Finite Group Databasebeta

← #41014394

Subgroups — #41014394

Order 27008.

496 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 95 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×#20269951C211yesmaximal, characteristic, complemented
3C2×#20269898C211yesmaximal, characteristic, complemented
4C2×#20269897C211yesmaximal, characteristic, complemented
5C2×C22⋊(C211⋊C8)C211yesmaximal, characteristic, complemented
6#27212908C211yesmaximal, characteristic, complemented
7C2×#121×C211C211yesmaximal, characteristic, complemented
8C22×D844C2211yescharacteristic
9C22×C22⋊D211—21no—
10C2×#16809947—22no—
11C2×#16809947—22no—
12C2×#16809947—22no—
13C22×C211⋊C8C2211yescharacteristic
14C22×C211⋊C8C2211yescharacteristic
15#20269951—22no—
16#20269951—22no—
17C2×C4⋊Dic211C2211yescharacteristic
18C2×(C2×D211)⋊C4—21no—
19#20269898—22no—
20#20269897—22no—
21C22⋊(C211⋊C8)—22no—
22C22×D4×C211—21no—
23C22×D4×C211C2211yescharacteristic
24C23×C4×C211C2211yescharacteristic
25#121×C211—22no—
26#121×C211—22no—
27C2×C4⋊C4×C211C2211yescharacteristic
28C2×C22⋊C4×C211—21no—
29C23×D211—21no—
30C2×D844—21no—
31C2×D844—21no—
32C2×D844—21no—
33C2×D844—21no—
34C2×D844—21no—
35C2×D844D412yescomplemented
36C2×C22⋊D211—24no—
37C2×C22⋊D211—42no—
38C2×C22⋊D211—42no—
39C22×Dic211—21no—
40#16809947—28no—
41C2×C211⋊C8—21no—
42C2×C211⋊C8—21no—
43C2×C211⋊C8—21no—
44C2×C211⋊C8—21no—
45C2×C211⋊C8—21no—
46C2×C211⋊C8D412yescomplemented
47C4⋊Dic211—21no—
48C4⋊Dic211D412yescomplemented
49(C2×D211)⋊C4—24no—
50C24×C211—21no—
51C24×C211D411yescharacteristic
52C2×D4×C211—24no—
53C2×D4×C211—21no—
54C2×D4×C211—21no—
55C2×D4×C211—21no—
56C2×D4×C211—21no—
57C2×D4×C211—21no—
58C2×D4×C211—42no—
59C2×D4×C211—42no—
60C2×D4×C211D412yescomplemented
61C22×C4×C211—21no—
62C22×C4×C211—21no—
63C22×C4×C211—21no—
64C22×C4×C211—21no—
65C22×C4×C211—21no—
66C22×C4×C211—21no—
67C22×C4×C211—21no—
68C22×C4×C211D411yescharacteristic
69C22×C4×C211C2311yescharacteristic
70C4⋊C4×C211—21no—
71C4⋊C4×C211D412yescomplemented
72C22⋊C4×C211—24no—
73C22×D211—22no—
74C22×D211—41no—
75C22×D211—41no—
76C22×D211—41no—
77C22×D211—41no—
78C22×D211—41no—
79C22×D211—41no—
80D844—22no—
81D844—22no—
82D844—41no—
83D844—41no—
84C22⋊D211—48no—
85C2×Dic211—22no—
86C2×Dic211—41no—
87C2×Dic211—41no—
88C211⋊C8—22no—
89C211⋊C8—41no—
90C23×C211—21no—
91C23×C211—21no—
92C23×C211—21no—
93C23×C211—21no—
94C23×C211—21no—
95C23×C211—21no—
96C23×C211—22no—
97C23×C211—41no—
98C23×C211—41no—
99C23×C211—41no—
100C23×C211—41no—
101C23×C211—41no—
102C23×C211—41no—
103C23×C211D812yescomplemented
104C23×C211C2×D411yescharacteristic
105D4×C211—22no—
106D4×C211—22no—
107D4×C211—48no—
108D4×C211—41no—
109D4×C211—41no—
110C2×C4×C211—21no—
111C2×C4×C211—22no—
112C2×C4×C211—21no—
113C2×C4×C211—21no—
114C2×C4×C211—41no—
115C2×C4×C211—41no—
116C2×C4×C211—41no—
117C2×C4×C211—41no—
118C2×C4×C211—41no—
119C2×C4×C211—41no—
120C2×C4×C211C2×D411yescharacteristic
121C2×C4×C211C2×D411yescharacteristic
122C2×C4×C211C2×D411yescharacteristic
123C2×C4×C211C2×D411yescharacteristic
124C2×C4×C211C2×D411yescharacteristic
125C2×C4×C211#4111yescharacteristic
126C22×C211—22no—
127C22×C211—22no—
128C22×C211—22no—
129C22×C211—41no—
130C22×C211—41no—
131C22×C211—41no—
132C22×C211—41no—
133C22×C211—42no—
134C22×C211—42no—
135C22×C211—42no—
136C22×C211—81no—
137C22×C211—81no—
138C22×C211—81no—
139C22×C211—81no—
140C22×C211#12011yescharacteristic
141C22×C211#12011yescharacteristic
142C22×C211#12111yescharacteristic
143C22×C211C2×D811yescharacteristic
144C22×C211#13511yescharacteristic
145C22×C211AGL(1,ℤ/8)11yescharacteristic
146C22×C211AGL(1,ℤ/8)11yescharacteristic
147C2×D211—42no—
148C2×D211—42no—
149C2×D211—42no—
150C2×D211—81no—
151C2×D211—81no—
152C2×D211—81no—
153C2×D211—81no—
154C4×C211—41no—
155C4×C211—42no—
156C4×C211#12011yescharacteristic
157C4×C211#12111yescharacteristic
158C4×C211#12111yescharacteristic
159C4×C211#12711yescharacteristic
160Dic211—42no—
161C2×C211—42no—
162C2×C211—82no—
163C2×C211#39211yescharacteristic
164C2×C211#44711yescharacteristic
165C2×C211#44711yescharacteristic
166C2×C211#44911yescharacteristic
167C2×C211#44911yescharacteristic
168C2×C211#46611yescharacteristic
169C2×C211#46911yescharacteristic
170D211—82no—
171C211#199911yescharacteristic, complemented, Sylow
172#1999—2111nomaximal, Sylow
173C22×D8—2111no—
174C2×#121—2111no—
175C2×#121D21111yescharacteristic, complemented
176C2×#102—2111no—
177C2×#102—2111no—
178C2×C22⋊C8—2111no—
179#340—2111no—
180C22×D4—21no—
181C22×D4—4221no—
182C22×D4—2111no—
183C22×D4C2×D21111yescharacteristic
184C23×C4C2×D21111yescharacteristic
185C2×D8—4222no—
186C2×D8—4222no—
187C2×D8—4222no—
188C22×C8—2111no—
189C22×C8—2111no—
190#121—22no—
191#121—22no—
192#121—4222no—
193#121—4222no—
194C2×C4⋊C4—2111no—
195C2×C4⋊C4C2×D21111yescharacteristic
196C2×C22⋊C4—21no—
197C2×C22⋊C4—4221no—
198#102—4222no—
199#102—4222no—
200C22⋊C8—4222no—
201C24—21no—
202C24—4221no—
203C24C22⋊D21111yescharacteristic
204C2×D4—24no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—21no—
209C2×D4—21no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—8442no—
213C2×D4—8442no—
214C2×D4—4224no—
215C2×D4—4221no—
216C2×D4—4221no—
217C2×D4—4221no—
218C2×D4—4221no—
219C2×D4—4221no—
220C2×D4—2112no—
221C2×D4C22⋊D21112yescomplemented
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—4221no—
230C22×C4C22⋊D21111yescharacteristic
231C22×C4C22×D21111yescharacteristic
232D8—4228no—
233C2×C8—4221no—
234C2×C8—4221no—
235C2×C8—4221no—
236C2×C8—4221no—
237C2×C8—4221no—
238C2×C8—2112no—
239C4⋊C4—21no—
240C4⋊C4—4221no—
241C4⋊C4—2112no—
242C4⋊C4C22⋊D21112yescomplemented
243C22⋊C4—24no—
244C22⋊C4—4224no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—22no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—8441no—
259C23—8441no—
260C23—8441no—
261C23—8441no—
262C23—8441no—
263C23—8441no—
264C23—4222no—
265C23#1680994712yescomplemented
266C23C2×C22⋊D21111yescharacteristic
267D4—22no—
268D4—22no—
269D4—48no—
270D4—41no—
271D4—41no—
272D4—8448no—
273D4—8441no—
274D4—8441no—
275D4—4222no—
276D4—4222no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—8441no—
288C2×C4—8441no—
289C2×C4—4222no—
290C2×C4D4×D21111yescharacteristic
291C2×C4D4×D21111yescharacteristic
292C2×C4D4×D21111yescharacteristic
293C2×C4C2×C22⋊D21111yescharacteristic
294C2×C4C2×C22⋊D21111yescharacteristic
295C2×C4#1680996111yescharacteristic
296C8—8441no—
297C8—4222no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—42no—
306C22—42no—
307C22—42no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—8442no—
313C22—8442no—
314C22—8442no—
315C22—16881no—
316C22—16881no—
317C22—16881no—
318C22—16881no—
319C22#2026994511yescharacteristic
320C22#2026994611yescharacteristic
321C22#2026994911yescharacteristic
322C22C2×#1680994711yescharacteristic
323C22#2026999711yescharacteristic
324C22#2027000211yescharacteristic
325C22#2027000311yescharacteristic
326C4—42no—
327C4—41no—
328C4—8442no—
329C4#2026994511yescharacteristic
330C4#2026994811yescharacteristic
331C4#2026994911yescharacteristic
332C4#2026997311yescharacteristic
333C2—42no—
334C2—82no—
335C2—16882no—
336C2#2721305411yescharacteristic
337C2#2721327911yescharacteristic
338C2#2721328011yescharacteristic
339C2#2721328911yescharacteristic
340C2#2721329111yescharacteristic
341C2#2721338011yescharacteristic
342C2#2721339711yescharacteristic
subgroupG/H
centerC23C2×C22⋊D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#121×C211C2
socleC23×C211C2×D4
hypercenterC2×#121D211
hypocenterC211#1999
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#16809947C2semidirect
2. C2×#20269951C2semidirect
3. C2×#20269898C2semidirect
4. C2×#20269897C2semidirect
5. C2×C22⋊(C211⋊C8)C2semidirect
6. #27212908C2semidirect
7. C2×#121×C211C2semidirect
35. C2×D844D4semidirect
46. C2×C211⋊C8D4semidirect
48. C4⋊Dic211D4semidirect
60. C2×D4×C211D4semidirect
71. C4⋊C4×C211D4semidirect
103. C23×C211D8semidirect
171. C211#1999semidirect
175. C2×#121D211semidirect
221. C2×D4C22⋊D211semidirect
242. C4⋊C4C22⋊D211semidirect
265. C23#16809947semidirect

Maximal subgroups

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

subgroupindex
C22×#168099472
C2×#202699512
C2×#202698982
C2×#202698972
C2×C22⋊(C211⋊C8)2
#272129082
C2×#121×C2112
#1999211