Finite Group Databasebeta

← #41014528

Subgroups — #41014528

Order 27008.

430 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 73 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#27213705C211yesmaximal, characteristic
2C2×#20269961C211yesmaximal, characteristic, complemented
3C2×#20269958C211yesmaximal, characteristic, complemented
4C2×#20269891C211yesmaximal, characteristic, complemented
5#27213007C211yesmaximal, characteristic, complemented
6#27212912C211yesmaximal, characteristic, complemented
7C2×#99×C211C211yesmaximal, characteristic, complemented
8C22×C22⋊D211C2211yescharacteristic
9C23×Dic211C2211yescharacteristic
10#20269961—22no—
11#20269961—22no—
12#20269960—21no—
13#20269958—21no—
14#20269958—22no—
15#20269958—22no—
16#20269949—22no—
17#20269946—21no—
18#20269945—21no—
19C2×C4⋊Dic211—21no—
20C2×Dic211⋊C4—21no—
21C2×C22⋊Dic211—21no—
22C2×C22⋊Dic211C2211yescharacteristic
23C2×C22⋊Dic211C2211yescharacteristic
24C2×(C2×D211)⋊C4—21no—
25#20269891—22no—
26C211⋊#95—22no—
27C22×D4×C211C2211yescharacteristic
28C2×C22⋊C4×C211C2211yescharacteristic
29C2×C22⋊C4×C211C2211yescharacteristic
30#99×C211—22no—
31C23×D211D411yescharacteristic
32C2×C22⋊D211—22no—
33C2×C22⋊D211—21no—
34C2×C22⋊D211—21no—
35C2×C22⋊D211—21no—
36C2×C22⋊D211—21no—
37C2×C22⋊D211—21no—
38C2×C22⋊D211—21no—
39C22×Dic211—21no—
40C22×Dic211—21no—
41C22×Dic211—21no—
42C22×Dic211—21no—
43C22×Dic211—21no—
44C22×Dic211—21no—
45C22×Dic211—21no—
46C22×Dic211D411yescharacteristic
47C22×Dic211D412yes—
48C22×Dic211D412yescomplemented
49C4⋊Dic211—24no—
50Dic211⋊C4—24no—
51Dic211⋊C4—22no—
52C22⋊Dic211—24no—
53C22⋊Dic211—22no—
54C22⋊Dic211—21no—
55C22⋊Dic211—21no—
56C22⋊Dic211—21no—
57C22⋊Dic211D412yescomplemented
58(C2×D211)⋊C4—24no—
59(C2×D211)⋊C4—22no—
60C24×C211D411yescharacteristic
61C24×C211C2311yescharacteristic
62C2×D4×C211—21no—
63C2×D4×C211—21no—
64C2×D4×C211—21no—
65C2×D4×C211—21no—
66C2×D4×C211—21no—
67C2×D4×C211—21no—
68C22×C4×C211—21no—
69C22×C4×C211—21no—
70C22×C4×C211D411yescharacteristic
71C22⋊C4×C211—21no—
72C22⋊C4×C211—21no—
73C22⋊C4×C211—21no—
74C22⋊C4×C211D412yescomplemented
75C22×D211—21no—
76C22×D211—21no—
77C22×D211—21no—
78C22×D211—42no—
79C22⋊D211—41no—
80C22⋊D211—41no—
81C22⋊D211—41no—
82C22⋊D211—41no—
83C22⋊D211—82no—
84C2×Dic211—22no—
85C2×Dic211—22no—
86C2×Dic211—22no—
87C2×Dic211—22no—
88C2×Dic211—21no—
89C2×Dic211—21no—
90C2×Dic211—22no—
91C2×Dic211—21no—
92C2×Dic211—42no—
93C2×Dic211—41no—
94C2×Dic211—41no—
95C2×Dic211—41no—
96C2×Dic211—41no—
97C2×Dic211—42no—
98C23×C211—21no—
99C23×C211—21no—
100C23×C211—21no—
101C23×C211—42no—
102C23×C211—42no—
103C23×C211C2×D411yescharacteristic
104C23×C211C2×D411yescharacteristic
105C23×C211C2×D411yescharacteristic
106C23×C211C2×D411yescharacteristic
107C23×C211C2×D411yescharacteristic
108C23×C211C2×D411yescharacteristic
109C23×C211#4111yescharacteristic
110D4×C211—41no—
111D4×C211—41no—
112D4×C211—41no—
113D4×C211—41no—
114C2×C4×C211—22no—
115C2×C4×C211—21no—
116C2×C4×C211—21no—
117C2×C4×C211—21no—
118C2×C4×C211—42no—
119C2×C4×C211—42no—
120C2×C4×C211—41no—
121C22×C211—22no—
122C22×C211—22no—
123C22×C211—22no—
124C22×C211—41no—
125C22×C211—41no—
126C22×C211—41no—
127C22×C211—41no—
128C22×C211—42no—
129C22×C211—41no—
130C22×C211—42no—
131C22×C211—81no—
132C22×C211#12011yescharacteristic
133C22×C211#12011yescharacteristic
134C22×C211#12011yescharacteristic
135C22×C211#12111yescharacteristic
136C22×C211#12111yescharacteristic
137C22×C211#12111yescharacteristic
138C22×C211#12711yescharacteristic
139C2×D211—42no—
140C2×D211—41no—
141C2×D211—42no—
142C2×D211—81no—
143C4×C211—42no—
144C4×C211—41no—
145C4×C211—81no—
146Dic211—42no—
147Dic211—42no—
148Dic211—41no—
149Dic211—42no—
150C2×C211—21no—
151C2×C211—21no—
152C2×C211—21no—
153C2×C211—21no—
154C2×C211—21no—
155C2×C211—21no—
156C2×C211—81no—
157C2×C211#39211yescharacteristic
158C2×C211#45812yes—
159D211—81no—
160C211#202211yescharacteristic, complemented, Sylow
161#2022—2111nomaximal, Sylow
162#534—2111no—
163C2×#123—2111no—
164C2×#123—2111no—
165C2×#99—2111no—
166C2×#99D21111yescharacteristic, complemented
167#381—2111no—
168#342—2111no—
169C22×D4—2111no—
170C22×D4C2×D21111yescharacteristic
171C23×C4—2111no—
172#123—4222no—
173#123—4222no—
174#123—4221no—
175#123—4221no—
176#123—4222no—
177#123—4222no—
178#121—4222no—
179#120—4221no—
180#120—4221no—
181C2×C4⋊C4—4221no—
182C2×C4⋊C4—4221no—
183C2×C22⋊C4—4221no—
184C2×C22⋊C4—4221no—
185C2×C22⋊C4—2111no—
186C2×C22⋊C4—2111no—
187C2×C22⋊C4C2×D21111yescharacteristic
188C2×C22⋊C4C2×D21111yescharacteristic
189#99—22no—
190#99—4222no—
191#95—4222no—
192C24—2111no—
193C24D84411yescharacteristic
194C24C22×D21111yescharacteristic
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—4221no—
202C2×D4—4221no—
203C2×D4—4221no—
204C2×D4—4221no—
205C2×D4—4221no—
206C2×D4—4221no—
207C2×D4—4222no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—4221no—
211C22×C4—4221no—
212C22×C4—4221no—
213C22×C4—4221no—
214C22×C4—4221no—
215C22×C4—4221no—
216C22×C4—4221no—
217C22×C4—2112no—
218C22×C4—2111no—
219C22×C4—2112no—
220C22×C4D84411yescharacteristic
221C4⋊C4—4224no—
222C4⋊C4—4224no—
223C4⋊C4—4222no—
224C22⋊C4—21no—
225C22⋊C4—21no—
226C22⋊C4—21no—
227C22⋊C4—4224no—
228C22⋊C4—4222no—
229C22⋊C4—4224no—
230C22⋊C4—4222no—
231C22⋊C4—4221no—
232C22⋊C4—4221no—
233C22⋊C4—4221no—
234C22⋊C4—2112no—
235C22⋊C4C22⋊D21112yescomplemented
236C23—21no—
237C23—21no—
238C23—21no—
239C23—42no—
240C23—42no—
241C23—8442no—
242C23—4221no—
243C23—4221no—
244C23—4221no—
245C23D4×D21111yescharacteristic
246C23D4×D21111yescharacteristic
247C23D4×D21111yescharacteristic
248C23D4×D21111yescharacteristic
249C23C2×C22⋊D21111yescharacteristic
250C23C2×D84411yescharacteristic
251C23#1680996211yescharacteristic
252D4—41no—
253D4—41no—
254D4—41no—
255D4—41no—
256D4—8441no—
257D4—8441no—
258D4—8441no—
259D4—8441no—
260D4—16882no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—8442no—
269C2×C4—8441no—
270C2×C4—8441no—
271C2×C4—8441no—
272C2×C4—8441no—
273C2×C4—8442no—
274C2×C4—4222no—
275C2×C4—4221no—
276C2×C4—4222no—
277C2×C4—4222no—
278C2×C4—4222no—
279C2×C4—4221no—
280C2×C4—4222no—
281C2×C4—4221no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—42no—
290C22—42no—
291C22—41no—
292C22—81no—
293C22—8442no—
294C22—8442no—
295C22—8441no—
296C22—16881no—
297C22#2026994511yescharacteristic
298C22#2026994711yescharacteristic
299C22#2026994711yescharacteristic
300C22#2026995011yescharacteristic
301C22#2026995011yescharacteristic
302C22#2026995111yescharacteristic
303C22#2026997311yescharacteristic
304C4—42no—
305C4—41no—
306C4—81no—
307C4—8442no—
308C4—8442no—
309C4—8442no—
310C4—8441no—
311C2—21no—
312C2—21no—
313C2—21no—
314C2—21no—
315C2—21no—
316C2—21no—
317C2—81no—
318C2—16881no—
319C2#2721305511yescharacteristic
320C2#2721333712yes—
subgroupG/H
centerC22#20269947
commutator, cosocleC24×C211C23
FrattiniC24C22×D211
FittingC2×#99×C211C2
socleC22×C211#120
prime radical#27213705C2
hypercenterC2×#99D211
hypocenterC211#2022
LCSG ▷ C24×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. C2×#20269961C2semidirect
3. C2×#20269958C2semidirect
4. C2×#20269891C2semidirect
5. #27213007C2semidirect
6. #27212912C2semidirect
7. C2×#99×C211C2semidirect
48. C22×Dic211D4semidirect
57. C22⋊Dic211D4semidirect
74. C22⋊C4×C211D4semidirect
160. C211#2022semidirect
166. C2×#99D211semidirect
235. C22⋊C4C22⋊D211semidirect

Maximal subgroups

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

subgroupindex
#272137052
C2×#202699612
C2×#202699582
C2×#202698912
#272130072
#272129122
C2×#99×C2112
#2022211