Finite Group Databasebeta

← #41014315

Subgroups — #41014315

Order 27008.

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

normal subgroups only

This group has 336 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
1#27213692C211yesmaximal, characteristic
2C2×#20269905C211yesmaximal, characteristic, complemented
3C2×#20269904C211yesmaximal, characteristic, complemented
4C2×#20269903C211yesmaximal, characteristic, complemented
5M16×Dic211C211yesmaximal, characteristic, complemented
6C4×Dic211⋊C4C211yesmaximal, characteristic, complemented
7C2×#104×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10#20269975C411yescharacteristic, complemented
11#20269973C411yescharacteristic, complemented
12#20269966C411yescharacteristic, complemented
13#20269964C411yescharacteristic, complemented
14#20269949—21no—
15#20269949—21no—
16C4×C22⋊D211—21no—
17C4×C22⋊D211—21no—
18C2×Dic211⋊C4—21no—
19C2×C4×Dic211C2211yescharacteristic
20C2×C4×Dic211C2211yescharacteristic
21#20269905—22no—
22#20269904—22no—
23#20269903—22no—
24#20269887—21no—
25C8×Dic211—21no—
26C211⋊#95—21no—
27C2×#41×C211C2211yescharacteristic
28C2×M16×C211C2211yescharacteristic
29C2×C42×C211C2211yescharacteristic
30#104×C211—22no—
31#16809962—21no—
32#16809962—21no—
33#16809962—21no—
34#16809962—21no—
35C2×Dic422C2×C411yescharacteristic
36C2×D844C2×C411yescharacteristic
37C2×C22⋊D211—21no—
38C2×C22⋊D211—21no—
39C2×C22⋊D211—21no—
40C22×Dic211—21no—
41C22×Dic211D411yescharacteristic
42C22×Dic211D411yescharacteristic
43C2×C4×D211—21no—
44#16809946—21no—
45#16809946D412yescomplemented
46C2×C211⋊C8—21no—
47Dic211⋊C4—24no—
48Dic211⋊C4—21no—
49Dic211⋊C4—21no—
50C22⋊Dic211—21no—
51C22⋊Dic211—21no—
52(C2×D211)⋊C4—21no—
53(C2×D211)⋊C4—21no—
54C4×Dic211—21no—
55C4×Dic211—21no—
56C4×Dic211C2×C411yescharacteristic
57C4×Dic211C2×C411yescharacteristic
58C4×Dic211D411yescharacteristic
59C4×Dic211D411yescharacteristic
60#41×C211—21no—
61#41×C211—21no—
62#41×C211—21no—
63#41×C211—21no—
64C2×Q8×C211C2×C411yescharacteristic
65C2×D4×C211—21no—
66C2×D4×C211C2×C411yescharacteristic
67C22×C4×C211—21no—
68C22×C4×C211—21no—
69C22×C4×C211C2311yescharacteristic
70M16×C211—21no—
71M16×C211D412yescomplemented
72C2×C8×C211—21no—
73C42×C211—21no—
74C42×C211—21no—
75C22×D211—21no—
76Dic422—21no—
77Dic422—21no—
78D844—21no—
79D844—21no—
80C22⋊D211—42no—
81C22⋊D211—41no—
82C22⋊D211—41no—
83C22⋊D211—42no—
84C2×Dic211—22no—
85C2×Dic211—22no—
86C2×Dic211—21no—
87C2×Dic211—41no—
88C2×Dic211—41no—
89C2×Dic211—41no—
90C2×Dic211C22⋊C411yescharacteristic
91C2×Dic211C22⋊C411yescharacteristic
92C2×Dic211C22⋊C411yescharacteristic
93C2×Dic211C22⋊C411yescharacteristic
94C4×D211—41no—
95C4×D211—41no—
96C211⋊C8—22no—
97C23×C211—21no—
98C23×C211C2×D411yescharacteristic
99Q8×C211—21no—
100Q8×C211—21no—
101D4×C211—21no—
102D4×C211—21no—
103D4×C211—41no—
104D4×C211—41no—
105C2×C4×C211—22no—
106C2×C4×C211—21no—
107C2×C4×C211—41no—
108C2×C4×C211—41no—
109C2×C4×C211—41no—
110C2×C4×C211—41no—
111C2×C4×C211C22×C411yescharacteristic
112C2×C4×C211C2×D411yescharacteristic
113C2×C4×C211C2×D411yescharacteristic
114C2×C4×C211C2×D411yescharacteristic
115C2×C4×C211#4111yescharacteristic
116C2×C4×C211#4111yescharacteristic
117C8×C211—22no—
118C22×C211—21no—
119C22×C211—21no—
120C22×C211—41no—
121C22×C211—41no—
122C22×C211—41no—
123C22×C211C2×C22⋊C411yescharacteristic
124C22×C211#12411yescharacteristic
125C22×C211#12711yescharacteristic
126C2×D211—41no—
127C2×D211—41no—
128C2×D211—41no—
129C4×C211—42no—
130C4×C211—41no—
131C4×C211C4×D411yescharacteristic
132C4×C211C4×D411yescharacteristic
133C4×C211#12111yescharacteristic
134C4×C211#12111yescharacteristic
135Dic211—21no—
136Dic211—21no—
137Dic211—21no—
138Dic211—42no—
139Dic211—41no—
140Dic211#10412yescomplemented
141C2×C211—21no—
142C2×C211—21no—
143C2×C211—81no—
144C2×C211#39011yescharacteristic
145C2×C211C2×#10411yescharacteristic
146C2×C211#42111yescharacteristic
147D211—81no—
148C211#198011yescharacteristic, complemented, Sylow
149#1980—2111nomaximal, Sylow
150#532—2111no—
151C2×#104—2111no—
152C2×#104—2111no—
153C2×#104—2111no—
154C2×#104D21111yescharacteristic, complemented
155C4×M16—2111no—
156C4×C4⋊C4—2111no—
157C2×#41—2111no—
158C2×#41C2×D21111yescharacteristic
159C2×M16—2111no—
160C2×M16C2×D21111yescharacteristic
161C4⋊Q8—2111no—
162#127—2111no—
163#124—2111no—
164#124—2111no—
165#121—4221no—
166#121—4221no—
167C4×D4—4221no—
168C4×D4—4221no—
169C2×C4⋊C4—4221no—
170C2×C42—2111no—
171C2×C42—2111no—
172C2×C42C2×D21111yescharacteristic
173#104—22no—
174#104—4222no—
175#104—4222no—
176#104—4222no—
177#97—4221no—
178C4×C8—4221no—
179#95—4221no—
180#41—21no—
181#41—21no—
182#41—21no—
183#41—21no—
184#41—4221no—
185#41—4221no—
186#41—4221no—
187#41—4221no—
188C2×Q8—2111no—
189C2×Q8C4×D21111yescharacteristic
190C2×D4—21no—
191C2×D4—4221no—
192C2×D4—4221no—
193C2×D4—4221no—
194C2×D4—2111no—
195C2×D4C4×D21111yescharacteristic
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—4221no—
199C22×C4—4221no—
200C22×C4—2111no—
201C22×C4—2111no—
202C22×C4C22×D21111yescharacteristic
203M16—21no—
204M16—4221no—
205M16—2112no—
206M16C22⋊D21112yescomplemented
207C2×C8—21no—
208C2×C8—4221no—
209C4⋊C4—4224no—
210C4⋊C4—4221no—
211C4⋊C4—4221no—
212C22⋊C4—4221no—
213C22⋊C4—4221no—
214C22⋊C4—4221no—
215C22⋊C4—4221no—
216C42—21no—
217C42—21no—
218C42—4221no—
219C42—4221no—
220C42—2111no—
221C42—2111no—
222C42—2111no—
223C42—2111no—
224C23—21no—
225C23—4221no—
226C23D4×D21111yescharacteristic
227Q8—21no—
228Q8—21no—
229Q8—4221no—
230Q8—4221no—
231D4—21no—
232D4—21no—
233D4—41no—
234D4—41no—
235D4—8441no—
236D4—8441no—
237D4—8442no—
238D4—8442no—
239D4—4221no—
240D4—4221no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—8441no—
248C2×C4—8441no—
249C2×C4—8441no—
250C2×C4—8441no—
251C2×C4—8441no—
252C2×C4—4222no—
253C2×C4—4221no—
254C2×C4—4222no—
255C2×C4—2111no—
256C2×C4—2111no—
257C2×C4—2111no—
258C2×C4—2111no—
259C2×C4C2×C4×D21111yescharacteristic
260C2×C4D4×D21111yescharacteristic
261C2×C4D4×D21111yescharacteristic
262C2×C4C2×C22⋊D21111yescharacteristic
263C2×C4#1680996111yescharacteristic
264C2×C4#1680996211yescharacteristic
265C8—22no—
266C8—4222no—
267C22—21no—
268C22—21no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—8441no—
273C22—8441no—
274C22—8441no—
275C22C22⋊C4×D21111yescharacteristic
276C22#2026996311yescharacteristic
277C22#2026997311yescharacteristic
278C4—42no—
279C4—41no—
280C4—8442no—
281C4—8441no—
282C4—4221no—
283C4—4221no—
284C4—4221no—
285C4—2112no—
286C4(C22⋊D211)⋊C411yescharacteristic
287C4C4×C22⋊D21111yescharacteristic
288C4#2026994911yescharacteristic
289C4#2026995011yescharacteristic
290C2—21no—
291C2—21no—
292C2—81no—
293C2—16881no—
294C2#2721304611yescharacteristic
295C2#104×D21111yescharacteristic
296C2#2721316111yescharacteristic
subgroupG/H
centerC2×C4D4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#104×C211C2
socleC22×C211C2×C22⋊C4
prime radical#27213692C2
cosocleC22×C4×C211C23
hypercenterC2×#104D211
hypocenterC211#1980
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
2. C2×#20269905C2semidirect
3. C2×#20269904C2semidirect
4. C2×#20269903C2semidirect
5. M16×Dic211C2semidirect
6. C4×Dic211⋊C4C2semidirect
7. C2×#104×C211C2semidirect
10. #20269975C4semidirect
11. #20269973C4semidirect
12. #20269966C4semidirect
13. #20269964C4semidirect
45. #16809946D4semidirect
71. M16×C211D4semidirect
140. Dic211#104semidirect
148. C211#1980semidirect
154. C2×#104D211semidirect
206. M16C22⋊D211semidirect

Maximal subgroups

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

subgroupindex
#272136922
C2×#202699052
C2×#202699042
C2×#202699032
M16×Dic2112
C4×Dic211⋊C42
C2×#104×C2112
#1980211