Finite Group Databasebeta

← #22572688

Subgroups — #22572688

Order 8576.

356 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16864092C211yesmaximal, characteristic
2#16863802C211yesmaximal, characteristic, complemented
3#16863734C211yesmaximal, characteristic, complemented
4#16863729C211yesmaximal, characteristic, complemented
5#16863714C211yesmaximal, characteristic, complemented
6#16863709C211yesmaximal, characteristic, complemented
7#16863703C211yesmaximal, characteristic, complemented
8#16863673C211yesmaximal, characteristic, complemented
9#16863644C211yesmaximal, characteristic, complemented
10#16863627C211yesmaximal, characteristic, complemented
11#16863494C211yesmaximal, characteristic, complemented
12#16863468C211yesmaximal, characteristic, complemented
13#16863436C211yesmaximal, characteristic, complemented
14C2×#14523976C211yesmaximal, characteristic, complemented
15#554×C67C211yesmaximal, characteristic, complemented
16C2×#11761615C2211yescharacteristic
17C2×#11761612C2211yescharacteristic
18C22×C67⋊C8C2211yescharacteristic
19#14524038—21no—
20#14524035C2211yescharacteristic
21#14524025—21no—
22#14524017C2211yescharacteristic
23#14524017C2211yescharacteristic
24#14524015—21no—
25C4×D268C2211yescharacteristic
26(C22⋊D67)⋊C4—21no—
27#14524002C2211yescharacteristic
28C2×C4⋊Dic67C2211yescharacteristic
29#14523976C2211yescharacteristic
30#14523976C2211yescharacteristic
31#14523976C2211yescharacteristic
32#14523976C2211yescharacteristic
33#14523976C2211yescharacteristic
34#14523976C2211yescharacteristic
35#14523974C2211yescharacteristic
36#14523974C2211yescharacteristic
37#14523974C2211yescharacteristic
38C4⋊(C67⋊C8)C2211yescharacteristic
39#14523967C2211yescharacteristic
40#14523967C2211yescharacteristic
41#14523967C2211yescharacteristic
42#14523963C2211yescharacteristic
43#14523963C2211yescharacteristic
44#14523963C2211yescharacteristic
45C22⋊(C67⋊C8)C2211yescharacteristic
46C22⋊(C67⋊C8)C2211yescharacteristic
47#14523954C2211yescharacteristic
48C4⋊Q8×C67C2211yescharacteristic
49#125×C67—21no—
50C22⋊Q8×C67—21no—
51C22⋊Q8×C67—21no—
52C22⋊Q8×C67C2211yescharacteristic
53C22⋊Q8×C67C2211yescharacteristic
54C4×Q8×C67C2211yescharacteristic
55C4×D4×C67—21no—
56C4×D4×C67C2211yescharacteristic
57C2×C4⋊C4×C67C2211yescharacteristic
58C2×C4⋊C4×C67C2211yescharacteristic
59C2×D268C2311yescharacteristic
60C2×C22⋊D67—21no—
61C2×C22⋊D67—21no—
62C22×Dic67—21no—
63C2×C4×D67—21no—
64#11761615—22no—
65#11761612—22no—
66C2×C67⋊C8—22no—
67C2×C67⋊C8C2311yescharacteristic
68C2×C67⋊C8C2311yescharacteristic
69C2×C67⋊C8C2311yescharacteristic
70C2×C67⋊C8C2311yescharacteristic
71C4⋊Dic67—21no—
72C4⋊Dic67C2311yescharacteristic
73C4⋊Dic67C2311yescharacteristic
74C4⋊Dic67C2311yescharacteristic
75Dic67⋊C4—21no—
76Dic67⋊C4—21no—
77C22⋊Dic67—21no—
78(C2×D67)⋊C4—21no—
79(C2×D67)⋊C4—21no—
80(C2×D67)⋊C4—21no—
81C4×Dic67—21no—
82C2×Q8×C67—21no—
83C2×Q8×C67C2311yescharacteristic
84C2×D4×C67D411yescharacteristic
85C22×C4×C67—21no—
86C22×C4×C67—21no—
87C22×C4×C67C2311yescharacteristic
88C22×C4×C67C2311yescharacteristic
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
99C22⋊C4×C67—21no—
100C22⋊C4×C67—21no—
101C22⋊C4×C67D411yescharacteristic
102C22⋊C4×C67D411yescharacteristic
103C42×C67—21no—
104C42×C67C2311yescharacteristic
105C23×C67C2×D411yescharacteristic
106C23×C67C2×D411yescharacteristic
107C22×D67—21no—
108Q8×C67—21no—
109Q8×C67—21no—
110Q8×C67—81no—
111D4×C67—22no—
112C2×C4×C67—21no—
113C2×C4×C67—21no—
114C2×C4×C67—21no—
115C2×C4×C67—21no—
116C2×C4×C67—21no—
117C2×C4×C67—21no—
118C2×C4×C67—21no—
119C2×C4×C67—21no—
120C2×C4×C67—81no—
121C2×C4×C67—81no—
122C2×C4×C67C2×D411yescharacteristic
123C2×C4×C67C2×D411yescharacteristic
124C2×C4×C67C2×D411yescharacteristic
125C2×C4×C67C2×D411yescharacteristic
126C2×C4×C67C2411yescharacteristic
127C22⋊D67—42no—
128C22⋊D67—81no—
129C2×Dic67—21no—
130C2×Dic67—21no—
131C2×Dic67—21no—
132C2×Dic67—81no—
133D268—21no—
134D268—21no—
135C4×D67—81no—
136C67⋊C8—21no—
137C67⋊C8—21no—
138C67⋊C8—21no—
139C67⋊C8#4112yes—
140C22×C67—21no—
141C22×C67—21no—
142C22×C67—21no—
143C22×C67—21no—
144C22×C67—21no—
145C22×C67#13712yes—
146C22×C67C22×D411yescharacteristic
147C2×D67—41no—
148C2×D67—41no—
149C2×D67—41no—
150C4×C67—21no—
151C4×C67—21no—
152C4×C67—21no—
153C4×C67—21no—
154C4×C67—41no—
155C4×C67—41no—
156C4×C67—41no—
157C4×C67—41no—
158C4×C67C2×#4111yescharacteristic
159C4×C672−1+411yescharacteristic
160Dic67—41no—
161Dic67—41no—
162Dic67—41no—
163C2×C67—22no—
164C2×C67—41no—
165C2×C67#54711yescharacteristic
166C2×C67C2×#13711yescharacteristic
167C2×C67#57611yescharacteristic
168D67—81no—
169#3346—671nomaximal, Sylow
170C67#334611yescharacteristic, complemented, Sylow
171#554D6711yescharacteristic, complemented
172#553—671no—
173#501—671no—
174#485—671no—
175#484—671no—
176#481—671no—
177#480—671no—
178#479—671no—
179#474—671no—
180#468—671no—
181#465—671no—
182#440—671no—
183#435—671no—
184#428—671no—
185C2×#107—671no—
186C2×SD16—671no—
187C2×M16—671no—
188C22×C8—671no—
189C4⋊Q8C2×D6711yescharacteristic
190#126—1341no—
191#125—21no—
192#125—671no—
193#123—1341no—
194C22⋊Q8—21no—
195C22⋊Q8—21no—
196C22⋊Q8C2×D6711yescharacteristic
197C22⋊Q8C2×D6711yescharacteristic
198#121—671no—
199#121—671no—
200#121—1341no—
201C4×Q8C2×D6711yescharacteristic
202C4×D4—21no—
203C4×D4—671no—
204C4×D4—1341no—
205C4×D4C2×D6711yescharacteristic
206#117—671no—
207C2×C4⋊C4—671no—
208C2×C4⋊C4C2×D6711yescharacteristic
209C2×C4⋊C4C2×D6711yescharacteristic
210#107—671no—
211#107—671no—
212#107—671no—
213#107—671no—
214#107—671no—
215#107—671no—
216#106—671no—
217#106—671no—
218#106—671no—
219C4⋊C8—671no—
220#103—671no—
221#103—671no—
222#103—671no—
223#102—671no—
224#102—671no—
225#102—671no—
226C22⋊C8—671no—
227C22⋊C8—671no—
228#97—671no—
229C2×Q8—21no—
230C2×Q8C22×D6711yescharacteristic
231C2×D4—671no—
232C2×D4—1341no—
233C2×D4—1341no—
234C2×D4C22⋊D6711yescharacteristic
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—1341no—
238C22×C4—1341no—
239C22×C4C22×D6711yescharacteristic
240C22×C4C22×D6711yescharacteristic
241SD16—1342no—
242M16—1342no—
243C2×C8—671no—
244C2×C8—671no—
245C2×C8—671no—
246C2×C8—671no—
247C2×C8—1342no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—671no—
255C4⋊C4—671no—
256C4⋊C4—671no—
257C4⋊C4—1341no—
258C4⋊C4—1341no—
259C4⋊C4—1341no—
260C4⋊C4C22⋊D6711yescharacteristic
261C4⋊C4C22×D6711yescharacteristic
262C4⋊C4C22×D6711yescharacteristic
263C4⋊C4C22×D6711yescharacteristic
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—1341no—
267C22⋊C4—1341no—
268C22⋊C4—1341no—
269C22⋊C4—1341no—
270C22⋊C4C22⋊D6711yescharacteristic
271C22⋊C4C22⋊D6711yescharacteristic
272C42—21no—
273C42—1341no—
274C42C22×D6711yescharacteristic
275C23—1341no—
276C23C2×C22⋊D6711yescharacteristic
277C23C2×C22⋊D6711yescharacteristic
278Q8—21no—
279Q8—21no—
280Q8—81no—
281D4—22no—
282D4—2682no—
283D4—5361no—
284D4—1341no—
285D4—1341no—
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—81no—
295C2×C4—81no—
296C2×C4—5361no—
297C2×C4—5361no—
298C2×C4—1341no—
299C2×C4—1341no—
300C2×C4—1341no—
301C2×C4C2×C22⋊D6711yescharacteristic
302C2×C4C2×C22⋊D6711yescharacteristic
303C2×C4C2×C22⋊D6711yescharacteristic
304C2×C4C2×C22⋊D6711yescharacteristic
305C2×C4C23×D6711yescharacteristic
306C8—672no—
307C8—1341no—
308C8—1341no—
309C8—1341no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—2681no—
316C22—2681no—
317C22—2681no—
318C22#1452407412yes—
319C22C22×C22⋊D6711yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—2681no—
329C4—2681no—
330C4—2681no—
331C4#41×D6711yescharacteristic
332C4#1452409011yescharacteristic
333C2—22no—
334C2—41no—
335C2—5361no—
336C2#1686404511yescharacteristic
337C2C2×#1452407411yescharacteristic
338C2#1686419911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#554×C67C2
socleC22×C67C22×D4
hypercenter#554D67
hypocenterC67#3346
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
2. #16863802C2semidirect
3. #16863734C2semidirect
4. #16863729C2semidirect
5. #16863714C2semidirect
6. #16863709C2semidirect
7. #16863703C2semidirect
8. #16863673C2semidirect
9. #16863644C2semidirect
10. #16863627C2semidirect
11. #16863494C2semidirect
12. #16863468C2semidirect
13. #16863436C2semidirect
14. C2×#14523976C2semidirect
15. #554×C67C2semidirect
170. C67#3346semidirect
171. #554D67semidirect