Finite Group Databasebeta

← #41014395

Subgroups — #41014395

Order 27008.

496 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 91 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, complemented
2C2×#20269952C211yesmaximal, characteristic, complemented
3C2×#20269948C211yesmaximal, characteristic, complemented
4C2×#20269899C211yesmaximal, characteristic, complemented
5C2×(C2×D211)⋊C8C211yesmaximal, characteristic, complemented
6#27212908C211yesmaximal, characteristic, complemented
7C2×#102×C211C211yesmaximal, characteristic, complemented
8C22×D844—21no—
9C22×D844C2211yescharacteristic
10C22×C22⋊D211—21no—
11C22×C4×D211C2211yescharacteristic
12C2×#16809947—22no—
13C2×#16809947—22no—
14C2×#16809947—22no—
15C22×C211⋊C8C2211yescharacteristic
16#20269952—22no—
17#20269952—22no—
18#20269948—22no—
19#20269948—22no—
20C2×C4⋊Dic211C2211yescharacteristic
21C2×C22⋊Dic211—21no—
22C2×(C2×D211)⋊C4—21no—
23#20269899—22no—
24(C2×D211)⋊C8—22no—
25C22×D4×C211C2211yescharacteristic
26C22×C8×C211C2211yescharacteristic
27C2×C4⋊C4×C211C2211yescharacteristic
28#102×C211—22no—
29C23×D211—21no—
30C23×D211D411yescharacteristic
31C2×D844—24no—
32C2×D844—21no—
33C2×D844—21no—
34C2×D844—21no—
35C2×D844—21no—
36C2×D844—21no—
37C2×D844—42no—
38C2×D844—42no—
39C2×D844D412yescomplemented
40C2×C22⋊D211—24no—
41C2×C22⋊D211—42no—
42C2×C22⋊D211—42no—
43C22×Dic211—21no—
44C22×Dic211D411yescharacteristic
45C2×C4×D211—21no—
46C2×C4×D211—21no—
47C2×C4×D211—21no—
48C2×C4×D211—21no—
49C2×C4×D211—21no—
50C2×C4×D211—21no—
51#16809947—28no—
52C2×C211⋊C8—21no—
53C2×C211⋊C8—21no—
54C2×C211⋊C8D412yescomplemented
55C4⋊Dic211—21no—
56C4⋊Dic211D412yescomplemented
57C22⋊Dic211—24no—
58(C2×D211)⋊C4—24no—
59C24×C211—21no—
60C2×D4×C211—21no—
61C2×D4×C211—21no—
62C2×D4×C211—21no—
63C2×D4×C211—21no—
64C2×D4×C211—21no—
65C2×D4×C211D412yescomplemented
66C22×C4×C211—21no—
67C22×C4×C211C2311yescharacteristic
68C2×C8×C211—21no—
69C2×C8×C211—21no—
70C2×C8×C211—21no—
71C4⋊C4×C211—21no—
72C4⋊C4×C211D412yescomplemented
73C22×D211—21no—
74C22×D211—21no—
75C22×D211—21no—
76C22×D211—21no—
77C22×D211—21no—
78C22×D211—21no—
79C22×D211—22no—
80C22×D211—41no—
81C22×D211—41no—
82C22×D211—41no—
83C22×D211—41no—
84C22×D211—41no—
85C22×D211—41no—
86C22×D211D812yescomplemented
87D844—22no—
88D844—22no—
89D844—48no—
90D844—41no—
91D844—41no—
92C22⋊D211—48no—
93C2×Dic211—21no—
94C2×Dic211—21no—
95C2×Dic211—22no—
96C2×Dic211—21no—
97C2×Dic211—41no—
98C2×Dic211—41no—
99C4×D211—41no—
100C4×D211—41no—
101C4×D211—41no—
102C4×D211—41no—
103C211⋊C8—22no—
104C23×C211—22no—
105C23×C211—41no—
106C23×C211—41no—
107C23×C211—41no—
108C23×C211—41no—
109C23×C211—41no—
110C23×C211—41no—
111C23×C211C2×D411yescharacteristic
112D4×C211—22no—
113D4×C211—22no—
114D4×C211—41no—
115D4×C211—41no—
116C2×C4×C211—22no—
117C2×C4×C211—41no—
118C2×C4×C211—41no—
119C2×C4×C211C2×D411yescharacteristic
120C2×C4×C211C2×D411yescharacteristic
121C2×C4×C211C2×D411yescharacteristic
122C2×C4×C211C2×D411yescharacteristic
123C2×C4×C211C2×D411yescharacteristic
124C2×C4×C211#4111yescharacteristic
125C8×C211—41no—
126C22×C211—42no—
127C22×C211—42no—
128C22×C211—42no—
129C22×C211—81no—
130C22×C211—81no—
131C22×C211—81no—
132C22×C211—81no—
133C22×C211#12011yescharacteristic
134C22×C211#12011yescharacteristic
135C22×C211#12111yescharacteristic
136C22×C211C2×D811yescharacteristic
137C22×C211#13511yescharacteristic
138C22×C211AGL(1,ℤ/8)11yescharacteristic
139C22×C211AGL(1,ℤ/8)11yescharacteristic
140C2×D211—22no—
141C2×D211—22no—
142C2×D211—22no—
143C2×D211—41no—
144C2×D211—41no—
145C2×D211—41no—
146C2×D211—41no—
147C2×D211—42no—
148C2×D211—42no—
149C2×D211—42no—
150C2×D211—81no—
151C2×D211—81no—
152C2×D211—81no—
153C2×D211—81no—
154C4×C211—42no—
155C4×C211#12011yescharacteristic
156C4×C211#12111yescharacteristic
157C4×C211#12111yescharacteristic
158C4×C211#12711yescharacteristic
159Dic211—41no—
160Dic211—42no—
161C2×C211—82no—
162C2×C211#39211yescharacteristic
163C2×C211#44711yescharacteristic
164C2×C211#44711yescharacteristic
165C2×C211#44911yescharacteristic
166C2×C211#44911yescharacteristic
167C2×C211#46611yescharacteristic
168C2×C211#46911yescharacteristic
169D211—42no—
170D211—82no—
171C211#199911yescharacteristic, complemented, Sylow
172#1999—2111nomaximal, Sylow
173C22×D8—2111no—
174C2×#121—2111no—
175C2×#121—2111no—
176C2×#102—2111no—
177C2×#102D21111yescharacteristic, complemented
178C2×C22⋊C8—2111no—
179#340—2111no—
180C22×D4—4221no—
181C22×D4—4221no—
182C22×D4—2111no—
183C22×D4C2×D21111yescharacteristic
184C23×C4—2111no—
185C2×D8—4222no—
186C2×D8—4222no—
187C2×D8—4222no—
188C22×C8—2111no—
189C22×C8C2×D21111yescharacteristic
190#121—4222no—
191#121—4222no—
192#121—4222no—
193#121—4222no—
194C2×C4⋊C4—2111no—
195C2×C4⋊C4C2×D21111yescharacteristic
196C2×C22⋊C4—4221no—
197C2×C22⋊C4—4221no—
198#102—22no—
199#102—4222no—
200C22⋊C8—4222no—
201C24—21no—
202C24—4221no—
203C24—2111no—
204C2×D4—21no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—21no—
209C2×D4—8442no—
210C2×D4—8442no—
211C2×D4—8442no—
212C2×D4—8442no—
213C2×D4—4224no—
214C2×D4—4224no—
215C2×D4—4221no—
216C2×D4—4221no—
217C2×D4—4221no—
218C2×D4—4221no—
219C2×D4—4221no—
220C2×D4—2112no—
221C2×D4D84412yescomplemented
222C22×C4—21no—
223C22×C4—4221no—
224C22×C4—4221no—
225C22×C4—4221no—
226C22×C4—4221no—
227C22×C4—4221no—
228C22×C4—4221no—
229C22×C4—4221no—
230C22×C4—2111no—
231C22×C4C22×D21111yescharacteristic
232D8—4228no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—4221no—
237C2×C8—4221no—
238C2×C8—2112no—
239C4⋊C4—21no—
240C4⋊C4—4221no—
241C4⋊C4—2112no—
242C4⋊C4C22⋊D21112yescomplemented
243C22⋊C4—4224no—
244C22⋊C4—4224no—
245C23—22no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—8441no—
253C23—8441no—
254C23—8441no—
255C23—8441no—
256C23—8441no—
257C23—8441no—
258C23—4222no—
259C23—4221no—
260C23—4221no—
261C23—4221no—
262C23—4221no—
263C23—4221no—
264C23—4221no—
265C23—2112no—
266C23D4×D21111yescharacteristic
267D4—22no—
268D4—22no—
269D4—41no—
270D4—41no—
271D4—8448no—
272D4—8448no—
273D4—8441no—
274D4—8441no—
275D4—4222no—
276D4—4222no—
277C2×C4—22no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—8441no—
281C2×C4—8441no—
282C2×C4—8441no—
283C2×C4—8441no—
284C2×C4—8441no—
285C2×C4—8441no—
286C2×C4—4221no—
287C2×C4—4221no—
288C2×C4—4222no—
289C2×C4—4221no—
290C2×C4D4×D21111yescharacteristic
291C2×C4D4×D21111yescharacteristic
292C2×C4D4×D21111yescharacteristic
293C2×C4C2×C22⋊D21111yescharacteristic
294C2×C4C2×D84411yescharacteristic
295C2×C4#1680996211yescharacteristic
296C8—41no—
297C8—4222no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22—8441no—
306C22—8441no—
307C22—8441no—
308C22—8441no—
309C22—8442no—
310C22—8442no—
311C22—8442no—
312C22—16881no—
313C22—16881no—
314C22—16881no—
315C22—16881no—
316C22—4222no—
317C22—4222no—
318C22—4222no—
319C22#2026994511yescharacteristic
320C22#2026994711yescharacteristic
321C22#2026995011yescharacteristic
322C22D8×D21111yescharacteristic
323C22#2026999811yescharacteristic
324C22#2027000111yescharacteristic
325C22#2027000411yescharacteristic
326C4—42no—
327C4—8442no—
328C4—8441no—
329C4#2026994711yescharacteristic
330C4#2026995011yescharacteristic
331C4#2026995111yescharacteristic
332C4#2026997311yescharacteristic
333C2—82no—
334C2—8442no—
335C2—16882no—
336C2#2721305511yescharacteristic
337C2#2721327811yescharacteristic
338C2#2721328111yescharacteristic
339C2#2721329011yescharacteristic
340C2#2721329211yescharacteristic
341C2#2721338111yescharacteristic
342C2#2721339811yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#102×C211C2
socleC23×C211C2×D4
hypercenterC2×#102D211
hypocenterC211#1999
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C22×#16809947C2semidirect
2. C2×#20269952C2semidirect
3. C2×#20269948C2semidirect
4. C2×#20269899C2semidirect
5. C2×(C2×D211)⋊C8C2semidirect
6. #27212908C2semidirect
7. C2×#102×C211C2semidirect
39. C2×D844D4semidirect
54. C2×C211⋊C8D4semidirect
56. C4⋊Dic211D4semidirect
65. C2×D4×C211D4semidirect
72. C4⋊C4×C211D4semidirect
86. C22×D211D8semidirect
171. C211#1999semidirect
177. C2×#102D211semidirect
221. C2×D4D844semidirect
242. C4⋊C4C22⋊D211semidirect

Maximal subgroups

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

subgroupindex
C22×#168099472
C2×#202699522
C2×#202699482
C2×#202698992
C2×(C2×D211)⋊C82
#272129082
C2×#102×C2112
#1999211