Finite Group Databasebeta

← #11741010

Subgroups — #11741010

Order 896.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×D4×Q8C211yesmaximal, characteristic, complemented
2C22×#1144C211yesmaximal, characteristic, complemented
3C2×#6505C211yesmaximal, characteristic, complemented
4#90243C211yesmaximal, characteristic
5#90000C211yesmaximal, characteristic, complemented
6#89999C211yesmaximal, characteristic, complemented
7#89862C211yesmaximal, characteristic, complemented
8#89858C211yesmaximal, characteristic, complemented
9#89824C211yesmaximal, characteristic, complemented
10#89822C211yesmaximal, characteristic, complemented
11#89820C211yesmaximal, characteristic, complemented
12#89819C211yesmaximal, characteristic, complemented
13#89802C211yesmaximal, characteristic, complemented
14#89800C211yesmaximal, characteristic, complemented
15#89783C211yesmaximal, characteristic, complemented
16C22×C7×Q8C2211yescharacteristic
17C22×C7×Q8C2211yescharacteristic
18C2×#1157C2211yescharacteristic
19C22×Dic14C2211yescharacteristic
20C7×C4⋊Q8—21no—
21C7×C4⋊Q8C2211yescharacteristic
22C7×C22⋊Q8—21no—
23C7×C22⋊Q8—21no—
24C7×C22⋊Q8C2211yescharacteristic
25C7×C22⋊Q8C2211yescharacteristic
26C4×C7×Q8C2211yescharacteristic
27C4×C7×D4—21no—
28C4×C7×D4C2211yescharacteristic
29C2×#1144—24no—
30C2×#1144C2211yescharacteristic
31C2×#1144C2211yescharacteristic
32C2×#1144C2211yescharacteristic
33C2×#1144C2211yescharacteristic
34C2×#1144C2211yescharacteristic
35C2×#1144C2211yescharacteristic
36#6505—24no—
37C2×#1143C2211yescharacteristic
38C2×#1143C2211yescharacteristic
39C2×#1143C2211yescharacteristic
40#6502—21no—
41#6493C2211yescharacteristic
42#6487C2211yescharacteristic
43C2×#1136C2211yescharacteristic
44C22×C7⋊C8C2211yescharacteristic
45#6459—21no—
46#6448—21no—
47(C22⋊D7)⋊C4—21no—
48#6438C2211yescharacteristic
49C4×Dic14C2211yescharacteristic
50#6409C2211yescharacteristic
51#6409C2211yescharacteristic
52#6409C2211yescharacteristic
53C22⋊(C7⋊C8)C2211yescharacteristic
54C22⋊(C7⋊C8)C2211yescharacteristic
55#6384C2211yescharacteristic
56#6384C2211yescharacteristic
57#6383C2211yescharacteristic
58#6382C2211yescharacteristic
59C4⋊(C7⋊C8)C2211yescharacteristic
60#6377C2211yescharacteristic
61#3270—71nomaximal, Sylow
62C2×C7×Q8—21no—
63C2×C7×Q8—21no—
64C2×C7×Q8—22no—
65C2×C7×Q8—22no—
66C2×C7×Q8—41no—
67C2×C7×Q8—41no—
68C2×C7×Q8C2311yescharacteristic
69C2×C7×Q8C2311yescharacteristic
70C2×C7×Q8C2311yescharacteristic
71C2×C7×D4D411yescharacteristic
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7C2311yescharacteristic
75C22×C4×C7C2311yescharacteristic
76C2×C22⋊D7—21no—
77C2×C22⋊D7—21no—
78C22×Dic7—21no—
79#1157—22no—
80#1157—41no—
81C2×D28C2311yescharacteristic
82C2×C4×D7—21no—
83C2×Dic14—22no—
84C2×Dic14—21no—
85C2×Dic14—41no—
86C2×Dic14C2311yescharacteristic
87C2×Dic14C2311yescharacteristic
88C7×C4⋊C4—21no—
89C7×C4⋊C4—21no—
90C7×C4⋊C4—21no—
91C7×C4⋊C4—21no—
92C7×C4⋊C4—21no—
93C7×C4⋊C4D411yescharacteristic
94C7×C4⋊C4C2311yescharacteristic
95C7×C22⋊C4—21no—
96C7×C22⋊C4—21no—
97C7×C22⋊C4D411yescharacteristic
98C7×C22⋊C4D411yescharacteristic
99C42×C7—21no—
100C42×C7C2311yescharacteristic
101C22⋊Dic7—21no—
102#1144—22no—
103#1144—22no—
104#1144—22no—
105#1144—22no—
106#1144—41no—
107#1144D414yescomplemented
108#1143—22no—
109#1143—22no—
110#1143—41no—
111(C2×D7)⋊C4—21no—
112(C2×D7)⋊C4—21no—
113(C2×D7)⋊C4—21no—
114C4⋊Dic7C2311yescharacteristic
115Dic7⋊C4—21no—
116Dic7⋊C4—21no—
117C4×Dic7—21no—
118#1136—22no—
119C2×C7⋊C8—22no—
120C2×C7⋊C8C2311yescharacteristic
121C2×C7⋊C8C2311yescharacteristic
122C2×C7⋊C8C2311yescharacteristic
123C2×C7⋊C8C2311yescharacteristic
124C2×#137—71no—
125C22×Q16—71no—
126D4×Q8D711yescharacteristic, complemented
127#548—71no—
128#497—71no—
129#470—71no—
130#465—71no—
131#464—71no—
132#462—71no—
133#452—71no—
134#451—71no—
135#451—71no—
136#448—71no—
137#441—71no—
138#435—71no—
139C23×C7C2×D411yescharacteristic
140C23×C7C2×D411yescharacteristic
141C22×D7—21no—
142C7×Q8—21no—
143C7×Q8—21no—
144C7×Q8—21no—
145C7×Q8—21no—
146C7×Q8—21no—
147C7×Q8—42no—
148C7×Q8—42no—
149C7×Q8C2×D412yes—
150C7×D4—22no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7—21no—
158C2×C4×C7—21no—
159C2×C4×C7—42no—
160C2×C4×C7—42no—
161C2×C4×C7C2×D411yescharacteristic
162C2×C4×C7C2×D411yescharacteristic
163C2×C4×C7C2×D411yescharacteristic
164C2×C4×C7C2×D411yescharacteristic
165C2×C4×C7C2411yescharacteristic
166C22⋊D7—42no—
167C22⋊D7—42no—
168C2×Dic7—21no—
169C2×Dic7—21no—
170C2×Dic7—21no—
171C2×Dic7—42no—
172D28—21no—
173D28—21no—
174C4×D7—42no—
175Dic14—21no—
176Dic14—21no—
177Dic14—21no—
178Dic14—42no—
179Dic14C2×D412yes—
180C7⋊C8—21no—
181C7⋊C8—21no—
182C7⋊C8—21no—
183C7⋊C8C2×D412yes—
184C2×#41—71no—
185C22×Q8—71no—
186C22×Q8C2×D711yescharacteristic
187C22×Q8C2×D711yescharacteristic
188#137—144no—
189C2×Q16—71no—
190C2×Q16—71no—
191C2×Q16—71no—
192C2×Q16—71no—
193C2×Q16—71no—
194C2×Q16—71no—
195C2×Q16—144no—
196C2×SD16—71no—
197C2×SD16—71no—
198C2×SD16—71no—
199C2×M16—71no—
200C22×C8—71no—
201C4⋊Q8—21no—
202C4⋊Q8C2×D711yescharacteristic
203#124—71no—
204#124—141no—
205C22⋊Q8—21no—
206C22⋊Q8—21no—
207C22⋊Q8—71no—
208C22⋊Q8—141no—
209C22⋊Q8C2×D711yescharacteristic
210C22⋊Q8C2×D711yescharacteristic
211#121—71no—
212#121—141no—
213C4×Q8—71no—
214C4×Q8C2×D711yescharacteristic
215C4×D4—21no—
216C4×D4—141no—
217C4×D4C2×D711yescharacteristic
218#106—71no—
219C4⋊C8—71no—
220#103—71no—
221#103—71no—
222#103—71no—
223#103—71no—
224#103—71no—
225#102—71no—
226C22⋊C8—71no—
227C22⋊C8—71no—
228#97—71no—
229C22×C7—21no—
230C22×C7—21no—
231C22×C7—21no—
232C22×C7—21no—
233C22×C7—21no—
234C22×C7#13712yescomplemented
235C22×C7C22×D411yescharacteristic
236C2×D7—41no—
237C2×D7—41no—
238C2×D7—41no—
239C4×C7—21no—
240C4×C7—21no—
241C4×C7—22no—
242C4×C7—21no—
243C4×C7—21no—
244C4×C7—41no—
245C4×C7—41no—
246C4×C7—41no—
247C4×C7C22×D411yescharacteristic
248C4×C7H(5,𝔽2)11yescharacteristic
249Dic7—22no—
250Dic7—41no—
251Dic7—41no—
252#41—142no—
253#41—281no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—71no—
261C2×Q8—71no—
262C2×Q8—142no—
263C2×Q8—141no—
264C2×Q8—281no—
265C2×Q8C22×D711yescharacteristic
266C2×Q8C22×D711yescharacteristic
267C2×Q8C22×D711yescharacteristic
268C2×D4—71no—
269C2×D4—141no—
270C2×D4—141no—
271C2×D4C22⋊D711yescharacteristic
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—141no—
275C22×C4—141no—
276C22×C4C22×D711yescharacteristic
277C22×C4C22×D711yescharacteristic
278Q16—74no—
279Q16—142no—
280Q16—142no—
281Q16—142no—
282Q16—142no—
283Q16—281no—
284SD16—142no—
285SD16—142no—
286SD16—281no—
287M16—142no—
288C2×C8—71no—
289C2×C8—71no—
290C2×C8—71no—
291C2×C8—71no—
292C2×C8—142no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—71no—
299C4⋊C4—141no—
300C4⋊C4—141no—
301C4⋊C4C22⋊D711yescharacteristic
302C4⋊C4C22×D711yescharacteristic
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—141no—
306C22⋊C4—141no—
307C22⋊C4—141no—
308C22⋊C4—141no—
309C22⋊C4C22⋊D711yescharacteristic
310C22⋊C4C22⋊D711yescharacteristic
311C42—21no—
312C42—141no—
313C42C22×D711yescharacteristic
314C2×C7—22no—
315C2×C7—41no—
316C2×C7D4211yescharacteristic
317C2×C7C2×#13711yescharacteristic
318C2×C7#57811yescharacteristic
319D7—81no—
320C23—141no—
321C23C2×C22⋊D711yescharacteristic
322C23C2×C22⋊D711yescharacteristic
323Q8—21no—
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—42no—
329Q8—42no—
330Q8—72no—
331Q8—141no—
332Q8—141no—
333Q8—141no—
334Q8—282no—
335Q8D4×D712yes—
336D4—22no—
337D4—141no—
338D4—141no—
339D4—282no—
340D4—282no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—141no—
352C2×C4—141no—
353C2×C4—141no—
354C2×C4—282no—
355C2×C4—282no—
356C2×C4C2×C22⋊D711yescharacteristic
357C2×C4C2×C22⋊D711yescharacteristic
358C2×C4C2×C22⋊D711yescharacteristic
359C2×C4C2×C22⋊D711yescharacteristic
360C2×C4C23×D711yescharacteristic
361C8—72no—
362C8—141no—
363C8—141no—
364C8—141no—
365C7#327011yescharacteristic, complemented, Sylow
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—281no—
372C22—281no—
373C22—281no—
374C22#650512yescomplemented
375C22C22×C22⋊D711yescharacteristic
376C4—21no—
377C4—21no—
378C4—22no—
379C4—21no—
380C4—21no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—142no—
385C4—281no—
386C4—281no—
387C4C2×D4×D711yescharacteristic
388C4#654811yescharacteristic
389C2—22no—
390C2—41no—
391C2—561no—
392C2D4×C22⋊D711yescharacteristic
393C2C2×#650511yescharacteristic
394C2#9052911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×D4×Q8C2
socleC22×C7C22×D4
prime radical#90243C2
hypercenterD4×Q8D7
hypocenterC7#3270
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. C22×#1144C2semidirect
3. C2×#6505C2semidirect
5. #90000C2semidirect
6. #89999C2semidirect
7. #89862C2semidirect
8. #89858C2semidirect
9. #89824C2semidirect
10. #89822C2semidirect
11. #89820C2semidirect
12. #89819C2semidirect
13. #89802C2semidirect
14. #89800C2semidirect
15. #89783C2semidirect
107. #1144D4semidirect
126. D4×Q8D7semidirect
234. C22×C7#137semidirect
365. C7#3270semidirect
374. 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
C22×#11442
C2×#65052
#902432
#900002
#899992
#898622
#898582
#898242
#898222
#898202
#898192
#898022
#898002
#897832
#32707