Finite Group Databasebeta

← #41013697

Subgroups — #41013697

Order 27008.

376 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 85 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×M16×D211C211yesmaximal, characteristic, complemented
2#27213653C211yesmaximal, characteristic
3C2×#20269904C211yesmaximal, characteristic, complemented
4C4×(C2×D211)⋊C4C211yesmaximal, characteristic, complemented
5#27212907C211yesmaximal, characteristic, complemented
6#27212906C211yesmaximal, characteristic, complemented
7C2×#104×C211C211yesmaximal, characteristic, complemented
8C22×C4×D211C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10C2×#16809945—21no—
11M16×D211—22no—
12M16×D211—22no—
13C2×C8×D211—21no—
14#20269958—21no—
15#20269953C411yescharacteristic, complemented
16#20269953C411yescharacteristic, complemented
17#20269948C411yescharacteristic, complemented
18#20269948C411yescharacteristic, complemented
19#20269945—21no—
20C4×C22⋊D211—21no—
21C4×C22⋊D211—21no—
22#20269936C2211yescharacteristic
23C2×(C2×D211)⋊C4—21no—
24C2×C4×Dic211C2211yescharacteristic
25#20269904—22no—
26C211⋊#95—21no—
27C2×#41×C211C2211yescharacteristic
28C2×M16×C211C2211yescharacteristic
29C2×C42×C211C2211yescharacteristic
30#104×C211—22no—
31C23×D211D411yescharacteristic
32C2×C22⋊D211—21no—
33C2×C22⋊D211—21no—
34C22×Dic211—21no—
35C22×Dic211D411yescharacteristic
36C2×C4×D211—21no—
37C2×C4×D211—21no—
38C2×C4×D211—21no—
39C2×C4×D211—21no—
40C2×C4×D211C2×C411yescharacteristic
41C2×C4×D211C2×C411yescharacteristic
42C2×C4×D211D411yescharacteristic
43C2×C4×D211D411yescharacteristic
44#16809946—21no—
45#16809946D412yescomplemented
46#16809945—24no—
47C2×C211⋊C8—21no—
48C8×D211—24no—
49C4⋊Dic211C2×C411yescharacteristic
50C4⋊Dic211C2×C411yescharacteristic
51Dic211⋊C4—21no—
52Dic211⋊C4—21no—
53C22⋊Dic211—21no—
54C22⋊Dic211—21no—
55C22⋊Dic211—21no—
56(C2×D211)⋊C4—24no—
57(C2×D211)⋊C4—21no—
58(C2×D211)⋊C4—21no—
59C4×Dic211—21no—
60C4×Dic211—21no—
61C4×Dic211—21no—
62#41×C211—21no—
63#41×C211—21no—
64#41×C211—21no—
65#41×C211—21no—
66C2×Q8×C211C2×C411yescharacteristic
67C2×D4×C211—21no—
68C2×D4×C211C2×C411yescharacteristic
69C22×C4×C211—21no—
70C22×C4×C211—21no—
71C22×C4×C211C2311yescharacteristic
72M16×C211—21no—
73M16×C211D412yescomplemented
74C2×C8×C211—21no—
75C42×C211—21no—
76C42×C211—21no—
77C22×D211—21no—
78C22×D211—21no—
79C22×D211—22no—
80C22×D211—41no—
81C22×D211C22⋊C411yescharacteristic
82C22×D211C22⋊C411yescharacteristic
83C22⋊D211—42no—
84C22⋊D211—42no—
85C2×Dic211—21no—
86C2×Dic211—21no—
87C2×Dic211—22no—
88C2×Dic211—21no—
89C2×Dic211—21no—
90C2×Dic211—41no—
91C2×Dic211—41no—
92C2×Dic211C22⋊C411yescharacteristic
93C2×Dic211C22⋊C411yescharacteristic
94C4×D211—21no—
95C4×D211—21no—
96C4×D211—21no—
97C4×D211—21no—
98C4×D211—21no—
99C4×D211—21no—
100C4×D211—21no—
101C4×D211—21no—
102C211⋊C8—22no—
103C23×C211—21no—
104C23×C211C2×D411yescharacteristic
105Q8×C211—21no—
106Q8×C211—21no—
107D4×C211—21no—
108D4×C211—21no—
109D4×C211—41no—
110D4×C211—41no—
111C2×C4×C211—22no—
112C2×C4×C211—21no—
113C2×C4×C211—41no—
114C2×C4×C211—41no—
115C2×C4×C211—41no—
116C2×C4×C211—41no—
117C2×C4×C211C22×C411yescharacteristic
118C2×C4×C211C2×D411yescharacteristic
119C2×C4×C211C2×D411yescharacteristic
120C2×C4×C211C2×D411yescharacteristic
121C2×C4×C211#4111yescharacteristic
122C2×C4×C211#4111yescharacteristic
123C8×C211—22no—
124C22×C211—21no—
125C22×C211—21no—
126C22×C211—41no—
127C22×C211—41no—
128C22×C211—41no—
129C22×C211C2×C22⋊C411yescharacteristic
130C22×C211#12111yescharacteristic
131C22×C211#12111yescharacteristic
132C2×D211—21no—
133C2×D211—21no—
134C2×D211—21no—
135C2×D211—21no—
136C2×D211—21no—
137C2×D211—42no—
138C2×D211—42no—
139C2×D211#10412yescomplemented
140C4×C211—42no—
141C4×C211—41no—
142C4×C211C4×D411yescharacteristic
143C4×C211C4×D411yescharacteristic
144C4×C211#12011yescharacteristic
145C4×C211#12311yescharacteristic
146Dic211—21no—
147Dic211—21no—
148Dic211—42no—
149Dic211—41no—
150Dic211—41no—
151C2×C211—21no—
152C2×C211—21no—
153C2×C211—81no—
154C2×C211#38611yescharacteristic
155C2×C211C2×#10411yescharacteristic
156C2×C211#42111yescharacteristic
157D211—22no—
158D211—41no—
159C211#187111yescharacteristic, complemented, Sylow
160#1871—2111nomaximal, Sylow
161C22×M16—2111no—
162#525—2111no—
163C2×#104—2111no—
164C2×#104D21111yescharacteristic, complemented
165C4×C22⋊C4—2111no—
166#339—2111no—
167#339—2111no—
168C2×#41C2×D21111yescharacteristic
169C23×C4—2111no—
170C2×M16—4221no—
171C2×M16—4222no—
172C2×M16—4222no—
173C2×M16—2111no—
174C2×M16C2×D21111yescharacteristic
175C22×C8—4221no—
176#123—4221no—
177C22⋊Q8—2111no—
178C22⋊Q8—2111no—
179#121—2111no—
180#121—2111no—
181#120—4221no—
182C4×D4—4221no—
183C4×D4—4221no—
184#117—2111no—
185C2×C22⋊C4—4221no—
186C2×C42—2111no—
187C2×C42C2×D21111yescharacteristic
188#104—22no—
189#104—4222no—
190#95—4221no—
191C24—2111no—
192#41—21no—
193#41—21no—
194#41—21no—
195#41—21no—
196C2×Q8C4×D21111yescharacteristic
197C2×D4—21no—
198C2×D4—4221no—
199C2×D4—4221no—
200C2×D4C4×D21111yescharacteristic
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—4221no—
204C22×C4—4221no—
205C22×C4—4221no—
206C22×C4—4221no—
207C22×C4—4221no—
208C22×C4—2111no—
209C22×C4—2111no—
210C22×C4—2111no—
211C22×C4—2111no—
212C22×C4—2111no—
213C22×C4C22×D21111yescharacteristic
214M16—21no—
215M16—4224no—
216M16—4221no—
217M16—2112no—
218M16C22⋊D21112yescomplemented
219C2×C8—21no—
220C2×C8—4224no—
221C2×C8—4221no—
222C4⋊C4—4221no—
223C4⋊C4—4221no—
224C4⋊C4—2111no—
225C4⋊C4—2111no—
226C22⋊C4—4224no—
227C22⋊C4—4221no—
228C22⋊C4—4221no—
229C22⋊C4—4221no—
230C22⋊C4—4221no—
231C22⋊C4—4221no—
232C42—21no—
233C42—21no—
234C42—4221no—
235C42—4221no—
236C42—4221no—
237C23—21no—
238C23—8441no—
239C23—4222no—
240C23—4221no—
241C23—4221no—
242C23—2111no—
243C23—2111no—
244C23D4×D21111yescharacteristic
245Q8—21no—
246Q8—21no—
247D4—21no—
248D4—21no—
249D4—41no—
250D4—41no—
251D4—8442no—
252D4—8442no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—8441no—
260C2×C4—8441no—
261C2×C4—4222no—
262C2×C4—4221no—
263C2×C4—4221no—
264C2×C4—4221no—
265C2×C4—4221no—
266C2×C4—4221no—
267C2×C4—4221no—
268C2×C4—4221no—
269C2×C4—4221no—
270C2×C4—4221no—
271C2×C4—4221no—
272C2×C4—4221no—
273C2×C4—4221no—
274C2×C4—2111no—
275C2×C4—2111no—
276C2×C4C2×C4×D21111yescharacteristic
277C2×C4D4×D21111yescharacteristic
278C2×C4D4×D21111yescharacteristic
279C2×C4C2×C22⋊D21111yescharacteristic
280C2×C4#1680996111yescharacteristic
281C2×C4#1680996211yescharacteristic
282C8—22no—
283C8—4222no—
284C22—21no—
285C22—21no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—8442no—
290C22—8442no—
291C22—4221no—
292C22—4221no—
293C22—4221no—
294C22—4221no—
295C22—4221no—
296C22—2112no—
297C22C22⋊C4×D21111yescharacteristic
298C22#2026994811yescharacteristic
299C22#2026995011yescharacteristic
300C4—42no—
301C4—41no—
302C4—8442no—
303C4—8441no—
304C4—8441no—
305C4—4221no—
306C4—4221no—
307C4(C22⋊D211)⋊C411yescharacteristic
308C4C4×C22⋊D21111yescharacteristic
309C4#2026994511yescharacteristic
310C4#2026995911yescharacteristic
311C2—21no—
312C2—21no—
313C2—81no—
314C2—8441no—
315C2—4222no—
316C2#2721302211yescharacteristic
317C2#104×D21111yescharacteristic
318C2#2721316111yescharacteristic
subgroupG/H
centerC2×C4D4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#104×C211C2
socleC22×C211C2×C22⋊C4
prime radical#27213653C2
cosocleC22×C4×C211C23
hypercenterC2×#104D211
hypocenterC211#1871
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
1. C2×M16×D211C2semidirect
3. C2×#20269904C2semidirect
4. C4×(C2×D211)⋊C4C2semidirect
5. #27212907C2semidirect
6. #27212906C2semidirect
7. C2×#104×C211C2semidirect
15. #20269953C4semidirect
16. #20269953C4semidirect
17. #20269948C4semidirect
18. #20269948C4semidirect
45. #16809946D4semidirect
73. M16×C211D4semidirect
139. C2×D211#104semidirect
159. C211#1871semidirect
164. C2×#104D211semidirect
218. M16C22⋊D211semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D2112
#272136532
C2×#202699042
C4×(C2×D211)⋊C42
#272129072
#272129062
C2×#104×C2112
#1871211