Finite Group Databasebeta

← #41013900

Subgroups — #41013900

Order 27008.

332 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 79 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#27213946C211yesmaximal, characteristic
2#27213614C211yesmaximal, characteristic, complemented
3#27213162C211yesmaximal, characteristic, complemented
4C2×#20269903C211yesmaximal, characteristic, complemented
5#27213119C211yesmaximal, characteristic, complemented
6#27212907C211yesmaximal, characteristic, complemented
7#344×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9C2×#16809962C2211yescharacteristic
10#20270010—21no—
11#20270010C412yescomplemented
12#20270005—21no—
13#20270005C412yescomplemented
14#20269999—22no—
15#20269942—21no—
16#20269939—21no—
17(C22⋊D211)⋊C4—22no—
18#20269936C2211yescharacteristic
19#20269934—21no—
20C2×C4×Dic211C2211yescharacteristic
21C42×D211—21no—
22#20269903—22no—
23#20269903—21no—
24#20269903—21no—
25#20269891—22no—
26C2×M16×C211C2211yescharacteristic
27C2×M16×C211C2211yescharacteristic
28#117×C211C2211yescharacteristic
29#16809962—21no—
30#16809962—21no—
31#16809962—21no—
32#16809962—21no—
33#16809962—21no—
34#16809962—21no—
35#16809962C2×C412yes—
36#16809962D412yes—
37C2×Dic422C2×C411yescharacteristic, complemented
38C2×Dic422D411yescharacteristic
39C2×D844C2×C411yescharacteristic, complemented
40C2×D844D411yescharacteristic
41C2×C22⋊D211—21no—
42C2×C22⋊D211—21no—
43C22×Dic211—21no—
44C2×C4×D211—22no—
45C2×C4×D211—21no—
46C2×C4×D211—21no—
47Dic844—22no—
48#16809951—22no—
49#16809951—22no—
50D1688—22no—
51C4⋊Dic211—21no—
52Dic211⋊C4—22no—
53C22⋊Dic211—21no—
54(C2×D211)⋊C4—22no—
55C4×Dic211—22no—
56C4×Dic211—21no—
57C4×Dic211—21no—
58C4×Dic211D411yescharacteristic
59C4×Dic211D411yescharacteristic
60C22×C4×C211C2311yescharacteristic
61M16×C211—21no—
62M16×C211—21no—
63M16×C211—21no—
64M16×C211C2×C412yescomplemented
65C2×C8×C211—21no—
66C2×C8×C211—21no—
67C4⋊C4×C211—21no—
68C22⋊C4×C211D412yes—
69C42×C211—21no—
70C22×D211—21no—
71C22×D211—21no—
72Dic422—21no—
73Dic422—21no—
74Dic422—21no—
75Dic422C22⋊C412yes—
76D844—21no—
77D844—21no—
78D844—21no—
79D844C22⋊C412yes—
80C22⋊D211—22no—
81C22⋊D211—22no—
82C22⋊D211—41no—
83C22⋊D211—41no—
84C2×Dic211—21no—
85C2×Dic211—21no—
86C2×Dic211—21no—
87C2×Dic211—21no—
88C2×Dic211—42no—
89C2×Dic211—41no—
90C4×D211—22no—
91C4×D211—22no—
92C4×D211—44no—
93C4×D211—41no—
94C4×D211—41no—
95C23×C211C2×D411yescharacteristic
96C2×C4×C211—22no—
97C2×C4×C211C22×C411yescharacteristic
98C2×C4×C211C2×D411yescharacteristic
99C2×C4×C211C2×D411yescharacteristic
100C2×C4×C211C2×D411yescharacteristic
101C2×C4×C211#4111yescharacteristic
102C2×C4×C211#4111yescharacteristic
103C8×C211—22no—
104C8×C211—41no—
105C22×C211—41no—
106C22×C211C4×D411yescharacteristic
107C22×C211#12011yescharacteristic
108C22×C211#12111yescharacteristic
109C2×D211—22no—
110C2×D211—42no—
111C2×D211—41no—
112C2×D211—81no—
113C4×C211—42no—
114C4×C211C2×C22⋊C411yescharacteristic
115C4×C211C4×D411yescharacteristic
116C4×C211#12111yescharacteristic
117C4×C211#12311yescharacteristic
118Dic211—22no—
119Dic211—22no—
120Dic211—41no—
121Dic211—41no—
122Dic211—81no—
123C2×C211—21no—
124C2×C211—21no—
125C2×C211—21no—
126C2×C211#38611yescharacteristic
127D211—42no—
128D211—81no—
129C211#190611yescharacteristic, complemented, Sylow
130#1906—2111nomaximal, Sylow
131#575—2111no—
132C4×#41—2111no—
133#421—2111no—
134C2×#104—2111no—
135#410—2111no—
136#344D21111yescharacteristic, complemented
137#339—2111no—
138C2×#41—2111no—
139C2×#41—2111no—
140#137—4221no—
141#137—2112no—
142AGL(1,ℤ/8)—4221no—
143AGL(1,ℤ/8)—2112no—
144#135—4222no—
145C2×M16C2×D21111yescharacteristic
146C2×M16C2×D21111yescharacteristic
147C4×Q8—4221no—
148C4×D4—4221no—
149C4×D4—4222no—
150#117—4221no—
151#117—2111no—
152#117C2×D21111yescharacteristic
153C2×C42—4221no—
154C2×C42—2111no—
155#104—4222no—
156#104—4221no—
157#104—4221no—
158#99—4222no—
159#41—4221no—
160#41—4221no—
161#41—4221no—
162#41—4221no—
163#41—4221no—
164#41—4221no—
165#41—2112no—
166#41—2112no—
167C2×Q8—2111no—
168C2×Q8—2111no—
169C2×D4—4221no—
170C2×D4—4221no—
171C2×D4—2111no—
172C2×D4—2111no—
173C22×C4—4222no—
174C22×C4—4221no—
175C22×C4—4221no—
176C22×C4—4221no—
177C22×C4C22×D21111yescharacteristic
178Q16—4222no—
179SD16—4222no—
180SD16—4222no—
181D8—4222no—
182M16—21no—
183M16—21no—
184M16—21no—
185M16C4×D21112yescomplemented
186C2×C8—21no—
187C2×C8—21no—
188C4⋊C4—21no—
189C4⋊C4—4222no—
190C4⋊C4—4221no—
191C22⋊C4—4222no—
192C22⋊C4—4221no—
193C22⋊C4D84412yes—
194C42—21no—
195C42—4222no—
196C42—4221no—
197C42—4221no—
198C42—2111no—
199C42—2111no—
200C23—4221no—
201C23—4221no—
202C23C2×D84411yescharacteristic
203Q8—4221no—
204Q8—4221no—
205Q8—4221no—
206Q8—2112no—
207D4—8441no—
208D4—8441no—
209D4—4222no—
210D4—4222no—
211D4—4221no—
212D4—4221no—
213D4—4221no—
214D4—2112no—
215C2×C4—22no—
216C2×C4—8444no—
217C2×C4—8442no—
218C2×C4—8441no—
219C2×C4—8441no—
220C2×C4—8441no—
221C2×C4—4221no—
222C2×C4—4221no—
223C2×C4—4221no—
224C2×C4—4221no—
225C2×C4—4222no—
226C2×C4—4222no—
227C2×C4C2×C4×D21111yescharacteristic
228C2×C4D4×D21111yescharacteristic
229C2×C4D4×D21111yescharacteristic
230C2×C4D4×D21111yescharacteristic
231C2×C4#1680996211yescharacteristic
232C2×C4#1680996311yescharacteristic
233C8—22no—
234C8—41no—
235C22—41no—
236C22—8442no—
237C22—8441no—
238C22—16881no—
239C22—4222no—
240C22C4×D84411yescharacteristic
241C22#2026994711yescharacteristic
242C22#2026995211yescharacteristic
243C4—42no—
244C4—8441no—
245C4—8441no—
246C4—16881no—
247C4—4222no—
248C4—4222no—
249C4C22⋊C4×D21111yescharacteristic
250C4#2026993911yescharacteristic
251C4#2026995011yescharacteristic
252C4#2026996211yescharacteristic
253C2—21no—
254C2—21no—
255C2—21no—
256C2—8442no—
257C2—16881no—
258C2#2721302411yescharacteristic
subgroupG/H
centerC4#20269950
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#344×C211C2
socleC2×C211#386
prime radical#27213946C2
cosocleC22×C4×C211C23
hypercenter#344D211
hypocenterC211#1906
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
2. #27213614C2semidirect
3. #27213162C2semidirect
4. C2×#20269903C2semidirect
5. #27213119C2semidirect
6. #27212907C2semidirect
7. #344×C211C2semidirect
11. #20270010C4semidirect
13. #20270005C4semidirect
37. C2×Dic422C2×C4semidirect
39. C2×D844C2×C4semidirect
64. M16×C211C2×C4semidirect
129. C211#1906semidirect
136. #344D211semidirect
185. M16C4×D211semidirect

Maximal subgroups

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

subgroupindex
#272139462
#272136142
#272131622
C2×#202699032
#272131192
#272129072
#344×C2112
#1906211