Finite Group Databasebeta

← #85309

Subgroups — #85309

Order 384.

354 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 40 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
1C2×C4⋊S4C211yesmaximal, characteristic
2C2×C4×S4C211yesmaximal, characteristic, complemented
3A4×C4⋊C4C211yesmaximal, characteristic, complemented
4(C2×S4)⋊C4C211yesmaximal, characteristic, complemented
5(C2×S4)⋊C4C211yesmaximal, characteristic, complemented
6(C2×S4)⋊C4C211yesmaximal, characteristic, complemented
7SL(2,ℤ/4)⋊C4C211yesmaximal, characteristic, complemented
8#2443—31nomaximal, Sylow
9C22×S4C2211yescharacteristic
10C22×S4C2211yescharacteristic
11C2×C4×A4C2211yescharacteristic
12C2×C4×A4C2211yescharacteristic
13C2×C4×A4C2211yescharacteristic
14C2×SL(2,ℤ/4)C2211yescharacteristic
15C2×SL(2,ℤ/4)C2211yescharacteristic
16C4⋊S4—22no—
17C4×S4—22no—
18#894—41nomaximal
19C2×#123—31no—
20C2×#121—31no—
21C2×C4×D4—31no—
22C22×C4⋊C4D311yescharacteristic, complemented
23#399—31no—
24#398—31no—
25#397—31no—
26#394—31no—
27#388—31no—
28#387—31no—
29#386—31no—
30#386—31no—
31#386—31no—
32#385—31no—
33#385—31no—
34C22×A4C2311yescharacteristic
35C2×S4—21no—
36C2×S4—21no—
37C2×S4—21no—
38C2×S4—21no—
39C2×S4—21no—
40C2×S4D412yes—
41C2×D12—41no—
42C2×C4×D3—41no—
43C4×A4—21no—
44C4×A4—21no—
45C4×A4—21no—
46SL(2,ℤ/4)—21no—
47SL(2,ℤ/4)—21no—
48C3×C4⋊C4—41no—
49(C2×D3)⋊C4—41no—
50(C2×D3)⋊C4—41no—
51(C2×D3)⋊C4—41no—
52Dic3⋊C4—41no—
53C22×D4—31no—
54C22×D4—31no—
55C22×D4—31no—
56C23×C4—31no—
57C23×C4C2×D311yescharacteristic
58C23×C4C2×D311yescharacteristic
59C23×C4C2×D311yescharacteristic
60#123—61no—
61#123—61no—
62#123—61no—
63#123—61no—
64#121—62no—
65#121—62no—
66C4×D4—62no—
67C4×D4—62no—
68C2×C4⋊C4—31no—
69C2×C4⋊C4—31no—
70C2×C4⋊C4—31no—
71C2×C4⋊C4—31no—
72C2×C4⋊C4—31no—
73C2×C4⋊C4—31no—
74C2×C4⋊C4—31no—
75C2×C4⋊C4—31no—
76C2×C22⋊C4—31no—
77C2×C22⋊C4—31no—
78C2×C22⋊C4—31no—
79C2×C22⋊C4—31no—
80C2×C22⋊C4—31no—
81C2×C22⋊C4—31no—
82C2×C22⋊C4—31no—
83C2×C22⋊C4—31no—
84C2×C22⋊C4—31no—
85C2×C22⋊C4—31no—
86C2×C22⋊C4—31no—
87C2×C42—31no—
88#95—31no—
89#95—31no—
90#95—31no—
91#95—31no—
92#95—31no—
93#95—31no—
94#95—31no—
95#95—31no—
96C22×D3—41no—
97C22×D3—41no—
98C2×A4C2×D411yescharacteristic
99C2×A4#4111yescharacteristic
100C2×A4#4111yescharacteristic
101S4—22no—
102S4—41no—
103C2×C3×C4—41no—
104C2×C3×C4—41no—
105C2×C3×C4—41no—
106C2×Dic3—41no—
107C2×Dic3—41no—
108D12—82no—
109C4×D3—82no—
110C24—31no—
111C24—31no—
112C24C22×D311yescharacteristic
113C2×D4—32no—
114C2×D4—32no—
115C2×D4—61no—
116C2×D4—61no—
117C2×D4—61no—
118C2×D4—61no—
119C2×D4—61no—
120C2×D4—62no—
121C2×D4—61no—
122C2×D4—61no—
123C2×D4—61no—
124C2×D4—61no—
125C2×D4—61no—
126C2×D4—61no—
127C2×D4—61no—
128C2×D4—121no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—31no—
133C22×C4—31no—
134C22×C4—31no—
135C22×C4—31no—
136C22×C4—31no—
137C22×C4—31no—
138C22×C4—31no—
139C22×C4—31no—
140C22×C4—31no—
141C22×C4—31no—
142C22×C4—31no—
143C22×C4—31no—
144C22×C4—61no—
145C22×C4—61no—
146C22×C4—61no—
147C22×C4—61no—
148C22×C4—62no—
149C22×C4—61no—
150C22×C4—121no—
151C4⋊C4—31no—
152C4⋊C4—31no—
153C4⋊C4—31no—
154C4⋊C4—61no—
155C4⋊C4—62no—
156C4⋊C4—62no—
157C4⋊C4—61no—
158C4⋊C4—61no—
159C4⋊C4—121no—
160C4⋊C4S411yescharacteristic, complemented
161C22⋊C4—62no—
162C22⋊C4—62no—
163C22⋊C4—62no—
164C22⋊C4—61no—
165C22⋊C4—61no—
166C22⋊C4—61no—
167C22⋊C4—61no—
168C22⋊C4—62no—
169C22⋊C4—61no—
170C22⋊C4—62no—
171C22⋊C4—61no—
172C22⋊C4—61no—
173C22⋊C4—61no—
174C22⋊C4—61no—
175C22⋊C4—61no—
176C22⋊C4—121no—
177C42—62no—
178C22×C3—41no—
179C2×D3—42no—
180C2×D3—81no—
181C2×D3—81no—
182C2×D3—81no—
183C2×D3—81no—
184C2×D3—81no—
185A4#12311yescharacteristic, complemented
186C3×C4—81no—
187C3×C4—81no—
188C3×C4—81no—
189Dic3—81no—
190Dic3—81no—
191C23—31no—
192C23—31no—
193C23—31no—
194C23—31no—
195C23—32no—
196C23—61no—
197C23—61no—
198C23—61no—
199C23—61no—
200C23—61no—
201C23—61no—
202C23—61no—
203C23—61no—
204C23—61no—
205C23—61no—
206C23—61no—
207C23—61no—
208C23—61no—
209C23#23511yescharacteristic
210C23D3×D411yescharacteristic
211C23#23911yescharacteristic
212D4—62no—
213D4—62no—
214D4—62no—
215D4—62no—
216D4—122no—
217D4—122no—
218D4—121no—
219D4—121no—
220D4—121no—
221D4—121no—
222C2×C4—32no—
223C2×C4—32no—
224C2×C4—31no—
225C2×C4—31no—
226C2×C4—31no—
227C2×C4—61no—
228C2×C4—61no—
229C2×C4—61no—
230C2×C4—61no—
231C2×C4—61no—
232C2×C4—61no—
233C2×C4—61no—
234C2×C4—61no—
235C2×C4—61no—
236C2×C4—61no—
237C2×C4—61no—
238C2×C4—61no—
239C2×C4—61no—
240C2×C4—61no—
241C2×C4—61no—
242C2×C4—61no—
243C2×C4—61no—
244C2×C4—61no—
245C2×C4—61no—
246C2×C4—61no—
247C2×C4—61no—
248C2×C4—61no—
249C2×C4—61no—
250C2×C4—61no—
251C2×C4—61no—
252C2×C4—61no—
253C2×C4—61no—
254C2×C4—61no—
255C2×C4—122no—
256C2×C4—122no—
257C2×C4C2×S411yescharacteristic
258C2×C4C2×S411yescharacteristic
259C2×C4C2×S411yescharacteristic
260C2×C3—41no—
261C2×C3—41no—
262C2×C3—41no—
263D3—82no—
264D3—161no—
265C22—31no—
266C22—31no—
267C22—31no—
268C22—31no—
269C22—31no—
270C22—31no—
271C22—31no—
272C22—31no—
273C22—31no—
274C22—61no—
275C22—62no—
276C22—62no—
277C22—62no—
278C22—121no—
279C22—121no—
280C22—121no—
281C22—121no—
282C22—121no—
283C22—121no—
284C22—121no—
285C22—121no—
286C22—121no—
287C22—121no—
288C22—121no—
289C22#89411yescharacteristic, complemented
290C22C22×S411yescharacteristic
291C4—21no—
292C4—21no—
293C4—21no—
294C4—62no—
295C4—61no—
296C4—61no—
297C4—61no—
298C4—121no—
299C4—121no—
300C4—121no—
301C4—121no—
302C4—121no—
303C3—41noSylow
304C2—31no—
305C2—31no—
306C2—31no—
307C2—31no—
308C2—122no—
309C2—241no—
310C2#591211yescharacteristic
311C2D4×S411yescharacteristic
312C2#591911yescharacteristic
subgroupG/H
center, FrattiniC22C22×S4
commutator, cosocleC22×A4C23
FittingC22×C4⋊C4D3
socleC24C22×D3
prime radicalC2×C4⋊S4C2
hypercenterC4⋊C4S4
hypocenterA4#123
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×S4C2semidirect
3. A4×C4⋊C4C2semidirect
4. (C2×S4)⋊C4C2semidirect
5. (C2×S4)⋊C4C2semidirect
6. (C2×S4)⋊C4C2semidirect
7. SL(2,ℤ/4)⋊C4C2semidirect
22. C22×C4⋊C4D3semidirect
160. C4⋊C4S4semidirect
185. A4#123semidirect
289. C22#894semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊S42
C2×C4×S42
A4×C4⋊C42
(C2×S4)⋊C42
SL(2,ℤ/4)⋊C42
#24433
#8944