Finite Group Databasebeta

← #41013727

Subgroups — #41013727

Order 27008.

324 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#20269984C211yesmaximal, characteristic, complemented
2C2×#20269904C211yesmaximal, characteristic, complemented
3C2×#20269904C211yesmaximal, characteristic, complemented
4#27213113C211yesmaximal, characteristic, complemented
5#27212911C211yesmaximal, characteristic, complemented
6#27212900C211yesmaximal, characteristic, complemented
7C2×#135×C211C211yesmaximal, characteristic
8#20269984—22no—
9#20269984—22no—
10C2×#16809946—21no—
11C2×#16809946C2211yescharacteristic
12C2×#16809946C2211yescharacteristic
13C22×C211⋊C8—21no—
14C22×C211⋊C8C2211yescharacteristic
15C2×C4×Dic211C2211yescharacteristic
16#20269904—22no—
17#20269904—22no—
18C22⋊(C211⋊C8)—21no—
19C22⋊(C211⋊C8)—21no—
20C2×#41×C211C2211yescharacteristic
21C2×#41×C211C2211yescharacteristic
22#135×C211—22no—
23#135×C211—22no—
24C2×Q16×C211C411yescharacteristic, complemented
25C2×SD16×C211C411yescharacteristic, complemented
26C2×SD16×C211C411yescharacteristic, complemented
27C2×D8×C211C411yescharacteristic, complemented
28C22×C8×C211C2211yescharacteristic
29C22×Dic211—21no—
30#16809946—24no—
31#16809946—21no—
32#16809946—21no—
33#16809946—21no—
34#16809946D412yes—
35C2×C211⋊C8—24no—
36C2×C211⋊C8—21no—
37C2×C211⋊C8—21no—
38C2×C211⋊C8—21no—
39C2×C211⋊C8—21no—
40C2×C211⋊C8D411yescharacteristic
41C2×C211⋊C8D411yescharacteristic
42C4×Dic211—21no—
43C4×Dic211—21no—
44#41×C211—21no—
45#41×C211—21no—
46#41×C211—21no—
47#41×C211—21no—
48#41×C211—21no—
49#41×C211—21no—
50#41×C211D412yes—
51#41×C211D412yes—
52C2×Q8×C211C2×C411yescharacteristic
53C2×Q8×C211C2×C411yescharacteristic
54C2×D4×C211—21no—
55C2×D4×C211—21no—
56C2×D4×C211C2×C411yescharacteristic
57C2×D4×C211C2×C411yescharacteristic
58C22×C4×C211—21no—
59C22×C4×C211—21no—
60C22×C4×C211C2311yescharacteristic
61Q16×C211—22no—
62SD16×C211—22no—
63SD16×C211—22no—
64D8×C211—22no—
65C2×C8×C211—21no—
66C2×C8×C211—21no—
67C2×C8×C211C2×C411yescharacteristic
68C2×C8×C211C2×C411yescharacteristic
69C2×Dic211—22no—
70C2×Dic211—41no—
71C2×Dic211—41no—
72C211⋊C8—21no—
73C211⋊C8—22no—
74C211⋊C8—21no—
75C211⋊C8—41no—
76C23×C211—21no—
77C23×C211—21no—
78C23×C211#4111yescharacteristic
79Q8×C211—21no—
80Q8×C211—21no—
81Q8×C211—21no—
82Q8×C211C22⋊C412yes—
83D4×C211—22no—
84D4×C211—22no—
85D4×C211—21no—
86D4×C211—21no—
87D4×C211—21no—
88D4×C211—41no—
89D4×C211—41no—
90D4×C211C22⋊C412yes—
91C2×C4×C211—21no—
92C2×C4×C211—22no—
93C2×C4×C211—22no—
94C2×C4×C211—21no—
95C2×C4×C211—41no—
96C2×C4×C211—41no—
97C2×C4×C211C22×C411yescharacteristic
98C2×C4×C211C2×D411yescharacteristic
99C2×C4×C211C2×D411yescharacteristic
100C2×C4×C211C2×D411yescharacteristic
101C2×C4×C211C2×D411yescharacteristic
102C2×C4×C211#4111yescharacteristic
103C8×C211—21no—
104C8×C211—21no—
105C22×C211—21no—
106C22×C211—21no—
107C22×C211—22no—
108C22×C211—41no—
109C22×C211—41no—
110C22×C211—41no—
111C22×C211—41no—
112C22×C211—41no—
113C22×C211C4×D411yescharacteristic
114C22×C211#12111yescharacteristic
115C22×C211#12311yescharacteristic
116C4×C211—22no—
117C4×C211—41no—
118C4×C211C2×C22⋊C411yescharacteristic
119C4×C211C4×D411yescharacteristic
120C4×C211#12011yescharacteristic
121C4×C211#12111yescharacteristic
122Dic211—42no—
123C2×C211—21no—
124C2×C211—21no—
125C2×C211—42no—
126C2×C211—81no—
127C2×C211#38611yescharacteristic
128C2×C211#44311yescharacteristic
129C2×C211#44411yescharacteristic
130C211#187611yescharacteristic, complemented, Sylow
131#1876—2111nomaximal, Sylow
132C2×#135D21111yescharacteristic
133C2×#131—2111no—
134C2×#104—2111no—
135C2×#104—2111no—
136#408—2111no—
137#341—2111no—
138#336—2111no—
139C2×#41C2×D21111yescharacteristic
140C2×#41C2×D21111yescharacteristic
141#135—22no—
142#135—22no—
143C2×Q16Dic21111yescharacteristic, complemented
144C2×SD16Dic21111yescharacteristic, complemented
145C2×SD16Dic21111yescharacteristic, complemented
146C2×D8Dic21111yescharacteristic, complemented
147#131—4222no—
148#131—4222no—
149C2×M16—4221no—
150C2×M16—2111no—
151C2×M16—2111no—
152C22×C8—4221no—
153C22×C8—2111no—
154C22×C8C2×D21111yescharacteristic
155C2×C42—2111no—
156#104—4222no—
157#104—4222no—
158C22⋊C8—4221no—
159C22⋊C8—4221no—
160#41—21no—
161#41—21no—
162#41—21no—
163#41—21no—
164#41—21no—
165#41—21no—
166#41C22⋊D21112yes—
167#41C22⋊D21112yes—
168C2×Q8C2×Dic21111yescharacteristic
169C2×Q8C2×Dic21111yescharacteristic
170C2×D4—21no—
171C2×D4—21no—
172C2×D4C2×Dic21111yescharacteristic
173C2×D4C2×Dic21111yescharacteristic
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—4221no—
177C22×C4C22×D21111yescharacteristic
178Q16—22no—
179SD16—22no—
180SD16—22no—
181D8—22no—
182M16—4221no—
183M16—4221no—
184M16—4224no—
185M16—4221no—
186M16—2112no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—4221no—
190C2×C8—4224no—
191C2×C8—4221no—
192C2×C8—4221no—
193C2×C8—4221no—
194C2×C8—2111no—
195C2×C8—2111no—
196C2×C8C2×Dic21111yescharacteristic
197C2×C8C2×Dic21111yescharacteristic
198C42—4221no—
199C42—4221no—
200C23—21no—
201C23—21no—
202C23#1680996111yescharacteristic
203Q8—21no—
204Q8—21no—
205Q8—21no—
206Q8C22⋊Dic21112yes—
207D4—22no—
208D4—22no—
209D4—21no—
210D4—21no—
211D4—21no—
212D4—41no—
213D4—41no—
214D4C22⋊Dic21112yes—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—8441no—
222C2×C4—8441no—
223C2×C4—4222no—
224C2×C4C22×Dic21111yescharacteristic
225C2×C4D4×D21111yescharacteristic
226C2×C4D4×D21111yescharacteristic
227C2×C4C2×C22⋊D21111yescharacteristic
228C2×C4C2×C22⋊D21111yescharacteristic
229C2×C4#1680996111yescharacteristic
230C8—21no—
231C8—21no—
232C8—8441no—
233C8—4221no—
234C8—4222no—
235C8—4221no—
236C22—21no—
237C22—21no—
238C22—22no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—41no—
243C22—41no—
244C22D4×Dic21111yescharacteristic
245C22#2026994911yescharacteristic
246C22#2026995811yescharacteristic
247C4—22no—
248C4—41no—
249C4—8442no—
250C4C2×C22⋊Dic21111yescharacteristic
251C4D4×Dic21111yescharacteristic
252C4#2026994511yescharacteristic
253C4#2026994911yescharacteristic
254C2—21no—
255C2—21no—
256C2—42no—
257C2—81no—
258C2#2721302611yescharacteristic
259C2#2721326311yescharacteristic
260C2#2721326811yescharacteristic
subgroupG/H
centerC2×C4D4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting, prime radicalC2×#135×C211C2
socleC22×C211C4×D4
cosocleC22×C4×C211C23
hypercenterC2×#135D211
hypocenterC211#1876
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#20269984C2semidirect
2. C2×#20269904C2semidirect
3. C2×#20269904C2semidirect
4. #27213113C2semidirect
5. #27212911C2semidirect
6. #27212900C2semidirect
24. C2×Q16×C211C4semidirect
25. C2×SD16×C211C4semidirect
26. C2×SD16×C211C4semidirect
27. C2×D8×C211C4semidirect
130. C211#1876semidirect
143. C2×Q16Dic211semidirect
144. C2×SD16Dic211semidirect
145. C2×SD16Dic211semidirect
146. C2×D8Dic211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699842
C2×#202699042
#272131132
#272129112
#272129002
C2×#135×C2112
#1876211