Finite Group Databasebeta

← #11736942

Subgroups — #11736942

Order 896.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6471C211yesmaximal, characteristic, complemented
2C2×#6467C211yesmaximal, characteristic, complemented
3#90244C211yesmaximal, characteristic, complemented
4#90241C211yesmaximal, characteristic, complemented
5C7×#435C211yesmaximal, characteristic, complemented
6#89906C211yesmaximal, characteristic, complemented
7#89887C211yesmaximal, characteristic, complemented
8#89617C211yesmaximal, characteristic, complemented
9#89615C211yesmaximal, characteristic, complemented
10#89506C211yesmaximal, characteristic, complemented
11#89506C211yesmaximal, characteristic, complemented
12#89505C211yesmaximal, characteristic, complemented
13#89504C211yesmaximal, characteristic, complemented
14#89484C211yesmaximal, characteristic, complemented
15#89482C211yesmaximal, characteristic, complemented
16C2×#1157C2211yescharacteristic
17C2×#1157C2211yescharacteristic
18C2×#1157C2211yescharacteristic, complemented
19C22×D28C2211yescharacteristic
20C2×C7×M16C2211yescharacteristic
21C22×C7×C8C2211yescharacteristic, complemented
22C4×C7×D4C2211yescharacteristic
23#6503—21no—
24#6500—21no—
25#6493C2211yescharacteristic
26#6493C2211yescharacteristic
27#6493C2211yescharacteristic, complemented
28#6487C2211yescharacteristic
29#6471—24no—
30C2×Dic28C2211yescharacteristic, complemented
31#6467—24no—
32C2×D56C2211yescharacteristic
33C2×D56C2211yescharacteristic
34C2×D56C2211yescharacteristic
35C2×D56C2211yescharacteristic
36C2×#1132C2211yescharacteristic
37C2×#1132C2211yescharacteristic
38C2×#1132C2211yescharacteristic, complemented
39C2×#1132C2211yescharacteristic, complemented
40#6457—21no—
41#6456—21no—
42#6447—21no—
43#6447—21no—
44#6447—21no—
45D7×C22⋊C4—21no—
46#6438C2211yescharacteristic
47#6437C2211yescharacteristic, complemented
48C4×D28C2211yescharacteristic
49C4×Dic14C2211yescharacteristic, complemented
50C7×C4⋊C8C2211yescharacteristic, complemented
51C7×C22⋊C8C2211yescharacteristic
52C7×C22⋊C8C2211yescharacteristic, complemented
53C7×#97C2211yescharacteristic, complemented
54#6395C2211yescharacteristic
55#6395C2211yescharacteristic
56#6395C2211yescharacteristic, complemented
57#6395C2211yescharacteristic, complemented
58#6392C2211yescharacteristic, complemented
59#6390C2211yescharacteristic, complemented
60#6390C2211yescharacteristic, complemented
61#3268—71nomaximal, Sylow
62C23×D7—21no—
63C2×C7×D4D411yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C22×C4×C7C2311yescharacteristic
66C2×C22⋊D7—21no—
67C2×C22⋊D7—21no—
68C2×C22⋊D7—21no—
69C2×C22⋊D7—21no—
70C2×C22⋊D7—21no—
71C2×C22⋊D7—21no—
72#1157—22no—
73#1157—22no—
74#1157—22no—
75#1157—41no—
76#1157—41no—
77#1157—41no—
78C2×D28—22no—
79C2×D28—21no—
80C2×D28—21no—
81C2×D28—21no—
82C2×D28—41no—
83C2×D28C2311yescharacteristic
84C2×D28C2311yescharacteristic
85C2×D28C2311yescharacteristic
86C2×D28C2311yescharacteristic
87C2×C4×D7—21no—
88C2×C4×D7—21no—
89C2×C4×D7—21no—
90C2×C4×D7—21no—
91C2×Dic14C2311yescharacteristic
92C2×Dic14C2311yescharacteristic
93C7×M16—22no—
94C2×C7×C8—22no—
95C2×C7×C8C2311yescharacteristic
96C2×C7×C8C2311yescharacteristic
97C2×C7×C8C2311yescharacteristic
98C2×C7×C8C2311yescharacteristic
99C7×C4⋊C4D411yescharacteristic
100C7×C22⋊C4D411yescharacteristic
101C7×C22⋊C4D411yescharacteristic
102C42×C7C2311yescharacteristic
103C22⋊Dic7—21no—
104(C2×D7)⋊C4—21no—
105(C2×D7)⋊C4—21no—
106(C2×D7)⋊C4—21no—
107(C2×D7)⋊C4—21no—
108(C2×D7)⋊C4—21no—
109(C2×D7)⋊C4—21no—
110C4⋊Dic7C2311yescharacteristic
111C4⋊Dic7C2311yescharacteristic
112Dic7⋊C4—21no—
113Dic7⋊C4—21no—
114C4×Dic7—21no—
115Dic28—22no—
116D56—22no—
117D56—22no—
118D56—22no—
119D56—41no—
120#1132—22no—
121#1132—22no—
122#1132—41no—
123#1132D414yescomplemented
124C2×AGL(1,ℤ/8)—71no—
125C2×#135—71no—
126#550—71no—
127#546—71no—
128#496—71no—
129#468—71no—
130#466—71no—
131#464—71no—
132#459—71no—
133#452—71no—
134#449—71no—
135#449—71no—
136#447—71no—
137#440—71no—
138#435D711yescharacteristic, complemented
139C23×C7C2×D411yescharacteristic
140C23×C7C2×D411yescharacteristic
141C22×D7—21no—
142C22×D7—21no—
143C22×D7—21no—
144C22×D7—21no—
145C22×D7—42no—
146C22×D7—42no—
147C22×D7—81no—
148C7×D4—41no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7C2×D411yescharacteristic
154C2×C4×C7C2×D411yescharacteristic
155C2×C4×C7C2×D411yescharacteristic
156C2×C4×C7C2×D411yescharacteristic
157C2×C4×C7C2411yescharacteristic
158C22⋊D7—42no—
159C22⋊D7—42no—
160C22⋊D7—42no—
161C22⋊D7—81no—
162C22⋊D7—81no—
163C22⋊D7—81no—
164C2×Dic7—21no—
165C2×Dic7—21no—
166C2×Dic7—21no—
167C2×Dic7—21no—
168D28—21no—
169D28—21no—
170D28—21no—
171D28—21no—
172D28—21no—
173D28—21no—
174D28—21no—
175D28—42no—
176D28—42no—
177D28—81no—
178D28C2×D412yes—
179C4×D7—42no—
180C4×D7—42no—
181C4×D7—42no—
182C4×D7—81no—
183Dic14—21no—
184Dic14—21no—
185Dic14—21no—
186Dic14C2×D412yes—
187C7×C8—21no—
188C7×C8—21no—
189C7×C8—21no—
190C7×C8C2×D412yes—
191C2×#41—71no—
192C2×#41—71no—
193C2×#41—71no—
194C22×D4—71no—
195AGL(1,ℤ/8)—144no—
196#135—144no—
197C2×Q16—71no—
198C2×SD16—71no—
199C2×SD16—71no—
200C2×SD16—71no—
201C2×SD16—71no—
202C2×D8—71no—
203C2×D8—71no—
204C2×D8—71no—
205C2×D8—71no—
206C2×M16C2×D711yescharacteristic
207C22×C8C2×D711yescharacteristic, complemented
208#127—71no—
209#127—141no—
210#124—71no—
211#123—141no—
212C22⋊Q8—71no—
213#121—71no—
214#121—71no—
215#121—71no—
216#121—141no—
217#121—141no—
218#121—141no—
219#120—141no—
220C4×Q8—71no—
221C4×D4—71no—
222C4×D4—141no—
223C4×D4C2×D711yescharacteristic
224C2×C22⋊C4—141no—
225#107—71no—
226C4⋊C8C2×D711yescharacteristic, complemented
227#103—71no—
228#103—71no—
229#102—71no—
230#102—71no—
231#102—71no—
232#102—71no—
233C22⋊C8C2×D711yescharacteristic
234C22⋊C8C2×D711yescharacteristic, complemented
235#97C2×D711yescharacteristic, complemented
236C22×C7—21no—
237C22×C7—21no—
238C22×C7—21no—
239C22×C7—21no—
240C22×C7—21no—
241C22×C7—21no—
242C22×C7C22×D411yescharacteristic
243C2×D7—22no—
244C2×D7—41no—
245C2×D7—41no—
246C2×D7—41no—
247C2×D7—41no—
248C2×D7—41no—
249C2×D7—41no—
250C2×D7—41no—
251C2×D7—41no—
252C2×D7—41no—
253C2×D7—41no—
254C2×D7—41no—
255C2×D7—84no—
256C4×C7—21no—
257C4×C7—21no—
258C4×C7—21no—
259C4×C7—21no—
260C4×C7C22×D411yescharacteristic
261C4×C7H(5,𝔽2)11yescharacteristic
262Dic7—22no—
263Dic7—41no—
264Dic7—41no—
265Dic7—41no—
266C24—141no—
267#41—142no—
268#41—142no—
269#41—142no—
270#41—281no—
271#41—281no—
272#41—281no—
273C2×Q8—71no—
274C2×Q8—71no—
275C2×D4—71no—
276C2×D4—71no—
277C2×D4—71no—
278C2×D4—71no—
279C2×D4—142no—
280C2×D4—141no—
281C2×D4—141no—
282C2×D4—141no—
283C2×D4—141no—
284C2×D4—141no—
285C2×D4—141no—
286C2×D4—141no—
287C2×D4—141no—
288C2×D4—141no—
289C2×D4—281no—
290C2×D4D2811yescharacteristic
291C22×C4—141no—
292C22×C4—141no—
293C22×C4—141no—
294C22×C4—141no—
295C22×C4C22×D711yescharacteristic
296C22×C4C22×D711yescharacteristic
297Q16—142no—
298SD16—74no—
299SD16—142no—
300SD16—142no—
301SD16—281no—
302D8—142no—
303D8—142no—
304D8—142no—
305D8—281no—
306M16—22no—
307C2×C8—22no—
308C2×C8C22×D711yescharacteristic
309C2×C8C22×D711yescharacteristic
310C2×C8C22×D711yescharacteristic
311C2×C8C22×D711yescharacteristic
312C4⋊C4—71no—
313C4⋊C4—71no—
314C4⋊C4—141no—
315C4⋊C4—141no—
316C4⋊C4D2811yescharacteristic
317C22⋊C4—141no—
318C22⋊C4—141no—
319C22⋊C4—141no—
320C22⋊C4—141no—
321C22⋊C4—141no—
322C22⋊C4—141no—
323C22⋊C4—141no—
324C22⋊C4D2811yescharacteristic
325C22⋊C4D2811yescharacteristic
326C42—141no—
327C42C22×D711yescharacteristic
328C2×C7—41no—
329C2×C7—41no—
330C2×C7D4211yescharacteristic
331C2×C7#57511yescharacteristic
332C2×C7#57611yescharacteristic
333D7—42no—
334D7—81no—
335D7—81no—
336D7—81no—
337C23—141no—
338C23—141no—
339C23—141no—
340C23—141no—
341C23—282no—
342C23—282no—
343C23—561no—
344C23C2×D2811yescharacteristic
345C23C2×D2811yescharacteristic
346Q8—72no—
347Q8—141no—
348Q8—141no—
349Q8—141no—
350D4—41no—
351D4—72no—
352D4—141no—
353D4—141no—
354D4—141no—
355D4—141no—
356D4—141no—
357D4—141no—
358D4—141no—
359D4—282no—
360D4—282no—
361D4—282no—
362D4—282no—
363D4—282no—
364D4—561no—
365D4—561no—
366D4—561no—
367D4—561no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—141no—
373C2×C4—141no—
374C2×C4—141no—
375C2×C4—141no—
376C2×C4—282no—
377C2×C4—282no—
378C2×C4—282no—
379C2×C4—561no—
380C2×C4C2×D2811yescharacteristic
381C2×C4C2×D2811yescharacteristic
382C2×C4C2×D2811yescharacteristic
383C2×C4C2×D2811yescharacteristic
384C2×C4C23×D711yescharacteristic
385C8—21no—
386C8—21no—
387C8—21no—
388C8D4×D712yes—
389C7#326811yescharacteristic, complemented, Sylow
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—142no—
397C22—281no—
398C22—281no—
399C22—281no—
400C22—281no—
401C22—281no—
402C22—281no—
403C22—281no—
404C22—281no—
405C22—281no—
406C22—281no—
407C22—281no—
408C22—564no—
409C22C22×D2811yescharacteristic
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—142no—
415C4—281no—
416C4—281no—
417C4—281no—
418C4C2×D4×D711yescharacteristic
419C4#655311yescharacteristic
420C2—41no—
421C2—41no—
422C2—282no—
423C2—561no—
424C2—561no—
425C2—561no—
426C2D4×D2811yescharacteristic
427C2#9043911yescharacteristic
428C2#9044311yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#435C2
socleC22×C7C22×D4
hypercenter#435D7
hypocenterC7#3268
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#6471C2semidirect
2. C2×#6467C2semidirect
3. #90244C2semidirect
4. #90241C2semidirect
5. C7×#435C2semidirect
6. #89906C2semidirect
7. #89887C2semidirect
8. #89617C2semidirect
9. #89615C2semidirect
10. #89506C2semidirect
11. #89506C2semidirect
12. #89505C2semidirect
13. #89504C2semidirect
14. #89484C2semidirect
15. #89482C2semidirect
18. C2×#1157C22semidirect
21. C22×C7×C8C22semidirect
27. #6493C22semidirect
30. C2×Dic28C22semidirect
38. C2×#1132C22semidirect
39. C2×#1132C22semidirect
47. #6437C22semidirect
49. C4×Dic14C22semidirect
50. C7×C4⋊C8C22semidirect
52. C7×C22⋊C8C22semidirect
53. C7×#97C22semidirect
56. #6395C22semidirect
57. #6395C22semidirect
58. #6392C22semidirect
59. #6390C22semidirect
60. #6390C22semidirect
123. #1132D4semidirect
138. #435D7semidirect
207. C22×C8C2×D7semidirect
226. C4⋊C8C2×D7semidirect
234. C22⋊C8C2×D7semidirect
235. #97C2×D7semidirect
389. C7#3268semidirect

Maximal subgroups

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

subgroupindex
C2×#64712
C2×#64672
#902442
#902412
C7×#4352
#899062
#898872
#896172
#896152
#895062
#895052
#895042
#894842
#894822
#32687