Finite Group Databasebeta

← #41014613

Subgroups — #41014613

Order 27008.

278 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 67 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#27213717C211yesmaximal, characteristic
2#27213186C211yesmaximal, characteristic, complemented
3C2×#20269904C211yesmaximal, characteristic
4C2×#20269895C211yesmaximal, characteristic, complemented
5#27213051C211yesmaximal, characteristic, complemented
6#27212907C211yesmaximal, characteristic, complemented
7#417×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10#20269975—21no—
11#20269964—21no—
12#20269958—22no—
13#20269956—21no—
14#20269956—21no—
15#20269936C2211yescharacteristic
16C2×C4×Dic211C2211yescharacteristic
17#20269910—21no—
18#20269908—21no—
19#20269904—22no—
20#20269895—22no—
21C211⋊#95—22no—
22C2×#41×C211C2211yescharacteristic
23C2×M16×C211C2211yescharacteristic
24C2×C4⋊C4×C211C2211yescharacteristic
25#103×C211—21no—
26#102×C211—21no—
27C2×Dic422—21no—
28C2×Dic422—21no—
29C2×Dic422—21no—
30C2×Dic422—21no—
31C2×Dic422—21no—
32C2×Dic422—21no—
33C22×Dic211—21no—
34C22×Dic211D412yes—
35#16809946—21no—
36#16809946D412yes—
37C2×C211⋊C8—21no—
38C4⋊Dic211D411yescharacteristic
39C4⋊Dic211D411yescharacteristic
40Dic211⋊C4—22no—
41Dic211⋊C4—22no—
42C22⋊Dic211—22no—
43C22⋊Dic211—22no—
44C22⋊Dic211—21no—
45C4×Dic211—21no—
46C4×Dic211—21no—
47C4×Dic211—21no—
48#41×C211—21no—
49#41×C211—21no—
50#41×C211—21no—
51#41×C211—21no—
52C2×Q8×C211D411yescharacteristic
53C2×D4×C211—21no—
54C2×D4×C211D411yescharacteristic
55C22×C4×C211—21no—
56C22×C4×C211—21no—
57C22×C4×C211C2311yescharacteristic
58M16×C211—21no—
59M16×C211—21no—
60C2×C8×C211—21no—
61C4⋊C4×C211—21no—
62C4⋊C4×C211—21no—
63Dic422—41no—
64Dic422—41no—
65Dic422—41no—
66Dic422—41no—
67C2×Dic211—22no—
68C2×Dic211—21no—
69C2×Dic211—21no—
70C2×Dic211—22no—
71C2×Dic211—22no—
72C2×Dic211—22no—
73C2×Dic211—41no—
74C2×Dic211—41no—
75C211⋊C8—22no—
76C23×C211—21no—
77C23×C211C2×D411yescharacteristic
78Q8×C211—21no—
79Q8×C211—21no—
80D4×C211—21no—
81D4×C211—21no—
82D4×C211—41no—
83D4×C211—41no—
84C2×C4×C211—21no—
85C2×C4×C211—41no—
86C2×C4×C211—41no—
87C2×C4×C211—42no—
88C2×C4×C211—41no—
89C2×C4×C211C2×D411yescharacteristic
90C2×C4×C211C2×D411yescharacteristic
91C2×C4×C211C2×D411yescharacteristic
92C2×C4×C211#4111yescharacteristic
93C2×C4×C211#4111yescharacteristic
94C2×C4×C211#4111yescharacteristic
95C8×C211—41no—
96C22×C211—21no—
97C22×C211—21no—
98C22×C211—41no—
99C22×C211—41no—
100C22×C211—41no—
101C22×C211#12011yescharacteristic
102C22×C211#12111yescharacteristic
103C22×C211#12311yescharacteristic
104C4×C211—41no—
105C4×C211—81no—
106C4×C211#12111yescharacteristic
107C4×C211#12111yescharacteristic
108C4×C211#12311yescharacteristic
109C4×C211#12411yescharacteristic
110Dic211—42no—
111Dic211—42no—
112Dic211—41no—
113Dic211—41no—
114C2×C211—21no—
115C2×C211—21no—
116C2×C211—81no—
117C2×C211#39411yescharacteristic
118C2×C211#45411yescharacteristic
119C2×C211#45611yescharacteristic
120C211#203811yescharacteristic, complemented, Sylow
121#2038—2111nomaximal, Sylow
122#536—2111no—
123#428—2111no—
124C2×#104—2111no—
125#417D21111yescharacteristic
126C2×#101—2111no—
127#391—2111no—
128#339—2111no—
129C2×#41C2×D21111yescharacteristic
130C22×Q8—2111no—
131C2×M16—2111no—
132C2×M16C2×D21111yescharacteristic
133C4⋊Q8—4221no—
134#124—4221no—
135#123—4222no—
136C22⋊Q8—4221no—
137C22⋊Q8—4221no—
138#117—2111no—
139C2×C4⋊C4C2×D21111yescharacteristic
140C2×C42—2111no—
141#107—4221no—
142#106—4221no—
143#104—4222no—
144#103—21no—
145#102—21no—
146#101—4222no—
147#95—4222no—
148#41—21no—
149#41—21no—
150#41—21no—
151#41—21no—
152C2×Q8—4221no—
153C2×Q8—4221no—
154C2×Q8—4221no—
155C2×Q8—4221no—
156C2×Q8—4221no—
157C2×Q8—4221no—
158C2×Q8C22⋊D21111yescharacteristic
159C2×D4—21no—
160C2×D4C22⋊D21111yescharacteristic
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—4221no—
164C22×C4—2112no—
165C22×C4C22×D21111yescharacteristic
166M16—21no—
167M16—21no—
168M16—4221no—
169M16—2112no—
170C2×C8—21no—
171C2×C8—4221no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—4222no—
175C4⋊C4—4222no—
176C4⋊C4—2111no—
177C4⋊C4—2111no—
178C22⋊C4—4222no—
179C22⋊C4—4222no—
180C22⋊C4—4221no—
181C42—4221no—
182C42—4221no—
183C42—4221no—
184C23—21no—
185C23D4×D21111yescharacteristic
186Q8—21no—
187Q8—21no—
188Q8—8441no—
189Q8—8441no—
190Q8—8441no—
191Q8—8441no—
192D4—21no—
193D4—21no—
194D4—41no—
195D4—41no—
196C2×C4—21no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—8441no—
202C2×C4—8441no—
203C2×C4—4222no—
204C2×C4—4221no—
205C2×C4—4221no—
206C2×C4—4222no—
207C2×C4—4222no—
208C2×C4—4222no—
209C2×C4D4×D21111yescharacteristic
210C2×C4D4×D21111yescharacteristic
211C2×C4C2×C22⋊D21111yescharacteristic
212C2×C4#1680996111yescharacteristic
213C2×C4#1680996211yescharacteristic
214C2×C4#1680996211yescharacteristic
215C8—41no—
216C8—4222no—
217C22—21no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22#2026994511yescharacteristic
223C22#2026995011yescharacteristic
224C22#2026995911yescharacteristic
225C4—41no—
226C4—81no—
227C4—8442no—
228C4—8442no—
229C4—8441no—
230C4—8441no—
231C4#2026994911yescharacteristic
232C4#2026995011yescharacteristic
233C4#2026996011yescharacteristic
234C4#2026996311yescharacteristic
235C2—21no—
236C2—21no—
237C2—81no—
238C2#2721306111yescharacteristic
239C2#2721331711yescharacteristic
240C2#2721332711yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#417×C211C2
socleC22×C211#120
prime radicalC2×#41×C211C22
hypercenter#417D211
hypocenterC211#2038
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. #27213186C2semidirect
4. C2×#20269895C2semidirect
5. #27213051C2semidirect
6. #27212907C2semidirect
120. C211#2038semidirect

Maximal subgroups

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

subgroupindex
#272137172
#272131862
C2×#202699042
C2×#202698952
#272130512
#272129072
#417×C2112
#2038211