Finite Group Databasebeta

← #46837492

Subgroups — #46837492

Order 34688.

430 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774456C211yesmaximal, characteristic
2C2×#22576143C211yesmaximal, characteristic, complemented
3C2×#22576075C211yesmaximal, characteristic, complemented
4C2×#22576074C211yesmaximal, characteristic, complemented
5#31773760C211yesmaximal, characteristic, complemented
6#31773663C211yesmaximal, characteristic, complemented
7C2×#123×C271C211yesmaximal, characteristic, complemented
8C22×C22⋊D271C2211yescharacteristic
9#22576143—22no—
10#22576143—21no—
11#22576143—22no—
12#22576141—21no—
13#22576132—22no—
14#22576129—21no—
15#22576128—21no—
16C2×Dic271⋊C4—21no—
17C2×C22⋊Dic271C2211yescharacteristic
18C2×C22⋊Dic271C2211yescharacteristic
19C2×C22⋊Dic271C2211yescharacteristic
20C2×(C2×D271)⋊C4—21no—
21#22576075—22no—
22#22576074—22no—
23C271⋊#95—22no—
24C22×D4×C271C2211yescharacteristic
25C23×C4×C271C2211yescharacteristic
26#123×C271—22no—
27#123×C271—22no—
28C2×C4⋊C4×C271—21no—
29C2×C22⋊C4×C271—21no—
30C2×C22⋊C4×C271C2211yescharacteristic
31C23×D271D411yescharacteristic
32C2×C22⋊D271—22no—
33C2×C22⋊D271—21no—
34C2×C22⋊D271—21no—
35C2×C22⋊D271—21no—
36C2×C22⋊D271—21no—
37C2×C22⋊D271—21no—
38C2×C22⋊D271—21no—
39C22×Dic271—21no—
40C22×Dic271—21no—
41C22×Dic271D411yescharacteristic
42C22×Dic271D412yes—
43Dic271⋊C4—22no—
44Dic271⋊C4—24no—
45C22⋊Dic271—22no—
46C22⋊Dic271—21no—
47C22⋊Dic271—21no—
48C22⋊Dic271—21no—
49C22⋊Dic271—21no—
50C22⋊Dic271—21no—
51C22⋊Dic271D412yescomplemented
52(C2×D271)⋊C4—22no—
53(C2×D271)⋊C4—24no—
54C24×C271D411yescharacteristic
55C24×C271C2311yescharacteristic
56C2×D4×C271—21no—
57C2×D4×C271—21no—
58C2×D4×C271—21no—
59C2×D4×C271—21no—
60C2×D4×C271—21no—
61C2×D4×C271—21no—
62C22×C4×C271—21no—
63C22×C4×C271—21no—
64C22×C4×C271—21no—
65C22×C4×C271—21no—
66C22×C4×C271—21no—
67C22×C4×C271—21no—
68C22×C4×C271—21no—
69C22×C4×C271D411yescharacteristic
70C22×C4×C271D412yescomplemented
71C4⋊C4×C271—24no—
72C22⋊C4×C271—24no—
73C22⋊C4×C271—21no—
74C22⋊C4×C271D412yescomplemented
75C22×D271—21no—
76C22×D271—21no—
77C22×D271—21no—
78C22×D271—42no—
79C22⋊D271—41no—
80C22⋊D271—41no—
81C22⋊D271—41no—
82C22⋊D271—41no—
83C22⋊D271—82no—
84C2×Dic271—22no—
85C2×Dic271—22no—
86C2×Dic271—21no—
87C2×Dic271—21no—
88C2×Dic271—21no—
89C2×Dic271—42no—
90C2×Dic271—42no—
91C2×Dic271—42no—
92C2×Dic271—41no—
93C23×C271—21no—
94C23×C271—21no—
95C23×C271—21no—
96C23×C271—42no—
97C23×C271—42no—
98C23×C271C2×D411yescharacteristic
99C23×C271C2×D411yescharacteristic
100C23×C271C2×D411yescharacteristic
101C23×C271C2×D411yescharacteristic
102C23×C271C2×D411yescharacteristic
103C23×C271C2×D411yescharacteristic
104C23×C271#4111yescharacteristic
105D4×C271—41no—
106D4×C271—41no—
107D4×C271—41no—
108D4×C271—41no—
109C2×C4×C271—22no—
110C2×C4×C271—22no—
111C2×C4×C271—21no—
112C2×C4×C271—22no—
113C2×C4×C271—21no—
114C2×C4×C271—22no—
115C2×C4×C271—21no—
116C2×C4×C271—42no—
117C2×C4×C271—41no—
118C2×C4×C271—41no—
119C2×C4×C271—41no—
120C2×C4×C271—41no—
121C22×C271—22no—
122C22×C271—22no—
123C22×C271—22no—
124C22×C271—41no—
125C22×C271—41no—
126C22×C271—41no—
127C22×C271—41no—
128C22×C271—42no—
129C22×C271—41no—
130C22×C271—42no—
131C22×C271—81no—
132C22×C271#12011yescharacteristic
133C22×C271#12011yescharacteristic
134C22×C271#12011yescharacteristic
135C22×C271#12111yescharacteristic
136C22×C271#12111yescharacteristic
137C22×C271#12111yescharacteristic
138C22×C271#12711yescharacteristic
139C2×D271—42no—
140C2×D271—41no—
141C2×D271—42no—
142C2×D271—81no—
143C4×C271—42no—
144C4×C271—42no—
145C4×C271—41no—
146Dic271—42no—
147Dic271—41no—
148Dic271—42no—
149Dic271—81no—
150C2×C271—21no—
151C2×C271—21no—
152C2×C271—21no—
153C2×C271—21no—
154C2×C271—21no—
155C2×C271—21no—
156C2×C271—81no—
157C2×C271#39211yescharacteristic
158C2×C271#45812yes—
159D271—81no—
160C271#202211yescharacteristic, complemented, Sylow
161#2022—2711nomaximal, Sylow
162#534—2711no—
163C2×#123—2711no—
164C2×#123D27111yescharacteristic, complemented
165C2×#99—2711no—
166C2×#99—2711no—
167#381—2711no—
168#342—2711no—
169C22×D4—2711no—
170C22×D4C2×D27111yescharacteristic
171C23×C4C2×D27111yescharacteristic
172#123—22no—
173#123—22no—
174#123—5422no—
175#123—5421no—
176#123—5421no—
177#123—5422no—
178#121—5422no—
179#120—5421no—
180#120—5421no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—5421no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—2711no—
185C2×C22⋊C4—2711no—
186C2×C22⋊C4—2711no—
187C2×C22⋊C4—5421no—
188C2×C22⋊C4C2×D27111yescharacteristic
189#99—5422no—
190#99—5422no—
191#95—5422no—
192C24—2711no—
193C24C22⋊D27111yescharacteristic
194C24C22×D27111yescharacteristic
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—5421no—
202C2×D4—5421no—
203C2×D4—5421no—
204C2×D4—5421no—
205C2×D4—5421no—
206C2×D4—5421no—
207C2×D4—5422no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—2711no—
216C22×C4—2712no—
217C22×C4—5421no—
218C22×C4—5421no—
219C22×C4C22⋊D27111yescharacteristic
220C22×C4C22⋊D27112yescomplemented
221C4⋊C4—24no—
222C4⋊C4—5424no—
223C4⋊C4—5422no—
224C22⋊C4—24no—
225C22⋊C4—21no—
226C22⋊C4—2712no—
227C22⋊C4—5424no—
228C22⋊C4—5422no—
229C22⋊C4—5422no—
230C22⋊C4—5421no—
231C22⋊C4—5421no—
232C22⋊C4—5421no—
233C22⋊C4—5421no—
234C22⋊C4—5421no—
235C22⋊C4C22⋊D27112yescomplemented
236C23—21no—
237C23—21no—
238C23—21no—
239C23—42no—
240C23—42no—
241C23—5421no—
242C23—5421no—
243C23—5421no—
244C23—10842no—
245C23D4×D27111yescharacteristic
246C23D4×D27111yescharacteristic
247C23D4×D27111yescharacteristic
248C23C2×C22⋊D27111yescharacteristic
249C23C2×C22⋊D27111yescharacteristic
250C23C2×C22⋊D27111yescharacteristic
251C23#1686465311yescharacteristic
252D4—41no—
253D4—41no—
254D4—41no—
255D4—41no—
256D4—10841no—
257D4—10841no—
258D4—10841no—
259D4—10841no—
260D4—21682no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—5422no—
274C2×C4—5421no—
275C2×C4—5422no—
276C2×C4—5421no—
277C2×C4—5421no—
278C2×C4—10842no—
279C2×C4—10842no—
280C2×C4—10842no—
281C2×C4—10841no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—42no—
290C22—41no—
291C22—42no—
292C22—81no—
293C22—10842no—
294C22—10841no—
295C22—10842no—
296C22—21681no—
297C22#2257612811yescharacteristic
298C22#2257612811yescharacteristic
299C22#2257612911yescharacteristic
300C22#2257613111yescharacteristic
301C22#2257613211yescharacteristic
302C22#2257613211yescharacteristic
303C22#2257615611yescharacteristic
304C4—42no—
305C4—42no—
306C4—41no—
307C4—10842no—
308C4—10841no—
309C4—10842no—
310C4—21681no—
311C2—21no—
312C2—21no—
313C2—21no—
314C2—21no—
315C2—21no—
316C2—21no—
317C2—81no—
318C2—21681no—
319C2#3177380511yescharacteristic
320C2#3177408612yes—
subgroupG/H
centerC22#22576128
commutator, cosocleC24×C271C23
FrattiniC24C22×D271
FittingC2×#123×C271C2
socleC22×C271#120
prime radical#31774456C2
hypercenterC2×#123D271
hypocenterC271#2022
LCSG ▷ C24×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C24 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×#22576143C2semidirect
3. C2×#22576075C2semidirect
4. C2×#22576074C2semidirect
5. #31773760C2semidirect
6. #31773663C2semidirect
7. C2×#123×C271C2semidirect
51. C22⋊Dic271D4semidirect
70. C22×C4×C271D4semidirect
74. C22⋊C4×C271D4semidirect
160. C271#2022semidirect
164. C2×#123D271semidirect
220. C22×C4C22⋊D271semidirect
235. C22⋊C4C22⋊D271semidirect

Maximal subgroups

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

subgroupindex
#317744562
C2×#225761432
C2×#225760752
C2×#225760742
#317737602
#317736632
C2×#123×C2712
#2022271