Finite Group Databasebeta

← #16846839

Subgroups — #16846839

Order 3968.

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#14521608C211yesmaximal, characteristic
2C2×#11758506C211yesmaximal, characteristic, complemented
3C2×#11758422C211yesmaximal, characteristic, complemented
4C2×#11758394C211yesmaximal, characteristic, complemented
5#14520912C211yesmaximal, characteristic, complemented
6#14520815C211yesmaximal, characteristic, complemented
7C2×C31×#123C211yesmaximal, characteristic, complemented
8C22×D4×C31C2211yescharacteristic
9C23×C4×C31C2211yescharacteristic
10C22×C22⋊D31C2211yescharacteristic
11C31×#123—22no—
12C31×#123—22no—
13C2×C4⋊C4×C31—21no—
14C2×C22⋊C4×C31—21no—
15C2×C22⋊C4×C31C2211yescharacteristic
16#11758530—21no—
17C2×C22⋊Dic31C2211yescharacteristic
18C2×C22⋊Dic31C2211yescharacteristic
19C2×C22⋊Dic31C2211yescharacteristic
20#11758516—22no—
21#11758514—21no—
22#11758512—21no—
23#11758506—21no—
24#11758506—22no—
25#11758506—22no—
26C2×(C2×D31)⋊C4—21no—
27C2×Dic31⋊C4—21no—
28#11758422—22no—
29C31⋊#95—22no—
30#11758394—22no—
31C24×C31D411yescharacteristic
32C24×C31C2311yescharacteristic
33C23×D31D411yescharacteristic
34C2×D4×C31—21no—
35C2×D4×C31—21no—
36C2×D4×C31—21no—
37C2×D4×C31—21no—
38C2×D4×C31—21no—
39C2×D4×C31—21no—
40C22×C4×C31—21no—
41C22×C4×C31—21no—
42C22×C4×C31—21no—
43C22×C4×C31—21no—
44C22×C4×C31—21no—
45C22×C4×C31—21no—
46C22×C4×C31—21no—
47C22×C4×C31D411yescharacteristic
48C22×C4×C31D412yescomplemented
49C2×C22⋊D31—21no—
50C2×C22⋊D31—21no—
51C2×C22⋊D31—21no—
52C2×C22⋊D31—21no—
53C2×C22⋊D31—21no—
54C2×C22⋊D31—21no—
55C2×C22⋊D31—22no—
56C22×Dic31—21no—
57C22×Dic31—21no—
58C22×Dic31D411yescharacteristic
59C22×Dic31D412yes—
60C4⋊C4×C31—24no—
61C22⋊C4×C31—24no—
62C22⋊C4×C31—21no—
63C22⋊C4×C31D412yescomplemented
64C22⋊Dic31—22no—
65C22⋊Dic31—21no—
66C22⋊Dic31—21no—
67C22⋊Dic31—21no—
68C22⋊Dic31—21no—
69C22⋊Dic31—21no—
70C22⋊Dic31D412yescomplemented
71(C2×D31)⋊C4—24no—
72(C2×D31)⋊C4—22no—
73Dic31⋊C4—24no—
74Dic31⋊C4—22no—
75C23×C31—21no—
76C23×C31—21no—
77C23×C31—21no—
78C23×C31—42no—
79C23×C31—42no—
80C23×C31C2×D411yescharacteristic
81C23×C31C2×D411yescharacteristic
82C23×C31C2×D411yescharacteristic
83C23×C31C2×D411yescharacteristic
84C23×C31C2×D411yescharacteristic
85C23×C31C2×D411yescharacteristic
86C23×C31#4111yescharacteristic
87C22×D31—21no—
88C22×D31—21no—
89C22×D31—21no—
90C22×D31—42no—
91D4×C31—41no—
92D4×C31—41no—
93D4×C31—41no—
94D4×C31—41no—
95C2×C4×C31—22no—
96C2×C4×C31—21no—
97C2×C4×C31—22no—
98C2×C4×C31—22no—
99C2×C4×C31—21no—
100C2×C4×C31—22no—
101C2×C4×C31—21no—
102C2×C4×C31—42no—
103C2×C4×C31—41no—
104C2×C4×C31—41no—
105C2×C4×C31—41no—
106C2×C4×C31—41no—
107C22⋊D31—41no—
108C22⋊D31—41no—
109C22⋊D31—41no—
110C22⋊D31—41no—
111C22⋊D31—82no—
112C2×Dic31—21no—
113C2×Dic31—22no—
114C2×Dic31—21no—
115C2×Dic31—22no—
116C2×Dic31—21no—
117C2×Dic31—42no—
118C2×Dic31—42no—
119C2×Dic31—42no—
120C2×Dic31—41no—
121#2022—311nomaximal, Sylow
122C22×C31—22no—
123C22×C31—22no—
124C22×C31—22no—
125C22×C31—41no—
126C22×C31—41no—
127C22×C31—41no—
128C22×C31—41no—
129C22×C31—42no—
130C22×C31—41no—
131C22×C31—42no—
132C22×C31—81no—
133C22×C31#12011yescharacteristic
134C22×C31#12011yescharacteristic
135C22×C31#12011yescharacteristic
136C22×C31#12111yescharacteristic
137C22×C31#12111yescharacteristic
138C22×C31#12111yescharacteristic
139C22×C31#12711yescharacteristic
140C2×D31—42no—
141C2×D31—41no—
142C2×D31—42no—
143C2×D31—81no—
144C4×C31—42no—
145C4×C31—42no—
146C4×C31—41no—
147Dic31—42no—
148Dic31—41no—
149Dic31—42no—
150Dic31—81no—
151#534—311no—
152C2×#123—311no—
153C2×#123D3111yescharacteristic, complemented
154C2×#99—311no—
155C2×#99—311no—
156#381—311no—
157#342—311no—
158C2×C31—21no—
159C2×C31—21no—
160C2×C31—21no—
161C2×C31—21no—
162C2×C31—21no—
163C2×C31—21no—
164C2×C31—81no—
165C2×C31#39211yescharacteristic
166C2×C31#45812yes—
167D31—81no—
168C22×D4—311no—
169C22×D4C2×D3111yescharacteristic
170C23×C4C2×D3111yescharacteristic
171#123—22no—
172#123—22no—
173#123—622no—
174#123—621no—
175#123—621no—
176#123—622no—
177#121—622no—
178#120—621no—
179#120—621no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—621no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—311no—
184C2×C22⋊C4—311no—
185C2×C22⋊C4—311no—
186C2×C22⋊C4—621no—
187C2×C22⋊C4C2×D3111yescharacteristic
188#99—622no—
189#99—622no—
190#95—622no—
191C31#202211yescharacteristic, complemented, Sylow
192C24—311no—
193C24C22⋊D3111yescharacteristic
194C24C22×D3111yescharacteristic
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—621no—
202C2×D4—621no—
203C2×D4—621no—
204C2×D4—621no—
205C2×D4—621no—
206C2×D4—621no—
207C2×D4—622no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—311no—
216C22×C4—312no—
217C22×C4—621no—
218C22×C4—621no—
219C22×C4C22⋊D3111yescharacteristic
220C22×C4C22⋊D3112yescomplemented
221C4⋊C4—24no—
222C4⋊C4—624no—
223C4⋊C4—622no—
224C22⋊C4—24no—
225C22⋊C4—21no—
226C22⋊C4—312no—
227C22⋊C4—624no—
228C22⋊C4—622no—
229C22⋊C4—622no—
230C22⋊C4—621no—
231C22⋊C4—621no—
232C22⋊C4—621no—
233C22⋊C4—621no—
234C22⋊C4—621no—
235C22⋊C4C22⋊D3112yescomplemented
236C23—21no—
237C23—21no—
238C23—21no—
239C23—42no—
240C23—42no—
241C23—621no—
242C23—621no—
243C23—621no—
244C23—1242no—
245C23D4×D3111yescharacteristic
246C23D4×D3111yescharacteristic
247C23D4×D3111yescharacteristic
248C23#9249711yescharacteristic
249C23C2×C22⋊D3111yescharacteristic
250C23C2×C22⋊D3111yescharacteristic
251C23C2×C22⋊D3111yescharacteristic
252D4—41no—
253D4—41no—
254D4—41no—
255D4—41no—
256D4—1241no—
257D4—1241no—
258D4—1241no—
259D4—1241no—
260D4—2482no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—622no—
274C2×C4—621no—
275C2×C4—622no—
276C2×C4—621no—
277C2×C4—621no—
278C2×C4—1242no—
279C2×C4—1242no—
280C2×C4—1242no—
281C2×C4—1241no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—42no—
290C22—41no—
291C22—42no—
292C22—81no—
293C22—1242no—
294C22—1241no—
295C22—1242no—
296C22—2481no—
297C22#1175851411yescharacteristic
298C22#1175851411yescharacteristic
299C22#1175851511yescharacteristic
300C22#1175851611yescharacteristic
301C22#1175851611yescharacteristic
302C22#1175851711yescharacteristic
303C22#1175853011yescharacteristic
304C4—42no—
305C4—42no—
306C4—41no—
307C4—1242no—
308C4—1241no—
309C4—1242no—
310C4—2481no—
311C2—21no—
312C2—21no—
313C2—21no—
314C2—21no—
315C2—21no—
316C2—21no—
317C2—81no—
318C2—2481no—
319C2#1452095711yescharacteristic
320C2#1452123812yes—
subgroupG/H
centerC22#11758514
commutator, cosocleC24×C31C23
FrattiniC24C22×D31
FittingC2×C31×#123C2
socleC22×C31#120
prime radical#14521608C2
hypercenterC2×#123D31
hypocenterC31#2022
LCSG ▷ C24×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C24 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×#11758506C2semidirect
3. C2×#11758422C2semidirect
4. C2×#11758394C2semidirect
5. #14520912C2semidirect
6. #14520815C2semidirect
7. C2×C31×#123C2semidirect
48. C22×C4×C31D4semidirect
63. C22⋊C4×C31D4semidirect
70. C22⋊Dic31D4semidirect
153. C2×#123D31semidirect
191. C31#2022semidirect
220. C22×C4C22⋊D31semidirect
235. C22⋊C4C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
#145216082
C2×#117585062
C2×#117584222
C2×#117583942
#145209122
#145208152
C2×C31×#1232
#202231