Finite Group Databasebeta

← #22592146

Subgroups — #22592146

Order 9088.

648 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14556104C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊D71C211yesmaximal, characteristic
3C2×C4⋊C4×D71C211yesmaximal, characteristic, complemented
4#17985935C211yesmaximal, characteristic, complemented
5#17985929C211yesmaximal, characteristic, complemented
6#17985922C211yesmaximal, characteristic, complemented
7#17985918C211yesmaximal, characteristic, complemented
8#17985894C211yesmaximal, characteristic, complemented
9#17985885C211yesmaximal, characteristic, complemented
10#17985875C211yesmaximal, characteristic
11#17985875C211yesmaximal, characteristic
12#17985872C211yesmaximal, characteristic, complemented
13#17985870C211yesmaximal, characteristic, complemented
14#17985870C211yesmaximal, characteristic, complemented
15C2×C22⋊Q8×C71C211yesmaximal, characteristic, complemented
16C22×C22⋊D71C2211yescharacteristic
17C22×C4×D71C2211yescharacteristic
18C22×C4×D71C2211yescharacteristic
19C22×C4×D71C2211yescharacteristic
20#14556104—24no—
21C4×C22⋊D71—24no—
22C2×C4⋊Dic71C2211yescharacteristic
23C2×C4⋊Dic71C2211yescharacteristic
24C2×Dic71⋊C4C2211yescharacteristic
25C2×Dic71⋊C4C2211yescharacteristic
26C2×Dic71⋊C4C2211yescharacteristic
27C2×Dic71⋊C4C2211yescharacteristic
28C2×Dic71⋊C4C2211yescharacteristic
29C2×Dic71⋊C4C2211yescharacteristic
30C2×Dic71⋊C4C2211yescharacteristic
31C4⋊C4×D71—24no—
32C2×C22⋊Dic71C2211yescharacteristic
33C2×C22⋊Dic71C2211yescharacteristic
34C2×C22⋊Dic71C2211yescharacteristic
35C2×(C2×D71)⋊C4C2211yescharacteristic
36C2×(C2×D71)⋊C4C2211yescharacteristic
37C2×(C2×D71)⋊C4C2211yescharacteristic
38C2×C4×Dic71C2211yescharacteristic
39C71⋊#95C2211yescharacteristic
40C71⋊#95C2211yescharacteristic
41C71⋊#95C2211yescharacteristic
42C71⋊#95C2211yescharacteristic
43C71⋊#95C2211yescharacteristic
44C71⋊#95C2211yescharacteristic
45C71⋊#95C2211yescharacteristic
46C71⋊#95C2211yescharacteristic
47C22×Q8×C71C2211yescharacteristic
48C23×C4×C71C2211yescharacteristic
49C22⋊Q8×C71—24no—
50C2×C4⋊C4×C71C2211yescharacteristic
51C2×C4⋊C4×C71C2211yescharacteristic
52C2×C4⋊C4×C71C2211yescharacteristic
53C2×C22⋊C4×C71C2211yescharacteristic
54C2×C22⋊C4×C71C2211yescharacteristic
55C23×D71C2311yescharacteristic
56C2×C22⋊D71—22no—
57C2×C22⋊D71—22no—
58C2×C22⋊D71Q814yescomplemented
59C22×Dic71C2311yescharacteristic
60C22×Dic71C2311yescharacteristic
61C22×Dic71C2311yescharacteristic
62C22×Dic71C2311yescharacteristic
63C22×Dic71C2311yescharacteristic
64C22×Dic71C2311yescharacteristic
65C22×Dic71C2311yescharacteristic
66C2×C4×D71—22no—
67C2×C4×D71—22no—
68C2×C4×D71—22no—
69C2×C4×D71—22no—
70C2×C4×D71—22no—
71C2×C4×D71—22no—
72C2×C4×D71—22no—
73C2×C4×D71—41no—
74C2×C4×D71—41no—
75C4⋊Dic71—22no—
76C4⋊Dic71—22no—
77Dic71⋊C4—22no—
78Dic71⋊C4—22no—
79Dic71⋊C4—22no—
80Dic71⋊C4—22no—
81Dic71⋊C4—22no—
82Dic71⋊C4—22no—
83Dic71⋊C4—41no—
84C22⋊Dic71—22no—
85C22⋊Dic71—22no—
86C22⋊Dic71—41no—
87(C2×D71)⋊C4—22no—
88(C2×D71)⋊C4—22no—
89(C2×D71)⋊C4D414yes—
90C4×Dic71—22no—
91C24×C71C2311yescharacteristic
92C2×Q8×C71—22no—
93C2×Q8×C71—22no—
94C2×Q8×C71—41no—
95C22×C4×C71—22no—
96C22×C4×C71—22no—
97C22×C4×C71—41no—
98C22×C4×C71C2311yescharacteristic
99C22×C4×C71C2311yescharacteristic
100C22×C4×C71C2311yescharacteristic
101C22×C4×C71C2311yescharacteristic
102C22×C4×C71C2311yescharacteristic
103C22×C4×C71C2311yescharacteristic
104C4⋊C4×C71—22no—
105C4⋊C4×C71—22no—
106C4⋊C4×C71D414yescomplemented
107C22⋊C4×C71—22no—
108C22⋊C4×C71—22no—
109C23×C71—21no—
110C23×C71—21no—
111C23×C71—21no—
112C23×C71—21no—
113C23×C71—21no—
114C23×C71—21no—
115C23×C71C2×Q812yes—
116C23×C71C2411yescharacteristic
117C22×D71—21no—
118C22×D71—21no—
119C22×D71—21no—
120C22×D71—21no—
121C22×D71C2×D412yes—
122C22×D71C2×Q812yes—
123C22×D71#4112yes—
124Q8×C71—44no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—21no—
133C2×C4×C71—21no—
134C2×C4×C71—21no—
135C2×C4×C71—21no—
136C2×C4×C71—21no—
137C2×C4×C71—21no—
138C2×C4×C71—21no—
139C2×C4×C71—44no—
140C2×C4×C71C2×D412yes—
141C2×C4×C71C2×D412yes—
142C2×C4×C71C2×D412yes—
143C2×C4×C71C2×D412yes—
144C22⋊D71—28no—
145C2×Dic71—21no—
146C2×Dic71—21no—
147C2×Dic71—21no—
148C2×Dic71—21no—
149C2×Dic71—21no—
150C2×Dic71—21no—
151C2×Dic71—21no—
152C2×Dic71—21no—
153C2×Dic71—21no—
154C2×Dic71—21no—
155C2×Dic71—21no—
156C2×Dic71—21no—
157C2×Dic71—21no—
158C2×Dic71—21no—
159C2×Dic71—21no—
160C2×Dic71—21no—
161C2×Dic71—21no—
162C2×Dic71—21no—
163C2×Dic71C2×D412yes—
164C2×Dic71C2×Q812yes—
165C2×Dic71#4112yes—
166C4×D71—28no—
167C4×D71—44no—
168C4×D71—44no—
169C22×C71—22no—
170C22×C71—22no—
171C22×C71—22no—
172C22×C71—41no—
173C22×C71—41no—
174C22×C71—41no—
175C22×C71—41no—
176C22×C71C22×D411yescharacteristic
177C22×C71C22×D411yescharacteristic
178C22×C71C22×Q811yescharacteristic
179C22×C71C2×#4111yescharacteristic
180C22×C71C2×#4111yescharacteristic
181C22×C71H(5,𝔽2)11yescharacteristic
182C22×C712−1+411yescharacteristic
183C2×D71—22no—
184C2×D71—22no—
185C2×D71—22no—
186C2×D71—22no—
187C2×D71—22no—
188C2×D71—22no—
189C2×D71C22⋊Q814yescomplemented
190C4×C71—22no—
191C4×C71—22no—
192C4×C71—22no—
193C4×C71—22no—
194C4×C71—41no—
195C4×C71—41no—
196Dic71—22no—
197Dic71—22no—
198Dic71—22no—
199Dic71—41no—
200Dic71—41no—
201Dic71—41no—
202Dic71—41no—
203C2×C71—42no—
204C2×C71C2×C22⋊Q811yescharacteristic
205C2×C71#52511yescharacteristic
206C2×C71#53911yescharacteristic
207C2×C71D4211yescharacteristic
208C2×C71#54711yescharacteristic
209C2×C71D4×Q811yescharacteristic
210C2×C71#55411yescharacteristic
211D71—24no—
212#2449—711nomaximal, Sylow
213C71#244911yescharacteristic, complemented, Sylow
214C2×C22⋊Q8—711no—
215C2×C22⋊Q8D7111yescharacteristic, complemented
216C2×C4×D4—711no—
217C22×C4⋊C4—711no—
218#399—711no—
219#398—711no—
220#396—711no—
221#395—711no—
222#389—711no—
223#387—711no—
224#386—711no—
225#386—711no—
226#385—711no—
227#385—711no—
228#385—711no—
229C22×Q8C2×D7111yescharacteristic
230C22×D4—711no—
231C23×C4—711no—
232C23×C4—711no—
233C23×C4—711no—
234C23×C4C2×D7111yescharacteristic
235C22⋊Q8—24no—
236C22⋊Q8—1424no—
237C4×D4—1424no—
238C2×C4⋊C4—711no—
239C2×C4⋊C4—711no—
240C2×C4⋊C4—711no—
241C2×C4⋊C4—711no—
242C2×C4⋊C4—711no—
243C2×C4⋊C4—711no—
244C2×C4⋊C4—711no—
245C2×C4⋊C4—711no—
246C2×C4⋊C4—711no—
247C2×C4⋊C4—1424no—
248C2×C4⋊C4C2×D7111yescharacteristic
249C2×C4⋊C4C2×D7111yescharacteristic
250C2×C4⋊C4C2×D7111yescharacteristic
251C2×C22⋊C4—711no—
252C2×C22⋊C4—711no—
253C2×C22⋊C4—711no—
254C2×C22⋊C4—711no—
255C2×C22⋊C4—711no—
256C2×C22⋊C4—711no—
257C2×C22⋊C4C2×D7111yescharacteristic
258C2×C22⋊C4C2×D7111yescharacteristic
259C2×C42—711no—
260#95—711no—
261#95—711no—
262#95—711no—
263#95—711no—
264#95—711no—
265#95—711no—
266#95—711no—
267#95—711no—
268C24—711no—
269C24C22×D7111yescharacteristic
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—41no—
273C2×D4—714no—
274C2×D4—1422no—
275C2×D4—1422no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—41no—
279C22×C4—2841no—
280C22×C4—2841no—
281C22×C4—711no—
282C22×C4—711no—
283C22×C4—711no—
284C22×C4—711no—
285C22×C4—711no—
286C22×C4—711no—
287C22×C4—711no—
288C22×C4—1422no—
289C22×C4—1422no—
290C22×C4—1422no—
291C22×C4—1422no—
292C22×C4—1422no—
293C22×C4—1422no—
294C22×C4—1422no—
295C22×C4C22×D7111yescharacteristic
296C22×C4C22×D7111yescharacteristic
297C22×C4C22×D7111yescharacteristic
298C22×C4C22×D7111yescharacteristic
299C22×C4C22×D7111yescharacteristic
300C22×C4C22×D7111yescharacteristic
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—2841no—
304C4⋊C4—1422no—
305C4⋊C4—1422no—
306C4⋊C4—1422no—
307C4⋊C4—1422no—
308C4⋊C4—1422no—
309C4⋊C4—1422no—
310C4⋊C4—1422no—
311C4⋊C4—1422no—
312C4⋊C4C22⋊D7114yescomplemented
313C22⋊C4—22no—
314C22⋊C4—22no—
315C22⋊C4—2841no—
316C22⋊C4—714no—
317C22⋊C4—1422no—
318C22⋊C4—1422no—
319C22⋊C4—1422no—
320C22⋊C4—1422no—
321C42—1422no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—712no—
329C23—712no—
330C23—712no—
331C23—1421no—
332C23—1421no—
333C23—1421no—
334C23—1421no—
335C23Q8×D7112yes—
336C23C23×D7111yescharacteristic
337Q8—44no—
338D4—1428no—
339C2×C4—21no—
340C2×C4—21no—
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—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—44no—
354C2×C4—2844no—
355C2×C4—2844no—
356C2×C4—712no—
357C2×C4—712no—
358C2×C4—712no—
359C2×C4—1428no—
360C2×C4—1421no—
361C2×C4—1421no—
362C2×C4—1421no—
363C2×C4—1421no—
364C2×C4—1421no—
365C2×C4—1421no—
366C2×C4—1421no—
367C2×C4—1421no—
368C2×C4—1421no—
369C2×C4—1421no—
370C2×C4—1421no—
371C2×C4—1421no—
372C2×C4—1421no—
373C2×C4—1421no—
374C2×C4—1421no—
375C2×C4—1421no—
376C2×C4—1421no—
377C2×C4—1421no—
378C2×C4D4×D7112yes—
379C2×C4C2×C22⋊D7112yes—
380C2×C4C2×C22⋊D7112yes—
381C2×C4C2×C22⋊D7112yes—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—714no—
390C22—1422no—
391C22—1422no—
392C22—1422no—
393C22—1422no—
394C22—1422no—
395C22—1422no—
396C22C2×D4×D7111yescharacteristic
397C22C22×C22⋊D7111yescharacteristic
398C22C2×Q8×D7111yescharacteristic
399C22#41×D7111yescharacteristic
400C22#41×D7111yescharacteristic
401C22#1455617311yescharacteristic
402C22#1455617611yescharacteristic
403C4—22no—
404C4—22no—
405C4—22no—
406C4—22no—
407C4—41no—
408C4—41no—
409C4—2841no—
410C4—2841no—
411C4—2841no—
412C4—2841no—
413C4—1422no—
414C4—1422no—
415C4—1422no—
416C2—42no—
417C2—1424no—
418C2C22⋊Q8×D7111yescharacteristic
419C2#1798650611yescharacteristic
420C2#1798659511yescharacteristic
421C2D4×C22⋊D7111yescharacteristic
422C2#1798665111yescharacteristic
423C2Q8×C22⋊D7111yescharacteristic
424C2#1798670311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
FittingC2×C22⋊Q8×C71C2
hypercenterC2×C22⋊Q8D71
hypocenterC71#2449
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#14556104C2semidirect
3. C2×C4⋊C4×D71C2semidirect
4. #17985935C2semidirect
5. #17985929C2semidirect
6. #17985922C2semidirect
7. #17985918C2semidirect
8. #17985894C2semidirect
9. #17985885C2semidirect
12. #17985872C2semidirect
13. #17985870C2semidirect
14. #17985870C2semidirect
15. C2×C22⋊Q8×C71C2semidirect
58. C2×C22⋊D71Q8semidirect
106. C4⋊C4×C71D4semidirect
189. C2×D71C22⋊Q8semidirect
213. C71#2449semidirect
215. C2×C22⋊Q8D71semidirect
312. C4⋊C4C22⋊D71semidirect