Finite Group Databasebeta

← #41014307

Subgroups — #41014307

Order 27008.

316 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C22×#16809947C211yesmaximal, characteristic
2C2×#20269912C211yesmaximal, characteristic, complemented
3#27213150C211yesmaximal, characteristic, complemented
4C2×#20269894C211yesmaximal, characteristic, complemented
5C2×#20269893C211yesmaximal, characteristic, complemented
6#27213102C211yesmaximal, characteristic, complemented
7#418×C211C211yesmaximal, characteristic, complemented
8C22×D844C2211yescharacteristic
9C2×#16809947—21no—
10C2×#16809947—21no—
11C2×#16809947—22no—
12C2×#16809947C414yescomplemented
13C2×#16809946C2211yescharacteristic
14C2×#16809946C2211yescharacteristic
15C22×C211⋊C8C2211yescharacteristic
16#20269912—22no—
17#20269897—22no—
18#20269894—22no—
19#20269893—22no—
20#20269888—21no—
21C4×C211⋊C8—21no—
22C22×D4×C211C2211yescharacteristic
23C2×M16×C211C2211yescharacteristic
24#117×C211C2211yescharacteristic
25#102×C211—22no—
26C23×D211—21no—
27C2×D844—21no—
28C2×D844—21no—
29C2×D844—21no—
30C2×D844—21no—
31C2×D844—21no—
32C2×D844C2×C412yes—
33#16809947—24no—
34#16809947—24no—
35#16809946—21no—
36#16809946—21no—
37#16809946—21no—
38#16809946D412yes—
39C2×C211⋊C8—21no—
40C2×C211⋊C8—21no—
41C2×C211⋊C8C2×C412yes—
42C2×C211⋊C8D411yescharacteristic
43C2×C211⋊C8D411yescharacteristic
44C2×C211⋊C8D412yes—
45C24×C211—21no—
46C2×D4×C211—21no—
47C2×D4×C211—21no—
48C2×D4×C211—21no—
49C2×D4×C211—21no—
50C2×D4×C211—21no—
51C2×D4×C211C2×C412yes—
52C22×C4×C211C2311yescharacteristic
53M16×C211—21no—
54M16×C211—21no—
55C2×C8×C211—21no—
56C4⋊C4×C211D412yes—
57C22⋊C4×C211—21no—
58C42×C211—21no—
59C22×D211—22no—
60C22×D211—41no—
61C22×D211—41no—
62C22×D211—82no—
63D844—22no—
64D844—22no—
65D844—41no—
66D844—41no—
67C211⋊C8—22no—
68C211⋊C8—41no—
69C211⋊C8C22⋊C412yes—
70C211⋊C8C22⋊C412yes—
71C23×C211—22no—
72C23×C211—41no—
73C23×C211—41no—
74C23×C211—82no—
75C23×C211#4111yescharacteristic
76D4×C211—22no—
77D4×C211—22no—
78D4×C211—41no—
79D4×C211—41no—
80C2×C4×C211—22no—
81C2×C4×C211C22×C411yescharacteristic
82C2×C4×C211C2×D411yescharacteristic
83C2×C4×C211C2×D411yescharacteristic
84C2×C4×C211C2×D411yescharacteristic
85C2×C4×C211C2×D411yescharacteristic
86C2×C4×C211#4111yescharacteristic
87C8×C211—41no—
88C22×C211—22no—
89C22×C211—42no—
90C22×C211—82no—
91C22×C211—82no—
92C22×C211—82no—
93C22×C211C4×D411yescharacteristic
94C22×C211#12111yescharacteristic
95C22×C211#12411yescharacteristic
96C2×D211—42no—
97C2×D211—82no—
98C2×D211—82no—
99C2×D211—82no—
100C4×C211—42no—
101C4×C211C2×C22⋊C411yescharacteristic
102C4×C211C4×D411yescharacteristic
103C4×C211#12111yescharacteristic
104C4×C211#12711yescharacteristic
105C2×C211—21no—
106C2×C211—21no—
107C2×C211—82no—
108C2×C211#39011yescharacteristic
109C2×C211#47212yes—
110D211—82no—
111C211#197811yescharacteristic, complemented, Sylow
112#1978—2111nomaximal, Sylow
113C22×D8—2111no—
114C2×#108—2111no—
115#418—2111no—
116#418D21111yescharacteristic, complemented
117C2×#100—2111no—
118C2×#100—2111no—
119#405—2111no—
120C22×D4—2111no—
121C22×D4C2×D21111yescharacteristic
122C2×D8—4221no—
123C2×D8—4221no—
124C2×D8—4222no—
125C2×D8—2114no—
126C2×M16—2111no—
127C2×M16—2111no—
128C2×M16C2×D21111yescharacteristic
129C22×C8—2111no—
130#117C2×D21111yescharacteristic
131#108—4222no—
132#102—22no—
133#102—4222no—
134#100—4222no—
135#100—4222no—
136#97—4221no—
137C4×C8—4221no—
138C24—21no—
139C24—4221no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—4221no—
146C2×D4—4221no—
147C2×D4—4221no—
148C2×D4—4221no—
149C2×D4—4221no—
150C2×D4—2112no—
151C2×D4C4×D21112yes—
152C22×C4C22×D21111yescharacteristic
153D8—4224no—
154D8—4224no—
155M16—21no—
156M16—21no—
157M16—4221no—
158M16—4221no—
159M16—4221no—
160M16—2112no—
161C2×C8—21no—
162C2×C8—4221no—
163C2×C8—4221no—
164C2×C8—2112no—
165C2×C8—2111no—
166C2×C8—2111no—
167C2×C8—2112no—
168C4⋊C4C22⋊D21112yes—
169C22⋊C4—21no—
170C42—21no—
171C23—22no—
172C23—41no—
173C23—41no—
174C23—82no—
175C23—8441no—
176C23—8441no—
177C23—16882no—
178C23—4222no—
179C23#1680996111yescharacteristic
180D4—22no—
181D4—22no—
182D4—41no—
183D4—41no—
184D4—8441no—
185D4—8441no—
186D4—4222no—
187D4—4222no—
188C2×C4—22no—
189C2×C4C2×C4×D21111yescharacteristic
190C2×C4D4×D21111yescharacteristic
191C2×C4D4×D21111yescharacteristic
192C2×C4D4×D21111yescharacteristic
193C2×C4C2×C22⋊D21111yescharacteristic
194C2×C4#1680996211yescharacteristic
195C8—41no—
196C8—8441no—
197C8—4222no—
198C8—2112no—
199C8—2112no—
200C22—22no—
201C22—42no—
202C22—82no—
203C22—82no—
204C22—82no—
205C22—8442no—
206C22—16882no—
207C22—16882no—
208C22—16882no—
209C22(C22⋊D211)⋊C411yescharacteristic
210C22#2026994911yescharacteristic
211C22#2026996311yescharacteristic
212C4—42no—
213C4C22⋊C4×D21111yescharacteristic
214C4C4×C22⋊D21111yescharacteristic
215C4#2026995011yescharacteristic
216C4#2026997311yescharacteristic
217C2—21no—
218C2—21no—
219C2—82no—
220C2—16882no—
221C2#2721304611yescharacteristic
222C2#2721341412yes—
subgroupG/H
centerC22#20269949
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#418×C211C2
socleC22×C211#121
prime radicalC22×#16809947C2
cosocleC22×C4×C211C23
hypercenter#418D211
hypocenterC211#1978
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. C2×#20269912C2semidirect
3. #27213150C2semidirect
4. C2×#20269894C2semidirect
5. C2×#20269893C2semidirect
6. #27213102C2semidirect
7. #418×C211C2semidirect
12. C2×#16809947C4semidirect
111. C211#1978semidirect
116. #418D211semidirect

Maximal subgroups

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

subgroupindex
C22×#168099472
C2×#202699122
#272131502
C2×#202698942
C2×#202698932
#272131022
#418×C2112
#1978211