Finite Group Databasebeta

← #46847928

Subgroups — #46847928

Order 34688.

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
1C2×Q16×D271C211yesmaximal, characteristic, complemented
2Q8×C22⋊D271C211yesmaximal, characteristic, complemented
3#31774554C211yesmaximal, characteristic
4#31774288C211yesmaximal, characteristic, complemented
5#31774155C211yesmaximal, characteristic, complemented
6#31774123C211yesmaximal, characteristic, complemented
7#31774117C211yesmaximal, characteristic, complemented
8#31774105C211yesmaximal, characteristic, complemented
9#31774059C211yesmaximal, characteristic, complemented
10#31774053C211yesmaximal, characteristic, complemented
11#31774051C211yesmaximal, characteristic, complemented
12#31774034C211yesmaximal, characteristic, complemented
13#31774004C211yesmaximal, characteristic, complemented
14#31773966C211yesmaximal, characteristic, complemented
15C2×#137×C271C211yesmaximal, characteristic, complemented
16C2×Q8×D271C2211yescharacteristic
17C2×Q8×D271C2211yescharacteristic
18C2×Dic1084C2211yescharacteristic
19C2×#16864644C2211yescharacteristic
20C2×#16864644C2211yescharacteristic
21C2×#16864644C2211yescharacteristic
22Q16×D271—24no—
23C2×#16864642C2211yescharacteristic
24C2×C8×D271C2211yescharacteristic
25#22576158—21no—
26#22576158C2211yescharacteristic
27#22576147—21no—
28#22576147C2211yescharacteristic
29#22576139—21no—
30#22576139—21no—
31#22576139C2211yescharacteristic
32#22576139C2211yescharacteristic
33#22576136—21no—
34#22576136C2211yescharacteristic
35#22576131—21no—
36#22576131C2211yescharacteristic
37Q8×Dic271C2211yescharacteristic
38Q8×Dic271C2211yescharacteristic
39C4×C22⋊D271—21no—
40C4×C22⋊D271—21no—
41C4×C22⋊D271C2211yescharacteristic
42#22576092C2211yescharacteristic
43Dic271⋊C8C2211yescharacteristic
44#22576085C2211yescharacteristic
45#22576085C2211yescharacteristic
46#22576084C2211yescharacteristic
47#22576084C2211yescharacteristic
48#22576084C2211yescharacteristic
49#22576081C2211yescharacteristic
50C22⋊(C271⋊C8)C2211yescharacteristic
51(C2×D271)⋊C8C2211yescharacteristic
52#22576070C2211yescharacteristic
53C2×#41×C271C2211yescharacteristic
54C22×Q8×C271C2211yescharacteristic
55#137×C271—24no—
56C2×Q16×C271C2211yescharacteristic
57C2×Q16×C271C2211yescharacteristic
58C2×SD16×C271C2211yescharacteristic
59C2×SD16×C271C2211yescharacteristic
60C2×M16×C271C2211yescharacteristic
61C2×Dic542—21no—
62C2×Dic542—21no—
63C2×Dic542C2311yescharacteristic
64C2×Dic542C2311yescharacteristic
65Q8×D271—22no—
66Q8×D271—22no—
67Q8×D271—41no—
68Q8×D271—41no—
69C2×C22⋊D271—21no—
70C2×C22⋊D271D411yescharacteristic
71C2×C4×D271—21no—
72C2×C4×D271—21no—
73C2×C4×D271C2311yescharacteristic
74Dic1084—22no—
75#16864644—22no—
76#16864644—22no—
77#16864644—41no—
78#16864642—41no—
79C2×C271⋊C8C2311yescharacteristic
80C2×C271⋊C8C2311yescharacteristic
81C8×D271—22no—
82C4⋊Dic271—21no—
83C4⋊Dic271—21no—
84C4⋊Dic271C2311yescharacteristic
85C4⋊Dic271C2311yescharacteristic
86Dic271⋊C4—21no—
87Dic271⋊C4—21no—
88Dic271⋊C4—21no—
89Dic271⋊C4—21no—
90Dic271⋊C4—21no—
91Dic271⋊C4D411yescharacteristic
92C22⋊Dic271—21no—
93C22⋊Dic271—21no—
94C22⋊Dic271—21no—
95C22⋊Dic271—21no—
96C22⋊Dic271—21no—
97C22⋊Dic271D411yescharacteristic
98(C2×D271)⋊C4—21no—
99(C2×D271)⋊C4D411yescharacteristic
100C4×Dic271—21no—
101C4×Dic271—21no—
102C4×Dic271C2311yescharacteristic
103#41×C271—22no—
104#41×C271—41no—
105C2×Q8×C271—21no—
106C2×Q8×C271—22no—
107C2×Q8×C271—41no—
108C2×Q8×C271C2311yescharacteristic
109C2×Q8×C271C2311yescharacteristic
110C2×Q8×C271C2311yescharacteristic
111C2×D4×C271—21no—
112C2×D4×C271C2311yescharacteristic
113C22×C4×C271—21no—
114C22×C4×C271—21no—
115C22×C4×C271C2311yescharacteristic
116Q16×C271—22no—
117Q16×C271D414yescomplemented
118SD16×C271—22no—
119SD16×C271—22no—
120M16×C271—22no—
121C2×C8×C271C2311yescharacteristic
122C2×C8×C271C2311yescharacteristic
123C22×D271C2×D411yescharacteristic
124Dic542—21no—
125Dic542—21no—
126Dic542—21no—
127Dic542—21no—
128Dic542—42no—
129Dic542—42no—
130C22⋊D271—22no—
131C22⋊D271—42no—
132C2×Dic271—21no—
133C2×Dic271—21no—
134C2×Dic271—21no—
135C2×Dic271—21no—
136C2×Dic271C2×D411yescharacteristic
137C2×Dic271C2×D411yescharacteristic
138C2×Dic271C2×D411yescharacteristic
139C4×D271—21no—
140C4×D271—21no—
141C4×D271—42no—
142C4×D271—42no—
143C271⋊C8—21no—
144C271⋊C8—21no—
145C23×C271—21no—
146C23×C271C2×D411yescharacteristic
147Q8×C271—21no—
148Q8×C271—21no—
149Q8×C271—21no—
150Q8×C271—21no—
151Q8×C271—42no—
152Q8×C271C2×D412yes—
153Q8×C271C2×D412yes—
154D4×C271—21no—
155D4×C271—21no—
156D4×C271—42no—
157C2×C4×C271—21no—
158C2×C4×C271—21no—
159C2×C4×C271—21no—
160C2×C4×C271—21no—
161C2×C4×C271—21no—
162C2×C4×C271—42no—
163C2×C4×C271—42no—
164C2×C4×C271C2×D411yescharacteristic
165C2×C4×C271C2411yescharacteristic
166C8×C271—21no—
167C8×C271C2×D412yes—
168C22×C271—21no—
169C22×C271—21no—
170C22×C271—21no—
171C22×C271—41no—
172C22×C271—41no—
173C22×C271—41no—
174C22×C271C22×D411yescharacteristic
175C2×D271—21no—
176C2×D271—21no—
177C2×D271#13712yescomplemented
178C4×C271—21no—
179C4×C271—22no—
180C4×C271—22no—
181C4×C271—41no—
182C4×C271C22×D411yescharacteristic
183C4×C271H(5,𝔽2)11yescharacteristic
184Dic271—21no—
185Dic271—21no—
186Dic271—21no—
187Dic271—41no—
188Dic271—41no—
189Dic271—41no—
190Dic271—41no—
191C2×C271—41no—
192C2×C271—81no—
193C2×C271D4211yescharacteristic
194C2×C271C2×#13711yescharacteristic
195C2×C271#57811yescharacteristic
196D271—22no—
197C271#327011yescharacteristic, complemented, Sylow
198#3270—2711nomaximal, Sylow
199C2×#137D27111yescharacteristic, complemented
200C22×Q16—2711no—
201D4×Q8—2711no—
202#548—2711no—
203#497—2711no—
204#470—2711no—
205#465—2711no—
206#464—2711no—
207#462—2711no—
208#452—2711no—
209#451—2711no—
210#451—2711no—
211#448—2711no—
212#441—2711no—
213#435—2711no—
214C2×#41C2×D27111yescharacteristic
215C22×Q8—2711no—
216C22×Q8—2711no—
217C22×Q8C2×D27111yescharacteristic
218#137—24no—
219C2×Q16—2711no—
220C2×Q16—2711no—
221C2×Q16—2711no—
222C2×Q16—2711no—
223C2×Q16—5424no—
224C2×Q16C2×D27111yescharacteristic
225C2×Q16C2×D27111yescharacteristic
226C2×SD16—2711no—
227C2×SD16C2×D27111yescharacteristic
228C2×SD16C2×D27111yescharacteristic
229C2×M16C2×D27111yescharacteristic
230C22×C8—2711no—
231C4⋊Q8—2711no—
232C4⋊Q8—5421no—
233#124—2711no—
234#124—5421no—
235C22⋊Q8—2711no—
236C22⋊Q8—2711no—
237C22⋊Q8—2711no—
238C22⋊Q8—5421no—
239C22⋊Q8—5421no—
240C22⋊Q8—5421no—
241#121—2711no—
242#121—5421no—
243C4×Q8—2711no—
244C4×Q8—2711no—
245C4×D4—2711no—
246C4×D4—5421no—
247C4×D4—5421no—
248#106—2711no—
249C4⋊C8—2711no—
250#103—2711no—
251#103—2711no—
252#103—2711no—
253#103—2711no—
254#103—2711no—
255#102—2711no—
256C22⋊C8—2711no—
257C22⋊C8—2711no—
258#97—2711no—
259#41—22no—
260#41—41no—
261C2×Q8—21no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—2711no—
265C2×Q8—2711no—
266C2×Q8—5421no—
267C2×Q8—5421no—
268C2×Q8—5422no—
269C2×Q8—5422no—
270C2×Q8—10841no—
271C2×Q8—10841no—
272C2×Q8C22×D27111yescharacteristic
273C2×Q8C22×D27111yescharacteristic
274C2×Q8C22×D27111yescharacteristic
275C2×D4—21no—
276C2×D4—2711no—
277C2×D4—5421no—
278C2×D4C22×D27111yescharacteristic
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—2711no—
282C22×C4—5421no—
283C22×C4—5421no—
284C22×C4C22×D27111yescharacteristic
285Q16—22no—
286Q16—5422no—
287Q16—5422no—
288Q16—5422no—
289Q16—10841no—
290Q16C22⋊D27114yescomplemented
291SD16—22no—
292SD16—22no—
293SD16—10841no—
294M16—22no—
295C2×C8—2711no—
296C2×C8—2711no—
297C2×C8—5422no—
298C2×C8C22×D27111yescharacteristic
299C2×C8C22×D27111yescharacteristic
300C4⋊C4—2711no—
301C4⋊C4—2711no—
302C4⋊C4—2711no—
303C4⋊C4—5421no—
304C4⋊C4—5421no—
305C4⋊C4—5421no—
306C4⋊C4—5421no—
307C4⋊C4—5421no—
308C4⋊C4—5421no—
309C4⋊C4—5421no—
310C22⋊C4—2711no—
311C22⋊C4—2711no—
312C22⋊C4—5421no—
313C22⋊C4—5421no—
314C22⋊C4—5421no—
315C22⋊C4—5421no—
316C22⋊C4—5421no—
317C22⋊C4—5421no—
318C42—2711no—
319C42—5421no—
320C42—5421no—
321C23—21no—
322C23—2711no—
323C23D4×D27111yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—42no—
329Q8—5421no—
330Q8—5421no—
331Q8—5421no—
332Q8—5421no—
333Q8—10842no—
334Q8—10842no—
335Q8C2×C22⋊D27112yes—
336Q8C2×C22⋊D27112yes—
337D4—21no—
338D4—21no—
339D4—42no—
340D4—5422no—
341D4—10842no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—2711no—
350C2×C4—2711no—
351C2×C4—2711no—
352C2×C4—5421no—
353C2×C4—5421no—
354C2×C4—5421no—
355C2×C4—5421no—
356C2×C4—5421no—
357C2×C4—5421no—
358C2×C4—10842no—
359C2×C4—10842no—
360C2×C4D4×D27111yescharacteristic
361C2×C4C23×D27111yescharacteristic
362C8—21no—
363C8—5421no—
364C8—5421no—
365C8C2×C22⋊D27112yes—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—2712no—
373C22—5421no—
374C22—5421no—
375C22C2×D4×D27111yescharacteristic
376C4—21no—
377C4—22no—
378C4—22no—
379C4—41no—
380C4—5421no—
381C4—5421no—
382C4—5421no—
383C4—10841no—
384C4—10841no—
385C4—10841no—
386C4—10841no—
387C4C22×C22⋊D27111yescharacteristic
388C4#2257620511yescharacteristic
389C2—41no—
390C2—81no—
391C2—5422no—
392C2D4×C22⋊D27111yescharacteristic
393C2#137×D27111yescharacteristic
394C2#3177471311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC2×#137×C271C2
socleC22×C271C22×D4
prime radical#31774554C2
hypercenterC2×#137D271
hypocenterC271#3270
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D271C2semidirect
2. Q8×C22⋊D271C2semidirect
4. #31774288C2semidirect
5. #31774155C2semidirect
6. #31774123C2semidirect
7. #31774117C2semidirect
8. #31774105C2semidirect
9. #31774059C2semidirect
10. #31774053C2semidirect
11. #31774051C2semidirect
12. #31774034C2semidirect
13. #31774004C2semidirect
14. #31773966C2semidirect
15. C2×#137×C271C2semidirect
117. Q16×C271D4semidirect
177. C2×D271#137semidirect
197. C271#3270semidirect
199. C2×#137D271semidirect
290. Q16C22⋊D271semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D2712
Q8×C22⋊D2712
#317745542
#317742882
#317741552
#317741232
#317741172
#317741052
#317740592
#317740532
#317740512
#317740342
#317740042
#317739662
C2×#137×C2712
#3270271