Finite Group Databasebeta

← #46848273

Subgroups — #46848273

Order 34688.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774597C211yesmaximal, characteristic, complemented
2#31774555C211yesmaximal, characteristic, complemented
3#31774251C211yesmaximal, characteristic, complemented
4#31774237C211yesmaximal, characteristic, complemented
5#31774214C211yesmaximal, characteristic, complemented
6#31774205C211yesmaximal, characteristic, complemented
7#31774130C211yesmaximal, characteristic, complemented
8#31774123C211yesmaximal, characteristic, complemented
9#31774039C211yesmaximal, characteristic, complemented
10#31774034C211yesmaximal, characteristic, complemented
11C4×#16864642C211yesmaximal, characteristic, complemented
12#31773962C211yesmaximal, characteristic, complemented
13#31773890C211yesmaximal, characteristic, complemented
14#103×D271C211yesmaximal, characteristic, complemented
15#463×C271C211yesmaximal, characteristic, complemented
16C2×#16864653C2211yescharacteristic
17C2×Q8×D271C2211yescharacteristic
18C2×#16864642C2211yescharacteristic
19C2×C8×D271C2211yescharacteristic
20C2×C8×D271C2211yescharacteristic
21#22576153—21no—
22#22576152C2211yescharacteristic
23#22576146—21no—
24#22576143—21no—
25#22576137—21no—
26#22576136C2211yescharacteristic
27#22576133—21no—
28#22576133—21no—
29#22576131C2211yescharacteristic
30#22576131C2211yescharacteristic
31#22576131C2211yescharacteristic
32C4×Dic542C2211yescharacteristic
33C4×D1084C2211yescharacteristic
34C4×C22⋊D271—21no—
35(C22⋊D271)⋊C4—21no—
36#22576118C2211yescharacteristic
37C4⋊C4×D271C2211yescharacteristic
38#22576094C2211yescharacteristic
39#22576092C2211yescharacteristic
40#22576091C2211yescharacteristic
41#22576085C2211yescharacteristic
42#22576085C2211yescharacteristic
43#22576084C2211yescharacteristic
44#22576084C2211yescharacteristic
45#22576083C2211yescharacteristic
46#22576081C2211yescharacteristic
47#22576081C2211yescharacteristic
48#22576081C2211yescharacteristic
49(C2×D271)⋊C8C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51C4×C271⋊C8C2211yescharacteristic
52C2×SD16×C271C2211yescharacteristic
53C2×D8×C271C2211yescharacteristic
54#124×C271C2211yescharacteristic
55C4×D4×C271C2211yescharacteristic
56C4⋊C8×C271C2211yescharacteristic
57#103×C271C2211yescharacteristic
58#102×C271C2211yescharacteristic
59#16864653—22no—
60#16864653—41no—
61C2×Dic542—21no—
62C2×Dic542C2311yescharacteristic
63Q8×D271—22no—
64Q8×D271—41no—
65C2×D1084D411yescharacteristic
66C2×C22⋊D271—21no—
67C2×C22⋊D271—21no—
68C2×C22⋊D271—21no—
69C2×C22⋊D271—21no—
70C22×Dic271—21no—
71C2×C4×D271—21no—
72C2×C4×D271—21no—
73C2×C4×D271C2311yescharacteristic
74C2×C4×D271C2311yescharacteristic
75#16864642—22no—
76C2×C271⋊C8C2311yescharacteristic
77C2×C271⋊C8C2311yescharacteristic
78C8×D271—41no—
79C8×D271—41no—
80C4⋊Dic271D411yescharacteristic
81C4⋊Dic271C2311yescharacteristic
82C4⋊Dic271C2311yescharacteristic
83C4⋊Dic271C2311yescharacteristic
84Dic271⋊C4—21no—
85Dic271⋊C4—21no—
86Dic271⋊C4—21no—
87Dic271⋊C4—21no—
88Dic271⋊C4—21no—
89C22⋊Dic271—21no—
90C22⋊Dic271—21no—
91C22⋊Dic271—21no—
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4—21no—
94(C2×D271)⋊C4D411yescharacteristic
95(C2×D271)⋊C4D411yescharacteristic
96C4×Dic271—21no—
97C4×Dic271—21no—
98C2×Q8×C271C2311yescharacteristic
99C2×D4×C271C2311yescharacteristic
100C2×D4×C271C2311yescharacteristic
101C22×C4×C271—21no—
102SD16×C271—41no—
103D8×C271—41no—
104C2×C8×C271C2311yescharacteristic
105C2×C8×C271C2311yescharacteristic
106C4⋊C4×C271C2311yescharacteristic
107C22⋊C4×C271—21no—
108C22⋊C4×C271—21no—
109C22⋊C4×C271—21no—
110C42×C271C2311yescharacteristic
111C22×D271C2×D411yescharacteristic
112C22×D271C2×D411yescharacteristic
113Dic542—21no—
114Dic542—42no—
115Dic542#4112yes—
116D1084—22no—
117C22⋊D271—42no—
118C22⋊D271—42no—
119C22⋊D271—42no—
120C22⋊D271—81no—
121C2×Dic271—21no—
122C2×Dic271—21no—
123C2×Dic271—21no—
124C2×Dic271—21no—
125C2×Dic271—42no—
126C2×Dic271C2×D411yescharacteristic
127C2×Dic271C2×D411yescharacteristic
128C4×D271—21no—
129C4×D271—21no—
130C4×D271—21no—
131C4×D271—21no—
132C4×D271—42no—
133C4×D271—81no—
134C271⋊C8—21no—
135C271⋊C8—21no—
136C23×C271—21no—
137C23×C271—21no—
138Q8×C271—21no—
139Q8×C271—21no—
140D4×C271—21no—
141D4×C271—21no—
142D4×C271—21no—
143D4×C271—21no—
144C2×C4×C271—21no—
145C2×C4×C271—21no—
146C2×C4×C271—81no—
147C2×C4×C271C2×D411yescharacteristic
148C2×C4×C271C2×D411yescharacteristic
149C2×C4×C271C2411yescharacteristic
150C8×C271—21no—
151C8×C271—21no—
152C22×C271—41no—
153C22×C271—41no—
154C22×C271—41no—
155C22×C271—41no—
156C22×C271—41no—
157C22×C271—41no—
158C22×C271C22×D411yescharacteristic
159C2×D271—21no—
160C2×D271—21no—
161C2×D271—21no—
162C2×D271—21no—
163C2×D271—21no—
164C2×D271#13512yes—
165C4×C271—21no—
166C4×C271—21no—
167C4×C271—41no—
168C4×C271—41no—
169C4×C271C2×#4111yescharacteristic
170C4×C271H(5,𝔽2)11yescharacteristic
171Dic271—21no—
172Dic271—21no—
173Dic271—22no—
174Dic271—41no—
175Dic271—41no—
176Dic271—41no—
177C2×C271—81no—
178C2×C271—81no—
179C2×C271#54611yescharacteristic
180C2×C271C2×#13511yescharacteristic
181C2×C271#57711yescharacteristic
182D271—22no—
183D271—41no—
184C271#330111yescharacteristic, complemented, Sylow
185#3301—2711nomaximal, Sylow
186#553—2711no—
187#548—2711no—
188#488—2711no—
189#485—2711no—
190#481—2711no—
191#479—2711no—
192#466—2711no—
193#465—2711no—
194#463D27111yescharacteristic, complemented
195#449—2711no—
196#448—2711no—
197C4×SD16—2711no—
198C8×D4—2711no—
199#416—2711no—
200C2×#103—2711no—
201C2×#41—2711no—
202C22×Q8—2711no—
203C2×SD16—2711no—
204C2×SD16C2×D27111yescharacteristic
205C2×D8C2×D27111yescharacteristic
206C22×C8—2711no—
207C22×C8—2711no—
208#126—5421no—
209#125—2711no—
210#124—5421no—
211#124C2×D27111yescharacteristic
212#123—5421no—
213C22⋊Q8—2711no—
214C22⋊Q8—5421no—
215#121—2711no—
216#121—2711no—
217#121—2711no—
218#121—5421no—
219#121—5421no—
220C4×Q8—2711no—
221C4×D4—2711no—
222C4×D4—5421no—
223C4×D4—5421no—
224C4×D4C2×D27111yescharacteristic
225#117—2711no—
226C2×C4⋊C4—2711no—
227#107—2711no—
228#106—2711no—
229#106—2711no—
230C4⋊C8C2×D27111yescharacteristic
231#103—2711no—
232#103—2711no—
233#103—2711no—
234#103—2711no—
235#103—2711no—
236#103C2×D27111yescharacteristic
237#102—2711no—
238#102—2711no—
239#102—2711no—
240#102C2×D27111yescharacteristic
241C22⋊C8—2711no—
242C22⋊C8—2711no—
243C4×C8—2711no—
244#41—5422no—
245#41—10841no—
246C2×Q8—2711no—
247C2×Q8—5421no—
248C2×Q8—5422no—
249C2×Q8—10841no—
250C2×Q8C22×D27111yescharacteristic
251C2×D4—2711no—
252C2×D4—5421no—
253C2×D4—5421no—
254C2×D4—5421no—
255C2×D4—5421no—
256C2×D4C22×D27111yescharacteristic
257C2×D4C22×D27111yescharacteristic
258C22×C4—21no—
259C22×C4—2711no—
260C22×C4—2711no—
261C22×C4—5421no—
262C22×C4—5421no—
263C22×C4—5421no—
264SD16—41no—
265SD16—5422no—
266D8—41no—
267C2×C8—2711no—
268C2×C8—2711no—
269C2×C8—10841no—
270C2×C8—10841no—
271C2×C8C22×D27111yescharacteristic
272C2×C8C22×D27111yescharacteristic
273C4⋊C4—2711no—
274C4⋊C4—2711no—
275C4⋊C4—2711no—
276C4⋊C4—2711no—
277C4⋊C4—5421no—
278C4⋊C4—5421no—
279C4⋊C4—5421no—
280C4⋊C4—5421no—
281C4⋊C4—5421no—
282C4⋊C4C22×D27111yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—2711no—
287C22⋊C4—2711no—
288C22⋊C4—5421no—
289C22⋊C4—5421no—
290C22⋊C4—5421no—
291C22⋊C4—5421no—
292C22⋊C4—5421no—
293C42—5421no—
294C42—5421no—
295C42C22×D27111yescharacteristic
296C23—21no—
297C23—21no—
298C23—2711no—
299C23—2711no—
300Q8—21no—
301Q8—21no—
302Q8—2712no—
303Q8—5421no—
304Q8—10842no—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—21no—
309D4—5422no—
310D4—10842no—
311D4—10842no—
312D4—10842no—
313D4—21681no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—81no—
317C2×C4—2711no—
318C2×C4—2711no—
319C2×C4—5421no—
320C2×C4—5421no—
321C2×C4—5421no—
322C2×C4—5421no—
323C2×C4—5421no—
324C2×C4—5421no—
325C2×C4—5421no—
326C2×C4—5421no—
327C2×C4—10842no—
328C2×C4—10842no—
329C2×C4—21681no—
330C2×C4D4×D27111yescharacteristic
331C2×C4D4×D27111yescharacteristic
332C2×C4C23×D27111yescharacteristic
333C8—21no—
334C8—21no—
335C8—5421no—
336C8—5421no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—2712no—
344C22—5421no—
345C22—5421no—
346C22—5421no—
347C22—5421no—
348C22—5421no—
349C22C2×D4×D27111yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—5421no—
355C4—5421no—
356C4—5422no—
357C4—10841no—
358C4—10841no—
359C4—10841no—
360C4#41×D27111yescharacteristic
361C4#2257620511yescharacteristic
362C2—81no—
363C2—81no—
364C2—5422no—
365C2—10841no—
366C2#3177454411yescharacteristic
367C2#135×D27111yescharacteristic
368C2#3177470711yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#463×C271C2
socleC22×C271C22×D4
hypercenter#463D271
hypocenterC271#3301
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #31774597C2semidirect
2. #31774555C2semidirect
3. #31774251C2semidirect
4. #31774237C2semidirect
5. #31774214C2semidirect
6. #31774205C2semidirect
7. #31774130C2semidirect
8. #31774123C2semidirect
9. #31774039C2semidirect
10. #31774034C2semidirect
11. C4×#16864642C2semidirect
12. #31773962C2semidirect
13. #31773890C2semidirect
14. #103×D271C2semidirect
15. #463×C271C2semidirect
184. C271#3301semidirect
194. #463D271semidirect

Maximal subgroups

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

subgroupindex
#317745972
#317745552
#317742512
#317742372
#317742142
#317742052
#317741302
#317741232
#317740392
#317740342
C4×#168646422
#317739622
#317738902
#103×D2712
#463×C2712
#3301271