Finite Group Databasebeta

← #41013759

Subgroups — #41013759

Order 27008.

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

normal subgroups only

This group has 310 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#27213912C211yesmaximal, characteristic, complemented
3#27213164C211yesmaximal, characteristic, complemented
4#27213140C211yesmaximal, characteristic, complemented
5#27213114C211yesmaximal, characteristic, complemented
6#27212910C211yesmaximal, 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#20269985—22no—
16#20269983—22no—
17C2×#16809946C2211yescharacteristic
18M16×D211—22no—
19C22×C211⋊C8C2211yescharacteristic
20#20269905—22no—
21#20269900—21no—
22#20269897—21no—
23(C2×D211)⋊C8—22no—
24C2×M16×C211C2211yescharacteristic
25C2×M16×C211C2211yescharacteristic
26#117×C211C2211yescharacteristic
27#16809962—21no—
28#16809962—21no—
29#16809962—21no—
30#16809962—21no—
31#16809962—21no—
32#16809962—21no—
33#16809962C2×C412yes—
34#16809962D412yes—
35C2×Dic422C2×C411yescharacteristic
36C2×Dic422D411yescharacteristic
37C2×D844C2×C411yescharacteristic
38C2×D844D411yescharacteristic
39C2×C22⋊D211—21no—
40C2×C22⋊D211—21no—
41C2×C4×D211—21no—
42C2×C4×D211—21no—
43Dic844—22no—
44#16809951—22no—
45#16809951—22no—
46D1688—22no—
47#16809946—21no—
48#16809946D412yes—
49#16809945—24no—
50C2×C211⋊C8—21no—
51C2×C211⋊C8—21no—
52C2×C211⋊C8—21no—
53C2×C211⋊C8—21no—
54C8×D211—24no—
55C22×C4×C211C2311yescharacteristic
56M16×C211—21no—
57M16×C211—21no—
58M16×C211—21no—
59M16×C211C2×C412yes—
60C2×C8×C211—21no—
61C2×C8×C211D412yes—
62C4⋊C4×C211—21no—
63C22⋊C4×C211—21no—
64C42×C211—21no—
65C22×D211—21no—
66C22×D211—21no—
67Dic422—21no—
68Dic422—21no—
69Dic422—21no—
70Dic422C22⋊C412yes—
71D844—21no—
72D844—21no—
73D844—21no—
74D844C22⋊C412yes—
75C22⋊D211—22no—
76C22⋊D211—22no—
77C22⋊D211—41no—
78C22⋊D211—41no—
79C2×Dic211—21no—
80C2×Dic211—21no—
81C4×D211—22no—
82C4×D211—22no—
83C4×D211—41no—
84C4×D211—41no—
85C211⋊C8—22no—
86C211⋊C8—41no—
87C23×C211#4111yescharacteristic
88C2×C4×C211—41no—
89C2×C4×C211C22×C411yescharacteristic
90C2×C4×C211C2×D411yescharacteristic
91C2×C4×C211C2×D411yescharacteristic
92C2×C4×C211C2×D411yescharacteristic
93C2×C4×C211C2×D411yescharacteristic
94C2×C4×C211#4111yescharacteristic
95C8×C211—22no—
96C8×C211—22no—
97C22×C211—41no—
98C22×C211C4×D411yescharacteristic
99C22×C211#12111yescharacteristic
100C22×C211#12311yescharacteristic
101C2×D211—22no—
102C2×D211—42no—
103C2×D211—41no—
104C2×D211—81no—
105C4×C211—81no—
106C4×C211C2×C22⋊C411yescharacteristic
107C4×C211C4×D411yescharacteristic
108C4×C211#12011yescharacteristic
109C4×C211#12111yescharacteristic
110Dic211—22no—
111Dic211—41no—
112C2×C211—21no—
113C2×C211—21no—
114C2×C211—21no—
115C2×C211#38611yescharacteristic
116D211—42no—
117D211—81no—
118C211#188111yescharacteristic, complemented, Sylow
119#1881—2111nomaximal, Sylow
120#575—2111no—
121#568—2111no—
122#421—2111no—
123#416—2111no—
124#408—2111no—
125#344D21111yescharacteristic, complemented
126#341—2111no—
127C2×#41—2111no—
128C2×#41—2111no—
129#137—4221no—
130#137—2112no—
131AGL(1,ℤ/8)—4221no—
132AGL(1,ℤ/8)—2112no—
133#135—4222no—
134#131—4222no—
135#131—4222no—
136C2×M16—4222no—
137C2×M16—2111no—
138C2×M16C2×D21111yescharacteristic
139C2×M16C2×D21111yescharacteristic
140C22×C8—2111no—
141#117C2×D21111yescharacteristic
142#104—4222no—
143#103—4221no—
144#102—4221no—
145C22⋊C8—4222no—
146#41—4221no—
147#41—4221no—
148#41—4221no—
149#41—4221no—
150#41—4221no—
151#41—4221no—
152#41—2112no—
153#41—2112no—
154C2×Q8—2111no—
155C2×Q8—2111no—
156C2×D4—4221no—
157C2×D4—4221no—
158C2×D4—2111no—
159C2×D4—2111no—
160C22×C4—4221no—
161C22×C4—4221no—
162C22×C4C22×D21111yescharacteristic
163Q16—4222no—
164SD16—4222no—
165SD16—4222no—
166D8—4222no—
167M16—21no—
168M16—21no—
169M16—21no—
170M16—4224no—
171M16—4221no—
172M16—2112no—
173M16C4×D21112yes—
174C2×C8—21no—
175C2×C8—4224no—
176C2×C8—4221no—
177C2×C8—4221no—
178C2×C8—4221no—
179C2×C8—4221no—
180C2×C8D84412yes—
181C4⋊C4—21no—
182C22⋊C4—21no—
183C42—21no—
184C23—4221no—
185C23—4221no—
186C23#1680996211yescharacteristic
187Q8—4221no—
188Q8—4221no—
189Q8—4221no—
190Q8—2112no—
191D4—8441no—
192D4—8441no—
193D4—4222no—
194D4—4222no—
195D4—4221no—
196D4—4221no—
197D4—4221no—
198D4—2112no—
199C2×C4—41no—
200C2×C4—8441no—
201C2×C4—8441no—
202C2×C4—4221no—
203C2×C4—4221no—
204C2×C4—4222no—
205C2×C4—4222no—
206C2×C4C2×C4×D21111yescharacteristic
207C2×C4D4×D21111yescharacteristic
208C2×C4D4×D21111yescharacteristic
209C2×C4D4×D21111yescharacteristic
210C2×C4C2×D84411yescharacteristic
211C2×C4#1680996311yescharacteristic
212C8—22no—
213C8—22no—
214C8—8441no—
215C8—4222no—
216C22—41no—
217C22—8442no—
218C22—8441no—
219C22—16881no—
220C22—4222no—
221C22C4×D84411yescharacteristic
222C22#2026995011yescharacteristic
223C22#2026996211yescharacteristic
224C4—81no—
225C4—8441no—
226C4—4222no—
227C4C22⋊C4×D21111yescharacteristic
228C4#2026993911yescharacteristic
229C4#2026994711yescharacteristic
230C4#2026995211yescharacteristic
231C2—21no—
232C2—21no—
233C2—21no—
234C2—8442no—
235C2—16881no—
236C2#2721302411yescharacteristic
subgroupG/H
centerC4#20269947
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#344×C211C2
socleC2×C211#386
prime radical#27213946C2
cosocleC22×C4×C211C23
hypercenter#344D211
hypocenterC211#1881
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. #27213912C2semidirect
3. #27213164C2semidirect
4. #27213140C2semidirect
5. #27213114C2semidirect
6. #27212910C2semidirect
7. #344×C211C2semidirect
11. #20270010C4semidirect
13. #20270005C4semidirect
118. C211#1881semidirect
125. #344D211semidirect

Maximal subgroups

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

subgroupindex
#272139462
#272139122
#272131642
#272131402
#272131142
#272129102
#344×C2112
#1881211