Finite Group Databasebeta

← #22570886

Subgroups — #22570886

Order 8576.

374 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16864001C211yesmaximal, characteristic, complemented
2#16863802C211yesmaximal, characteristic, complemented
3#16863786C211yesmaximal, characteristic, complemented
4#16863750C211yesmaximal, characteristic, complemented
5#16863744C211yesmaximal, characteristic, complemented
6#16863647C211yesmaximal, characteristic, complemented
7#16863644C211yesmaximal, characteristic, complemented
8#16863630C211yesmaximal, characteristic, complemented
9#16863627C211yesmaximal, characteristic, complemented
10#16863506C211yesmaximal, characteristic, complemented
11#16863494C211yesmaximal, characteristic, complemented
12C4×#11761615C211yesmaximal, characteristic, complemented
13C4×#11761613C211yesmaximal, characteristic, complemented
14#16863352C211yesmaximal, characteristic, complemented
15#542×C67C211yesmaximal, characteristic, complemented
16C2×#11761615C2211yescharacteristic
17C2×#11761615C2211yescharacteristic
18C2×#11761613C2211yescharacteristic
19C2×#11761613C2211yescharacteristic
20C2×#11761612C2211yescharacteristic
21C22×C67⋊C8C2211yescharacteristic
22#14524040C2211yescharacteristic
23#14524035C2211yescharacteristic
24#14524028—21no—
25#14524028—21no—
26#14524017C2211yescharacteristic
27#14524017C2211yescharacteristic
28#14524016—21no—
29#14524016—21no—
30C4×D268C2211yescharacteristic
31C4×D268C2211yescharacteristic
32#14523976C2211yescharacteristic
33#14523976C2211yescharacteristic
34#14523974C2211yescharacteristic
35#14523974C2211yescharacteristic
36#14523967C2211yescharacteristic
37#14523967C2211yescharacteristic
38#14523964C2211yescharacteristic
39#14523964C2211yescharacteristic
40#14523963C2211yescharacteristic
41#14523963C2211yescharacteristic
42#14523963C2211yescharacteristic
43#14523963C2211yescharacteristic
44#14523954C2211yescharacteristic
45#14523954C2211yescharacteristic
46C4×C67⋊C8C2211yescharacteristic
47C4×C67⋊C8C2211yescharacteristic
48C4⋊Q8×C67C2211yescharacteristic
49#126×C67—21no—
50#124×C67—21no—
51#124×C67C2211yescharacteristic
52#123×C67—21no—
53C22⋊Q8×C67—21no—
54C22⋊Q8×C67C2211yescharacteristic
55#121×C67C2211yescharacteristic
56C4×Q8×C67C2211yescharacteristic
57C4×D4×C67C2211yescharacteristic
58#117×C67C2211yescharacteristic
59C2×D268—21no—
60C2×D268—21no—
61C2×D268C2311yescharacteristic
62C2×D268C2311yescharacteristic
63C2×C22⋊D67—21no—
64C2×C22⋊D67—21no—
65C2×C4×D67—21no—
66C2×C4×D67—21no—
67#11761615—22no—
68#11761615—22no—
69#11761613—22no—
70#11761613—22no—
71#11761612—22no—
72C2×C67⋊C8—22no—
73C2×C67⋊C8C2311yescharacteristic
74C2×C67⋊C8C2311yescharacteristic
75C2×C67⋊C8C2311yescharacteristic
76C2×C67⋊C8C2311yescharacteristic
77C4⋊Dic67C2311yescharacteristic
78C4⋊Dic67C2311yescharacteristic
79Dic67⋊C4—21no—
80Dic67⋊C4—21no—
81(C2×D67)⋊C4—21no—
82(C2×D67)⋊C4—21no—
83(C2×D67)⋊C4—21no—
84(C2×D67)⋊C4—21no—
85C2×Q8×C67—21no—
86C2×Q8×C67C2311yescharacteristic
87C2×D4×C67—21no—
88C2×D4×C67C2311yescharacteristic
89C22×C4×C67—21no—
90C22×C4×C67C2311yescharacteristic
91C4⋊C4×C67—21no—
92C4⋊C4×C67—21no—
93C4⋊C4×C67—21no—
94C4⋊C4×C67D411yescharacteristic
95C4⋊C4×C67D411yescharacteristic
96C4⋊C4×C67C2311yescharacteristic
97C4⋊C4×C67C2311yescharacteristic
98C22⋊C4×C67—21no—
99C22⋊C4×C67—21no—
100C22⋊C4×C67—21no—
101C22⋊C4×C67—21no—
102C22⋊C4×C67—21no—
103C22⋊C4×C67D411yescharacteristic
104C22⋊C4×C67D411yescharacteristic
105C42×C67—21no—
106C42×C67C2311yescharacteristic
107C42×C67C2311yescharacteristic
108C23×C67—21no—
109C23×C67C2×D411yescharacteristic
110C22×D67—21no—
111C22×D67—21no—
112Q8×C67—21no—
113Q8×C67—21no—
114Q8×C67—81no—
115D4×C67—21no—
116D4×C67—21no—
117D4×C67—81no—
118C2×C4×C67—21no—
119C2×C4×C67—21no—
120C2×C4×C67—21no—
121C2×C4×C67—21no—
122C2×C4×C67—21no—
123C2×C4×C67—81no—
124C2×C4×C67C2×D411yescharacteristic
125C2×C4×C67C2×D411yescharacteristic
126C2×C4×C67C2×D411yescharacteristic
127C2×C4×C67C2×D411yescharacteristic
128C2×C4×C67C2×D411yescharacteristic
129C2×C4×C67C2411yescharacteristic
130C22⋊D67—81no—
131C22⋊D67—81no—
132C2×Dic67—21no—
133C2×Dic67—21no—
134D268—21no—
135D268—21no—
136D268—21no—
137D268—21no—
138D268—81no—
139D268—81no—
140C4×D67—81no—
141C4×D67—81no—
142C67⋊C8—21no—
143C67⋊C8—21no—
144C67⋊C8#4112yes—
145C67⋊C8#4112yes—
146C22×C67—21no—
147C22×C67—21no—
148C22×C67—21no—
149C22×C67—41no—
150C22×C67—41no—
151C22×C67—41no—
152C22×C67C22×D411yescharacteristic
153C2×D67—41no—
154C2×D67—41no—
155C2×D67—41no—
156C2×D67—41no—
157C2×D67—41no—
158C2×D67—41no—
159C4×C67—21no—
160C4×C67—21no—
161C4×C67—21no—
162C4×C67—21no—
163C4×C67—21no—
164C4×C67—41no—
165C4×C67—41no—
166C4×C67—41no—
167C4×C67C2×#4111yescharacteristic
168C4×C67C2×#4111yescharacteristic
169Dic67—41no—
170Dic67—41no—
171C2×C67—41no—
172C2×C67—81no—
173C2×C67#53211yescharacteristic
174C2×C67#57611yescharacteristic
175C2×C67#57711yescharacteristic
176D67—81no—
177D67—81no—
178#3179—671nomaximal, Sylow
179C67#317911yescharacteristic, complemented, Sylow
180#542D6711yescharacteristic, complemented
181#540—671no—
182#501—671no—
183#496—671no—
184#488—671no—
185#486—671no—
186#469—671no—
187#468—671no—
188#466—671no—
189#465—671no—
190#442—671no—
191#440—671no—
192C4×SD16—671no—
193C4×D8—671no—
194#405—671no—
195C2×SD16—671no—
196C2×SD16—671no—
197C2×D8—671no—
198C2×D8—671no—
199C2×M16—671no—
200C22×C8—671no—
201C4⋊Q8C2×D6711yescharacteristic
202#127—671no—
203#126—21no—
204#125—671no—
205#124—21no—
206#124C2×D6711yescharacteristic
207#123—21no—
208#123—1341no—
209#123—1341no—
210C22⋊Q8—21no—
211C22⋊Q8C2×D6711yescharacteristic
212#121—671no—
213#121—671no—
214#121—1341no—
215#121—1341no—
216#121C2×D6711yescharacteristic
217C4×Q8C2×D6711yescharacteristic
218C4×D4—671no—
219C4×D4—671no—
220C4×D4C2×D6711yescharacteristic
221#117C2×D6711yescharacteristic
222#107—671no—
223#107—671no—
224#106—671no—
225#106—671no—
226#103—671no—
227#103—671no—
228#102—671no—
229#102—671no—
230#102—671no—
231#102—671no—
232#102—671no—
233#102—671no—
234#97—671no—
235#97—671no—
236C4×C8—671no—
237C4×C8—671no—
238C2×Q8—21no—
239C2×Q8C22×D6711yescharacteristic
240C2×D4—21no—
241C2×D4—671no—
242C2×D4—671no—
243C2×D4—1341no—
244C2×D4—1341no—
245C2×D4—1341no—
246C2×D4—1341no—
247C2×D4C22×D6711yescharacteristic
248C22×C4—21no—
249C22×C4—1341no—
250C22×C4—1341no—
251C22×C4C22×D6711yescharacteristic
252SD16—1342no—
253SD16—1342no—
254D8—1342no—
255D8—1342no—
256M16—1342no—
257C2×C8—671no—
258C2×C8—671no—
259C2×C8—671no—
260C2×C8—671no—
261C2×C8—1342no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—671no—
266C4⋊C4—671no—
267C4⋊C4—1341no—
268C4⋊C4—1341no—
269C4⋊C4C22⋊D6711yescharacteristic
270C4⋊C4C22⋊D6711yescharacteristic
271C4⋊C4C22×D6711yescharacteristic
272C4⋊C4C22×D6711yescharacteristic
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—1341no—
279C22⋊C4—1341no—
280C22⋊C4—1341no—
281C22⋊C4—1341no—
282C22⋊C4C22⋊D6711yescharacteristic
283C22⋊C4C22⋊D6711yescharacteristic
284C42—21no—
285C42C22×D6711yescharacteristic
286C42C22×D6711yescharacteristic
287C23—21no—
288C23—1341no—
289C23—1341no—
290C23C2×C22⋊D6711yescharacteristic
291Q8—21no—
292Q8—21no—
293Q8—81no—
294D4—21no—
295D4—21no—
296D4—81no—
297D4—5361no—
298D4—5361no—
299D4—5361no—
300D4—5361no—
301D4—1341no—
302D4—1341no—
303D4—1341no—
304D4—1341no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—81no—
311C2×C4—5361no—
312C2×C4—5361no—
313C2×C4—1341no—
314C2×C4—1341no—
315C2×C4C2×C22⋊D6711yescharacteristic
316C2×C4C2×C22⋊D6711yescharacteristic
317C2×C4C2×C22⋊D6711yescharacteristic
318C2×C4C2×C22⋊D6711yescharacteristic
319C2×C4C2×C22⋊D6711yescharacteristic
320C2×C4C23×D6711yescharacteristic
321C8—672no—
322C8—672no—
323C8—1341no—
324C8—1341no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—2681no—
332C22—2681no—
333C22—2681no—
334C22—2681no—
335C22—2681no—
336C22—2681no—
337C22C22×C22⋊D6711yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—2681no—
347C4—2681no—
348C4#41×D6711yescharacteristic
349C4#41×D6711yescharacteristic
350C2—41no—
351C2—81no—
352C2—5361no—
353C2—5361no—
354C2#1686394211yescharacteristic
355C2#1686419911yescharacteristic
356C2#1686420411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#542×C67C2
socleC22×C67C22×D4
hypercenter#542D67
hypocenterC67#3179
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #16864001C2semidirect
2. #16863802C2semidirect
3. #16863786C2semidirect
4. #16863750C2semidirect
5. #16863744C2semidirect
6. #16863647C2semidirect
7. #16863644C2semidirect
8. #16863630C2semidirect
9. #16863627C2semidirect
10. #16863506C2semidirect
11. #16863494C2semidirect
12. C4×#11761615C2semidirect
13. C4×#11761613C2semidirect
14. #16863352C2semidirect
15. #542×C67C2semidirect
179. C67#3179semidirect
180. #542D67semidirect