Finite Group Databasebeta

← #46848253

Subgroups — #46848253

Order 34688.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774587C211yesmaximal, characteristic, complemented
2#31774556C211yesmaximal, characteristic
3#31774252C211yesmaximal, characteristic, complemented
4#31774239C211yesmaximal, characteristic, complemented
5#31774211C211yesmaximal, characteristic, complemented
6#31774205C211yesmaximal, characteristic, complemented
7#31774138C211yesmaximal, characteristic, complemented
8#31774126C211yesmaximal, characteristic, complemented
9#31774120C211yesmaximal, characteristic, complemented
10#31774061C211yesmaximal, characteristic, complemented
11#31774055C211yesmaximal, characteristic, complemented
12C4×Dic1084C211yesmaximal, characteristic, complemented
13#31773894C211yesmaximal, characteristic, complemented
14C2×#22576085C211yesmaximal, characteristic, complemented
15C8×D4×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C22×Dic542C2211yescharacteristic
18C2×Dic1084C2211yescharacteristic
19C2×Dic1084C2211yescharacteristic
20C2×#16864643C2211yescharacteristic
21#22576153—21no—
22#22576152C2211yescharacteristic
23#22576150—21no—
24#22576148C2211yescharacteristic
25#22576146—21no—
26#22576141—21no—
27#22576139C2211yescharacteristic
28#22576139C2211yescharacteristic
29#22576139C2211yescharacteristic
30#22576138—21no—
31#22576134C2211yescharacteristic
32#22576132—21no—
33C4×Dic542C2211yescharacteristic
34C4×Dic542C2211yescharacteristic
35(C22⋊D271)⋊C4—21no—
36#22576119C2211yescharacteristic
37#22576116—21no—
38C2×C4⋊Dic271C2211yescharacteristic
39#22576094C2211yescharacteristic
40#22576094C2211yescharacteristic
41#22576092C2211yescharacteristic
42#22576085C2211yescharacteristic
43#22576085C2211yescharacteristic
44#22576085C2211yescharacteristic
45#22576085C2211yescharacteristic
46#22576085C2211yescharacteristic
47#22576085C2211yescharacteristic
48#22576085C2211yescharacteristic
49#22576082C2211yescharacteristic
50#22576082C2211yescharacteristic
51#22576082C2211yescharacteristic
52C22×C8×C271C2211yescharacteristic
53C22×C8×C271C2211yescharacteristic
54C4×D4×C271C2211yescharacteristic
55C4⋊C8×C271C2211yescharacteristic
56C22⋊C8×C271C2211yescharacteristic
57C22⋊C8×C271C2211yescharacteristic
58C4×C8×C271C2211yescharacteristic
59#16864654—22no—
60#16864654—41no—
61C2×Dic542—22no—
62C2×Dic542—21no—
63C2×Dic542—41no—
64C2×Dic542C2311yescharacteristic
65C2×Dic542C2311yescharacteristic
66C2×Dic542C2311yescharacteristic
67C2×D1084C2311yescharacteristic
68C2×C22⋊D271—21no—
69C2×C22⋊D271—21no—
70C22×Dic271—21no—
71C22×Dic271—21no—
72C2×C4×D271—21no—
73Dic1084—22no—
74Dic1084—41no—
75#16864643—41no—
76C4⋊Dic271—21no—
77C4⋊Dic271C2311yescharacteristic
78C4⋊Dic271C2311yescharacteristic
79C4⋊Dic271C2311yescharacteristic
80C4⋊Dic271C2311yescharacteristic
81Dic271⋊C4—21no—
82Dic271⋊C4—21no—
83Dic271⋊C4—21no—
84Dic271⋊C4—21no—
85Dic271⋊C4—21no—
86Dic271⋊C4—21no—
87Dic271⋊C4—21no—
88C22⋊Dic271—21no—
89C22⋊Dic271—21no—
90C22⋊Dic271—21no—
91C22⋊Dic271—21no—
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4—21no—
94(C2×D271)⋊C4—21no—
95C4×Dic271—21no—
96C4×Dic271—21no—
97C4×Dic271—21no—
98C2×D4×C271D411yescharacteristic
99C22×C4×C271C2311yescharacteristic
100C22×C4×C271C2311yescharacteristic
101C2×C8×C271—41no—
102C2×C8×C271—41no—
103C2×C8×C271C2311yescharacteristic
104C2×C8×C271C2311yescharacteristic
105C2×C8×C271C2311yescharacteristic
106C2×C8×C271C2311yescharacteristic
107C4⋊C4×C271D411yescharacteristic
108C22⋊C4×C271D411yescharacteristic
109C22⋊C4×C271D411yescharacteristic
110C42×C271C2311yescharacteristic
111C22×D271—21no—
112Dic542—21no—
113Dic542—21no—
114Dic542—21no—
115Dic542—21no—
116Dic542—21no—
117Dic542—42no—
118Dic542#4112yes—
119D1084—21no—
120D1084—21no—
121C22⋊D271—42no—
122C22⋊D271—42no—
123C2×Dic271—21no—
124C2×Dic271—21no—
125C2×Dic271—21no—
126C2×Dic271—21no—
127C2×Dic271—21no—
128C2×Dic271—21no—
129C2×Dic271—21no—
130C2×Dic271—42no—
131C2×Dic271—42no—
132C4×D271—42no—
133C23×C271C2×D411yescharacteristic
134C23×C271C2×D411yescharacteristic
135D4×C271—22no—
136C2×C4×C271—21no—
137C2×C4×C271—21no—
138C2×C4×C271—21no—
139C2×C4×C271—21no—
140C2×C4×C271C2×D411yescharacteristic
141C2×C4×C271C2×D411yescharacteristic
142C2×C4×C271C2×D411yescharacteristic
143C2×C4×C271C2×D411yescharacteristic
144C2×C4×C271C2411yescharacteristic
145C8×C271—21no—
146C8×C271—21no—
147C8×C271—21no—
148C8×C271—21no—
149C22×C271—21no—
150C22×C271—21no—
151C22×C271—21no—
152C22×C271—21no—
153C22×C271—21no—
154C22×C271#13512yes—
155C22×C271C22×D411yescharacteristic
156C2×D271—41no—
157C2×D271—41no—
158C2×D271—41no—
159C4×C271—21no—
160C4×C271—21no—
161C4×C271—21no—
162C4×C271—21no—
163C4×C271C2×#4111yescharacteristic
164C4×C271H(5,𝔽2)11yescharacteristic
165Dic271—22no—
166Dic271—41no—
167Dic271—41no—
168Dic271—41no—
169Dic271—41no—
170Dic271—41no—
171Dic271—41no—
172C2×C271—22no—
173C2×C271—41no—
174C2×C271#54611yescharacteristic
175C2×C271C2×#13511yescharacteristic
176C2×C271#57811yescharacteristic
177D271—81no—
178C271#330011yescharacteristic, complemented, Sylow
179#3300—2711nomaximal, Sylow
180#552—2711no—
181#548—2711no—
182#488—2711no—
183#485—2711no—
184#480—2711no—
185#479—2711no—
186#467—2711no—
187#465—2711no—
188#464—2711no—
189#452—2711no—
190#451—2711no—
191C4×Q16—2711no—
192C8×D4D27111yescharacteristic, complemented
193#416—2711no—
194C2×#103—2711no—
195C2×#41—2711no—
196C22×Q8—2711no—
197C2×Q16—2711no—
198C2×Q16—2711no—
199C2×SD16—2711no—
200C22×C8C2×D27111yescharacteristic
201C22×C8C2×D27111yescharacteristic
202#126—5421no—
203#125—2711no—
204#125—5421no—
205#124—2711no—
206#124—5421no—
207#123—5421no—
208C22⋊Q8—2711no—
209C22⋊Q8—2711no—
210C22⋊Q8—2711no—
211C22⋊Q8—5421no—
212#121—2711no—
213#121—5421no—
214C4×Q8—2711no—
215C4×Q8—2711no—
216C4×D4—5421no—
217C4×D4C2×D27111yescharacteristic
218#117—2711no—
219#117—5421no—
220C2×C4⋊C4—2711no—
221#107—2711no—
222#107—2711no—
223#106—2711no—
224C4⋊C8C2×D27111yescharacteristic
225#103—2711no—
226#103—2711no—
227#103—2711no—
228#103—2711no—
229#103—2711no—
230#103—2711no—
231#103—2711no—
232#102—2711no—
233#102—2711no—
234#102—2711no—
235C22⋊C8C2×D27111yescharacteristic
236C22⋊C8C2×D27111yescharacteristic
237C4×C8C2×D27111yescharacteristic
238#41—5422no—
239#41—10841no—
240C2×Q8—2711no—
241C2×Q8—2711no—
242C2×Q8—2711no—
243C2×Q8—5422no—
244C2×Q8—5421no—
245C2×Q8—10841no—
246C2×D4—2711no—
247C2×D4—5421no—
248C2×D4—5421no—
249C2×D4D108411yescharacteristic
250C22×C4—5421no—
251C22×C4—5421no—
252C22×C4—5421no—
253C22×C4C22×D27111yescharacteristic
254C22×C4C22×D27111yescharacteristic
255Q16—5422no—
256Q16—10841no—
257SD16—10841no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8C22×D27111yescharacteristic
261C2×C8C22×D27111yescharacteristic
262C2×C8C22×D27111yescharacteristic
263C2×C8C22×D27111yescharacteristic
264C4⋊C4—2711no—
265C4⋊C4—2711no—
266C4⋊C4—2711no—
267C4⋊C4—2711no—
268C4⋊C4—5421no—
269C4⋊C4—5421no—
270C4⋊C4—5421no—
271C4⋊C4—5421no—
272C4⋊C4—5421no—
273C4⋊C4—5421no—
274C4⋊C4—5421no—
275C4⋊C4—5421no—
276C4⋊C4D108411yescharacteristic
277C22⋊C4—5421no—
278C22⋊C4—5421no—
279C22⋊C4—5421no—
280C22⋊C4—5421no—
281C22⋊C4—5421no—
282C22⋊C4—5421no—
283C22⋊C4—5421no—
284C22⋊C4D108411yescharacteristic
285C22⋊C4D108411yescharacteristic
286C42—5421no—
287C42—5421no—
288C42—5421no—
289C42C22×D27111yescharacteristic
290C23—5421no—
291C23C2×D108411yescharacteristic
292C23C2×D108411yescharacteristic
293Q8—2712no—
294Q8—5421no—
295Q8—5421no—
296Q8—5421no—
297Q8—5421no—
298Q8—5421no—
299Q8—10842no—
300D4—22no—
301D4—5421no—
302D4—5421no—
303D4—10842no—
304D4—10842no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—5421no—
310C2×C4—5421no—
311C2×C4—5421no—
312C2×C4—5421no—
313C2×C4—5421no—
314C2×C4—5421no—
315C2×C4—5421no—
316C2×C4—10842no—
317C2×C4—10842no—
318C2×C4—10842no—
319C2×C4C2×D108411yescharacteristic
320C2×C4C2×D108411yescharacteristic
321C2×C4C2×D108411yescharacteristic
322C2×C4C2×D108411yescharacteristic
323C2×C4C23×D27111yescharacteristic
324C8—21no—
325C8—21no—
326C8—21no—
327C8—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—10841no—
334C22—10841no—
335C22—10841no—
336C22#2257618212yes—
337C22C22×D108411yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—5422no—
343C4—10841no—
344C4—10841no—
345C4—10841no—
346C4—10841no—
347C4—10841no—
348C4—10841no—
349C4#41×D27111yescharacteristic
350C4#2257620511yescharacteristic
351C2—22no—
352C2—41no—
353C2—21681no—
354C2#3177454211yescharacteristic
355C2C2×#2257618211yescharacteristic
356C2#3177471411yescharacteristic
subgroupG/H
centerC22C22×D1084
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC8×D4×C271C2
socleC22×C271C22×D4
prime radical#31774556C2
hypercenterC8×D4D271
hypocenterC271#3300
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #31774587C2semidirect
3. #31774252C2semidirect
4. #31774239C2semidirect
5. #31774211C2semidirect
6. #31774205C2semidirect
7. #31774138C2semidirect
8. #31774126C2semidirect
9. #31774120C2semidirect
10. #31774061C2semidirect
11. #31774055C2semidirect
12. C4×Dic1084C2semidirect
13. #31773894C2semidirect
14. C2×#22576085C2semidirect
15. C8×D4×C271C2semidirect
178. C271#3300semidirect
192. C8×D4D271semidirect

Maximal subgroups

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

subgroupindex
#317745872
#317745562
#317742522
#317742392
#317742112
#317742052
#317741382
#317741262
#317741202
#317740612
#317740552
C4×Dic10842
#317738942
C2×#225760852
C8×D4×C2712
#3300271