Finite Group Databasebeta

← #85299

Subgroups — #85299

Order 384.

454 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 42 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, complemented
2C2×GL(2,ℤ/4)C211yesmaximal, characteristic, complemented
3C2×C4⋊S4C211yesmaximal, characteristic, complemented
4C2×C4×S4C211yesmaximal, characteristic, complemented
5A4×C22⋊C4C211yesmaximal, characteristic, complemented
6(C2×S4)⋊C4C211yesmaximal, characteristic, complemented
7SL(2,ℤ/4)⋊C4C211yesmaximal, characteristic, complemented
8#2424—31nomaximal, Sylow
9C23×A4C2211yescharacteristic
10C22×S4C2211yescharacteristic
11C22×S4C2211yescharacteristic
12C2×C4×A4C2211yescharacteristic
13C2×C4×A4C2211yescharacteristic
14GL(2,ℤ/4)—22no—
15GL(2,ℤ/4)—22no—
16C2×SL(2,ℤ/4)C2211yescharacteristic
17C2×SL(2,ℤ/4)C2211yescharacteristic
18C4⋊S4—22no—
19C4×S4—22no—
20#884—41nomaximal
21C2×#121—31no—
22C2×#121—31no—
23C2×#120—31no—
24C2×#120—31no—
25C2×C4×D4—31no—
26C22×C22⋊C4D311yescharacteristic, complemented
27#396—31no—
28#394—31no—
29#394—31no—
30#392—31no—
31#390—31no—
32#388—31no—
33#386—31no—
34#386—31no—
35#385—31no—
36C22×A4—21no—
37C22×A4—21no—
38C22×A4—21no—
39C22×A4C2311yescharacteristic
40C2×S4—21no—
41C2×S4—21no—
42C2×S4—21no—
43C2×S4—21no—
44C2×S4—21no—
45C2×S4D412yescomplemented
46C2×#67—41no—
47C2×#67—41no—
48C2×D12—41no—
49C2×C4×D3—41no—
50C4×A4—21no—
51C4×A4—21no—
52SL(2,ℤ/4)—21no—
53SL(2,ℤ/4)D412yescomplemented
54C3×C22⋊C4—41no—
55(C2×D3)⋊C4—41no—
56Dic3⋊C4—41no—
57C25C2×D311yescharacteristic
58C22×D4—31no—
59C22×D4—31no—
60C22×D4—31no—
61C22×D4—31no—
62C22×D4—31no—
63C22×D4—31no—
64C22×D4—31no—
65C23×C4—31no—
66C23×C4C2×D311yescharacteristic
67C23×C4C2×D311yescharacteristic
68#121—61no—
69#121—61no—
70#121—61no—
71#121—62no—
72#121—62no—
73#121—61no—
74#120—62no—
75#120—62no—
76#120—62no—
77#120—62no—
78C4×D4—62no—
79C4×D4—62no—
80C2×C4⋊C4—31no—
81C2×C4⋊C4—31no—
82C2×C4⋊C4—31no—
83C2×C4⋊C4—31no—
84C2×C22⋊C4—31no—
85C2×C22⋊C4—31no—
86C2×C22⋊C4—31no—
87C2×C22⋊C4—31no—
88C2×C22⋊C4—31no—
89C2×C22⋊C4—31no—
90C2×C22⋊C4—31no—
91C2×C22⋊C4—31no—
92C2×C22⋊C4—31no—
93C2×C22⋊C4—31no—
94C2×C22⋊C4—31no—
95C2×C22⋊C4—31no—
96C2×C22⋊C4—31no—
97C2×C22⋊C4—31no—
98C2×C22⋊C4—31no—
99C2×C42—31no—
100#95—31no—
101#95—31no—
102#95—31no—
103#95—31no—
104C23×C3—41no—
105C22×D3—41no—
106C22×D3—41no—
107C2×A4—41no—
108C2×A4C2×D411yescharacteristic
109C2×A4C2×D411yescharacteristic
110C2×A4#4111yescharacteristic
111S4—22no—
112S4—41no—
113C2×C3×C4—41no—
114C2×C3×C4—41no—
115#67—82no—
116#67—82no—
117C2×Dic3—41no—
118C2×Dic3—41no—
119D12—82no—
120C4×D3—82no—
121C24—21no—
122C24—21no—
123C24—21no—
124C24—31no—
125C24—31no—
126C24—31no—
127C24—31no—
128C24—61no—
129C24—61no—
130C24—61no—
131C24C22×D311yescharacteristic
132C2×D4—32no—
133C2×D4—32no—
134C2×D4—62no—
135C2×D4—62no—
136C2×D4—61no—
137C2×D4—61no—
138C2×D4—61no—
139C2×D4—61no—
140C2×D4—61no—
141C2×D4—62no—
142C2×D4—61no—
143C2×D4—62no—
144C2×D4—62no—
145C2×D4—61no—
146C2×D4—61no—
147C2×D4—61no—
148C2×D4—62no—
149C2×D4—61no—
150C2×D4—62no—
151C2×D4—61no—
152C2×D4—61no—
153C2×D4—61no—
154C2×D4—61no—
155C2×D4—61no—
156C2×D4—61no—
157C2×D4—121no—
158C2×D4—121no—
159C2×D4—121no—
160C2×D4—121no—
161C2×D4—121no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—31no—
165C22×C4—31no—
166C22×C4—31no—
167C22×C4—31no—
168C22×C4—31no—
169C22×C4—31no—
170C22×C4—31no—
171C22×C4—31no—
172C22×C4—31no—
173C22×C4—31no—
174C22×C4—62no—
175C22×C4—61no—
176C22×C4—61no—
177C22×C4—61no—
178C22×C4—61no—
179C22×C4—121no—
180C4⋊C4—62no—
181C4⋊C4—62no—
182C4⋊C4—61no—
183C4⋊C4—61no—
184C4⋊C4—121no—
185C22⋊C4—32no—
186C22⋊C4—32no—
187C22⋊C4—31no—
188C22⋊C4—31no—
189C22⋊C4—31no—
190C22⋊C4—61no—
191C22⋊C4—62no—
192C22⋊C4—62no—
193C22⋊C4—62no—
194C22⋊C4—62no—
195C22⋊C4—62no—
196C22⋊C4—62no—
197C22⋊C4—61no—
198C22⋊C4—61no—
199C22⋊C4—62no—
200C22⋊C4—61no—
201C22⋊C4—61no—
202C22⋊C4—121no—
203C22⋊C4S411yescharacteristic, complemented
204C42—62no—
205C22×C3—41no—
206C22×C3—81no—
207C22×C3—81no—
208C22×C3—81no—
209C2×D3—42no—
210C2×D3—81no—
211C2×D3—81no—
212C2×D3—81no—
213C2×D3—81no—
214C2×D3—81no—
215A4#12111yescharacteristic, complemented
216C3×C4—81no—
217C3×C4—81no—
218Dic3—42no—
219Dic3—81no—
220C23—31no—
221C23—31no—
222C23—31no—
223C23—31no—
224C23—32no—
225C23—32no—
226C23—31no—
227C23—41no—
228C23—61no—
229C23—61no—
230C23—61no—
231C23—61no—
232C23—61no—
233C23—61no—
234C23—61no—
235C23—61no—
236C23—61no—
237C23—61no—
238C23—61no—
239C23—61no—
240C23—61no—
241C23—61no—
242C23—61no—
243C23—61no—
244C23—61no—
245C23—61no—
246C23—61no—
247C23—61no—
248C23—61no—
249C23—61no—
250C23—61no—
251C23—61no—
252C23—121no—
253C23—121no—
254C23—121no—
255C23—122no—
256C23#23511yescharacteristic
257C23D3×D411yescharacteristic
258C23D3×D411yescharacteristic
259C23C2×S411yescharacteristic
260D4—62no—
261D4—62no—
262D4—62no—
263D4—62no—
264D4—122no—
265D4—122no—
266D4—124no—
267D4—122no—
268D4—122no—
269D4—122no—
270D4—121no—
271D4—121no—
272D4—121no—
273D4—124no—
274D4—122no—
275D4—121no—
276C2×C4—32no—
277C2×C4—32no—
278C2×C4—32no—
279C2×C4—31no—
280C2×C4—31no—
281C2×C4—61no—
282C2×C4—61no—
283C2×C4—61no—
284C2×C4—61no—
285C2×C4—61no—
286C2×C4—61no—
287C2×C4—61no—
288C2×C4—61no—
289C2×C4—61no—
290C2×C4—61no—
291C2×C4—61no—
292C2×C4—61no—
293C2×C4—61no—
294C2×C4—61no—
295C2×C4—61no—
296C2×C4—61no—
297C2×C4—61no—
298C2×C4—61no—
299C2×C4—61no—
300C2×C4—61no—
301C2×C4—61no—
302C2×C4—61no—
303C2×C4—61no—
304C2×C4—122no—
305C2×C4—122no—
306C2×C4C2×S411yescharacteristic
307C2×C4C2×S411yescharacteristic
308C2×C3—41no—
309C2×C3—41no—
310C2×C3—41no—
311C2×C3—161no—
312D3—82no—
313D3—161no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—31no—
318C22—31no—
319C22—31no—
320C22—31no—
321C22—31no—
322C22—31no—
323C22—31no—
324C22—31no—
325C22—31no—
326C22—61no—
327C22—62no—
328C22—62no—
329C22—62no—
330C22—62no—
331C22—61no—
332C22—62no—
333C22—61no—
334C22—61no—
335C22—121no—
336C22—121no—
337C22—121no—
338C22—121no—
339C22—121no—
340C22—121no—
341C22—121no—
342C22—121no—
343C22—121no—
344C22—121no—
345C22—121no—
346C22—121no—
347C22—121no—
348C22—121no—
349C22—121no—
350C22—121no—
351C22—121no—
352C22#88411yescharacteristic, complemented
353C22C22×S411yescharacteristic
354C4—21no—
355C4—21no—
356C4—62no—
357C4—62no—
358C4—61no—
359C4—61no—
360C4—62no—
361C4—121no—
362C4—121no—
363C4—121no—
364C3—41noSylow
365C2—31no—
366C2—31no—
367C2—31no—
368C2—31no—
369C2—41no—
370C2—121no—
371C2—122no—
372C2—241no—
373C2#591211yescharacteristic
374C2D4×S411yescharacteristic
375C2D4×S411yescharacteristic
subgroupG/H
center, FrattiniC22C22×S4
commutator, cosocleC22×A4C23
FittingC22×C22⋊C4D3
socleC24C22×D3
hypercenterC22⋊C4S4
hypocenterA4#121
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×GL(2,ℤ/4)C2semidirect
2. C2×GL(2,ℤ/4)C2semidirect
3. C2×C4⋊S4C2semidirect
4. C2×C4×S4C2semidirect
5. A4×C22⋊C4C2semidirect
6. (C2×S4)⋊C4C2semidirect
7. SL(2,ℤ/4)⋊C4C2semidirect
26. C22×C22⋊C4D3semidirect
45. C2×S4D4semidirect
53. SL(2,ℤ/4)D4semidirect
203. C22⋊C4S4semidirect
215. A4#121semidirect
352. C22#884semidirect

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
C2×C4×S42
A4×C22⋊C42
(C2×S4)⋊C42
SL(2,ℤ/4)⋊C42
#24243
#8844