Finite Group Databasebeta

← #22599945

Subgroups — #22599945

Order 9088.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14556152C211yesmaximal, characteristic, complemented
2#17986643C211yesmaximal, characteristic, complemented
3#17986637C211yesmaximal, characteristic, complemented
4#17986380C211yesmaximal, characteristic, complemented
5#17986245C211yesmaximal, characteristic, complemented
6#17986242C211yesmaximal, characteristic, complemented
7#17986216C211yesmaximal, characteristic, complemented
8#17986212C211yesmaximal, characteristic, complemented
9#17986161C211yesmaximal, characteristic, complemented
10#17986158C211yesmaximal, characteristic, complemented
11#17986133C211yesmaximal, characteristic, complemented
12#17986131C211yesmaximal, characteristic, complemented
13D8×Dic71C211yesmaximal, characteristic, complemented
14#17986075C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C71C211yesmaximal, characteristic, complemented
16C2×#11765864C2211yescharacteristic
17C2×D4×D71C2211yescharacteristic, complemented
18#14556152—24no—
19C2×#11765855C2211yescharacteristic, complemented
20C2×#11765854C2211yescharacteristic, complemented
21C2×#11765853C2211yescharacteristic, complemented
22C2×#11765850C2211yescharacteristic
23C2×#11765850C2211yescharacteristic
24C2×#11765848C2211yescharacteristic, complemented
25#14556117C2211yescharacteristic, complemented
26#14556115C2211yescharacteristic, complemented
27#14556111—21no—
28#14556109—21no—
29#14556107C2211yescharacteristic, complemented
30#14556104C2211yescharacteristic, complemented
31#14556102C2211yescharacteristic
32#14556100—21no—
33#14556100—21no—
34#14556099C2211yescharacteristic
35#14556097—21no—
36#14556096—21no—
37D4×Dic71C2211yescharacteristic, complemented
38D4×Dic71C2211yescharacteristic, complemented
39C4×C22⋊D71C2211yescharacteristic
40C2×C22⋊Dic71—21no—
41C2×(C2×D71)⋊C4—21no—
42#14556062C2211yescharacteristic, complemented
43Dic71⋊C8C2211yescharacteristic, complemented
44#14556053C2211yescharacteristic, complemented
45#14556052C2211yescharacteristic, complemented
46#14556050C2211yescharacteristic, complemented
47#14556049C2211yescharacteristic, complemented
48#14556049C2211yescharacteristic, complemented
49#14556049C2211yescharacteristic, complemented
50C22⋊(C71⋊C8)C2211yescharacteristic, complemented
51(C2×D71)⋊C8C2211yescharacteristic, complemented
52C8×Dic71C2211yescharacteristic, complemented
53C2×#41×C71C2211yescharacteristic
54C22×D4×C71C2211yescharacteristic, complemented
55AGL(1,ℤ/8)×C71—24no—
56C2×SD16×C71C2211yescharacteristic, complemented
57C2×SD16×C71C2211yescharacteristic, complemented
58C2×D8×C71C2211yescharacteristic
59C2×D8×C71C2211yescharacteristic
60C2×M16×C71C2211yescharacteristic, complemented
61C23×D71—21no—
62#11765864—22no—
63#11765864—41no—
64C2×Dic142C2311yescharacteristic
65C2×D284C2311yescharacteristic
66C2×C22⋊D71—21no—
67C2×C22⋊D71—21no—
68C2×C22⋊D71—21no—
69C2×C22⋊D71—21no—
70C2×C22⋊D71D411yescharacteristic
71D4×D71—22no—
72D4×D71—41no—
73C22×Dic71—21no—
74C22×Dic71—21no—
75C22×Dic71—21no—
76C2×C4×D71C2311yescharacteristic
77#11765855—41no—
78#11765854—22no—
79#11765853—22no—
80#11765850—22no—
81#11765850—41no—
82#11765848—22no—
83C2×C71⋊C8C2311yescharacteristic
84C2×C71⋊C8C2311yescharacteristic
85C4⋊Dic71C2311yescharacteristic
86C4⋊Dic71C2311yescharacteristic
87Dic71⋊C4—21no—
88Dic71⋊C4—21no—
89Dic71⋊C4D411yescharacteristic
90C22⋊Dic71—21no—
91C22⋊Dic71—21no—
92C22⋊Dic71—21no—
93C22⋊Dic71—21no—
94C22⋊Dic71—21no—
95C22⋊Dic71—21no—
96C22⋊Dic71D411yescharacteristic
97(C2×D71)⋊C4—21no—
98(C2×D71)⋊C4—21no—
99(C2×D71)⋊C4—21no—
100(C2×D71)⋊C4—21no—
101(C2×D71)⋊C4D411yescharacteristic
102C4×Dic71C2311yescharacteristic
103C24×C71—21no—
104#41×C71—22no—
105#41×C71—41no—
106C2×Q8×C71C2311yescharacteristic
107C2×D4×C71—22no—
108C2×D4×C71—21no—
109C2×D4×C71—21no—
110C2×D4×C71—41no—
111C2×D4×C71C2311yescharacteristic
112C2×D4×C71C2311yescharacteristic
113C2×D4×C71C2311yescharacteristic
114C22×C4×C71—21no—
115C22×C4×C71C2311yescharacteristic
116SD16×C71—22no—
117SD16×C71—22no—
118D8×C71—22no—
119D8×C71D414yescomplemented
120M16×C71—22no—
121C2×C8×C71C2311yescharacteristic
122C2×C8×C71C2311yescharacteristic
123C23×C71—21no—
124C23×C71—21no—
125C23×C71—21no—
126C23×C71—42no—
127C23×C71—42no—
128C23×C71—81no—
129C23×C71C2×D411yescharacteristic
130C22×D71—21no—
131C22×D71—42no—
132C22×D71—42no—
133C22×D71—81no—
134C22×D71C2×D411yescharacteristic
135Q8×C71—21no—
136Q8×C71—21no—
137D4×C71—21no—
138D4×C71—21no—
139D4×C71—21no—
140D4×C71—21no—
141D4×C71—42no—
142D4×C71—42no—
143D4×C71C2×D412yes—
144D4×C71C2×D412yes—
145C2×C4×C71—21no—
146C2×C4×C71—21no—
147C2×C4×C71—21no—
148C2×C4×C71—42no—
149C2×C4×C71C2×D411yescharacteristic
150C2×C4×C71C2411yescharacteristic
151C22⋊D71—41no—
152C22⋊D71—42no—
153C22⋊D71—42no—
154C22⋊D71—81no—
155C22⋊D71—81no—
156C2×Dic71—21no—
157C2×Dic71—21no—
158C2×Dic71—21no—
159C2×Dic71—42no—
160C2×Dic71—81no—
161C2×Dic71—81no—
162C2×Dic71C2×D411yescharacteristic
163C2×Dic71C2×D411yescharacteristic
164C2×Dic71C2×D411yescharacteristic
165D284—21no—
166D284—21no—
167C4×D71—21no—
168C4×D71—21no—
169Dic142—21no—
170Dic142—21no—
171C8×C71—21no—
172C8×C71C2×D412yes—
173C71⋊C8—21no—
174C71⋊C8—21no—
175C22×C71—22no—
176C22×C71—22no—
177C22×C71—21no—
178C22×C71—21no—
179C22×C71—21no—
180C22×C71—41no—
181C22×C71—41no—
182C22×C71—41no—
183C22×C71—41no—
184C22×C71—41no—
185C22×C71—41no—
186C22×C71—41no—
187C22×C71—84no—
188C22×C71C22×D411yescharacteristic
189C2×D71—21no—
190C2×D71—21no—
191C2×D71—21no—
192C2×D71—41no—
193C2×D71—41no—
194C2×D71—41no—
195C2×D71—84no—
196C4×C71—21no—
197C4×C71—41no—
198C4×C71C22×D411yescharacteristic
199C4×C71H(5,𝔽2)11yescharacteristic
200Dic71—21no—
201Dic71—21no—
202Dic71—21no—
203Dic71—41no—
204Dic71—41no—
205Dic71—41no—
206C2×C71—42no—
207C2×C71—41no—
208C2×C71—42no—
209C2×C71—81no—
210C2×C71D4211yescharacteristic
211C2×C71#57511yescharacteristic
212C2×C71#57611yescharacteristic
213D71—41no—
214D71—81no—
215#3273—711nomaximal, Sylow
216C71#327311yescharacteristic, complemented, Sylow
217C2×AGL(1,ℤ/8)—711no—
218C2×AGL(1,ℤ/8)D7111yescharacteristic, complemented
219#546—711no—
220#546—711no—
221#495—711no—
222#468—711no—
223#468—711no—
224#463—711no—
225#463—711no—
226#452—711no—
227#452—711no—
228#447—711no—
229#447—711no—
230C4×D8—711no—
231#436—711no—
232C2×#41—711no—
233C2×#41C2×D7111yescharacteristic
234C22×D4—711no—
235C22×D4C2×D7111yescharacteristic, complemented
236AGL(1,ℤ/8)—24no—
237AGL(1,ℤ/8)—1424no—
238C2×Q16—711no—
239C2×SD16—711no—
240C2×SD16—711no—
241C2×SD16C2×D7111yescharacteristic, complemented
242C2×SD16C2×D7111yescharacteristic, complemented
243C2×D8—711no—
244C2×D8—711no—
245C2×D8C2×D7111yescharacteristic
246C2×D8C2×D7111yescharacteristic
247C2×M16—711no—
248C2×M16C2×D7111yescharacteristic, complemented
249#124—711no—
250#124—711no—
251#123—1421no—
252#123—1421no—
253C22⋊Q8—711no—
254C22⋊Q8—711no—
255#121—711no—
256#121—711no—
257#121—1421no—
258#121—1421no—
259#120—1421no—
260#120—1421no—
261C4×D4—711no—
262C4×D4—711no—
263C4×D4—711no—
264C2×C22⋊C4—1421no—
265C2×C22⋊C4—1421no—
266#107—711no—
267C4⋊C8—711no—
268#103—711no—
269#103—711no—
270#102—711no—
271#102—711no—
272#102—711no—
273#102—711no—
274C22⋊C8—711no—
275C22⋊C8—711no—
276C4×C8—711no—
277C24—21no—
278C24—1421no—
279#41—22no—
280#41—41no—
281#41—2841no—
282#41—1422no—
283C2×Q8—711no—
284C2×Q8C22×D7111yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—2841no—
290C2×D4—711no—
291C2×D4—711no—
292C2×D4—1421no—
293C2×D4—1421no—
294C2×D4—1422no—
295C2×D4—1421no—
296C2×D4—1421no—
297C2×D4C22×D7111yescharacteristic
298C2×D4C22×D7111yescharacteristic
299C2×D4C22×D7111yescharacteristic
300C22×C4—21no—
301C22×C4—711no—
302C22×C4—1421no—
303C22×C4—1421no—
304C22×C4—1421no—
305C22×C4C22×D7111yescharacteristic
306Q16—2841no—
307SD16—22no—
308SD16—22no—
309SD16—1422no—
310SD16—1422no—
311D8—22no—
312D8—2841no—
313D8—1422no—
314D8C22⋊D7114yescomplemented
315M16—22no—
316M16—1422no—
317C2×C8—711no—
318C2×C8—711no—
319C2×C8C22×D7111yescharacteristic
320C2×C8C22×D7111yescharacteristic
321C4⋊C4—711no—
322C4⋊C4—711no—
323C4⋊C4—711no—
324C4⋊C4—1421no—
325C4⋊C4—1421no—
326C22⋊C4—711no—
327C22⋊C4—711no—
328C22⋊C4—1421no—
329C22⋊C4—1421no—
330C22⋊C4—1421no—
331C22⋊C4—1421no—
332C22⋊C4—1421no—
333C22⋊C4—1421no—
334C22⋊C4—1421no—
335C22⋊C4—1421no—
336C22⋊C4—1421no—
337C22⋊C4—1421no—
338C42—711no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—42no—
343C23—42no—
344C23—81no—
345C23—2842no—
346C23—2842no—
347C23—5681no—
348C23—711no—
349C23—1421no—
350C23D4×D7111yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—1421no—
354Q8—1421no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—21no—
359D4—42no—
360D4—42no—
361D4—2842no—
362D4—2842no—
363D4—2841no—
364D4—5681no—
365D4—5681no—
366D4—1421no—
367D4—1421no—
368D4C2×C22⋊D7112yes—
369D4C2×C22⋊D7112yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—2842no—
375C2×C4—5681no—
376C2×C4—5681no—
377C2×C4—711no—
378C2×C4—711no—
379C2×C4—711no—
380C2×C4—1421no—
381C2×C4—1421no—
382C2×C4—1421no—
383C2×C4—1421no—
384C2×C4—1421no—
385C2×C4D4×D7111yescharacteristic
386C2×C4C23×D7111yescharacteristic
387C8—21no—
388C8—1421no—
389C8—1421no—
390C8C2×C22⋊D7112yes—
391C22—21no—
392C22—22no—
393C22—22no—
394C22—21no—
395C22—21no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—2841no—
405C22—2841no—
406C22—2841no—
407C22—5684no—
408C22—1421no—
409C22—1421no—
410C22—1421no—
411C22C2×D4×D7111yescharacteristic
412C4—21no—
413C4—41no—
414C4—2841no—
415C4—2841no—
416C4—2841no—
417C4—1421no—
418C4—1421no—
419C4—1421no—
420C4C22×C22⋊D7111yescharacteristic
421C4#1455617311yescharacteristic
422C2—41no—
423C2—42no—
424C2—42no—
425C2—81no—
426C2—2841no—
427C2—5681no—
428C2D4×C22⋊D7111yescharacteristic
429C2#1798679511yescharacteristic
430C2#1798680311yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC2×AGL(1,ℤ/8)×C71C2
socleC22×C71C22×D4
hypercenterC2×AGL(1,ℤ/8)D71
hypocenterC71#3273
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#14556152C2semidirect
2. #17986643C2semidirect
3. #17986637C2semidirect
4. #17986380C2semidirect
5. #17986245C2semidirect
6. #17986242C2semidirect
7. #17986216C2semidirect
8. #17986212C2semidirect
9. #17986161C2semidirect
10. #17986158C2semidirect
11. #17986133C2semidirect
12. #17986131C2semidirect
13. D8×Dic71C2semidirect
14. #17986075C2semidirect
15. C2×AGL(1,ℤ/8)×C71C2semidirect
17. C2×D4×D71C22semidirect
19. C2×#11765855C22semidirect
20. C2×#11765854C22semidirect
21. C2×#11765853C22semidirect
24. C2×#11765848C22semidirect
25. #14556117C22semidirect
26. #14556115C22semidirect
29. #14556107C22semidirect
30. #14556104C22semidirect
37. D4×Dic71C22semidirect
38. D4×Dic71C22semidirect
42. #14556062C22semidirect
43. Dic71⋊C8C22semidirect
44. #14556053C22semidirect
45. #14556052C22semidirect
46. #14556050C22semidirect
47. #14556049C22semidirect
48. #14556049C22semidirect
49. #14556049C22semidirect
50. C22⋊(C71⋊C8)C22semidirect
51. (C2×D71)⋊C8C22semidirect
52. C8×Dic71C22semidirect
54. C22×D4×C71C22semidirect
56. C2×SD16×C71C22semidirect
57. C2×SD16×C71C22semidirect
60. C2×M16×C71C22semidirect
119. D8×C71D4semidirect
216. C71#3273semidirect
218. C2×AGL(1,ℤ/8)D71semidirect
235. C22×D4C2×D71semidirect
241. C2×SD16C2×D71semidirect
242. C2×SD16C2×D71semidirect
248. C2×M16C2×D71semidirect
314. D8C22⋊D71semidirect