C2211yescharacteristic
24C4⋊Q8×C67C2211yescharacteristic
25#124×C67—21no—
26#124×C67C2211yescharacteristic
27#123×C67—21no—
28#123×C67—21no—
29C22⋊Q8×C67—21no—
30C22⋊Q8×C67—21no—
31C22⋊Q8×C67C2211yescharacteristic
32#121×C67—21no—
33#121×C67C2211yescharacteristic
34#121×C67C2211yescharacteristic
35#121×C67C2211yescharacteristic
36C4×Q8×C67C2211yescharacteristic
37C4×D4×C67—21no—
38C4×D4×C67C2211yescharacteristic
39C4×D4×C67C2211yescharacteristic
40C2×C4⋊C4×C67C2211yescharacteristic
41C2×C4⋊C4×C67C2211yescharacteristic
42#107×C67C2211yescharacteristic
43#107×C67C2211yescharacteristic
44#107×C67C2211yescharacteristic
45C4⋊C8×C67C2211yescharacteristic
46#103×C67C2211yescharacteristic
47#103×C67C2211yescharacteristic
48#103×C67C2211yescharacteristic
49#103×C67C2211yescharacteristic
50#103×C67C2211yescharacteristic
51#103×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—22no—
60#41×C67—41no—
61#41×C67—44no—
62C2×Q8×C67—21no—
63C2×Q8×C67—21no—
64C2×Q8×C67—22no—
65C2×Q8×C67—41no—
66C2×Q8×C67C2311yescharacteristic
67C2×Q8×C67C2311yescharacteristic
68C2×D4×C67—21no—
69C2×D4×C67—21no—
70C2×D4×C67—21no—
71C2×D4×C67—21no—
72C2×D4×C67D411yescharacteristic
73C2×D4×C67C2311yescharacteristic
74C2×D4×C67C2311yescharacteristic
75C22×C4×C67—21no—
76C22×C4×C67—21no—
77C22×C4×C67—21no—
78C22×C4×C67—21no—
79C22×C4×C67—21no—
80C22×C4×C67C2311yescharacteristic
81C22×C4×C67C2311yescharacteristic
82SD16×C67—22no—
83SD16×C67—41no—
84D8×C67—41no—
85M16×C67—41no—
86C2×C8×C67—41no—
87C2×C8×C67C2311yescharacteristic
88C2×C8×C67C2311yescharacteristic
89C2×C8×C67C2311yescharacteristic
90C2×C8×C67C2311yescharacteristic
91C4⋊C4×C67—21no—
92C4⋊C4×C67—21no—
93C4⋊C4×C67—21no—
94C4⋊C4×C67—21no—
95C4⋊C4×C67—21no—
96C4⋊C4×C67D411yescharacteristic
97C4⋊C4×C67C2311yescharacteristic
98C4⋊C4×C67C2311yescharacteristic
99C4⋊C4×C67C2311yescharacteristic
100C4⋊C4×C67C2311yescharacteristic
101C22⋊C4×C67—21no—
102C22⋊C4×C67—21no—
103C22⋊C4×C67—21no—
104C22⋊C4×C67—21no—
105C22⋊C4×C67—21no—
106C22⋊C4×C67—21no—
107C22⋊C4×C67—21no—
108C22⋊C4×C67D411yescharacteristic
109C22⋊C4×C67D411yescharacteristic
110C42×C67—21no—
111C42×C67C2311yescharacteristic
112C23×C67—21no—
113C23×C67—21no—
114C23×C67C2×D411yescharacteristic
115C23×C67C2×D411yescharacteristic
116Q8×C67—21no—
117Q8×C67—21no—
118Q8×C67—21no—
119Q8×C67—42no—
120Q8×C67—42no—
121Q8×C67#4112yescomplemented
122D4×C67—22no—
123D4×C67—21no—
124D4×C67—21no—
125D4×C67—21no—
126D4×C67—21no—
127D4×C67—42no—
128D4×C67—42no—
129D4×C67—42no—
130D4×C67—42no—
131C2×C4×C67—21no—
132C2×C4×C67—21no—
133C2×C4×C67—21no—
134C2×C4×C67—21no—
135C2×C4×C67—21no—
136C2×C4×C67—21no—
137C2×C4×C67—21no—
138C2×C4×C67—21no—
139C2×C4×C67—21no—
140C2×C4×C67—21no—
141C2×C4×C67—42no—
142C2×C4×C67—42no—
143C2×C4×C67—42no—
144C2×C4×C67—42no—
145C2×C4×C67—42no—
146C2×C4×C67C2×D411yescharacteristic
147C2×C4×C67C2×D411yescharacteristic
148C2×C4×C67C2×D411yescharacteristic
149C2×C4×C67C2×D411yescharacteristic
150C2×C4×C67C2411yescharacteristic
151C8×C67—21no—
152C8×C67—21no—
153C8×C67—21no—
154C8×C67—21no—
155C22×C67—21no—
156C22×C67—21no—
157C22×C67—21no—
158C22×C67—21no—
159C22×C67—21no—
160C22×C67—41no—
161C22×C67—41no—
162C22×C67—41no—
163C22×C67—41no—
164C22×C67—41no—
165C22×C67—41no—
166C22×C67AGL(1,ℤ/8)12yes—
167C22×C67C22×D411yescharacteristic
168C4×C67—21no—
169C4×C67—21no—
170C4×C67—22no—
171C4×C67—21no—
172C4×C67—21no—
173C4×C67—41no—
174C4×C67—41no—
175C4×C67—41no—
176C4×C67—41no—
177C4×C67—41no—
178C4×C67C2×#4111yescharacteristic
179C4×C67H(5,𝔽2)11yescharacteristic
180C2×C67—22no—
181C2×C67—41no—
182C2×C67—81no—
183C2×C67—81no—
184C2×C67#54611yescharacteristic
185C2×C67C2×AGL(1,ℤ/8)11yescharacteristic
186C2×C67#57811yescharacteristic
187#3308C6711yesmaximal, characteristic, direct product factor, Sylow
188C67#330811yescharacteristic, direct product factor, Sylow
189#548C2×C6711yescharacteristic, complemented
190#547C2×C6711yescharacteristic, complemented
191#489C2×C6711yescharacteristic, complemented
192#484C2×C6711yescharacteristic, complemented
193#480C2×C6711yescharacteristic, complemented
194#477C2×C6711yescharacteristic, complemented
195#468C2×C6711yescharacteristic, complemented
196#466C2×C6711yescharacteristic, complemented
197#463C2×C6711yescharacteristic, complemented
198#449C2×C6711yescharacteristic, complemented
199#448C2×C6711yescharacteristic, complemented
200#440C2×C6711yescharacteristic, complemented
201#435C2×C6711yescharacteristic, complemented
202#417C2×C6711yescharacteristic, complemented
203C2×#103C2×C6711yescharacteristic, complemented
204C2×#41—21no—
205C2×#41C22×C6711yescharacteristic
206C22×Q8C22×C6711yescharacteristic
207C2×SD16C22×C6711yescharacteristic
208C2×SD16C22×C6711yescharacteristic
209C2×D8C22×C6711yescharacteristic
210C2×M16C22×C6711yescharacteristic
211C22×C8C22×C6711yescharacteristic
212C4⋊Q8C22×C6711yescharacteristic
213#124—21no—
214#124C22×C6711yescharacteristic
215#123—21no—
216#123—21no—
217C22⋊Q8—21no—
218C22⋊Q8—21no—
219C22⋊Q8C22×C6711yescharacteristic
220#121—21no—
221#121C22×C6711yescharacteristic
222#121C22×C6711yescharacteristic
223#121C22×C6711yescharacteristic
224C4×Q8C22×C6711yescharacteristic
225C4×D4—21no—
226C4×D4C22×C6711yescharacteristic
227C4×D4C22×C6711yescharacteristic
228C2×C4⋊C4C22×C6711yescharacteristic
229C2×C4⋊C4C22×C6711yescharacteristic
230#107C22×C6711yescharacteristic
231#107C22×C6711yescharacteristic
232#107C22×C6711yescharacteristic
233C4⋊C8C22×C6711yescharacteristic
234#103C22×C6711yescharacteristic
235#103C22×C6711yescharacteristic
236#103C22×C6711yescharacteristic
237#103C22×C6711yescharacteristic
238#103C22×C6711yescharacteristic
239#103C22×C6711yescharacteristic
240#102C22×C6711yescharacteristic
241#102C22×C6711yescharacteristic
242#102C22×C6711yescharacteristic
243#102C22×C6711yescharacteristic
244C22⋊C8C22×C6711yescharacteristic
245C22⋊C8C22×C6711yescharacteristic
246#97C22×C6711yescharacteristic
247#41—22no—
248#41—41no—
249#41—44no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—22no—
253C2×Q8—41no—
254C2×Q8C23×C6711yescharacteristic
255C2×Q8C23×C6711yescharacteristic
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4D4×C6711yescharacteristic
261C2×D4C23×C6711yescharacteristic
262C2×D4C23×C6711yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4C23×C6711yescharacteristic
269C22×C4C23×C6711yescharacteristic
270SD16—22no—
271SD16—41no—
272D8—41no—
273M16—41no—
274C2×C8—41no—
275C2×C8C23×C6711yescharacteristic
276C2×C8C23×C6711yescharacteristic
277C2×C8C23×C6711yescharacteristic
278C2×C8C23×C6711yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4D4×C6711yescharacteristic
285C4⋊C4C23×C6711yescharacteristic
286C4⋊C4C23×C6711yescharacteristic
287C4⋊C4C23×C6711yescharacteristic
288C4⋊C4C23×C6711yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—21no—
296C22⋊C4D4×C6711yescharacteristic
297C22⋊C4D4×C6711yescharacteristic
298C42—21no—
299C42C23×C6711yescharacteristic
300C23—21no—
301C23—21no—
302C23C2×D4×C6711yescharacteristic
303C23C2×D4×C6711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—42no—
308Q8—42no—
309Q8#41×C6712yescomplemented
310D4—22no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—21no—
315D4—42no—
316D4—42no—
317D4—42no—
318D4—42no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4C2×D4×C6711yescharacteristic
335C2×C4C2×D4×C6711yescharacteristic
336C2×C4C2×D4×C6711yescharacteristic
337C2×C4C2×D4×C6711yescharacteristic
338C2×C4C24×C6711yescharacteristic
339C8—21no—
340C8—21no—
341C8—21no—
342C8—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22AGL(1,ℤ/8)×C6712yes—
355C22C22×D4×C6711yescharacteristic
356C4—21no—
357C4—21no—
358C4—22no—
359C4—21no—
360C4—21no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4C2×#41×C6711yescharacteristic
367C4H(5,𝔽2)×C6711yescharacteristic
368C2—22no—
369C2—41no—
370C2—81no—
371C2—81no—
372C2#546×C6711yescharacteristic
373C2C2×AGL(1,ℤ/8)×C6711yescharacteristic
374C2#578×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. #548×C67C2semidirect
2. #547×C67C2semidirect
3. #489×C67C2semidirect
4. #484×C67C2semidirect
5. #480×C67C2semidirect
6. #477×C67C2semidirect
7. #468×C67C2semidirect
8. #466×C67C2semidirect
9. #463×C67C2semidirect
10. #449×C67C2semidirect
11. #448×C67C2semidirect
12. #440×C67C2semidirect
13. #435×C67C2semidirect
14. #417×C67C2semidirect
15. C2×#103×C67C2semidirect
121. Q8×C67#41semidirect
187. #3308C67direct (N × H)
188. C67#3308direct (N × H)
189. #548C2×C67semidirect
190. #547C2×C67semidirect
191. #489C2×C67semidirect
192. #484C2×C67semidirect
193. #480C2×C67semidirect
194. #477C2×C67semidirect
195. #468C2×C67semidirect
196. #466C2×C67semidirect
197. #463C2×C67semidirect
198. #449C2×C67semidirect
199. #448C2×C67semidirect
200. #440C2×C67semidirect
201. #435C2×C67semidirect
202. #417C2×C67semidirect
203. C2×#103C2×C67semidirect
309. Q8#41×C67semidirect

Maximal subgroups

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

subgroupindex
#548×C672
#547×C672
#489×C672
#484×C672
#480×C672
#477×C672
#468×C672
#466×C672
#463×C672
#449×C672
#448×C672
#440×C672
#435×C672
#417×C672
C2×#103×C672
#330867