Finite Group Databasebeta

← GL(2,ℤ/4)⋊C4 #85296

Subgroups — GL(2,ℤ/4)⋊C4 #85296

Order 384.

457 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 457 conjugacy classes of subgroups — too many to draw. Only its 54 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
2C22×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
3C2×C4×S4C211yesmaximal, characteristic, complemented
4A4×C22⋊C4C211yesmaximal, characteristic, complemented
5(C2×S4)⋊C4C211yesmaximal, characteristic, complemented
6SL(2,ℤ/4)⋊C4C211yesmaximal, characteristic, complemented
7C4×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
8#2358—31nomaximal, Sylow
9C23×A4C2211yescharacteristic
10C22×S4C2211yescharacteristic
11C2×C4×A4C2211yescharacteristic
12C2×C4×A4C2211yescharacteristic
13GL(2,ℤ/4)C414yescomplemented
14C2×SL(2,ℤ/4)—22no—
15C2×SL(2,ℤ/4)C2211yescharacteristic
16C2×SL(2,ℤ/4)C2211yescharacteristic
17C2×SL(2,ℤ/4)C2211yescharacteristic
18C4×S4—22no—
19#67⋊C4—41nomaximal
20C2×#120—31no—
21C2×C4×D4—31no—
22C2×C4×D4—31no—
23C22×C22⋊C4—31no—
24C22×C22⋊C4D311yescharacteristic, complemented
25#390—31no—
26#388—31no—
27#388—31no—
28#386—31no—
29#386—31no—
30#386—31no—
31#385—31no—
32#385—31no—
33C4×C22⋊C4—31no—
34C4×C22⋊C4—31no—
35C22×A4—21no—
36C22×A4—21no—
37C22×A4C2×C412yescomplemented
38C22×A4C2311yescharacteristic
39C2×S4—21no—
40C2×S4—21no—
41C2×S4C2×C412yescomplemented
42C2×#67—41no—
43C22×Dic3—41no—
44C2×C4×D3—41no—
45C4×A4—21no—
46C4×A4—21no—
47SL(2,ℤ/4)—21no—
48SL(2,ℤ/4)C2×C412yescomplemented
49SL(2,ℤ/4)D412yescomplemented
50C3×C22⋊C4—41no—
51(C2×D3)⋊C4—41no—
52Dic3⋊C4—41no—
53C4×Dic3—41no—
54C25C2×D311yescharacteristic
55C22×D4—31no—
56C22×D4—31no—
57C22×D4—31no—
58C23×C4—31no—
59C23×C4—31no—
60C23×C4—31no—
61C23×C4C2×D311yescharacteristic
62C23×C4C2×D311yescharacteristic
63#120—34no—
64#120—34no—
65C4×D4—62no—
66C4×D4—62no—
67C4×D4—61no—
68C4×D4—61no—
69C4×D4—61no—
70C4×D4—61no—
71C2×C4⋊C4—31no—
72C2×C4⋊C4—31no—
73C2×C4⋊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—62no—
89C2×C22⋊C4—62no—
90C2×C42—31no—
91C2×C42—31no—
92C2×C42—31no—
93#95—31no—
94#95—31no—
95#95—31no—
96#95—31no—
97#95—31no—
98#95—31no—
99C23×C3—41no—
100C22×D3—41no—
101C2×A4—22no—
102C2×A4C22×C411yescharacteristic
103C2×A4C2×D411yescharacteristic
104C2×A4#4111yescharacteristic
105S4—22no—
106C2×C3×C4—41no—
107C2×C3×C4—41no—
108#67—44no—
109C2×Dic3—41no—
110C2×Dic3—41no—
111C2×Dic3—41no—
112C2×Dic3—82no—
113C4×D3—82no—
114C24—21no—
115C24—21no—
116C24—31no—
117C24—31no—
118C24—32no—
119C24—31no—
120C24—61no—
121C24—61no—
122C24C4×D312yescomplemented
123C24C22×D311yescharacteristic
124C2×D4—32no—
125C2×D4—34no—
126C2×D4—32no—
127C2×D4—34no—
128C2×D4—61no—
129C2×D4—61no—
130C2×D4—61no—
131C2×D4—62no—
132C2×D4—61no—
133C2×D4—61no—
134C2×D4—61no—
135C2×D4—121no—
136C2×D4—121no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—31no—
140C22×C4—31no—
141C22×C4—31no—
142C22×C4—31no—
143C22×C4—31no—
144C22×C4—31no—
145C22×C4—31no—
146C22×C4—31no—
147C22×C4—31no—
148C22×C4—31no—
149C22×C4—31no—
150C22×C4—62no—
151C22×C4—62no—
152C22×C4—62no—
153C22×C4—61no—
154C22×C4—61no—
155C22×C4—61no—
156C22×C4—61no—
157C22×C4—62no—
158C22×C4—61no—
159C22×C4—61no—
160C22×C4—121no—
161C22×C4—121no—
162C22×C4—121no—
163C4⋊C4—62no—
164C4⋊C4—61no—
165C4⋊C4—61no—
166C4⋊C4—121no—
167C22⋊C4—32no—
168C22⋊C4—32no—
169C22⋊C4—34no—
170C22⋊C4—34no—
171C22⋊C4—32no—
172C22⋊C4—32no—
173C22⋊C4—31no—
174C22⋊C4—31no—
175C22⋊C4—31no—
176C22⋊C4—61no—
177C22⋊C4—62no—
178C22⋊C4—62no—
179C22⋊C4—62no—
180C22⋊C4—61no—
181C22⋊C4—61no—
182C22⋊C4—61no—
183C22⋊C4—61no—
184C22⋊C4—121no—
185C22⋊C4S411yescharacteristic, complemented
186C42—62no—
187C42—61no—
188C42—61no—
189C42—121no—
190C22×C3—42no—
191C22×C3—41no—
192C22×C3—81no—
193C22×C3—81no—
194C2×D3—42no—
195C2×D3—81no—
196C2×D3—81no—
197A4C4×D411yescharacteristic, complemented
198C3×C4—81no—
199C3×C4—81no—
200Dic3—42no—
201Dic3—42no—
202Dic3—81no—
203C23—22no—
204C23—31no—
205C23—31no—
206C23—31no—
207C23—31no—
208C23—32no—
209C23—32no—
210C23—32no—
211C23—31no—
212C23—32no—
213C23—61no—
214C23—61no—
215C23—61no—
216C23—61no—
217C23—61no—
218C23—62no—
219C23—61no—
220C23—61no—
221C23—61no—
222C23—62no—
223C23—62no—
224C23—61no—
225C23—61no—
226C23—61no—
227C23—61no—
228C23—61no—
229C23—61no—
230C23—61no—
231C23—122no—
232C23C2×C4×D311yescharacteristic
233C23D3×D411yescharacteristic
234C23#23711yescharacteristic
235C23C2×S411yescharacteristic
236D4—62no—
237D4—62no—
238D4—62no—
239D4—64no—
240D4—62no—
241D4—64no—
242D4—124no—
243C2×C4—32no—
244C2×C4—32no—
245C2×C4—32no—
246C2×C4—32no—
247C2×C4—32no—
248C2×C4—31no—
249C2×C4—31no—
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—61no—
267C2×C4—61no—
268C2×C4—61no—
269C2×C4—61no—
270C2×C4—61no—
271C2×C4—61no—
272C2×C4—61no—
273C2×C4—61no—
274C2×C4—122no—
275C2×C4—122no—
276C2×C4—122no—
277C2×C4—124no—
278C2×C4—122no—
279C2×C4C2×S411yescharacteristic
280C2×C4C2×S411yescharacteristic
281C2×C3—41no—
282C2×C3—41no—
283C2×C3—41no—
284C2×C3—82no—
285D3—82no—
286C22—21no—
287C22—21no—
288C22—31no—
289C22—31no—
290C22—31no—
291C22—31no—
292C22—31no—
293C22—31no—
294C22—31no—
295C22—31no—
296C22—31no—
297C22—32no—
298C22—62no—
299C22—62no—
300C22—62no—
301C22—62no—
302C22—61no—
303C22—62no—
304C22—62no—
305C22—62no—
306C22—62no—
307C22—61no—
308C22—61no—
309C22—62no—
310C22—121no—
311C22—121no—
312C22—121no—
313C22—121no—
314C22—121no—
315C22—121no—
316C22#67⋊C411yescharacteristic, complemented
317C22C4×S412yescomplemented
318C22C22×S411yescharacteristic
319C4—21no—
320C4—21no—
321C4—62no—
322C4—62no—
323C4—62no—
324C4—62no—
325C4—61no—
326C4—62no—
327C4—61no—
328C4—121no—
329C4—121no—
330C3—41noSylow
331C2—22no—
332C2—31no—
333C2—31no—
334C2—31no—
335C2—31no—
336C2—62no—
337C2—122no—
338C2C2×C4×S411yescharacteristic
339C2D4×S411yescharacteristic
340C2#591411yescharacteristic
subgroupG/H
center, FrattiniC22C22×S4
commutatorC2×A4C22×C4
FittingC22×C22⋊C4D3
socleC24C22×D3
prime radicalC2×GL(2,ℤ/4)C2
cosocleC22×A4C23
hypercenterC22⋊C4S4
hypocenterA4C4×D4
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×SL(2,ℤ/4)C2semidirect
3. C2×C4×S4C2semidirect
4. A4×C22⋊C4C2semidirect
5. (C2×S4)⋊C4C2semidirect
6. SL(2,ℤ/4)⋊C4C2semidirect
7. C4×SL(2,ℤ/4)C2semidirect
13. GL(2,ℤ/4)C4semidirect
24. C22×C22⋊C4D3semidirect
37. C22×A4C2×C4semidirect
41. C2×S4C2×C4semidirect
48. SL(2,ℤ/4)C2×C4semidirect
49. SL(2,ℤ/4)D4semidirect
122. C24C4×D3semidirect
185. C22⋊C4S4semidirect
197. A4C4×D4semidirect
316. C22#67⋊C4semidirect
317. C22C4×S4semidirect

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
C22×SL(2,ℤ/4)2
C2×C4×S42
A4×C22⋊C42
(C2×S4)⋊C42
SL(2,ℤ/4)⋊C42
C4×SL(2,ℤ/4)2
#23583
#67⋊C44