Finite Group Databasebeta

← #16846836

Subgroups — #16846836

Order 3968.

638 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 73 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, complemented
2C2×#11758459C211yesmaximal, characteristic, complemented
3C2×#11758514C211yesmaximal, characteristic, complemented
4C2×#11758394C211yesmaximal, characteristic, complemented
5#14520904C211yesmaximal, characteristic, complemented
6#14520815C211yesmaximal, characteristic, complemented
7C2×C31×#99C211yesmaximal, characteristic, complemented
8C24×D31C2211yescharacteristic
9C22×D4×C31C2211yescharacteristic
10C22×C22⋊D31—21no—
11C22×C22⋊D31C2211yescharacteristic
12C22×D124—21no—
13C2×C22⋊C4×C31C2211yescharacteristic
14C2×C22⋊C4×C31C2211yescharacteristic
15#11758530—21no—
16C2×C22⋊Dic31C2211yescharacteristic
17C2×C22⋊Dic31C2211yescharacteristic
18#11758516—22no—
19#11758514—21no—
20#11758514—22no—
21#11758514—22no—
22#11758512—21no—
23#11758506—21no—
24C2×(C2×D31)⋊C4—21no—
25C2×(C2×D31)⋊C4—21no—
26C2×(C2×D31)⋊C4—22no—
27#11758459—22no—
28#11758459—22no—
29C31×#99—22no—
30#11758394—22no—
31C24×C31D411yescharacteristic, complemented
32C24×C31C2311yescharacteristic
33C23×D31—21no—
34C23×D31—21no—
35C23×D31—21no—
36C23×D31—21no—
37C23×D31—21no—
38C23×D31—21no—
39C23×D31—44no—
40C23×D31D412yescomplemented
41C23×D31D411yescharacteristic, complemented
42C2×D4×C31—21no—
43C2×D4×C31—21no—
44C2×D4×C31—21no—
45C2×D4×C31—21no—
46C2×D4×C31—21no—
47C2×D4×C31—21no—
48C22×C4×C31—21no—
49C22×C4×C31—21no—
50C22×C4×C31D411yescharacteristic, complemented
51C2×C22⋊D31—24no—
52C2×C22⋊D31—21no—
53C2×C22⋊D31—21no—
54C2×C22⋊D31—21no—
55C2×C22⋊D31—21no—
56C2×C22⋊D31—21no—
57C2×C22⋊D31—21no—
58C2×C22⋊D31—22no—
59C2×C22⋊D31—44no—
60C22×Dic31—21no—
61C22×Dic31D411yescharacteristic, complemented
62C22×Dic31D412yescomplemented
63C2×D124—24no—
64C2×D124—44no—
65C22⋊C4×C31—21no—
66C22⋊C4×C31—21no—
67C22⋊C4×C31—21no—
68C22⋊C4×C31D412yescomplemented
69C22⋊Dic31—22no—
70C22⋊Dic31—21no—
71C22⋊Dic31—21no—
72C22⋊Dic31—21no—
73C22⋊Dic31D412yescomplemented
74(C2×D31)⋊C4—24no—
75(C2×D31)⋊C4—24no—
76(C2×D31)⋊C4—22no—
77(C2×D31)⋊C4—44no—
78Dic31⋊C4—22no—
79C23×C31—21no—
80C23×C31—21no—
81C23×C31—21no—
82C23×C31—42no—
83C23×C31—42no—
84C23×C31C2×D411yescharacteristic
85C23×C31C2×D411yescharacteristic
86C23×C31C2×D411yescharacteristic
87C23×C31C2×D411yescharacteristic
88C23×C31C2×D411yescharacteristic
89C23×C31C2×D411yescharacteristic
90C23×C31#4111yescharacteristic
91C22×D31—24no—
92C22×D31—24no—
93C22×D31—24no—
94C22×D31—22no—
95C22×D31—22no—
96C22×D31—22no—
97C22×D31—21no—
98C22×D31—21no—
99C22×D31—21no—
100C22×D31—44no—
101C22×D31—44no—
102C22×D31—44no—
103C22×D31—44no—
104C22×D31—42no—
105C22×D31—42no—
106C22×D31—42no—
107C22×D31—41no—
108C22×D31—41no—
109C22×D31—41no—
110C22×D31—41no—
111C22×D31—42no—
112D4×C31—41no—
113D4×C31—41no—
114D4×C31—41no—
115D4×C31—41no—
116C2×C4×C31—21no—
117C2×C4×C31—21no—
118C2×C4×C31—22no—
119C2×C4×C31—21no—
120C2×C4×C31—42no—
121C2×C4×C31—42no—
122C2×C4×C31—41no—
123C22⋊D31—48no—
124C22⋊D31—41no—
125C22⋊D31—41no—
126C22⋊D31—41no—
127C22⋊D31—41no—
128C22⋊D31—82no—
129C2×Dic31—21no—
130C2×Dic31—22no—
131C2×Dic31—21no—
132C2×Dic31—22no—
133C2×Dic31—21no—
134C2×Dic31—42no—
135C2×Dic31—42no—
136D124—48no—
137#2021—311nomaximal, Sylow
138C22×C31—22no—
139C22×C31—22no—
140C22×C31—22no—
141C22×C31—41no—
142C22×C31—41no—
143C22×C31—41no—
144C22×C31—41no—
145C22×C31—42no—
146C22×C31—41no—
147C22×C31—42no—
148C22×C31—81no—
149C22×C31#12011yescharacteristic
150C22×C31#12011yescharacteristic
151C22×C31#12011yescharacteristic
152C22×C31#12111yescharacteristic
153C22×C31#12111yescharacteristic
154C22×C31#12111yescharacteristic
155C22×C31#12711yescharacteristic
156C2×D31—24no—
157C2×D31—24no—
158C2×D31—24no—
159C2×D31—42no—
160C2×D31—42no—
161C2×D31—44no—
162C2×D31—42no—
163C2×D31—44no—
164C2×D31—44no—
165C2×D31—42no—
166C2×D31—44no—
167C2×D31—42no—
168C2×D31—41no—
169C2×D31—42no—
170C2×D31—81no—
171C4×C31—42no—
172C4×C31—41no—
173C4×C31—81no—
174Dic31—42no—
175Dic31—41no—
176Dic31—42no—
177#534—311no—
178C2×#120—311no—
179C2×#120—311no—
180C2×#99—311no—
181C2×#99D3111yescharacteristic, complemented
182#379—311no—
183#342—311no—
184C2×C31—21no—
185C2×C31—21no—
186C2×C31—21no—
187C2×C31—21no—
188C2×C31—21no—
189C2×C31—21no—
190C2×C31—81no—
191C2×C31#39211yescharacteristic
192C2×C31B(4,𝔽2)12yes—
193D31—44no—
194D31—81no—
195C25—311no—
196C22×D4—311no—
197C22×D4—621no—
198C22×D4—621no—
199C22×D4C2×D3111yescharacteristic
200#123—621no—
201#123—621no—
202#121—622no—
203#120—622no—
204#120—622no—
205#120—622no—
206#120—621no—
207#120—621no—
208#120—622no—
209C2×C22⋊C4—311no—
210C2×C22⋊C4—311no—
211C2×C22⋊C4—621no—
212C2×C22⋊C4—621no—
213C2×C22⋊C4—622no—
214C2×C22⋊C4C2×D3111yescharacteristic
215C2×C22⋊C4C2×D3111yescharacteristic
216#99—22no—
217#99—622no—
218C31#202111yescharacteristic, complemented, Sylow
219C24—312no—
220C24—311no—
221C24—621no—
222C24—621no—
223C24—621no—
224C24—621no—
225C24—621no—
226C24—621no—
227C24—1244no—
228C24D12411yescharacteristic, complemented
229C24C22×D3111yescharacteristic
230C2×D4—21no—
231C2×D4—21no—
232C2×D4—21no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—21no—
236C2×D4—624no—
237C2×D4—624no—
238C2×D4—621no—
239C2×D4—621no—
240C2×D4—621no—
241C2×D4—621no—
242C2×D4—621no—
243C2×D4—621no—
244C2×D4—622no—
245C2×D4—1244no—
246C2×D4—1244no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—311no—
250C22×C4—312no—
251C22×C4—621no—
252C22×C4D12411yescharacteristic, complemented
253C4⋊C4—622no—
254C22⋊C4—21no—
255C22⋊C4—21no—
256C22⋊C4—21no—
257C22⋊C4—312no—
258C22⋊C4—624no—
259C22⋊C4—624no—
260C22⋊C4—622no—
261C22⋊C4—622no—
262C22⋊C4—621no—
263C22⋊C4—621no—
264C22⋊C4—621no—
265C22⋊C4—1244no—
266C22⋊C4C22⋊D3112yescomplemented
267C23—21no—
268C23—21no—
269C23—21no—
270C23—42no—
271C23—42no—
272C23—624no—
273C23—624no—
274C23—624no—
275C23—622no—
276C23—622no—
277C23—622no—
278C23—621no—
279C23—621no—
280C23—621no—
281C23—1242no—
282C23—1242no—
283C23—1242no—
284C23—1242no—
285C23—1244no—
286C23—1244no—
287C23—1244no—
288C23—1244no—
289C23—1241no—
290C23—1241no—
291C23—1241no—
292C23—1241no—
293C23C2×D12411yescharacteristic
294C23#9249511yescharacteristic
295C23D4×D3111yescharacteristic
296C23D4×D3111yescharacteristic
297C23D4×D3111yescharacteristic
298C23D4×D3111yescharacteristic
299C23C2×C22⋊D3111yescharacteristic
300D4—41no—
301D4—41no—
302D4—41no—
303D4—41no—
304D4—1248no—
305D4—1248no—
306D4—1241no—
307D4—1241no—
308D4—1241no—
309D4—1241no—
310D4—2482no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—622no—
319C2×C4—622no—
320C2×C4—621no—
321C2×C4—621no—
322C2×C4—621no—
323C2×C4—1242no—
324C2×C4—1242no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—42no—
333C22—41no—
334C22—42no—
335C22—81no—
336C22—624no—
337C22—624no—
338C22—624no—
339C22—1242no—
340C22—1244no—
341C22—1242no—
342C22—1244no—
343C22—1242no—
344C22—1244no—
345C22—1242no—
346C22—1242no—
347C22—1244no—
348C22—1241no—
349C22—1242no—
350C22—2481no—
351C22#1175845911yescharacteristic
352C22#1175845911yescharacteristic
353C22#1175846111yescharacteristic
354C22#1175846111yescharacteristic
355C22#1175850711yescharacteristic
356C22#1175851411yescharacteristic
357C22#1175851711yescharacteristic
358C4—42no—
359C4—41no—
360C4—81no—
361C4—1242no—
362C4—1241no—
363C4—1242no—
364C2—21no—
365C2—21no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—21no—
370C2—81no—
371C2—1244no—
372C2—2481no—
373C2#1452095811yescharacteristic
374C2#1452123712yes—
subgroupG/H
centerC22#11758459
commutator, cosocleC24×C31C23
FrattiniC24C22×D31
FittingC2×C31×#99C2
socleC22×C31#120
hypercenterC2×#99D31
hypocenterC31#2021
LCSG ▷ C24×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
1. #14521608C2semidirect
2. C2×#11758459C2semidirect
3. C2×#11758514C2semidirect
4. C2×#11758394C2semidirect
5. #14520904C2semidirect
6. #14520815C2semidirect
7. C2×C31×#99C2semidirect
31. C24×C31D4semidirect
40. C23×D31D4semidirect
41. C23×D31D4semidirect
50. C22×C4×C31D4semidirect
61. C22×Dic31D4semidirect
62. C22×Dic31D4semidirect
68. C22⋊C4×C31D4semidirect
73. C22⋊Dic31D4semidirect
181. C2×#99D31semidirect
218. C31#2021semidirect
228. C24D124semidirect
252. C22×C4D124semidirect
266. 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×#117584592
C2×#117585142
C2×#117583942
#145209042
#145208152
C2×C31×#992
#202131