Finite Group Databasebeta

← #46847934

Subgroups — #46847934

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×#22576192C211yesmaximal, characteristic, complemented
2Q8×C22⋊D271C211yesmaximal, characteristic, complemented
3#31774558C211yesmaximal, characteristic
4#31774290C211yesmaximal, characteristic, complemented
5#31774152C211yesmaximal, characteristic, complemented
6#31774126C211yesmaximal, characteristic, complemented
7#31774121C211yesmaximal, characteristic, complemented
8#31774105C211yesmaximal, characteristic, complemented
9#31774058C211yesmaximal, characteristic, complemented
10#31774053C211yesmaximal, characteristic, complemented
11#31774051C211yesmaximal, characteristic, complemented
12#31774036C211yesmaximal, characteristic, complemented
13#31774004C211yesmaximal, characteristic, complemented
14#31773968C211yesmaximal, characteristic, complemented
15C22×Q16×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×Q8×D271C2211yescharacteristic
18#22576192—24no—
19C2×#16864644C2211yescharacteristic
20C2×#16864644C2211yescharacteristic
21C2×#16864643C2211yescharacteristic
22C2×#16864641C2211yescharacteristic
23C2×#16864641C2211yescharacteristic
24C2×#16864637C2211yescharacteristic
25#22576158—21no—
26#22576158C2211yescharacteristic
27#22576149—21no—
28#22576149C2211yescharacteristic
29#22576139—21no—
30#22576139C2211yescharacteristic
31#22576136—21no—
32#22576136—21no—
33#22576136C2211yescharacteristic
34#22576136C2211yescharacteristic
35#22576134—21no—
36#22576134C2211yescharacteristic
37Q8×Dic271C2211yescharacteristic
38Q8×Dic271C2211yescharacteristic
39C4×C22⋊D271—21no—
40C4×C22⋊D271—21no—
41C4×C22⋊D271C2211yescharacteristic
42#22576092C2211yescharacteristic
43Dic271⋊C8C2211yescharacteristic
44#22576085C2211yescharacteristic
45#22576084C2211yescharacteristic
46#22576084C2211yescharacteristic
47#22576084C2211yescharacteristic
48#22576084C2211yescharacteristic
49#22576082C2211yescharacteristic
50C22⋊(C271⋊C8)C2211yescharacteristic
51(C2×D271)⋊C8C2211yescharacteristic
52#22576070C2211yescharacteristic
53C22×Q8×C271C2211yescharacteristic
54C22×Q8×C271C2211yescharacteristic
55C2×Q16×C271—24no—
56C2×Q16×C271C2211yescharacteristic
57C2×Q16×C271C2211yescharacteristic
58C2×Q16×C271C2211yescharacteristic
59C2×Q16×C271C2211yescharacteristic
60C22×C8×C271C2211yescharacteristic
61#16864655—22no—
62#16864655—41no—
63C2×Dic542—21no—
64C2×Dic542C2311yescharacteristic
65Q8×D271—22no—
66Q8×D271—41no—
67C2×D1084—21no—
68C2×D1084C2311yescharacteristic
69C2×C22⋊D271—21no—
70C2×C22⋊D271D411yescharacteristic
71C2×C4×D271—21no—
72C2×C4×D271—21no—
73C2×C4×D271C2311yescharacteristic
74#16864644—22no—
75#16864644—41no—
76#16864643—22no—
77#16864641—22no—
78#16864641—41no—
79#16864637—22no—
80C2×C271⋊C8C2311yescharacteristic
81C2×C271⋊C8C2311yescharacteristic
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⋊Dic271D411yescharacteristic
94(C2×D271)⋊C4—21no—
95(C2×D271)⋊C4—21no—
96(C2×D271)⋊C4—21no—
97(C2×D271)⋊C4—21no—
98(C2×D271)⋊C4—21no—
99(C2×D271)⋊C4D411yescharacteristic
100C4×Dic271—21no—
101C4×Dic271—21no—
102C4×Dic271C2311yescharacteristic
103C2×Q8×C271—21no—
104C2×Q8×C271—21no—
105C2×Q8×C271—22no—
106C2×Q8×C271—22no—
107C2×Q8×C271—41no—
108C2×Q8×C271—41no—
109C2×Q8×C271C2311yescharacteristic
110C2×Q8×C271C2311yescharacteristic
111C2×Q8×C271C2311yescharacteristic
112C2×Q8×C271C2311yescharacteristic
113C22×C4×C271—21no—
114C22×C4×C271—21no—
115C22×C4×C271C2311yescharacteristic
116Q16×C271—22no—
117Q16×C271—22no—
118Q16×C271—22no—
119Q16×C271D414yescomplemented
120C2×C8×C271—22no—
121C2×C8×C271C2311yescharacteristic
122C2×C8×C271C2311yescharacteristic
123C22×D271—21no—
124C22×D271C2×D411yescharacteristic
125Dic542—21no—
126Dic542—21no—
127Dic542—42no—
128D1084—21no—
129D1084—21no—
130D1084—42no—
131C22⋊D271—22no—
132C22⋊D271—42no—
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×C271C2×D411yescharacteristic
146Q8×C271—21no—
147Q8×C271—21no—
148Q8×C271—21no—
149Q8×C271—21no—
150Q8×C271—21no—
151Q8×C271—21no—
152Q8×C271—42no—
153Q8×C271—42no—
154Q8×C271C2×D412yes—
155Q8×C271C2×D412yes—
156C2×C4×C271—21no—
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#13712yescomplemented
171C22×C271C22×D411yescharacteristic
172C2×D271—21no—
173C2×D271—21no—
174C2×D271—21no—
175C2×D271—41no—
176C2×D271—41no—
177C2×D271—41no—
178C4×C271—21no—
179C4×C271—22no—
180C4×C271—22no—
181C4×C271—41no—
182C4×C271—41no—
183C4×C271C22×D411yescharacteristic
184C4×C271H(5,𝔽2)11yescharacteristic
185Dic271—21no—
186Dic271—21no—
187Dic271—21no—
188Dic271—41no—
189Dic271—41no—
190Dic271—41no—
191C2×C271—22no—
192C2×C271D4211yescharacteristic
193C2×C271C2×#13711yescharacteristic
194C2×C271#57811yescharacteristic
195D271—41no—
196D271—81no—
197C271#327011yescharacteristic, complemented, Sylow
198#3270—2711nomaximal, Sylow
199C2×#137—2711no—
200C22×Q16D27111yescharacteristic, complemented
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×#41—2711no—
215C22×Q8—2711no—
216C22×Q8C2×D27111yescharacteristic
217C22×Q8C2×D27111yescharacteristic
218#137—5424no—
219C2×Q16—24no—
220C2×Q16—2711no—
221C2×Q16—2711no—
222C2×Q16C2×D27111yescharacteristic
223C2×Q16C2×D27111yescharacteristic
224C2×Q16C2×D27111yescharacteristic
225C2×Q16C2×D27111yescharacteristic
226C2×SD16—2711no—
227C2×SD16—2711no—
228C2×SD16—2711no—
229C2×M16—2711no—
230C22×C8C2×D27111yescharacteristic
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—5422no—
260#41—10841no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—2711no—
268C2×Q8—5421no—
269C2×Q8—5422no—
270C2×Q8—10841no—
271C2×Q8C22×D27111yescharacteristic
272C2×Q8C22×D27111yescharacteristic
273C2×Q8C22×D27111yescharacteristic
274C2×Q8C22×D27111yescharacteristic
275C2×D4—2711no—
276C2×D4—2711no—
277C2×D4—5421no—
278C2×D4—5421no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—2711no—
282C22×C4—5421no—
283C22×C4—5421no—
284C22×C4C22×D27111yescharacteristic
285Q16—22no—
286Q16—22no—
287Q16—22no—
288Q16—5422no—
289Q16—10841no—
290Q16C22⋊D27114yescomplemented
291SD16—5422no—
292SD16—5422no—
293SD16—10841no—
294M16—5422no—
295C2×C8—22no—
296C2×C8—2711no—
297C2×C8—2711no—
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—2711no—
322C23—5421no—
323C23D4×D27111yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—21no—
330Q8—42no—
331Q8—42no—
332Q8—5421no—
333Q8—5421no—
334Q8—10842no—
335Q8C2×C22⋊D27112yes—
336Q8C2×C22⋊D27112yes—
337D4—5422no—
338D4—5421no—
339D4—5421no—
340D4—10842no—
341D4—10842no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—2711no—
351C2×C4—2711no—
352C2×C4—2711no—
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—5421no—
369C22—5421no—
370C22—5421no—
371C22—10841no—
372C22—10841no—
373C22—10841no—
374C22#2257619212yescomplemented
375C22C2×D4×D27111yescharacteristic
376C4—21no—
377C4—22no—
378C4—22no—
379C4—41no—
380C4—41no—
381C4—5421no—
382C4—5421no—
383C4—5421no—
384C4—10841no—
385C4—10841no—
386C4—10841no—
387C4C22×C22⋊D27111yescharacteristic
388C4#2257620511yescharacteristic
389C2—22no—
390C2—10841no—
391C2—21681no—
392C2D4×C22⋊D27111yescharacteristic
393C2C2×#2257619211yescharacteristic
394C2#3177471011yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC22×Q16×C271C2
socleC22×C271C22×D4
prime radical#31774558C2
hypercenterC22×Q16D271
hypocenterC271#3270
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
1. C2×#22576192C2semidirect
2. Q8×C22⋊D271C2semidirect
4. #31774290C2semidirect
5. #31774152C2semidirect
6. #31774126C2semidirect
7. #31774121C2semidirect
8. #31774105C2semidirect
9. #31774058C2semidirect
10. #31774053C2semidirect
11. #31774051C2semidirect
12. #31774036C2semidirect
13. #31774004C2semidirect
14. #31773968C2semidirect
15. C22×Q16×C271C2semidirect
119. Q16×C271D4semidirect
170. C22×C271#137semidirect
197. C271#3270semidirect
200. C22×Q16D271semidirect
290. Q16C22⋊D271semidirect
374. C22#22576192semidirect

Maximal subgroups

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

subgroupindex
C2×#225761922
Q8×C22⋊D2712
#317745582
#317742902
#317741522
#317741262
#317741212
#317741052
#317740582
#317740532
#317740512
#317740362
#317740042
#317739682
C22×Q16×C2712
#3270271