Finite Group Databasebeta

← #22599141

Subgroups — #22599141

Order 9088.

462 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22⋊Q8×D71C211yesmaximal, characteristic, complemented
2#121×D71C211yesmaximal, characteristic
3#17986336C211yesmaximal, characteristic, complemented
4#17986333C211yesmaximal, characteristic, complemented
5#17986314C211yesmaximal, characteristic, complemented
6#17986311C211yesmaximal, characteristic, complemented
7#17986247C211yesmaximal, characteristic, complemented
8#17986245C211yesmaximal, characteristic, complemented
9#17986243C211yesmaximal, characteristic, complemented
10#17986151C211yesmaximal, characteristic, complemented
11#17986148C211yesmaximal, characteristic, complemented
12#17986139C211yesmaximal, characteristic, complemented
13#17986136C211yesmaximal, characteristic, complemented
14#17985962C211yesmaximal, characteristic, complemented
15#468×C71C211yesmaximal, characteristic, complemented
16C2×Q8×D71C2211yescharacteristic, complemented
17C2×D4×D71—21no—
18C2×D4×D71C2211yescharacteristic
19C22×C4×D71C2211yescharacteristic
20C2×#11765854C2211yescharacteristic, complemented
21C2×#11765853C2211yescharacteristic, complemented
22C2×#11765852C2211yescharacteristic, complemented
23C2×#11765849C2211yescharacteristic, complemented
24#14556108C2211yescharacteristic
25#14556107C2211yescharacteristic, complemented
26#14556106—21no—
27#14556105—21no—
28#14556104C2211yescharacteristic
29#14556103C2211yescharacteristic
30#14556102C2211yescharacteristic
31#14556101—21no—
32#14556100—21no—
33#14556099C2211yescharacteristic
34C4⋊C4×D71—21no—
35C4⋊C4×D71C2211yescharacteristic
36C4⋊C4×D71C2211yescharacteristic, complemented
37C22⋊C4×D71—21no—
38C22⋊C4×D71—21no—
39#14556062C2211yescharacteristic, complemented
40#14556061C2211yescharacteristic, complemented
41#14556061C2211yescharacteristic, complemented
42#14556053C2211yescharacteristic, complemented
43#14556052C2211yescharacteristic, complemented
44#14556051C2211yescharacteristic, complemented
45#14556050C2211yescharacteristic, complemented
46#14556049C2211yescharacteristic, complemented
47#14556048C2211yescharacteristic, complemented
48(C2×D71)⋊C8C2211yescharacteristic
49(C2×D71)⋊C8C2211yescharacteristic
50(C2×D71)⋊C8C2211yescharacteristic, complemented
51(C2×D71)⋊C8C2211yescharacteristic, complemented
52C2×SD16×C71C2211yescharacteristic, complemented
53C2×M16×C71C2211yescharacteristic, complemented
54C22⋊Q8×C71C2211yescharacteristic, complemented
55#121×C71C2211yescharacteristic
56#107×C71C2211yescharacteristic, complemented
57#103×C71C2211yescharacteristic, complemented
58#102×C71C2211yescharacteristic, complemented
59C23×D71—21no—
60C23×D71D411yescharacteristic
61C2×Dic142—21no—
62C2×Dic142C2311yescharacteristic
63Q8×D71—41no—
64Q8×D71—41no—
65C2×D284—21no—
66C2×D284C2311yescharacteristic
67C2×C22⋊D71—21no—
68C2×C22⋊D71—21no—
69C2×C22⋊D71—21no—
70D4×D71—41no—
71D4×D71—41no—
72D4×D71—44no—
73C22×Dic71D411yescharacteristic
74C2×C4×D71—21no—
75C2×C4×D71—21no—
76C2×C4×D71—21no—
77C2×C4×D71—41no—
78C2×C4×D71—41no—
79C2×C4×D71D411yescharacteristic
80C2×C4×D71D411yescharacteristic
81C2×C4×D71C2311yescharacteristic
82C2×C4×D71C2311yescharacteristic
83#11765854—41no—
84#11765853—41no—
85#11765852—41no—
86#11765849—41no—
87C2×C71⋊C8C2311yescharacteristic
88C2×C71⋊C8C2311yescharacteristic
89C4⋊Dic71—21no—
90C4⋊Dic71C2311yescharacteristic
91C4⋊Dic71C2311yescharacteristic
92Dic71⋊C4—21no—
93Dic71⋊C4—21no—
94Dic71⋊C4—21no—
95Dic71⋊C4—21no—
96C22⋊Dic71—21no—
97C22⋊Dic71—21no—
98(C2×D71)⋊C4—21no—
99(C2×D71)⋊C4—21no—
100(C2×D71)⋊C4—21no—
101(C2×D71)⋊C4—21no—
102C2×Q8×C71C2311yescharacteristic
103C2×D4×C71—21no—
104C2×D4×C71C2311yescharacteristic
105C22×C4×C71C2311yescharacteristic
106SD16×C71—41no—
107M16×C71—41no—
108C2×C8×C71C2311yescharacteristic
109C2×C8×C71C2311yescharacteristic
110C4⋊C4×C71—21no—
111C4⋊C4×C71C2311yescharacteristic
112C4⋊C4×C71C2311yescharacteristic
113C22⋊C4×C71—21no—
114C22⋊C4×C71—21no—
115C23×C71—21no—
116C23×C71C2×D411yescharacteristic
117C22×D71—22no—
118C22×D71—22no—
119C22×D71—21no—
120C22×D71—41no—
121C22×D71—42no—
122C22×D71—42no—
123C22×D71—42no—
124C22×D71C2×D411yescharacteristic
125C22×D71C2×D411yescharacteristic
126Q8×C71—21no—
127Q8×C71—21no—
128D4×C71—21no—
129D4×C71—21no—
130D4×C71—42no—
131C2×C4×C71—21no—
132C2×C4×C71—21no—
133C2×C4×C71—21no—
134C2×C4×C71—21no—
135C2×C4×C71—21no—
136C2×C4×C71C2×D411yescharacteristic
137C2×C4×C71C2411yescharacteristic
138C22⋊D71—42no—
139C22⋊D71—42no—
140C22⋊D71—42no—
141C2×Dic71—21no—
142C2×Dic71—21no—
143C2×Dic71—21no—
144C2×Dic71—41no—
145C2×Dic71C2×D411yescharacteristic
146C2×Dic71C2×D411yescharacteristic
147D284—21no—
148D284—21no—
149D284—42no—
150C4×D71—21no—
151C4×D71—21no—
152C4×D71—21no—
153C4×D71—21no—
154C4×D71—41no—
155C4×D71—41no—
156C4×D71—42no—
157C4×D71—42no—
158C4×D71—42no—
159Dic142—21no—
160Dic142—21no—
161Dic142—42no—
162C8×C71—21no—
163C8×C71—21no—
164C71⋊C8—21no—
165C71⋊C8—21no—
166C22×C71—21no—
167C22×C71—21no—
168C22×C71—21no—
169C22×C71—41no—
170C22×C71—41no—
171C22×C71—41no—
172C22×C71C22×D411yescharacteristic
173C2×D71—21no—
174C2×D71—21no—
175C2×D71—21no—
176C2×D71—21no—
177C2×D71—44no—
178C2×D71—41no—
179C2×D71—41no—
180C2×D71—41no—
181C2×D71—84no—
182C2×D71AGL(1,ℤ/8)12yes—
183C2×D71#13712yes—
184C4×C71—21no—
185C4×C71—41no—
186C4×C71—41no—
187C4×C71—41no—
188C4×C71H(5,𝔽2)11yescharacteristic
189C4×C71H(5,𝔽2)11yescharacteristic
190Dic71—21no—
191Dic71—21no—
192Dic71—41no—
193Dic71—41no—
194Dic71—41no—
195C2×C71—41no—
196C2×C71—81no—
197C2×C71#53411yescharacteristic
198C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
199C2×C71C2×#13711yescharacteristic
200D71—22no—
201D71—22no—
202D71—81no—
203#3193—711nomaximal, Sylow
204C71#319311yescharacteristic, complemented, Sylow
205C2×C22⋊Q8—711no—
206C2×#121—711no—
207#484—711no—
208#484—711no—
209#480—711no—
210#480—711no—
211#468—711no—
212#468—711no—
213#468—711no—
214#468D7111yescharacteristic, complemented
215#450—711no—
216#450—711no—
217#448—711no—
218#448—711no—
219#407—711no—
220C22×Q8—711no—
221C22×D4—711no—
222C22×D4—1421no—
223C23×C4—711no—
224C2×SD16—711no—
225C2×SD16—711no—
226C2×SD16—711no—
227C2×SD16C2×D7111yescharacteristic, complemented
228C2×M16—711no—
229C2×M16C2×D7111yescharacteristic, complemented
230C22⋊Q8—711no—
231C22⋊Q8—711no—
232C22⋊Q8—711no—
233C22⋊Q8—1421no—
234C22⋊Q8—1421no—
235C22⋊Q8C2×D7111yescharacteristic, complemented
236#121—711no—
237#121—711no—
238#121—711no—
239#121—1421no—
240#121—1421no—
241#121C2×D7111yescharacteristic
242C2×C4⋊C4—711no—
243C2×C4⋊C4—711no—
244C2×C4⋊C4—1421no—
245C2×C22⋊C4—1421no—
246C2×C22⋊C4—1421no—
247#107—711no—
248#107—711no—
249#107—711no—
250#107C2×D7111yescharacteristic, complemented
251#103—711no—
252#103—711no—
253#103—711no—
254#103C2×D7111yescharacteristic, complemented
255#102—711no—
256#102—711no—
257#102—711no—
258#102C2×D7111yescharacteristic, complemented
259C22⋊C8—711no—
260C22⋊C8—711no—
261C22⋊C8—711no—
262C22⋊C8—711no—
263C24—711no—
264C24—1421no—
265C2×Q8—2841no—
266C2×Q8—2841no—
267C2×Q8—711no—
268C2×Q8—1421no—
269C2×Q8C22×D7111yescharacteristic
270C2×D4—21no—
271C2×D4—2844no—
272C2×D4—2841no—
273C2×D4—2841no—
274C2×D4—711no—
275C2×D4—1421no—
276C2×D4—1421no—
277C2×D4—1421no—
278C2×D4—1421no—
279C2×D4C22×D7111yescharacteristic
280C22×C4—2841no—
281C22×C4—2841no—
282C22×C4—711no—
283C22×C4—711no—
284C22×C4—711no—
285C22×C4—711no—
286C22×C4—711no—
287C22×C4—1421no—
288C22×C4—1421no—
289C22×C4—1421no—
290C22×C4C22×D7111yescharacteristic
291SD16—41no—
292SD16—2841no—
293SD16—2841no—
294SD16—2841no—
295M16—41no—
296M16—2841no—
297C2×C8—711no—
298C2×C8—711no—
299C2×C8C22×D7111yescharacteristic
300C2×C8C22×D7111yescharacteristic
301C4⋊C4—21no—
302C4⋊C4—711no—
303C4⋊C4—711no—
304C4⋊C4—1421no—
305C4⋊C4—1421no—
306C4⋊C4—1421no—
307C4⋊C4—1421no—
308C4⋊C4—1421no—
309C4⋊C4C22×D7111yescharacteristic
310C4⋊C4C22×D7111yescharacteristic
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—1421no—
314C22⋊C4—1421no—
315C22⋊C4—1421no—
316C22⋊C4—1421no—
317C22⋊C4—1421no—
318C22⋊C4—1421no—
319C23—21no—
320C23—2841no—
321C23—2842no—
322C23—2842no—
323C23—2842no—
324C23—711no—
325C23—711no—
326C23—1422no—
327C23—1422no—
328C23—1421no—
329C23D4×D7111yescharacteristic
330Q8—21no—
331Q8—21no—
332Q8—2842no—
333Q8—1421no—
334Q8—1421no—
335D4—21no—
336D4—21no—
337D4—42no—
338D4—2842no—
339D4—2842no—
340D4—2842no—
341D4—2842no—
342D4—1421no—
343D4—1421no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—2841no—
350C2×C4—2841no—
351C2×C4—2842no—
352C2×C4—2842no—
353C2×C4—2842no—
354C2×C4—2841no—
355C2×C4—711no—
356C2×C4—711no—
357C2×C4—1421no—
358C2×C4—1421no—
359C2×C4—1421no—
360C2×C4—1421no—
361C2×C4—1421no—
362C2×C4—1421no—
363C2×C4—1421no—
364C2×C4D4×D7111yescharacteristic
365C2×C4C23×D7111yescharacteristic
366C8—21no—
367C8—21no—
368C8—1421no—
369C8—1421no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—2844no—
377C22—2841no—
378C22—2841no—
379C22—2841no—
380C22—5684no—
381C22—712no—
382C22—712no—
383C22—1421no—
384C22—1421no—
385C22—1421no—
386C22—1421no—
387C22C2×D4×D7111yescharacteristic
388C4—21no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—2841no—
393C4—2841no—
394C4—2841no—
395C4—1421no—
396C4—1421no—
397C4#1455617311yescharacteristic
398C4#1455617411yescharacteristic
399C2—41no—
400C2—81no—
401C2—5681no—
402C2—1422no—
403C2—1422no—
404C2#1798655711yescharacteristic
405C2AGL(1,ℤ/8)×D7111yescharacteristic
406C2#137×D7111yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#468×C71C2
socleC22×C71C22×D4
hypercenter#468D71
hypocenterC71#3193
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C22⋊Q8×D71C2semidirect
3. #17986336C2semidirect
4. #17986333C2semidirect
5. #17986314C2semidirect
6. #17986311C2semidirect
7. #17986247C2semidirect
8. #17986245C2semidirect
9. #17986243C2semidirect
10. #17986151C2semidirect
11. #17986148C2semidirect
12. #17986139C2semidirect
13. #17986136C2semidirect
14. #17985962C2semidirect
15. #468×C71C2semidirect
16. C2×Q8×D71C22semidirect
20. C2×#11765854C22semidirect
21. C2×#11765853C22semidirect
22. C2×#11765852C22semidirect
23. C2×#11765849C22semidirect
25. #14556107C22semidirect
36. C4⋊C4×D71C22semidirect
39. #14556062C22semidirect
40. #14556061C22semidirect
41. #14556061C22semidirect
42. #14556053C22semidirect
43. #14556052C22semidirect
44. #14556051C22semidirect
45. #14556050C22semidirect
46. #14556049C22semidirect
47. #14556048C22semidirect
50. (C2×D71)⋊C8C22semidirect
51. (C2×D71)⋊C8C22semidirect
52. C2×SD16×C71C22semidirect
53. C2×M16×C71C22semidirect
54. C22⋊Q8×C71C22semidirect
56. #107×C71C22semidirect
57. #103×C71C22semidirect
58. #102×C71C22semidirect
204. C71#3193semidirect
214. #468D71semidirect
227. C2×SD16C2×D71semidirect
229. C2×M16C2×D71semidirect
235. C22⋊Q8C2×D71semidirect
250. #107C2×D71semidirect
254. #103C2×D71semidirect
258. #102C2×D71semidirect