Finite Group Databasebeta

← #11741004

Subgroups — #11741004

Order 896.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×D4×Q8C211yesmaximal, characteristic, complemented
2C2×#6505C211yesmaximal, characteristic, complemented
3C22×#1143C211yesmaximal, characteristic, complemented
4#90241C211yesmaximal, characteristic
5#90000C211yesmaximal, characteristic, complemented
6#89999C211yesmaximal, characteristic, complemented
7#89858C211yesmaximal, characteristic, complemented
8#89855C211yesmaximal, characteristic, complemented
9#89823C211yesmaximal, characteristic, complemented
10#89821C211yesmaximal, characteristic, complemented
11#89819C211yesmaximal, characteristic, complemented
12#89818C211yesmaximal, characteristic, complemented
13#89800C211yesmaximal, characteristic, complemented
14#89798C211yesmaximal, characteristic, complemented
15#89783C211yesmaximal, characteristic, complemented
16C22×C7×Q8C2211yescharacteristic
17C22×C7×Q8C2211yescharacteristic, complemented
18C2×#1157C2211yescharacteristic
19C22×D28C2211yescharacteristic
20C7×C4⋊Q8—21no—
21C7×C4⋊Q8C2211yescharacteristic, complemented
22C7×C22⋊Q8—21no—
23C7×C22⋊Q8—21no—
24C7×C22⋊Q8C2211yescharacteristic
25C7×C22⋊Q8C2211yescharacteristic, complemented
26C4×C7×Q8C2211yescharacteristic, complemented
27C4×C7×D4—21no—
28C4×C7×D4C2211yescharacteristic
29C2×#1144C2211yescharacteristic, complemented
30C2×#1144C2211yescharacteristic, complemented
31C2×#1144C2211yescharacteristic, complemented
32#6505—24no—
33C2×#1143—24no—
34C2×#1143C2211yescharacteristic
35C2×#1143C2211yescharacteristic
36C2×#1143C2211yescharacteristic
37C2×#1143C2211yescharacteristic
38C2×#1143C2211yescharacteristic
39C2×#1143C2211yescharacteristic
40#6500—21no—
41#6493C2211yescharacteristic
42#6487C2211yescharacteristic
43C2×#1136C2211yescharacteristic, complemented
44C22×C7⋊C8C2211yescharacteristic
45#6457—21no—
46#6447—21no—
47D7×C22⋊C4—21no—
48#6438C2211yescharacteristic
49C4×D28C2211yescharacteristic
50#6409C2211yescharacteristic, complemented
51#6409C2211yescharacteristic, complemented
52#6409C2211yescharacteristic, complemented
53C22⋊(C7⋊C8)C2211yescharacteristic
54C22⋊(C7⋊C8)C2211yescharacteristic, complemented
55#6384C2211yescharacteristic, complemented
56#6383C2211yescharacteristic
57#6383C2211yescharacteristic
58#6381C2211yescharacteristic, complemented
59C4⋊(C7⋊C8)C2211yescharacteristic, complemented
60#6377C2211yescharacteristic, complemented
61#3266—71nomaximal, Sylow
62C23×D7—21no—
63C2×C7×Q8—21no—
64C2×C7×Q8—21no—
65C2×C7×Q8—22no—
66C2×C7×Q8—22no—
67C2×C7×Q8—41no—
68C2×C7×Q8—41no—
69C2×C7×Q8C2311yescharacteristic
70C2×C7×Q8C2311yescharacteristic
71C2×C7×Q8C2311yescharacteristic
72C2×C7×D4D411yescharacteristic
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7C2311yescharacteristic
76C22×C4×C7C2311yescharacteristic
77C2×C22⋊D7—21no—
78C2×C22⋊D7—21no—
79#1157—22no—
80#1157—41no—
81C2×D28—22no—
82C2×D28—21no—
83C2×D28—41no—
84C2×D28C2311yescharacteristic
85C2×D28C2311yescharacteristic
86C2×C4×D7—21no—
87C2×C4×D7—21no—
88C2×Dic14C2311yescharacteristic
89C7×C4⋊C4—21no—
90C7×C4⋊C4—21no—
91C7×C4⋊C4—21no—
92C7×C4⋊C4—21no—
93C7×C4⋊C4—21no—
94C7×C4⋊C4D411yescharacteristic
95C7×C4⋊C4C2311yescharacteristic
96C7×C22⋊C4—21no—
97C7×C22⋊C4—21no—
98C7×C22⋊C4D411yescharacteristic
99C7×C22⋊C4D411yescharacteristic
100C42×C7—21no—
101C42×C7C2311yescharacteristic
102C22⋊Dic7—21no—
103#1144—22no—
104#1144—22no—
105#1144—41no—
106#1143—22no—
107#1143—22no—
108#1143—22no—
109#1143—22no—
110#1143—41no—
111#1143D414yescomplemented
112(C2×D7)⋊C4—21no—
113(C2×D7)⋊C4—21no—
114(C2×D7)⋊C4—21no—
115(C2×D7)⋊C4—21no—
116C4⋊Dic7C2311yescharacteristic
117Dic7⋊C4—21no—
118#1136—22no—
119C2×C7⋊C8—22no—
120C2×C7⋊C8C2311yescharacteristic
121C2×C7⋊C8C2311yescharacteristic
122C2×C7⋊C8C2311yescharacteristic
123C2×C7⋊C8C2311yescharacteristic
124C2×#137—71no—
125C22×SD16—71no—
126D4×Q8D711yescharacteristic, complemented
127#546—71no—
128#497—71no—
129#468—71no—
130#467—71no—
131#464—71no—
132#461—71no—
133#452—71no—
134#451—71no—
135#450—71no—
136#448—71no—
137#440—71no—
138#435—71no—
139C23×C7C2×D411yescharacteristic
140C23×C7C2×D411yescharacteristic
141C22×D7—21no—
142C22×D7—21no—
143C22×D7—42no—
144C22×D7—42no—
145C22×D7—81no—
146C7×Q8—21no—
147C7×Q8—21no—
148C7×Q8—21no—
149C7×Q8—21no—
150C7×Q8—21no—
151C7×Q8—42no—
152C7×Q8—42no—
153C7×Q8C2×D412yes—
154C7×D4—22no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7—21no—
158C2×C4×C7—21no—
159C2×C4×C7—21no—
160C2×C4×C7—21no—
161C2×C4×C7—21no—
162C2×C4×C7—21no—
163C2×C4×C7—42no—
164C2×C4×C7—42no—
165C2×C4×C7C2×D411yescharacteristic
166C2×C4×C7C2×D411yescharacteristic
167C2×C4×C7C2×D411yescharacteristic
168C2×C4×C7C2×D411yescharacteristic
169C2×C4×C7C2411yescharacteristic
170C22⋊D7—42no—
171C22⋊D7—81no—
172C2×Dic7—21no—
173C2×Dic7—21no—
174D28—21no—
175D28—21no—
176D28—21no—
177D28—42no—
178D28C2×D412yes—
179C4×D7—42no—
180C4×D7—81no—
181Dic14—21no—
182Dic14—21no—
183C7⋊C8—21no—
184C7⋊C8—21no—
185C7⋊C8—21no—
186C7⋊C8C2×D412yes—
187C2×#41—71no—
188C22×Q8C2×D711yescharacteristic
189C22×Q8C2×D711yescharacteristic, complemented
190C22×D4—71no—
191#137—144no—
192C2×Q16—71no—
193C2×Q16—71no—
194C2×Q16—71no—
195C2×SD16—71no—
196C2×SD16—71no—
197C2×SD16—71no—
198C2×SD16—71no—
199C2×SD16—71no—
200C2×SD16—71no—
201C2×SD16—144no—
202C2×M16—71no—
203C22×C8—71no—
204C4⋊Q8—21no—
205C4⋊Q8C2×D711yescharacteristic, complemented
206#124—71no—
207#123—141no—
208C22⋊Q8—21no—
209C22⋊Q8—21no—
210C22⋊Q8—71no—
211C22⋊Q8C2×D711yescharacteristic
212C22⋊Q8C2×D711yescharacteristic, complemented
213#121—71no—
214#121—141no—
215#120—141no—
216C4×Q8C2×D711yescharacteristic, complemented
217C4×D4—21no—
218C4×D4—71no—
219C4×D4C2×D711yescharacteristic
220C2×C22⋊C4—141no—
221#107—71no—
222C4⋊C8—71no—
223#103—71no—
224#103—71no—
225#103—71no—
226#103—71no—
227#102—71no—
228#102—71no—
229C22⋊C8—71no—
230C22⋊C8—71no—
231#97—71no—
232C22×C7—21no—
233C22×C7—21no—
234C22×C7—21no—
235C22×C7—21no—
236C22×C7—21no—
237C22×C7#13712yescomplemented
238C22×C7C22×D411yescharacteristic
239C2×D7—22no—
240C2×D7—41no—
241C2×D7—41no—
242C2×D7—41no—
243C2×D7—41no—
244C2×D7—41no—
245C2×D7—84no—
246C4×C7—21no—
247C4×C7—21no—
248C4×C7—22no—
249C4×C7—21no—
250C4×C7—21no—
251C4×C7—41no—
252C4×C7—41no—
253C4×C7—41no—
254C4×C7C22×D411yescharacteristic
255C4×C7H(5,𝔽2)11yescharacteristic
256Dic7—41no—
257Dic7—41no—
258C24—141no—
259#41—142no—
260#41—281no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—71no—
268C2×Q8C22×D711yescharacteristic
269C2×Q8C22×D711yescharacteristic
270C2×Q8C22×D711yescharacteristic
271C2×D4—71no—
272C2×D4—71no—
273C2×D4—142no—
274C2×D4—141no—
275C2×D4—141no—
276C2×D4—141no—
277C2×D4—281no—
278C2×D4C22⋊D711yescharacteristic
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—141no—
282C22×C4—141no—
283C22×C4C22×D711yescharacteristic
284C22×C4C22×D711yescharacteristic
285Q16—142no—
286Q16—142no—
287Q16—281no—
288SD16—74no—
289SD16—142no—
290SD16—142no—
291SD16—142no—
292SD16—142no—
293SD16—281no—
294M16—142no—
295C2×C8—71no—
296C2×C8—71no—
297C2×C8—71no—
298C2×C8—71no—
299C2×C8—142no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—71no—
306C4⋊C4—141no—
307C4⋊C4C22⋊D711yescharacteristic
308C4⋊C4C22×D711yescharacteristic
309C22⋊C4—21no—
310C22⋊C4—21no—
311C22⋊C4—141no—
312C22⋊C4—141no—
313C22⋊C4—141no—
314C22⋊C4—141no—
315C22⋊C4—141no—
316C22⋊C4C22⋊D711yescharacteristic
317C22⋊C4C22⋊D711yescharacteristic
318C42—21no—
319C42C22×D711yescharacteristic
320C2×C7—22no—
321C2×C7—41no—
322C2×C7D4211yescharacteristic
323C2×C7C2×#13711yescharacteristic
324C2×C7#57711yescharacteristic
325D7—42no—
326D7—81no—
327C23—141no—
328C23—141no—
329C23—282no—
330C23—282no—
331C23—561no—
332C23C2×C22⋊D711yescharacteristic
333C23C2×C22⋊D711yescharacteristic
334Q8—21no—
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—42no—
340Q8—42no—
341Q8—141no—
342Q8—141no—
343Q8D4×D712yes—
344D4—22no—
345D4—72no—
346D4—141no—
347D4—141no—
348D4—141no—
349D4—282no—
350D4—282no—
351D4—561no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—141no—
363C2×C4—141no—
364C2×C4—282no—
365C2×C4—561no—
366C2×C4C2×C22⋊D711yescharacteristic
367C2×C4C2×C22⋊D711yescharacteristic
368C2×C4C2×C22⋊D711yescharacteristic
369C2×C4C2×C22⋊D711yescharacteristic
370C2×C4C23×D711yescharacteristic
371C8—72no—
372C8—141no—
373C8—141no—
374C8—141no—
375C7#326611yescharacteristic, complemented, Sylow
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—142no—
382C22—281no—
383C22—281no—
384C22—281no—
385C22—281no—
386C22—281no—
387C22—564no—
388C22#650512yescomplemented
389C22C22×C22⋊D711yescharacteristic
390C4—21no—
391C4—21no—
392C4—22no—
393C4—21no—
394C4—21no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—281no—
399C4—281no—
400C4C2×D4×D711yescharacteristic
401C4#654811yescharacteristic
402C2—22no—
403C2—41no—
404C2—282no—
405C2—561no—
406C2D4×C22⋊D711yescharacteristic
407C2C2×#650511yescharacteristic
408C2#9052811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×D4×Q8C2
socleC22×C7C22×D4
prime radical#90241C2
hypercenterD4×Q8D7
hypocenterC7#3266
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C7×D4×Q8C2semidirect
2. C2×#6505C2semidirect
3. C22×#1143C2semidirect
5. #90000C2semidirect
6. #89999C2semidirect
7. #89858C2semidirect
8. #89855C2semidirect
9. #89823C2semidirect
10. #89821C2semidirect
11. #89819C2semidirect
12. #89818C2semidirect
13. #89800C2semidirect
14. #89798C2semidirect
15. #89783C2semidirect
17. C22×C7×Q8C22semidirect
21. C7×C4⋊Q8C22semidirect
25. C7×C22⋊Q8C22semidirect
26. C4×C7×Q8C22semidirect
29. C2×#1144C22semidirect
30. C2×#1144C22semidirect
31. C2×#1144C22semidirect
43. C2×#1136C22semidirect
50. #6409C22semidirect
51. #6409C22semidirect
52. #6409C22semidirect
54. C22⋊(C7⋊C8)C22semidirect
55. #6384C22semidirect
58. #6381C22semidirect
59. C4⋊(C7⋊C8)C22semidirect
60. #6377C22semidirect
111. #1143D4semidirect
126. D4×Q8D7semidirect
189. C22×Q8C2×D7semidirect
205. C4⋊Q8C2×D7semidirect
212. C22⋊Q8C2×D7semidirect
216. C4×Q8C2×D7semidirect
237. C22×C7#137semidirect
375. C7#3266semidirect
388. C22#6505semidirect

Maximal subgroups

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

subgroupindex
C7×D4×Q82
C2×#65052
C22×#11432
#902412
#900002
#899992
#898582
#898552
#898232
#898212
#898192
#898182
#898002
#897982
#897832
#32667