Finite Group Databasebeta

← C67×#3320 #22557751

Subgroups — C67×#3320 #22557751

Order 8576.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#553×C67C211yesmaximal, characteristic
2D4×Q8×C67C211yesmaximal, characteristic, complemented
3#491×C67C211yesmaximal, characteristic, complemented
4#485×C67C211yesmaximal, characteristic, complemented
5#484×C67C211yesmaximal, characteristic, complemented
6#479×C67C211yesmaximal, characteristic, complemented
7#468×C67C211yesmaximal, characteristic, complemented
8#465×C67C211yesmaximal, characteristic, complemented
9#461×C67C211yesmaximal, characteristic, complemented
10#448×C67C211yesmaximal, characteristic, complemented
11#448×C67C211yesmaximal, characteristic, complemented
12#440×C67C211yesmaximal, characteristic, complemented
13#435×C67C211yesmaximal, characteristic, complemented
14#419×C67C211yesmaximal, characteristic, complemented
15C2×#103×C67C211yesmaximal, characteristic, complemented
16C22×Q8×C67C2211yescharacteristic
17C22×Q8×C67C2211yescharacteristic
18C2×SD16×C67C2211yescharacteristic
19C2×SD16×C67C2211yescharacteristic
20C2×SD16×C67C2211yescharacteristic
21C2×M16×C67C2211yescharacteristic
22C22×C8×C67C2211yescharacteristic
23C4⋊Q8×C67—21no—
24C4⋊Q8×C67C2211yescharacteristic
25#126×C67—21no—
26#125×C67C2211yescharacteristic
27#123×C67—21no—
28C22⋊Q8×C67—21no—
29C22⋊Q8×C67—21no—
30C22⋊Q8×C67C2211yescharacteristic
31C22⋊Q8×C67C2211yescharacteristic
32#121×C67—21no—
33#121×C67C2211yescharacteristic
34#121×C67C2211yescharacteristic
35C4×Q8×C67C2211yescharacteristic
36C4×D4×C67—21no—
37C4×D4×C67—21no—
38C4×D4×C67C2211yescharacteristic
39C4×D4×C67C2211yescharacteristic
40#117×C67C2211yescharacteristic
41C2×C4⋊C4×C67C2211yescharacteristic
42#107×C67C2211yescharacteristic
43#107×C67C2211yescharacteristic
44#106×C67C2211yescharacteristic
45C4⋊C8×C67C2211yescharacteristic
46#103×C67C2211yescharacteristic
47#103×C67C2211yescharacteristic
48#103×C67C2211yescharacteristic
49#103×C67C2211yescharacteristic
50#103×C67C2211yescharacteristic
51#103×C67C2211yescharacteristic
52#103×C67C2211yescharacteristic
53#103×C67C2211yescharacteristic
54#103×C67C2211yescharacteristic
55#102×C67C2211yescharacteristic
56C22⋊C8×C67C2211yescharacteristic
57C22⋊C8×C67C2211yescharacteristic
58#97×C67C2211yescharacteristic
59C2×Q8×C67—21no—
60C2×Q8×C67—21no—
61C2×Q8×C67—22no—
62C2×Q8×C67—22no—
63C2×Q8×C67—41no—
64C2×Q8×C67—41no—
65C2×Q8×C67C2311yescharacteristic
66C2×Q8×C67C2311yescharacteristic
67C2×Q8×C67C2311yescharacteristic
68C2×D4×C67—21no—
69C2×D4×C67—21no—
70C2×D4×C67D411yescharacteristic
71C2×D4×C67C2311yescharacteristic
72C22×C4×C67—21no—
73C22×C4×C67—21no—
74C22×C4×C67—21no—
75C22×C4×C67—21no—
76C22×C4×C67C2311yescharacteristic
77C22×C4×C67C2311yescharacteristic
78SD16×C67—22no—
79SD16×C67—41no—
80SD16×C67—41no—
81M16×C67—41no—
82C2×C8×C67—41no—
83C2×C8×C67C2311yescharacteristic
84C2×C8×C67C2311yescharacteristic
85C2×C8×C67C2311yescharacteristic
86C2×C8×C67C2311yescharacteristic
87C4⋊C4×C67—21no—
88C4⋊C4×C67—21no—
89C4⋊C4×C67—21no—
90C4⋊C4×C67—21no—
91C4⋊C4×C67—21no—
92C4⋊C4×C67—21no—
93C4⋊C4×C67—21no—
94C4⋊C4×C67—21no—
95C4⋊C4×C67D411yescharacteristic
96C4⋊C4×C67C2311yescharacteristic
97C4⋊C4×C67C2311yescharacteristic
98C4⋊C4×C67C2311yescharacteristic
99C4⋊C4×C67C2311yescharacteristic
100C22⋊C4×C67—21no—
101C22⋊C4×C67—21no—
102C22⋊C4×C67—21no—
103C22⋊C4×C67—21no—
104C22⋊C4×C67—21no—
105C22⋊C4×C67—21no—
106C22⋊C4×C67D411yescharacteristic
107C22⋊C4×C67D411yescharacteristic
108C42×C67—21no—
109C42×C67—21no—
110C42×C67C2311yescharacteristic
111C23×C67—21no—
112C23×C67C2×D411yescharacteristic
113C23×C67C2×D411yescharacteristic
114Q8×C67—21no—
115Q8×C67—21no—
116Q8×C67—21no—
117Q8×C67—21no—
118Q8×C67—21no—
119Q8×C67—42no—
120Q8×C67—42no—
121Q8×C67#4112yes—
122D4×C67—22no—
123D4×C67—21no—
124D4×C67—21no—
125D4×C67—42no—
126D4×C67—81no—
127C2×C4×C67—21no—
128C2×C4×C67—21no—
129C2×C4×C67—21no—
130C2×C4×C67—21no—
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—42no—
139C2×C4×C67—42no—
140C2×C4×C67—81no—
141C2×C4×C67—81no—
142C2×C4×C67C2×D411yescharacteristic
143C2×C4×C67C2×D411yescharacteristic
144C2×C4×C67C2×D411yescharacteristic
145C2×C4×C67C2×D411yescharacteristic
146C2×C4×C67C2411yescharacteristic
147C8×C67—21no—
148C8×C67—21no—
149C8×C67—21no—
150C8×C67—21no—
151C22×C67—21no—
152C22×C67—21no—
153C22×C67—21no—
154C22×C67—21no—
155C22×C67—21no—
156C22×C67—41no—
157C22×C67—41no—
158C22×C67—41no—
159C22×C67#13712yes—
160C22×C67C22×D411yescharacteristic
161C4×C67—21no—
162C4×C67—21no—
163C4×C67—22no—
164C4×C67—21no—
165C4×C67—21no—
166C4×C67—41no—
167C4×C67—41no—
168C4×C67—41no—
169C4×C67—41no—
170C4×C67—41no—
171C4×C67—41no—
172C4×C67C2×#4111yescharacteristic
173C4×C67H(5,𝔽2)11yescharacteristic
174C2×C67—22no—
175C2×C67—41no—
176C2×C67—81no—
177C2×C67#54611yescharacteristic
178C2×C67C2×#13711yescharacteristic
179C2×C67#57711yescharacteristic
180#3320C6711yesmaximal, characteristic, direct product factor, Sylow
181C67#332011yescharacteristic, direct product factor, Sylow
182#553C2×C6711yescharacteristic
183D4×Q8C2×C6711yescharacteristic, complemented
184#491C2×C6711yescharacteristic, complemented
185#485C2×C6711yescharacteristic, complemented
186#484C2×C6711yescharacteristic, complemented
187#479C2×C6711yescharacteristic, complemented
188#468C2×C6711yescharacteristic, complemented
189#465C2×C6711yescharacteristic, complemented
190#461C2×C6711yescharacteristic, complemented
191#448C2×C6711yescharacteristic, complemented
192#448C2×C6711yescharacteristic, complemented
193#440C2×C6711yescharacteristic, complemented
194#435C2×C6711yescharacteristic, complemented
195#419C2×C6711yescharacteristic, complemented
196C2×#103C2×C6711yescharacteristic, complemented
197C22×Q8C22×C6711yescharacteristic
198C22×Q8C22×C6711yescharacteristic
199C2×SD16C22×C6711yescharacteristic
200C2×SD16C22×C6711yescharacteristic
201C2×SD16C22×C6711yescharacteristic
202C2×M16C22×C6711yescharacteristic
203C22×C8C22×C6711yescharacteristic
204C4⋊Q8—21no—
205C4⋊Q8C22×C6711yescharacteristic
206#126—21no—
207#125C22×C6711yescharacteristic
208#123—21no—
209C22⋊Q8—21no—
210C22⋊Q8—21no—
211C22⋊Q8C22×C6711yescharacteristic
212C22⋊Q8C22×C6711yescharacteristic
213#121—21no—
214#121C22×C6711yescharacteristic
215#121C22×C6711yescharacteristic
216C4×Q8C22×C6711yescharacteristic
217C4×D4—21no—
218C4×D4—21no—
219C4×D4C22×C6711yescharacteristic
220C4×D4C22×C6711yescharacteristic
221#117C22×C6711yescharacteristic
222C2×C4⋊C4C22×C6711yescharacteristic
223#107C22×C6711yescharacteristic
224#107C22×C6711yescharacteristic
225#106C22×C6711yescharacteristic
226C4⋊C8C22×C6711yescharacteristic
227#103C22×C6711yescharacteristic
228#103C22×C6711yescharacteristic
229#103C22×C6711yescharacteristic
230#103C22×C6711yescharacteristic
231#103C22×C6711yescharacteristic
232#103C22×C6711yescharacteristic
233#103C22×C6711yescharacteristic
234#103C22×C6711yescharacteristic
235#103C22×C6711yescharacteristic
236#102C22×C6711yescharacteristic
237C22⋊C8C22×C6711yescharacteristic
238C22⋊C8C22×C6711yescharacteristic
239#97C22×C6711yescharacteristic
240C2×Q8—21no—
241C2×Q8—21no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8C23×C6711yescharacteristic
247C2×Q8C23×C6711yescharacteristic
248C2×Q8C23×C6711yescharacteristic
249C2×D4—21no—
250C2×D4—21no—
251C2×D4D4×C6711yescharacteristic
252C2×D4C23×C6711yescharacteristic
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4C23×C6711yescharacteristic
258C22×C4C23×C6711yescharacteristic
259SD16—22no—
260SD16—41no—
261SD16—41no—
262M16—41no—
263C2×C8—41no—
264C2×C8C23×C6711yescharacteristic
265C2×C8C23×C6711yescharacteristic
266C2×C8C23×C6711yescharacteristic
267C2×C8C23×C6711yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4D4×C6711yescharacteristic
277C4⋊C4C23×C6711yescharacteristic
278C4⋊C4C23×C6711yescharacteristic
279C4⋊C4C23×C6711yescharacteristic
280C4⋊C4C23×C6711yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4D4×C6711yescharacteristic
288C22⋊C4D4×C6711yescharacteristic
289C42—21no—
290C42—21no—
291C42C23×C6711yescharacteristic
292C23—21no—
293C23C2×D4×C6711yescharacteristic
294C23C2×D4×C6711yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—21no—
299Q8—21no—
300Q8—42no—
301Q8—42no—
302Q8#41×C6712yes—
303D4—22no—
304D4—21no—
305D4—21no—
306D4—42no—
307D4—81no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4C2×D4×C6711yescharacteristic
324C2×C4C2×D4×C6711yescharacteristic
325C2×C4C2×D4×C6711yescharacteristic
326C2×C4C2×D4×C6711yescharacteristic
327C2×C4C24×C6711yescharacteristic
328C8—21no—
329C8—21no—
330C8—21no—
331C8—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22#137×C6712yes—
341C22C22×D4×C6711yescharacteristic
342C4—21no—
343C4—21no—
344C4—22no—
345C4—21no—
346C4—21no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4C2×#41×C6711yescharacteristic
354C4H(5,𝔽2)×C6711yescharacteristic
355C2—22no—
356C2—41no—
357C2—81no—
358C2#546×C6711yescharacteristic
359C2C2×#137×C6711yescharacteristic
360C2#577×C6711yescharacteristic
subgroupG/H
center, socleC22×C67C22×D4
commutator, Frattini, cosocleC2×C4C24×C67
prime radical#553×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. D4×Q8×C67C2semidirect
3. #491×C67C2semidirect
4. #485×C67C2semidirect
5. #484×C67C2semidirect
6. #479×C67C2semidirect
7. #468×C67C2semidirect
8. #465×C67C2semidirect
9. #461×C67C2semidirect
10. #448×C67C2semidirect
11. #448×C67C2semidirect
12. #440×C67C2semidirect
13. #435×C67C2semidirect
14. #419×C67C2semidirect
15. C2×#103×C67C2semidirect
180. #3320C67direct (N × H)
181. C67#3320direct (N × H)
183. D4×Q8C2×C67semidirect
184. #491C2×C67semidirect
185. #485C2×C67semidirect
186. #484C2×C67semidirect
187. #479C2×C67semidirect
188. #468C2×C67semidirect
189. #465C2×C67semidirect
190. #461C2×C67semidirect
191. #448C2×C67semidirect
192. #448C2×C67semidirect
193. #440C2×C67semidirect
194. #435C2×C67semidirect
195. #419C2×C67semidirect
196. C2×#103C2×C67semidirect

Maximal subgroups

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

subgroupindex
#553×C672
D4×Q8×C672
#491×C672
#485×C672
#484×C672
#479×C672
#468×C672
#465×C672
#461×C672
#448×C672
#440×C672
#435×C672
#419×C672
C2×#103×C672
#332067