Finite Group Databasebeta

← #11737300

Subgroups — #11737300

Order 896.

374 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90230C211yesmaximal, characteristic, complemented
2D4×Dic14C211yesmaximal, characteristic, complemented
3C7×#442C211yesmaximal, characteristic
4#89924C211yesmaximal, characteristic
5D8×Dic7C211yesmaximal, characteristic
6#89781C211yesmaximal, characteristic, complemented
7#89780C211yesmaximal, characteristic, complemented
8#89628C211yesmaximal, characteristic, complemented
9#89627C211yesmaximal, characteristic, complemented
10#89538C211yesmaximal, characteristic, complemented
11#89533C211yesmaximal, characteristic, complemented
12#89533C211yesmaximal, characteristic, complemented
13#89531C211yesmaximal, characteristic, complemented
14#89474C211yesmaximal, characteristic, complemented
15#89473C211yesmaximal, characteristic, complemented
16C22×Dic14—21no—
17C2×C7×D8C2211yescharacteristic
18C4×C7×D4C2211yescharacteristic
19C4×C7×D4C2211yescharacteristic
20D4×Dic7C2211yescharacteristic
21D4×Dic7C2211yescharacteristic
22D4×Dic7C2211yescharacteristic
23D4×Dic7C2211yescharacteristic
24#6487—21no—
25#6487—21no—
26C2×Dic7⋊C4—21no—
27#6453C2211yescharacteristic
28#6451C2211yescharacteristic
29#6451C2211yescharacteristic
30#6451C2211yescharacteristic
31#6441—21no—
32#6441—21no—
33#6441—21no—
34#6441—21no—
35#6433C2211yescharacteristic
36#6432C2211yescharacteristic
37C4×Dic14C2211yescharacteristic
38C7×#106C2211yescharacteristic
39C7×#102C2211yescharacteristic
40C7×#102C2211yescharacteristic
41C7×#97C2211yescharacteristic
42#6406C2211yescharacteristic
43#6406C2211yescharacteristic
44#6406C2211yescharacteristic
45#6406C2211yescharacteristic
46#6392C2211yescharacteristic
47#6392C2211yescharacteristic
48#6391C2211yescharacteristic
49#6391C2211yescharacteristic
50#6389C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52Dic7⋊C8C2211yescharacteristic
53C8×Dic7C2211yescharacteristic
54#6382C2211yescharacteristic
55#6382C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3384—71nomaximal, Sylow
60C2×C7×D4C2311yescharacteristic
61C2×C7×D4C2311yescharacteristic
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C22×Dic7—21no—
65C22×Dic7—21no—
66C22×Dic7—21no—
67C22×Dic7—21no—
68C2×Dic14—21no—
69C2×Dic14—21no—
70C2×Dic14—21no—
71C2×Dic14—21no—
72C2×Dic14—44no—
73C2×Dic14D411yescharacteristic
74C7×D8Q814yescomplemented
75C2×C7×C8C2311yescharacteristic
76C2×C7×C8C2311yescharacteristic
77C7×C4⋊C4C2311yescharacteristic
78C7×C4⋊C4C2311yescharacteristic
79C7×C22⋊C4—21no—
80C7×C22⋊C4—21no—
81C42×C7C2311yescharacteristic
82C22⋊Dic7—21no—
83C22⋊Dic7—21no—
84C22⋊Dic7—21no—
85C22⋊Dic7—21no—
86C4⋊Dic7—21no—
87C4⋊Dic7—21no—
88C4⋊Dic7D411yescharacteristic
89C4⋊Dic7C2311yescharacteristic
90C4⋊Dic7C2311yescharacteristic
91C4⋊Dic7C2311yescharacteristic
92C4⋊Dic7C2311yescharacteristic
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4D411yescharacteristic
100Dic7⋊C4D411yescharacteristic
101C4×Dic7C2311yescharacteristic
102C4×Dic7C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105#554—71no—
106D4×Q8—71no—
107#501—71no—
108#501—71no—
109#498—71no—
110#478—71no—
111#478—71no—
112#476—71no—
113#476—71no—
114#476—71no—
115#474—71no—
116#446—71no—
117#442—71no—
118#442D711yescharacteristic
119C4×D8—71no—
120C23×C7—21no—
121C23×C7—21no—
122C7×D4—21no—
123C7×D4—21no—
124C7×D4C2×Q812yes—
125C7×D4C2×Q812yescomplemented
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—42no—
129C2×C4×C7—81no—
130C2×C4×C7C2×D411yescharacteristic
131C2×C4×C7C2×D411yescharacteristic
132C2×C4×C7C2411yescharacteristic
133C2×Dic7—21no—
134C2×Dic7—21no—
135C2×Dic7—21no—
136C2×Dic7—21no—
137C2×Dic7—42no—
138C2×Dic7—42no—
139C2×Dic7—42no—
140C2×Dic7—81no—
141C2×Dic7C2×D411yescharacteristic
142C2×Dic7C2×D411yescharacteristic
143C2×Dic7C2×D411yescharacteristic
144C2×Dic7C2×D411yescharacteristic
145Dic14—22no—
146Dic14—42no—
147Dic14—42no—
148Dic14—42no—
149Dic14—81no—
150C7×C8—21no—
151C7×C8C2×Q812yescomplemented
152C7⋊C8—21no—
153C7⋊C8—21no—
154C22×Q8—141no—
155C2×D8C2×D711yescharacteristic
156C4⋊Q8—71no—
157C4⋊Q8—71no—
158C4⋊Q8—71no—
159C4⋊Q8—71no—
160#125—71no—
161#125—71no—
162C22⋊Q8—141no—
163C22⋊Q8—141no—
164C22⋊Q8—141no—
165C22⋊Q8—141no—
166C22⋊Q8—141no—
167C22⋊Q8—141no—
168C4×Q8—71no—
169C4×D4—71no—
170C4×D4—71no—
171C4×D4—71no—
172C4×D4—71no—
173C4×D4C2×D711yescharacteristic
174C4×D4C2×D711yescharacteristic
175C2×C4⋊C4—141no—
176#107—71no—
177#107—71no—
178#107—71no—
179#106—71no—
180#106—71no—
181#106—71no—
182#106—71no—
183#106—71no—
184#106C2×D711yescharacteristic
185C4⋊C8—71no—
186C4⋊C8—71no—
187C4⋊C8—71no—
188#102—71no—
189#102—71no—
190#102—71no—
191#102—71no—
192#102C2×D711yescharacteristic
193#102C2×D711yescharacteristic
194#97—71no—
195#97C2×D711yescharacteristic
196C4×C8—71no—
197C22×C7—22no—
198C22×C7—22no—
199C22×C7—41no—
200C22×C7—41no—
201C22×C7—41no—
202C22×C7—41no—
203C22×C7C22×D411yescharacteristic
204C4×C7—21no—
205C4×C7—21no—
206C4×C7—41no—
207C4×C7—41no—
208C4×C7C22×Q811yescharacteristic
209C4×C72−1+411yescharacteristic
210Dic7—21no—
211Dic7—21no—
212Dic7—21no—
213Dic7—41no—
214Dic7—41no—
215Dic7—41no—
216Dic7—41no—
217Dic7AGL(1,ℤ/8)12yes—
218C2×Q8—71no—
219C2×Q8—141no—
220C2×Q8—141no—
221C2×Q8—141no—
222C2×Q8—141no—
223C2×Q8—284no—
224C2×D4C22×D711yescharacteristic
225C2×D4C22×D711yescharacteristic
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—141no—
229C22×C4—141no—
230C22×C4—141no—
231C22×C4—141no—
232D8Dic1414yescomplemented
233C2×C8—71no—
234C2×C8—71no—
235C2×C8C22×D711yescharacteristic
236C2×C8C22×D711yescharacteristic
237C4⋊C4—71no—
238C4⋊C4—71no—
239C4⋊C4—71no—
240C4⋊C4—71no—
241C4⋊C4—71no—
242C4⋊C4—71no—
243C4⋊C4—71no—
244C4⋊C4—141no—
245C4⋊C4—141no—
246C4⋊C4—141no—
247C4⋊C4—141no—
248C4⋊C4—141no—
249C4⋊C4—141no—
250C4⋊C4—141no—
251C4⋊C4—141no—
252C4⋊C4C22×D711yescharacteristic
253C4⋊C4C22×D711yescharacteristic
254C22⋊C4—21no—
255C22⋊C4—21no—
256C22⋊C4—141no—
257C22⋊C4—141no—
258C22⋊C4—141no—
259C22⋊C4—141no—
260C42—71no—
261C42—71no—
262C42C22×D711yescharacteristic
263C2×C7—42no—
264C2×C7—42no—
265C2×C7D4×Q811yescharacteristic
266C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
267C2×C7#57711yescharacteristic
268C23—21no—
269C23—21no—
270Q8—142no—
271Q8—282no—
272Q8—282no—
273Q8—282no—
274Q8—561no—
275D4—21no—
276D4—21no—
277D4C2×Dic1412yes—
278D4C2×Dic1412yescomplemented
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—42no—
282C2×C4—71no—
283C2×C4—71no—
284C2×C4—71no—
285C2×C4—71no—
286C2×C4—81no—
287C2×C4—141no—
288C2×C4—141no—
289C2×C4—141no—
290C2×C4—141no—
291C2×C4—282no—
292C2×C4—282no—
293C2×C4—282no—
294C2×C4—561no—
295C2×C4D4×D711yescharacteristic
296C2×C4D4×D711yescharacteristic
297C2×C4C23×D711yescharacteristic
298C8—21no—
299C8—141no—
300C8—141no—
301C8C2×Dic1412yescomplemented
302C7#338411yescharacteristic, complemented, Sylow
303C22—22no—
304C22—22no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22C2×D4×D711yescharacteristic
310C4—21no—
311C4—21no—
312C4—41no—
313C4—41no—
314C4—72no—
315C4—141no—
316C4—141no—
317C4—141no—
318C4—281no—
319C4—281no—
320C4—281no—
321C4—281no—
322C4C22×Dic1411yescharacteristic
323C4#655411yescharacteristic
324C2—42no—
325C2—42no—
326C2D4×Dic1411yescharacteristic
327C2D7×AGL(1,ℤ/8)11yescharacteristic
328C2#9046611yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#442C2
socleC22×C7C22×D4
prime radicalC2×C7×D8C22
hypercenter#442D7
hypocenterC7#3384
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #90230C2semidirect
2. D4×Dic14C2semidirect
6. #89781C2semidirect
7. #89780C2semidirect
8. #89628C2semidirect
9. #89627C2semidirect
10. #89538C2semidirect
11. #89533C2semidirect
12. #89533C2semidirect
13. #89531C2semidirect
14. #89474C2semidirect
15. #89473C2semidirect
74. C7×D8Q8semidirect
125. C7×D4C2×Q8semidirect
151. C7×C8C2×Q8semidirect
232. D8Dic14semidirect
278. D4C2×Dic14semidirect
301. C8C2×Dic14semidirect
302. C7#3384semidirect

Maximal subgroups

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

subgroupindex
#902302
D4×Dic142
C7×#4422
#899242
D8×Dic72
#897812
#897802
#896282
#896272
#895382
#895332
#895312
#894742
#894732
#33847