Finite Group Databasebeta

← #44508573

Subgroups — #44508573

Order 30848.

430 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 75 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#29508276C211yesmaximal, characteristic
2C2×#21440625C211yesmaximal, characteristic, complemented
3C2×#21440557C211yesmaximal, characteristic, complemented
4C2×#21440556C211yesmaximal, characteristic, complemented
5#29507580C211yesmaximal, characteristic, complemented
6#29507483C211yesmaximal, characteristic, complemented
7C2×#123×C241C211yesmaximal, characteristic, complemented
8C22×C22⋊D241C2211yescharacteristic
9#21440625—22no—
10#21440625—21no—
11#21440625—22no—
12#21440623—21no—
13#21440614—22no—
14#21440611—21no—
15#21440610—21no—
16C2×#16838749—21no—
17C2×C22⋊Dic241C2211yescharacteristic
18C2×C22⋊Dic241C2211yescharacteristic
19C2×C22⋊Dic241C2211yescharacteristic
20C2×#16838747—21no—
21#21440557—22no—
22#21440556—22no—
23#21440549—22no—
24C22×D4×C241C2211yescharacteristic
25C23×C4×C241C2211yescharacteristic
26#123×C241—22no—
27#123×C241—22no—
28C2×C4⋊C4×C241—21no—
29C2×C22⋊C4×C241—21no—
30C2×C22⋊C4×C241C2211yescharacteristic
31C23×D241D411yescharacteristic
32C2×C22⋊D241—22no—
33C2×C22⋊D241—21no—
34C2×C22⋊D241—21no—
35C2×C22⋊D241—21no—
36C2×C22⋊D241—21no—
37C2×C22⋊D241—21no—
38C2×C22⋊D241—21no—
39C22×Dic241—21no—
40C22×Dic241—21no—
41C22×Dic241D411yescharacteristic
42C22×Dic241D412yes—
43#16838749—24no—
44#16838749—22no—
45C22⋊Dic241—22no—
46C22⋊Dic241—21no—
47C22⋊Dic241—21no—
48C22⋊Dic241—21no—
49C22⋊Dic241—21no—
50C22⋊Dic241—21no—
51C22⋊Dic241D412yescomplemented
52#16838747—24no—
53#16838747—22no—
54C24×C241D411yescharacteristic
55C24×C241C2311yescharacteristic
56C2×D4×C241—21no—
57C2×D4×C241—21no—
58C2×D4×C241—21no—
59C2×D4×C241—21no—
60C2×D4×C241—21no—
61C2×D4×C241—21no—
62C22×C4×C241—21no—
63C22×C4×C241—21no—
64C22×C4×C241—21no—
65C22×C4×C241—21no—
66C22×C4×C241—21no—
67C22×C4×C241—21no—
68C22×C4×C241—21no—
69C22×C4×C241D411yescharacteristic
70C22×C4×C241D412yescomplemented
71C4⋊C4×C241—24no—
72C22⋊C4×C241—24no—
73C22⋊C4×C241—21no—
74C22⋊C4×C241D412yescomplemented
75C22×D241—21no—
76C22×D241—21no—
77C22×D241—21no—
78C22×D241—42no—
79C22⋊D241—41no—
80C22⋊D241—41no—
81C22⋊D241—41no—
82C22⋊D241—41no—
83C22⋊D241—82no—
84C2×Dic241—22no—
85C2×Dic241—22no—
86C2×Dic241—21no—
87C2×Dic241—21no—
88C2×Dic241—21no—
89C2×Dic241—42no—
90C2×Dic241—42no—
91C2×Dic241—42no—
92C2×Dic241—41no—
93C23×C241—21no—
94C23×C241—21no—
95C23×C241—21no—
96C23×C241—42no—
97C23×C241—42no—
98C23×C241C2×D411yescharacteristic
99C23×C241C2×D411yescharacteristic
100C23×C241C2×D411yescharacteristic
101C23×C241C2×D411yescharacteristic
102C23×C241C2×D411yescharacteristic
103C23×C241C2×D411yescharacteristic
104C23×C241#4111yescharacteristic
105D4×C241—41no—
106D4×C241—41no—
107D4×C241—41no—
108D4×C241—41no—
109C2×C4×C241—22no—
110C2×C4×C241—22no—
111C2×C4×C241—21no—
112C2×C4×C241—22no—
113C2×C4×C241—21no—
114C2×C4×C241—22no—
115C2×C4×C241—21no—
116C2×C4×C241—42no—
117C2×C4×C241—41no—
118C2×C4×C241—41no—
119C2×C4×C241—41no—
120C2×C4×C241—41no—
121C22×C241—22no—
122C22×C241—22no—
123C22×C241—22no—
124C22×C241—41no—
125C22×C241—41no—
126C22×C241—41no—
127C22×C241—41no—
128C22×C241—42no—
129C22×C241—41no—
130C22×C241—42no—
131C22×C241—81no—
132C22×C241#12011yescharacteristic
133C22×C241#12011yescharacteristic
134C22×C241#12011yescharacteristic
135C22×C241#12111yescharacteristic
136C22×C241#12111yescharacteristic
137C22×C241#12111yescharacteristic
138C22×C241#12711yescharacteristic
139C2×D241—42no—
140C2×D241—41no—
141C2×D241—42no—
142C2×D241—81no—
143C4×C241—42no—
144C4×C241—42no—
145C4×C241—41no—
146Dic241—42no—
147Dic241—41no—
148Dic241—42no—
149Dic241—81no—
150C2×C241—21no—
151C2×C241—21no—
152C2×C241—21no—
153C2×C241—21no—
154C2×C241—21no—
155C2×C241—21no—
156C2×C241—81no—
157C2×C241#39211yescharacteristic
158C2×C241#45812yes—
159D241—81no—
160C241#202211yescharacteristic, complemented, Sylow
161#2022—2411nomaximal, Sylow
162#534—2411no—
163C2×#123—2411no—
164C2×#123D24111yescharacteristic, complemented
165C2×#99—2411no—
166C2×#99—2411no—
167#381—2411no—
168#342—2411no—
169C22×D4—2411no—
170C22×D4C2×D24111yescharacteristic
171C23×C4C2×D24111yescharacteristic
172#123—22no—
173#123—22no—
174#123—4821no—
175#123—4821no—
176#123—4822no—
177#123—4822no—
178#121—4822no—
179#120—4821no—
180#120—4821no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—4821no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—4821no—
185C2×C22⋊C4—2411no—
186C2×C22⋊C4—2411no—
187C2×C22⋊C4—2411no—
188C2×C22⋊C4C2×D24111yescharacteristic
189#99—4822no—
190#99—4822no—
191#95—4822no—
192C24—2411no—
193C24C22⋊D24111yescharacteristic
194C24C22×D24111yescharacteristic
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—4822no—
202C2×D4—4821no—
203C2×D4—4821no—
204C2×D4—4821no—
205C2×D4—4821no—
206C2×D4—4821no—
207C2×D4—4821no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—4821no—
216C22×C4—4821no—
217C22×C4—2412no—
218C22×C4—2411no—
219C22×C4C22⋊D24111yescharacteristic
220C22×C4C22⋊D24112yescomplemented
221C4⋊C4—24no—
222C4⋊C4—4824no—
223C4⋊C4—4822no—
224C22⋊C4—24no—
225C22⋊C4—21no—
226C22⋊C4—4824no—
227C22⋊C4—4822no—
228C22⋊C4—4821no—
229C22⋊C4—4821no—
230C22⋊C4—4822no—
231C22⋊C4—4821no—
232C22⋊C4—4821no—
233C22⋊C4—4821no—
234C22⋊C4—2412no—
235C22⋊C4C22⋊D24112yescomplemented
236C23—21no—
237C23—21no—
238C23—21no—
239C23—42no—
240C23—42no—
241C23—9642no—
242C23—4821no—
243C23—4821no—
244C23—4821no—
245C23D4×D24111yescharacteristic
246C23D4×D24111yescharacteristic
247C23D4×D24111yescharacteristic
248C23C2×C22⋊D24111yescharacteristic
249C23C2×C22⋊D24111yescharacteristic
250C23C2×C22⋊D24111yescharacteristic
251C23#1683876911yescharacteristic
252D4—41no—
253D4—41no—
254D4—41no—
255D4—41no—
256D4—19282no—
257D4—9641no—
258D4—9641no—
259D4—9641no—
260D4—9641no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—9642no—
274C2×C4—9641no—
275C2×C4—9642no—
276C2×C4—9642no—
277C2×C4—4822no—
278C2×C4—4821no—
279C2×C4—4822no—
280C2×C4—4821no—
281C2×C4—4821no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—42no—
290C22—41no—
291C22—42no—
292C22—81no—
293C22—19281no—
294C22—9642no—
295C22—9641no—
296C22—9642no—
297C22#2144061011yescharacteristic
298C22#2144061011yescharacteristic
299C22#2144061111yescharacteristic
300C22#2144061311yescharacteristic
301C22#2144061411yescharacteristic
302C22#2144061411yescharacteristic
303C22#2144063811yescharacteristic
304C4—42no—
305C4—42no—
306C4—41no—
307C4—19281no—
308C4—9642no—
309C4—9641no—
310C4—9642no—
311C2—21no—
312C2—21no—
313C2—21no—
314C2—21no—
315C2—21no—
316C2—21no—
317C2—81no—
318C2—19281no—
319C2#2950762511yescharacteristic
320C2#2950790612yes—
subgroupG/H
centerC22#21440610
commutator, cosocleC24×C241C23
FrattiniC24C22×D241
FittingC2×#123×C241C2
socleC22×C241#120
prime radical#29508276C2
hypercenterC2×#123D241
hypocenterC241#2022
LCSG ▷ C24×C241 ▷ C2×C241 ▷ C241
UCSC1 ◁ C22 ◁ C24 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×#21440625C2semidirect
3. C2×#21440557C2semidirect
4. C2×#21440556C2semidirect
5. #29507580C2semidirect
6. #29507483C2semidirect
7. C2×#123×C241C2semidirect
51. C22⋊Dic241D4semidirect
70. C22×C4×C241D4semidirect
74. C22⋊C4×C241D4semidirect
160. C241#2022semidirect
164. C2×#123D241semidirect
220. C22×C4C22⋊D241semidirect
235. C22⋊C4C22⋊D241semidirect

Maximal subgroups

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

subgroupindex
#295082762
C2×#214406252
C2×#214405572
C2×#214405562
#295075802
#295074832
C2×#123×C2412
#2022241