Finite Group Databasebeta

← #41014617

Subgroups — #41014617

Order 27008.

278 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 69 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#27213722C211yesmaximal, characteristic
2C2×#20269903C211yesmaximal, characteristic
3#27213145C211yesmaximal, characteristic
4C2×#20269895C211yesmaximal, characteristic, complemented
5#27213051C211yesmaximal, characteristic, complemented
6#27212905C211yesmaximal, characteristic, complemented
7#428×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9C22×Dic422C2211yescharacteristic
10C2×#16809946C2211yescharacteristic
11#20269977—21no—
12#20269965—21no—
13#20269961—22no—
14#20269957—21no—
15#20269957—21no—
16C2×C4×Dic211C2211yescharacteristic
17#20269903—22no—
18#20269900—21no—
19#20269897—21no—
20#20269895—22no—
21C211⋊#95—22no—
22C2×M16×C211C2211yescharacteristic
23#117×C211C2211yescharacteristic
24C2×C4⋊C4×C211C2211yescharacteristic
25#107×C211—21no—
26#106×C211—21no—
27#16809962—21no—
28#16809962—21no—
29#16809962—21no—
30#16809962—21no—
31C2×Dic422—21no—
32C2×Dic422—21no—
33C2×Dic422—21no—
34C2×Dic422—21no—
35C2×Dic422—21no—
36C2×Dic422—21no—
37C2×Dic422D411yescharacteristic
38C2×D844D411yescharacteristic
39C2×C22⋊D211—21no—
40C22×Dic211—21no—
41C22×Dic211D412yes—
42C2×C4×D211—21no—
43#16809946—21no—
44#16809946—21no—
45C2×C211⋊C8—21no—
46C4⋊Dic211—22no—
47C4⋊Dic211—22no—
48(C2×D211)⋊C4—22no—
49(C2×D211)⋊C4—22no—
50C4×Dic211—21no—
51C4×Dic211—21no—
52C22×C4×C211—21no—
53C22×C4×C211C2311yescharacteristic
54M16×C211—21no—
55M16×C211D412yes—
56C2×C8×C211—21no—
57C4⋊C4×C211—21no—
58C4⋊C4×C211—21no—
59C4⋊C4×C211D411yescharacteristic
60C4⋊C4×C211D411yescharacteristic
61C22⋊C4×C211—21no—
62C42×C211—21no—
63C22×D211—21no—
64Dic422—21no—
65Dic422—21no—
66Dic422—41no—
67Dic422—41no—
68Dic422—41no—
69Dic422—41no—
70D844—21no—
71D844—21no—
72C22⋊D211—41no—
73C22⋊D211—41no—
74C2×Dic211—22no—
75C2×Dic211—22no—
76C2×Dic211—21no—
77C2×Dic211—22no—
78C2×Dic211—22no—
79C2×Dic211—41no—
80C2×Dic211—41no—
81C4×D211—41no—
82C4×D211—41no—
83C211⋊C8—41no—
84C23×C211C2×D411yescharacteristic
85C2×C4×C211—21no—
86C2×C4×C211—21no—
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—22no—
96C22×C211—21no—
97C22×C211—21no—
98C22×C211#12011yescharacteristic
99C22×C211#12111yescharacteristic
100C22×C211#12311yescharacteristic
101C2×D211—41no—
102C2×D211—41no—
103C2×D211—41no—
104C4×C211—41no—
105C4×C211—41no—
106C4×C211—81no—
107C4×C211#12111yescharacteristic
108C4×C211#12111yescharacteristic
109C4×C211#12311yescharacteristic
110C4×C211#12411yescharacteristic
111Dic211—42no—
112Dic211—41no—
113Dic211—42no—
114C2×C211—21no—
115C2×C211—21no—
116C2×C211#39411yescharacteristic
117C2×C211#45411yescharacteristic
118C2×C211#45611yescharacteristic
119D211—81no—
120C211#203811yescharacteristic, complemented, Sylow
121#2038—2111nomaximal, Sylow
122#536—2111no—
123#428D21111yescharacteristic, complemented
124C2×#104—2111no—
125#417—2111no—
126C2×#101—2111no—
127#391—2111no—
128#339—2111no—
129C2×#41—2111no—
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#117C2×D21111yescharacteristic
139C2×C4⋊C4C2×D21111yescharacteristic
140C2×C42—2111no—
141#107—21no—
142#106—21no—
143#104—4222no—
144#103—4221no—
145#102—4221no—
146#101—4222no—
147#95—4222no—
148#41—4221no—
149#41—4221no—
150#41—4221no—
151#41—4221no—
152C2×Q8—4221no—
153C2×Q8—4221no—
154C2×Q8—4221no—
155C2×Q8—4221no—
156C2×Q8—4221no—
157C2×Q8—4221no—
158C2×Q8—2111no—
159C2×D4—4221no—
160C2×D4—2111no—
161C22×C4—21no—
162C22×C4—4221no—
163C22×C4—4221no—
164C22×C4—2112no—
165C22×C4C22×D21111yescharacteristic
166M16—21no—
167M16—4221no—
168M16—4221no—
169M16D84412yes—
170C2×C8—21no—
171C2×C8—4221no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—4222no—
175C4⋊C4—4222no—
176C4⋊C4C22⋊D21111yescharacteristic
177C4⋊C4C22⋊D21111yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—4222no—
180C22⋊C4—4222no—
181C42—21no—
182C42—4221no—
183C42—4221no—
184C23—4221no—
185C23D4×D21111yescharacteristic
186Q8—8441no—
187Q8—8441no—
188Q8—8441no—
189Q8—8441no—
190Q8—4221no—
191Q8—4221no—
192D4—8441no—
193D4—8441no—
194D4—4221no—
195D4—4221no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—8441no—
201C2×C4—8441no—
202C2×C4—8441no—
203C2×C4—8441no—
204C2×C4—4222no—
205C2×C4—4222no—
206C2×C4—4221no—
207C2×C4—4222no—
208C2×C4—4222no—
209C2×C4D4×D21111yescharacteristic
210C2×C4C2×C22⋊D21111yescharacteristic
211C2×C4C2×D84411yescharacteristic
212C2×C4#1680996211yescharacteristic
213C2×C4#1680996311yescharacteristic
214C2×C4#1680996311yescharacteristic
215C8—22no—
216C8—8441no—
217C22—21no—
218C22—21no—
219C22—8441no—
220C22—8441no—
221C22—8441no—
222C22#2026994511yescharacteristic
223C22#2026995211yescharacteristic
224C22#2026996211yescharacteristic
225C4—41no—
226C4—41no—
227C4—81no—
228C4—8442no—
229C4—8441no—
230C4—8442no—
231C4#2026995111yescharacteristic
232C4#2026995211yescharacteristic
233C4#2026996211yescharacteristic
234C4#2026996611yescharacteristic
235C2—21no—
236C2—21no—
237C2—16881no—
238C2#2721306311yescharacteristic
239C2#2721331711yescharacteristic
240C2#2721332711yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#428×C211C2
socleC22×C211#120
prime radicalC2×#16809962C22
hypercenter#428D211
hypocenterC211#2038
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
4. C2×#20269895C2semidirect
5. #27213051C2semidirect
6. #27212905C2semidirect
7. #428×C211C2semidirect
120. C211#2038semidirect
123. #428D211semidirect

Maximal subgroups

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

subgroupindex
#272137222
C2×#202699032
#272131452
C2×#202698952
#272130512
#272129052
#428×C2112
#2038211