Finite Group Databasebeta

← #41014293

Subgroups — #41014293

Order 27008.

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

normal subgroups only

This group has 284 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
1C2×#20269997C211yesmaximal, characteristic
2#27213182C211yesmaximal, characteristic, complemented
3#27213164C211yesmaximal, characteristic, complemented
4#27213163C211yesmaximal, characteristic, complemented
5#27213119C211yesmaximal, characteristic, complemented
6#27213102C211yesmaximal, characteristic, complemented
7#410×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9#20269997—21no—
10#20269997—21no—
11#20269997—21no—
12#20269997—21no—
13C2×#16809952C411yescharacteristic, complemented
14C2×#16809950C411yescharacteristic, complemented
15C2×#16809949C411yescharacteristic, complemented
16C2×#16809947C411yescharacteristic, complemented
17C2×#16809946C2211yescharacteristic
18C22×C211⋊C8C2211yescharacteristic
19#20269936C2211yescharacteristic
20#20269910—21no—
21#20269908—21no—
22#20269905—22no—
23#20269904—22no—
24#20269891—22no—
25#20269888—21no—
26C4×C211⋊C8—21no—
27C2×#41×C211C2211yescharacteristic
28#117×C211C2211yescharacteristic
29#117×C211C2211yescharacteristic
30#99×C211—22no—
31#16809962—21no—
32#16809962—21no—
33#16809962—21no—
34#16809962—21no—
35C2×Dic422C2×C411yescharacteristic
36C2×D844C2×C411yescharacteristic
37C2×C22⋊D211—21no—
38C2×C4×D211—21no—
39#16809952—21no—
40#16809952—21no—
41#16809950—21no—
42#16809950—21no—
43#16809949—21no—
44#16809949—21no—
45#16809947—21no—
46#16809947—21no—
47#16809946—21no—
48#16809946D412yes—
49C2×C211⋊C8—21no—
50C2×C211⋊C8C2×C411yescharacteristic
51C2×C211⋊C8C2×C411yescharacteristic
52C2×C211⋊C8D411yescharacteristic
53C2×C211⋊C8D411yescharacteristic
54C2×C211⋊C8D411yescharacteristic
55C2×C211⋊C8D411yescharacteristic
56C4⋊Dic211—21no—
57C22⋊Dic211—21no—
58C4×Dic211—21no—
59#41×C211—21no—
60#41×C211—21no—
61#41×C211—21no—
62#41×C211—21no—
63C2×Q8×C211C2×C411yescharacteristic
64C2×D4×C211—21no—
65C2×D4×C211C2×C411yescharacteristic
66C22×C4×C211—21no—
67C22×C4×C211C2311yescharacteristic
68C4⋊C4×C211—21no—
69C4⋊C4×C211—21no—
70C22⋊C4×C211—21no—
71C22⋊C4×C211D412yes—
72C42×C211—21no—
73C42×C211—21no—
74C22×D211—21no—
75Dic422—21no—
76Dic422—21no—
77D844—21no—
78D844—21no—
79C22⋊D211—41no—
80C22⋊D211—41no—
81C2×Dic211—21no—
82C2×Dic211—41no—
83C4×D211—41no—
84C4×D211—41no—
85C211⋊C8—22no—
86C211⋊C8C22⋊C411yescharacteristic
87C211⋊C8C22⋊C411yescharacteristic
88C211⋊C8C22⋊C411yescharacteristic
89C211⋊C8C22⋊C411yescharacteristic
90C23×C211—21no—
91C23×C211C2×D411yescharacteristic
92Q8×C211—21no—
93Q8×C211—21no—
94D4×C211—21no—
95D4×C211—21no—
96D4×C211—41no—
97D4×C211—41no—
98C2×C4×C211—21no—
99C2×C4×C211—22no—
100C2×C4×C211—41no—
101C2×C4×C211—41no—
102C2×C4×C211—41no—
103C2×C4×C211C22×C411yescharacteristic
104C2×C4×C211C2×D411yescharacteristic
105C2×C4×C211C2×D411yescharacteristic
106C2×C4×C211C2×D411yescharacteristic
107C2×C4×C211#4111yescharacteristic
108C2×C4×C211#4111yescharacteristic
109C22×C211—41no—
110C22×C211—41no—
111C22×C211—81no—
112C22×C211C4×D411yescharacteristic
113C22×C211#12111yescharacteristic
114C22×C211#12711yescharacteristic
115C2×D211—41no—
116C2×D211—81no—
117C4×C211—42no—
118C4×C211—41no—
119C4×C211—81no—
120C4×C211C2×C22⋊C411yescharacteristic
121C4×C211C4×D411yescharacteristic
122C4×C211#12111yescharacteristic
123C4×C211#12411yescharacteristic
124Dic211—41no—
125Dic211—81no—
126C2×C211—21no—
127C2×C211—21no—
128C2×C211—21no—
129C2×C211—81no—
130C2×C211#39011yescharacteristic
131D211—81no—
132C211#197611yescharacteristic, complemented, Sylow
133#1976—2111nomaximal, Sylow
134C2×#135—2111no—
135#427—2111no—
136#421—2111no—
137#421—2111no—
138#410—2111no—
139#410D21111yescharacteristic, complemented
140#405—2111no—
141C2×#41—2111no—
142C2×#41C2×D21111yescharacteristic
143#135—4221no—
144#135—4221no—
145#135—4221no—
146#135—4221no—
147C2×Q16—2111no—
148C2×SD16—2111no—
149C2×SD16—2111no—
150C2×D8—2111no—
151C2×M16—2111no—
152C22×C8—2111no—
153#117—2111no—
154#117C2×D21111yescharacteristic
155#117C2×D21111yescharacteristic
156#107—4221no—
157#106—4221no—
158#104—4222no—
159#104—4222no—
160#99—22no—
161#99—4222no—
162#97—4221no—
163C4×C8—4221no—
164#41—21no—
165#41—21no—
166#41—21no—
167#41—21no—
168#41—4221no—
169#41—4221no—
170#41—4221no—
171#41—4221no—
172C2×Q8—2111no—
173C2×Q8C4×D21111yescharacteristic
174C2×D4—21no—
175C2×D4—4221no—
176C2×D4—2111no—
177C2×D4C4×D21111yescharacteristic
178C22×C4—21no—
179C22×C4—4221no—
180C22×C4C22×D21111yescharacteristic
181Q16—4221no—
182Q16—4221no—
183SD16—4221no—
184SD16—4221no—
185SD16—4221no—
186SD16—4221no—
187D8—4221no—
188D8—4221no—
189M16—4221no—
190M16—2112no—
191C2×C8—4221no—
192C2×C8—2111no—
193C2×C8—2111no—
194C2×C8—2111no—
195C2×C8—2111no—
196C2×C8—2111no—
197C2×C8—2111no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—4221no—
201C22⋊C4—21no—
202C22⋊C4—4221no—
203C22⋊C4C22⋊D21112yes—
204C42—21no—
205C42—21no—
206C42—4221no—
207C23—21no—
208C23—4221no—
209C23C2×C22⋊D21111yescharacteristic
210Q8—21no—
211Q8—21no—
212Q8—4221no—
213Q8—4221no—
214D4—21no—
215D4—21no—
216D4—41no—
217D4—41no—
218D4—8441no—
219D4—8441no—
220D4—4221no—
221D4—4221no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—8441no—
228C2×C4—8441no—
229C2×C4—8441no—
230C2×C4—4221no—
231C2×C4C2×C4×D21111yescharacteristic
232C2×C4D4×D21111yescharacteristic
233C2×C4D4×D21111yescharacteristic
234C2×C4D4×D21111yescharacteristic
235C2×C4#1680996111yescharacteristic
236C2×C4#1680996211yescharacteristic
237C8—4222no—
238C8—2111no—
239C8—2111no—
240C8—2111no—
241C8—2111no—
242C22—41no—
243C22—41no—
244C22—81no—
245C22—8441no—
246C22—16881no—
247C22C4×C22⋊D21111yescharacteristic
248C22#2026994911yescharacteristic
249C22#2026997311yescharacteristic
250C4—42no—
251C4—41no—
252C4—81no—
253C4—8441no—
254C4—16881no—
255C4C22⋊C4×D21111yescharacteristic
256C4(C22⋊D211)⋊C411yescharacteristic
257C4#2026995011yescharacteristic
258C4#2026996311yescharacteristic
259C2—21no—
260C2—21no—
261C2—21no—
262C2—81no—
263C2—16881no—
264C2#2721304611yescharacteristic
subgroupG/H
centerC4#20269950
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#410×C211C2
socleC2×C211#390
prime radicalC2×#20269997C2
cosocleC22×C4×C211C23
hypercenter#410D211
hypocenterC211#1976
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #27213182C2semidirect
3. #27213164C2semidirect
4. #27213163C2semidirect
5. #27213119C2semidirect
6. #27213102C2semidirect
7. #410×C211C2semidirect
13. C2×#16809952C4semidirect
14. C2×#16809950C4semidirect
15. C2×#16809949C4semidirect
16. C2×#16809947C4semidirect
132. C211#1976semidirect
139. #410D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699972
#272131822
#272131642
#272131632
#272131192
#272131022
#410×C2112
#1976211