Finite Group Databasebeta

← #85298

Subgroups — #85298

Order 384.

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

normal subgroups only

This group has 377 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×GL(2,ℤ/4)C211yesmaximal, characteristic
2C2×C4×S4C211yesmaximal, characteristic, complemented
3A4×C22⋊C4C211yesmaximal, characteristic, complemented
4#5432C211yesmaximal, characteristic, complemented
5(C2×S4)⋊C4C211yesmaximal, characteristic, complemented
6C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
7SL(2,ℤ/4)⋊C4C211yesmaximal, characteristic, complemented
8#2418—31nomaximal, Sylow
9C23×A4C2211yescharacteristic
10C22×S4C2211yescharacteristic
11C2×C4×A4C2211yescharacteristic
12C2×C4×A4C2211yescharacteristic
13GL(2,ℤ/4)—22no—
14C2×SL(2,ℤ/4)C2211yescharacteristic
15C2×SL(2,ℤ/4)C2211yescharacteristic
16C2×SL(2,ℤ/4)C2211yescharacteristic
17C4×S4—22no—
18#883—41nomaximal
19C2×#123—31no—
20C2×#120—31no—
21C2×C4×D4—31no—
22C22×C22⋊C4D311yescharacteristic, complemented
23#399—31no—
24#397—31no—
25#394—31no—
26#394—31no—
27#388—31no—
28#388—31no—
29#386—31no—
30#386—31no—
31#385—31no—
32#380—31no—
33#379—31no—
34C22×A4—21no—
35C22×A4—21no—
36C22×A4—21no—
37C22×A4C2311yescharacteristic
38C2×S4—21no—
39C2×S4—21no—
40C2×S4D412yes—
41C2×#67—41no—
42C2×C4×D3—41no—
43C4×A4—21no—
44C4×A4—21no—
45SL(2,ℤ/4)—21no—
46SL(2,ℤ/4)—21no—
47SL(2,ℤ/4)—21no—
48C3×C22⋊C4—41no—
49#217—41no—
50(C2×D3)⋊C4—41no—
51C4⋊Dic3—41no—
52Dic3⋊C4—41no—
53C25C2×D311yescharacteristic
54C22×D4—31no—
55C22×D4—31no—
56C22×D4—31no—
57C23×C4—31no—
58C23×C4C2×D311yescharacteristic
59C23×C4C2×D311yescharacteristic
60#123—61no—
61#123—61no—
62#123—61no—
63#123—61no—
64#120—62no—
65#120—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×C22⋊C4—31no—
74C2×C22⋊C4—31no—
75C2×C22⋊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×C22⋊C4—31no—
88C2×C22⋊C4—31no—
89C2×C42—31no—
90#95—31no—
91#95—31no—
92#95—31no—
93#95—31no—
94#95—31no—
95#95—31no—
96C23×C3—41no—
97C22×D3—41no—
98C2×A4—41no—
99C2×A4C2×D411yescharacteristic
100C2×A4#4111yescharacteristic
101C2×A4#4111yescharacteristic
102S4—22no—
103C2×C3×C4—41no—
104C2×C3×C4—41no—
105#67—82no—
106C2×Dic3—41no—
107C2×Dic3—41no—
108C2×Dic3—41no—
109C4×D3—82no—
110C24—21no—
111C24—21no—
112C24—21no—
113C24—31no—
114C24—31no—
115C24—31no—
116C24—61no—
117C24—61no—
118C24—61no—
119C24C22×D311yescharacteristic
120C2×D4—32no—
121C2×D4—32no—
122C2×D4—61no—
123C2×D4—61no—
124C2×D4—61no—
125C2×D4—61no—
126C2×D4—62no—
127C2×D4—62no—
128C2×D4—61no—
129C2×D4—62no—
130C2×D4—61no—
131C2×D4—121no—
132C2×D4—121no—
133C22×C4—21no—
134C22×C4—21no—
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—31no—
145C22×C4—31no—
146C22×C4—61no—
147C22×C4—61no—
148C22×C4—61no—
149C22×C4—62no—
150C22×C4—61no—
151C22×C4—121no—
152C4⋊C4—62no—
153C4⋊C4—61no—
154C4⋊C4—61no—
155C4⋊C4—61no—
156C4⋊C4—61no—
157C4⋊C4—121no—
158C4⋊C4—121no—
159C22⋊C4—31no—
160C22⋊C4—31no—
161C22⋊C4—31no—
162C22⋊C4—61no—
163C22⋊C4—62no—
164C22⋊C4—62no—
165C22⋊C4—61no—
166C22⋊C4—61no—
167C22⋊C4—62no—
168C22⋊C4—62no—
169C22⋊C4—61no—
170C22⋊C4—61no—
171C22⋊C4—61no—
172C22⋊C4—61no—
173C22⋊C4—61no—
174C22⋊C4—61no—
175C22⋊C4—61no—
176C22⋊C4—61no—
177C22⋊C4—121no—
178C22⋊C4—121no—
179C22⋊C4—121no—
180C22⋊C4S411yescharacteristic, complemented
181C42—62no—
182C22×C3—41no—
183C22×C3—81no—
184C22×C3—81no—
185C22×C3—81no—
186C2×D3—42no—
187C2×D3—81no—
188C2×D3—81no—
189A4#12311yescharacteristic, complemented
190C3×C4—81no—
191C3×C4—81no—
192Dic3—81no—
193Dic3—81no—
194Dic3—81no—
195C23—31no—
196C23—31no—
197C23—31no—
198C23—31no—
199C23—32no—
200C23—32no—
201C23—31no—
202C23—41no—
203C23—61no—
204C23—61no—
205C23—61no—
206C23—61no—
207C23—61no—
208C23—61no—
209C23—61no—
210C23—61no—
211C23—61no—
212C23—61no—
213C23—61no—
214C23—61no—
215C23—61no—
216C23—61no—
217C23—61no—
218C23—61no—
219C23—61no—
220C23—121no—
221C23—121no—
222C23—121no—
223C23—122no—
224C23#23511yescharacteristic
225C23D3×D411yescharacteristic
226C23#23711yescharacteristic
227C23C2×S411yescharacteristic
228D4—62no—
229D4—62no—
230D4—62no—
231D4—62no—
232D4—122no—
233D4—124no—
234D4—122no—
235C2×C4—32no—
236C2×C4—31no—
237C2×C4—31no—
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—61no—
256C2×C4—61no—
257C2×C4—61no—
258C2×C4—61no—
259C2×C4—61no—
260C2×C4—61no—
261C2×C4—61no—
262C2×C4—61no—
263C2×C4—61no—
264C2×C4—61no—
265C2×C4—61no—
266C2×C4—122no—
267C2×C4—122no—
268C2×C4C2×S411yescharacteristic
269C2×C4C2×S411yescharacteristic
270C2×C3—41no—
271C2×C3—41no—
272C2×C3—41no—
273C2×C3—161no—
274D3—82no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—31no—
279C22—31no—
280C22—31no—
281C22—31no—
282C22—31no—
283C22—31no—
284C22—31no—
285C22—31no—
286C22—31no—
287C22—61no—
288C22—62no—
289C22—62no—
290C22—62no—
291C22—62no—
292C22—61no—
293C22—62no—
294C22—61no—
295C22—61no—
296C22—121no—
297C22—121no—
298C22—121no—
299C22—121no—
300C22—121no—
301C22—121no—
302C22—121no—
303C22—121no—
304C22—121no—
305C22—121no—
306C22#88311yescharacteristic, complemented
307C22C22×S411yescharacteristic
308C4—21no—
309C4—21no—
310C4—62no—
311C4—61no—
312C4—61no—
313C4—121no—
314C4—121no—
315C4—121no—
316C4—121no—
317C4—121no—
318C4—121no—
319C3—41noSylow
320C2—31no—
321C2—31no—
322C2—31no—
323C2—31no—
324C2—41no—
325C2—121no—
326C2—122no—
327C2#591211yescharacteristic
328C2D4×S411yescharacteristic
329C2#591411yescharacteristic
subgroupG/H
center, FrattiniC22C22×S4
commutator, cosocleC22×A4C23
FittingC22×C22⋊C4D3
socleC24C22×D3
prime radicalC2×GL(2,ℤ/4)C2
hypercenterC22⋊C4S4
hypocenterA4#123
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×S4C2semidirect
3. A4×C22⋊C4C2semidirect
4. #5432C2semidirect
5. (C2×S4)⋊C4C2semidirect
6. C4⋊SL(2,ℤ/4)C2semidirect
7. SL(2,ℤ/4)⋊C4C2semidirect
22. C22×C22⋊C4D3semidirect
180. C22⋊C4S4semidirect
189. A4#123semidirect
306. C22#883semidirect

Maximal subgroups

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

subgroupindex
C2×GL(2,ℤ/4)2
C2×C4×S42
A4×C22⋊C42
#54322
(C2×S4)⋊C42
C4⋊SL(2,ℤ/4)2
SL(2,ℤ/4)⋊C42
#24183
#8834