Finite Group Databasebeta

← #46848003

Subgroups — #46848003

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
2C2×#22576191C211yesmaximal, characteristic, complemented
3Q8×C22⋊D271C211yesmaximal, characteristic, complemented
4#31774549C211yesmaximal, characteristic
5#31774275C211yesmaximal, characteristic, complemented
6#31774154C211yesmaximal, characteristic, complemented
7#31774153C211yesmaximal, characteristic, complemented
8#31774105C211yesmaximal, characteristic, complemented
9#31774101C211yesmaximal, characteristic, complemented
10#31774061C211yesmaximal, characteristic, complemented
11#31774058C211yesmaximal, characteristic, complemented
12#31774054C211yesmaximal, characteristic, complemented
13#31773982C211yesmaximal, characteristic, complemented
14#31773972C211yesmaximal, characteristic, complemented
15#451×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864654C2211yescharacteristic
18C2×Q8×D271C2211yescharacteristic
19#22576192—24no—
20#22576191—24no—
21C2×Dic1084C2211yescharacteristic
22C2×#16864644C2211yescharacteristic
23C2×#16864644C2211yescharacteristic
24C2×#16864644C2211yescharacteristic
25C2×#16864643C2211yescharacteristic
26C2×#16864641C2211yescharacteristic
27C2×#16864641C2211yescharacteristic
28C2×#16864638C2211yescharacteristic
29C2×#16864637C2211yescharacteristic
30#22576158—21no—
31#22576158C2211yescharacteristic
32#22576158C2211yescharacteristic
33#22576145—21no—
34#22576142—21no—
35#22576140C2211yescharacteristic
36#22576139—21no—
37#22576139C2211yescharacteristic
38#22576136—21no—
39#22576136C2211yescharacteristic
40#22576133—21no—
41Q8×Dic271C2211yescharacteristic
42#22576122C2211yescharacteristic
43C4×C22⋊D271—21no—
44C4×C22⋊D271C2211yescharacteristic
45C4⋊C4×D271—21no—
46#22576091C2211yescharacteristic
47Dic271⋊C8C2211yescharacteristic
48#22576085C2211yescharacteristic
49#22576084C2211yescharacteristic
50#22576083C2211yescharacteristic
51#22576082C2211yescharacteristic
52(C2×D271)⋊C8C2211yescharacteristic
53C8×Dic271C2211yescharacteristic
54C22×Q8×C271C2211yescharacteristic
55C2×Q16×C271C2211yescharacteristic
56C2×Q16×C271C2211yescharacteristic
57C22⋊Q8×C271C2211yescharacteristic
58#103×C271C2211yescharacteristic
59#103×C271C2211yescharacteristic
60C22⋊C8×C271C2211yescharacteristic
61#16864655—22no—
62#16864655—41no—
63#16864654—22no—
64#16864654—41no—
65C2×Dic542—21no—
66C2×Dic542C2311yescharacteristic
67C2×Dic542C2311yescharacteristic
68Q8×D271—22no—
69Q8×D271—41no—
70C2×D1084—21no—
71C2×D1084C2311yescharacteristic
72C2×C22⋊D271—21no—
73C2×C22⋊D271D411yescharacteristic
74C2×C4×D271—21no—
75C2×C4×D271—21no—
76C2×C4×D271—21no—
77C2×C4×D271—21no—
78C2×C4×D271C2311yescharacteristic
79Dic1084—41no—
80#16864644—22no—
81#16864644—22no—
82#16864644—22no—
83#16864643—22no—
84#16864641—22no—
85#16864641D414yescomplemented
86#16864638—22no—
87#16864637—22no—
88C2×C271⋊C8C2311yescharacteristic
89C2×C271⋊C8C2311yescharacteristic
90C4⋊Dic271—21no—
91C4⋊Dic271—21no—
92C4⋊Dic271C2311yescharacteristic
93Dic271⋊C4—21no—
94Dic271⋊C4—21no—
95Dic271⋊C4—21no—
96Dic271⋊C4—21no—
97Dic271⋊C4—21no—
98Dic271⋊C4—21no—
99Dic271⋊C4D411yescharacteristic
100C22⋊Dic271—21no—
101C22⋊Dic271D411yescharacteristic
102(C2×D271)⋊C4—21no—
103(C2×D271)⋊C4—21no—
104(C2×D271)⋊C4—21no—
105(C2×D271)⋊C4D411yescharacteristic
106C4×Dic271—21no—
107C4×Dic271C2311yescharacteristic
108C2×Q8×C271—21no—
109C2×Q8×C271—22no—
110C2×Q8×C271—41no—
111C2×Q8×C271C2311yescharacteristic
112C2×Q8×C271C2311yescharacteristic
113C2×Q8×C271C2311yescharacteristic
114C22×C4×C271—21no—
115C22×C4×C271C2311yescharacteristic
116Q16×C271—22no—
117Q16×C271—41no—
118C2×C8×C271C2311yescharacteristic
119C2×C8×C271C2311yescharacteristic
120C4⋊C4×C271—21no—
121C4⋊C4×C271C2311yescharacteristic
122C22⋊C4×C271—21no—
123C22×D271—21no—
124C22×D271C2×D411yescharacteristic
125Dic542—21no—
126Dic542—21no—
127Dic542—21no—
128Dic542—21no—
129Dic542—42no—
130D1084—21no—
131D1084—42no—
132D1084C2×D412yes—
133C22⋊D271—22no—
134C22⋊D271—42no—
135C2×Dic271—21no—
136C2×Dic271—21no—
137C2×Dic271—21no—
138C2×Dic271C2×D411yescharacteristic
139C2×Dic271C2×D411yescharacteristic
140C2×Dic271C2×D411yescharacteristic
141C4×D271—21no—
142C4×D271—21no—
143C4×D271—42no—
144C4×D271—42no—
145C4×D271—42no—
146C4×D271—42no—
147C271⋊C8—21no—
148C271⋊C8C2×D412yes—
149C23×C271C2×D411yescharacteristic
150Q8×C271—21no—
151Q8×C271—21no—
152Q8×C271—21no—
153Q8×C271—21no—
154Q8×C271—21no—
155Q8×C271—42no—
156Q8×C271C2×D412yes—
157C2×C4×C271—21no—
158C2×C4×C271—21no—
159C2×C4×C271—21no—
160C2×C4×C271—21no—
161C2×C4×C271—21no—
162C2×C4×C271—21no—
163C2×C4×C271—42no—
164C2×C4×C271C2×D411yescharacteristic
165C2×C4×C271C2411yescharacteristic
166C8×C271—21no—
167C8×C271—21no—
168C22×C271—21no—
169C22×C271—21no—
170C22×C271—21no—
171C22×C271C22×D411yescharacteristic
172C2×D271—21no—
173C2×D271—22no—
174C2×D271—21no—
175C2×D271—21no—
176C2×D271—41no—
177C2×D271—41no—
178C4×C271—21no—
179C4×C271—22no—
180C4×C271—41no—
181C4×C271—41no—
182C4×C271—41no—
183C4×C271C22×D411yescharacteristic
184C4×C271H(5,𝔽2)11yescharacteristic
185Dic271—21no—
186Dic271—21no—
187Dic271—41no—
188Dic271—41no—
189Dic271—41no—
190Dic271#13712yescomplemented
191C2×C271—41no—
192C2×C271D4211yescharacteristic
193C2×C271C2×#13711yescharacteristic
194C2×C271#57811yescharacteristic
195D271—41no—
196D271—42no—
197C271#327611yescharacteristic, complemented, Sylow
198#3276—2711nomaximal, Sylow
199C2×#137—2711no—
200C2×#137—2711no—
201D4×Q8—2711no—
202#547—2711no—
203#494—2711no—
204#470—2711no—
205#470—2711no—
206#462—2711no—
207#461—2711no—
208#452—2711no—
209#452—2711no—
210#451—2711no—
211#451D27111yescharacteristic, complemented
212C4×SD16—2711no—
213#436—2711no—
214C2×#41—2711no—
215C2×#41—2711no—
216C22×Q8—2711no—
217C22×Q8C2×D27111yescharacteristic
218#137—5424no—
219#137—5424no—
220C2×Q16—2711no—
221C2×Q16—2711no—
222C2×Q16—2711no—
223C2×Q16—2711no—
224C2×Q16C2×D27111yescharacteristic
225C2×Q16C2×D27111yescharacteristic
226C2×SD16—2711no—
227C2×SD16—2711no—
228C2×SD16—2711no—
229C2×M16—2711no—
230C2×M16—2711no—
231C4⋊Q8—2711no—
232C4⋊Q8—2711no—
233C4⋊Q8—5421no—
234#123—5421no—
235#123—5421no—
236C22⋊Q8—2711no—
237C22⋊Q8—2711no—
238C22⋊Q8—2711no—
239C22⋊Q8—5421no—
240C22⋊Q8—5421no—
241C22⋊Q8C2×D27111yescharacteristic
242#121—5421no—
243C4×Q8—2711no—
244C4×D4—2711no—
245C4×D4—2711no—
246C4×D4—5421no—
247C2×C4⋊C4—5421no—
248#106—2711no—
249C4⋊C8—2711no—
250#103—2711no—
251#103—2711no—
252#103—2711no—
253#103C2×D27111yescharacteristic
254#103C2×D27111yescharacteristic
255#102—2711no—
256C22⋊C8—2711no—
257C22⋊C8C2×D27111yescharacteristic
258C4×C8—2711no—
259#41—5422no—
260#41—5422no—
261#41—10841no—
262#41—10841no—
263C2×Q8—21no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—2711no—
267C2×Q8—2711no—
268C2×Q8—5421no—
269C2×Q8—5422no—
270C2×Q8—10841no—
271C2×Q8C22×D27111yescharacteristic
272C2×Q8C22×D27111yescharacteristic
273C2×Q8C22×D27111yescharacteristic
274C2×D4—2711no—
275C2×D4—2711no—
276C2×D4—5421no—
277C2×D4—5421no—
278C22×C4—21no—
279C22×C4—2711no—
280C22×C4—5421no—
281C22×C4—5421no—
282C22×C4—5421no—
283C22×C4—5421no—
284C22×C4C22×D27111yescharacteristic
285Q16—22no—
286Q16—41no—
287Q16—5422no—
288Q16—5422no—
289Q16—5422no—
290Q16—10841no—
291SD16—2714no—
292SD16—5422no—
293SD16—5422no—
294M16—5422no—
295M16—5422no—
296C2×C8—2711no—
297C2×C8—2711no—
298C2×C8C22×D27111yescharacteristic
299C2×C8C22×D27111yescharacteristic
300C4⋊C4—21no—
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—
310C4⋊C4—5421no—
311C4⋊C4C22×D27111yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—2711no—
314C22⋊C4—2711no—
315C22⋊C4—5421no—
316C22⋊C4—5421no—
317C22⋊C4—5421no—
318C22⋊C4—5421no—
319C42—2711no—
320C42—5421no—
321C23—2711no—
322C23—5421no—
323C23D4×D27111yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—42no—
330Q8—5421no—
331Q8—5421no—
332Q8—5421no—
333Q8—5421no—
334Q8—10842no—
335Q8D4×D27112yes—
336D4—2712no—
337D4—5422no—
338D4—5421no—
339D4—10842no—
340D4—10842no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—42no—
348C2×C4—2711no—
349C2×C4—2711no—
350C2×C4—2711no—
351C2×C4—5421no—
352C2×C4—5421no—
353C2×C4—5421no—
354C2×C4—5421no—
355C2×C4—5421no—
356C2×C4—10842no—
357C2×C4—10842no—
358C2×C4—10842no—
359C2×C4—10842no—
360C2×C4D4×D27111yescharacteristic
361C2×C4C23×D27111yescharacteristic
362C8—21no—
363C8—21no—
364C8—2712no—
365C8—5421no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—5421no—
370C22—5422no—
371C22—5421no—
372C22—5421no—
373C22—10841no—
374C22—10841no—
375C22C2×D4×D27111yescharacteristic
376C4—21no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—2712no—
382C4—5421no—
383C4—5421no—
384C4—10841no—
385C4—10841no—
386C4—10841no—
387C4C2×D4×D27111yescharacteristic
388C4#2257620511yescharacteristic
389C2—41no—
390C2—10841no—
391C2—10842no—
392C2#3177452911yescharacteristic
393C2#137×D27111yescharacteristic
394C2#3177471011yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#451×C271C2
socleC22×C271C22×D4
prime radical#31774549C2
hypercenter#451D271
hypocenterC271#3276
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C2×#22576192C2semidirect
2. C2×#22576191C2semidirect
3. Q8×C22⋊D271C2semidirect
5. #31774275C2semidirect
6. #31774154C2semidirect
7. #31774153C2semidirect
8. #31774105C2semidirect
9. #31774101C2semidirect
10. #31774061C2semidirect
11. #31774058C2semidirect
12. #31774054C2semidirect
13. #31773982C2semidirect
14. #31773972C2semidirect
15. #451×C271C2semidirect
85. #16864641D4semidirect
190. Dic271#137semidirect
197. C271#3276semidirect
211. #451D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761922
C2×#225761912
Q8×C22⋊D2712
#317745492
#317742752
#317741542
#317741532
#317741052
#317741012
#317740612
#317740582
#317740542
#317739822
#317739722
#451×C2712
#3276271