Finite Group Databasebeta

← #46839767

Subgroups — #46839767

Order 34688.

610 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 161 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
1C2×#22576141C211yesmaximal, characteristic
2C2×#22576116C211yesmaximal, characteristic, complemented
3C22×Dic271⋊C4C211yesmaximal, characteristic, complemented
4#31773828C212yesmaximal, complemented
5#31773820C212yesmaximal, complemented
6#31773793C212yesmaximal, complemented
7#31773785C212yesmaximal, complemented
8#31773777C211yesmaximal, characteristic
9#31773772C211yesmaximal, characteristic, complemented
10#31773760C211yesmaximal, characteristic, complemented
11C2×#123×C271C211yesmaximal, characteristic
12C23×Dic271C2211yescharacteristic
13C23×Dic271C2211yescharacteristic
14#22576141—24no—
15#22576116—24no—
16C2×C4⋊Dic271C2212yes—
17C2×Dic271⋊C4—24no—
18C2×Dic271⋊C4C2212yes—
19C2×Dic271⋊C4C2211yescharacteristic
20C2×Dic271⋊C4C2212yes—
21C2×Dic271⋊C4C2212yes—
22C2×Dic271⋊C4C2211yescharacteristic
23C2×C22⋊Dic271C2211yescharacteristic
24C2×C22⋊Dic271C2211yescharacteristic
25C2×C22⋊Dic271C2212yes—
26C2×C22⋊Dic271C2211yescharacteristic
27C2×C22⋊Dic271C2211yescharacteristic
28C2×C4×Dic271C2212yes—
29C271⋊#95C2212yes—
30C271⋊#95C2212yes—
31C271⋊#95C2212yes—
32C271⋊#95C2212yes—
33C22×D4×C271C2211yescharacteristic
34C23×C4×C271C2211yescharacteristic
35#123×C271—24no—
36C2×C4⋊C4×C271C2212yes—
37C2×C22⋊C4×C271C2212yes—
38C2×C22⋊C4×C271C2211yescharacteristic
39C22×Dic271—24no—
40C22×Dic271—22no—
41C22×Dic271—22no—
42C22×Dic271—22no—
43C22×Dic271D414yes—
44C22×Dic271C2311yescharacteristic
45C22×Dic271C2312yes—
46C22×Dic271C2311yescharacteristic
47C22×Dic271C2311yescharacteristic
48C22×Dic271C2311yescharacteristic
49C22×Dic271C2312yes—
50C4⋊Dic271—24no—
51Dic271⋊C4—22no—
52Dic271⋊C4—22no—
53Dic271⋊C4—24no—
54Dic271⋊C4—24no—
55Dic271⋊C4—24no—
56C22⋊Dic271—22no—
57C22⋊Dic271—22no—
58C22⋊Dic271—24no—
59C22⋊Dic271—22no—
60C22⋊Dic271—22no—
61C4×Dic271—24no—
62C24×C271C2311yescharacteristic
63C24×C271C2311yescharacteristic
64C2×D4×C271—22no—
65C2×D4×C271—22no—
66C2×D4×C271—22no—
67C22×C4×C271—22no—
68C22×C4×C271—22no—
69C22×C4×C271—22no—
70C22×C4×C271C2312yes—
71C22×C4×C271C2311yescharacteristic
72C22×C4×C271C2312yes—
73C4⋊C4×C271—24no—
74C22⋊C4×C271—24no—
75C22⋊C4×C271D414yes—
76C2×Dic271—28no—
77C2×Dic271—28no—
78C2×Dic271—21no—
79C2×Dic271—22no—
80C2×Dic271—21no—
81C2×Dic271—22no—
82C2×Dic271—21no—
83C2×Dic271—21no—
84C2×Dic271—22no—
85C2×Dic271—21no—
86C2×Dic271—21no—
87C2×Dic271—22no—
88C2×Dic271—22no—
89C2×Dic271—22no—
90C2×Dic271—22no—
91C2×Dic271C2×D412yes—
92C2×Dic271C2×D412yes—
93C2×Dic271#4114yes—
94C23×C271—22no—
95C23×C271—22no—
96C23×C271—21no—
97C23×C271—21no—
98C23×C271—21no—
99C23×C271—22no—
100C23×C271—22no—
101C23×C271C2×D412yes—
102C23×C271C2×D412yes—
103C23×C271#4112yes—
104C23×C271C2411yescharacteristic
105D4×C271—44no—
106C2×C4×C271—22no—
107C2×C4×C271—22no—
108C2×C4×C271—22no—
109C2×C4×C271—21no—
110C2×C4×C271—22no—
111C2×C4×C271—21no—
112C2×C4×C271—22no—
113C2×C4×C271—21no—
114C2×C4×C271—44no—
115C2×C4×C271C2×D414yes—
116C22×C271—22no—
117C22×C271—22no—
118C22×C271—24no—
119C22×C271—24no—
120C22×C271—41no—
121C22×C271—41no—
122C22×C271—41no—
123C22×C271—42no—
124C22×C271—42no—
125C22×C271#12114yes—
126C22×C271C22×D411yescharacteristic
127C22×C271C22×D411yescharacteristic
128C22×C271C2×#4112yes—
129C22×C271C2×#4111yescharacteristic
130C22×C2712−1+412yes—
131C4×C271—24no—
132C4×C271—42no—
133C4×C271—41no—
134Dic271—22no—
135Dic271—22no—
136Dic271—24no—
137Dic271—41no—
138Dic271—41no—
139Dic271—42no—
140C2×C271—24no—
141C2×C271—81no—
142C2×C271C2×#12111yescharacteristic
143C2×C271#52511yescharacteristic
144C2×C271#53611yescharacteristic
145C2×C271#54712yes—
146C2×C271#55212yes—
147C271#241311yescharacteristic, complemented, Sylow
148#2413—2711nomaximal, Sylow
149C2×#123—2711no—
150C2×#123D27111yescharacteristic
151C2×#117—2711no—
152C22×C4⋊C4—2711no—
153#398—2712no—
154#396—2712no—
155#389—2712no—
156#388—2712no—
157#386—2711no—
158#385—2711no—
159#381—2711no—
160C22×D4C2×D27111yescharacteristic
161C23×C4—2711no—
162C23×C4—2711no—
163C23×C4C2×D27111yescharacteristic
164#123—24no—
165#123—5424no—
166#117—5424no—
167C2×C4⋊C4—2712no—
168C2×C4⋊C4—2712no—
169C2×C4⋊C4—2712no—
170C2×C4⋊C4—2711no—
171C2×C4⋊C4—2711no—
172C2×C4⋊C4—2712no—
173C2×C4⋊C4—5424no—
174C2×C4⋊C4C2×D27112yes—
175C2×C22⋊C4—2711no—
176C2×C22⋊C4—2711no—
177C2×C22⋊C4—2711no—
178C2×C22⋊C4—2711no—
179C2×C22⋊C4—2712no—
180C2×C22⋊C4C2×D27112yes—
181C2×C22⋊C4C2×D27111yescharacteristic
182C2×C42—2712no—
183#95—2712no—
184#95—2712no—
185#95—2712no—
186#95—2712no—
187C24C22×D27111yescharacteristic
188C24C22×D27111yescharacteristic
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—2714no—
196C22×C4—2712no—
197C22×C4—2711no—
198C22×C4—2711no—
199C22×C4—2711no—
200C22×C4—2712no—
201C22×C4—2711no—
202C22×C4—5424no—
203C22×C4—5422no—
204C22×C4—5422no—
205C22×C4—5422no—
206C22×C4C22×D27112yes—
207C22×C4C22×D27111yescharacteristic
208C22×C4C22×D27112yes—
209C4⋊C4—24no—
210C4⋊C4—5424no—
211C4⋊C4—5424no—
212C4⋊C4—5422no—
213C4⋊C4—5422no—
214C4⋊C4—5424no—
215C4⋊C4—5424no—
216C22⋊C4—24no—
217C22⋊C4—5424no—
218C22⋊C4—5422no—
219C22⋊C4—5422no—
220C22⋊C4—5422no—
221C22⋊C4—5422no—
222C22⋊C4C22⋊D27114yes—
223C42—5424no—
224C23—22no—
225C23—22no—
226C23—21no—
227C23—21no—
228C23—22no—
229C23—21no—
230C23—22no—
231C23D4×D27112yes—
232C23C2×C22⋊D27112yes—
233C23#1686465312yes—
234C23C23×D27111yescharacteristic
235D4—44no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—44no—
245C2×C4—2712no—
246C2×C4—2712no—
247C2×C4—2714no—
248C2×C4—5428no—
249C2×C4—5428no—
250C2×C4—5422no—
251C2×C4—5421no—
252C2×C4—5421no—
253C2×C4—5422no—
254C2×C4—5421no—
255C2×C4—5421no—
256C2×C4—5421no—
257C2×C4—5421no—
258C2×C4—5422no—
259C2×C4—5422no—
260C2×C4—5422no—
261C2×C4—5422no—
262C2×C4—5422no—
263C2×C4C2×C22⋊D27114yes—
264C22—22no—
265C22—24no—
266C22—24no—
267C22—22no—
268C22—41no—
269C22—42no—
270C22—42no—
271C22—41no—
272C22—41no—
273C22#2257613114yes—
274C22C2×D4×D27111yescharacteristic
275C22C22×C22⋊D27111yescharacteristic
276C22#41×D27112yes—
277C22C2×#1686465311yescharacteristic
278C22#2257620712yes—
279C4—24no—
280C4—42no—
281C4—41no—
282C4—5422no—
283C4—5422no—
284C4—5424no—
285C4—10841no—
286C4—10842no—
287C4—10841no—
288C2—24no—
289C2—81no—
290C2C2×#2257613111yescharacteristic
291C2#3177440511yescharacteristic
292C2#3177446911yescharacteristic
293C2#3177454612yes—
294C2#3177458412yes—
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
FittingC2×#123×C271C2
prime radicalC22×D4×C271C22
hypercenterC2×#123D271
hypocenterC271#2413
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×#22576116C2semidirect
3. C22×Dic271⋊C4C2semidirect
4. #31773828C2semidirect
5. #31773820C2semidirect
6. #31773793C2semidirect
7. #31773785C2semidirect
9. #31773772C2semidirect
10. #31773760C2semidirect
147. C271#2413semidirect

Maximal subgroups

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

subgroupindex
C2×#225761412
C2×#225761162
C22×Dic271⋊C42
#317738282
#317738202
#317737932
#317737852
#317737772
#317737722
#317737602
C2×#123×C2712
#2413271