Finite Group Databasebeta

← #11923051

Subgroups — #11923051

Order 1152.

814 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 103 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×C3×Q8×A4C211yesmaximal, characteristic
2C2×#64259C211yesmaximal, characteristic
3#10593645C214yesmaximal
4C4×C3⋊SL(2,ℤ/4)C211yesmaximal, characteristic
5#85502—33nomaximal
6#84065—31nomaximal
7C3×Q8×A4—22no—
8C2×C3×C4×A4C2212yes—
9C2×C3×C4×A4C2211yescharacteristic
10C2×C3⋊SL(2,ℤ/4)C2212yes—
11C2×C3⋊SL(2,ℤ/4)C2212yes—
12#64259—22no—
13#64153—41nomaximal
14C23×C3×Q8D311yescharacteristic
15C2×Q8×A4D313yes—
16C2×A4⋊Q8—33no—
17C2×#944—31no—
18C2×#944—31no—
19C2×#924—31no—
20SL(2,ℤ/4)⋊C4—312no—
21C4×SL(2,ℤ/4)—33no—
22#5230—34no—
23#5229—31no—
24#5229—31no—
25#5210—34no—
26C4×#217—31no—
27C22×C3×A4C2311yescharacteristic
28C2×C32×Q8—41no—
29C2×#634—41no—
30C3×C4×A4—22no—
31C3×C4×A4D412yes—
32C3⋊SL(2,ℤ/4)—22no—
33C3⋊SL(2,ℤ/4)Q814yescomplemented
34#3750—44no—
35C4×C3⋊Dic3—41no—
36#2434—91noSylow
37C22×C3×Q8—22no—
38C22×C3×Q8—31no—
39C22×C3×Q8—31no—
40C22×C3×Q8—32no—
41C22×C3×Q8—62no—
42C23×C3×C4C2×D312yes—
43C23×C3×C4C2×D311yescharacteristic
44C22×Dic6—31no—
45Q8×A4—26no—
46C2×C4×A4C2×D316yes—
47C2×C4×A4C2×D313yes—
48C2×SL(2,ℤ/4)—36no—
49C2×SL(2,ℤ/4)—36no—
50A4⋊Q8—66no—
51C2×#217—32no—
52C2×#217—32no—
53#944—64no—
54#944—62no—
55#944—61no—
56#944—61no—
57#944—123no—
58C2×C4⋊Dic3—31no—
59#924—62no—
60#924—62no—
61C2×Dic3⋊C4—34no—
62C2×Dic3⋊C4—34no—
63C2×Dic3⋊C4—32no—
64C2×C4×Dic3—31no—
65C2×C4×Dic3—31no—
66C3⋊#95—32no—
67C3⋊#95—34no—
68C3⋊#95—34no—
69C2×C3×A4C2×D411yescharacteristic
70C2×C3×A4C2×Q812yes—
71C32×Q8—82no—
72C2×C32×C4—41no—
73C2×C32×C4—42no—
74C2×C3⋊Dic3—42no—
75C2×C3⋊Dic3—42no—
76#634—82no—
77C23×Q8C3⋊D311yescharacteristic
78C2×C4⋊Q8—91no—
79C2×C4⋊Q8—91no—
80C2×C22⋊Q8—91no—
81#395—94no—
82#391—91no—
83#391—91no—
84#385—94no—
85C4×C22⋊C4—91no—
86C24×C3C22×D311yescharacteristic
87C22×A4C22×D313yes—
88C2×C3×Q8—34no—
89C2×C3×Q8—32no—
90C2×C3×Q8—31no—
91C2×C3×Q8—43no—
92C2×C3×Q8—61no—
93C2×C3×Q8—62no—
94C2×C3×Q8—62no—
95C2×C3×Q8—62no—
96C2×C3×Q8—64no—
97C2×C3×Q8—64no—
98C2×C3×Q8S411yescharacteristic
99C22×C3×C4—22no—
100C22×C3×C4—31no—
101C22×C3×C4—31no—
102C22×C3×C4—32no—
103C22×C3×C4—32no—
104C22×C3×C4—32no—
105C22×C3×C4—62no—
106C22×C3×C4#6712yes—
107C22×Dic3—32no—
108C22×Dic3—32no—
109C22×Dic3—32no—
110C22×Dic3—32no—
111C2×Dic6—62no—
112C2×Dic6—61no—
113C2×Dic6—62no—
114C2×Dic6—61no—
115C2×Dic6—123no—
116C4×A4—26no—
117C4×A4#6716yes—
118SL(2,ℤ/4)—312no—
119SL(2,ℤ/4)—66no—
120#217—34no—
121#217—34no—
122#217—62no—
123#217—62no—
124C4⋊Dic3—62no—
125Dic3⋊C4—64no—
126Dic3⋊C4—68no—
127Dic3⋊C4—64no—
128Dic3⋊C4—64no—
129Dic3⋊C4—1212no—
130C4×Dic3—62no—
131C4×Dic3—61no—
132C4×Dic3—61no—
133C4×Dic3—123no—
134C22×C32—41no—
135C3×A4C4⋊Q811yescharacteristic, complemented
136C32×C4—42no—
137C32×C4—82no—
138C3⋊Dic3—44no—
139C3⋊Dic3—82no—
140C22×Q8—22no—
141C22×Q8—31no—
142C22×Q8—31no—
143C22×Q8—32no—
144C22×Q8—62no—
145C22×Q8—91no—
146C23×C4C2×C3⋊D312yes—
147C23×C4C2×C3⋊D311yescharacteristic
148C4⋊Q8—184no—
149C4⋊Q8—182no—
150C4⋊Q8—181no—
151C4⋊Q8—181no—
152C22⋊Q8—182no—
153C22⋊Q8—182no—
154C2×C4⋊C4—91no—
155C2×C4⋊C4—92no—
156C2×C4⋊C4—94no—
157C2×C4⋊C4—94no—
158C2×C22⋊C4—92no—
159C2×C22⋊C4—92no—
160C2×C42—91no—
161C2×C42—91no—
162#95—92no—
163#95—94no—
164#95—94no—
165C23×C3—31no—
166C23×C3—32no—
167C23×C3—31no—
168C23×C3D3×Q812yes—
169C23×C3C2×#6711yescharacteristic
170C2×A4D3×Q816yes—
171C2×A4C2×#6713yes—
172C3×Q8—22no—
173C3×Q8—64no—
174C3×Q8—64no—
175C3×Q8—62no—
176C3×Q8—86no—
177C2×C3×C4—31no—
178C2×C3×C4—32no—
179C2×C3×C4—32no—
180C2×C3×C4—32no—
181C2×C3×C4—34no—
182C2×C3×C4—32no—
183C2×C3×C4—43no—
184C2×C3×C4—46no—
185C2×C3×C4—62no—
186C2×C3×C4—62no—
187C2×C3×C4—61no—
188C2×C3×C4—62no—
189C2×C3×C4C2×S412yes—
190C2×C3×C4C2×S411yescharacteristic
191C2×Dic3—34no—
192C2×Dic3—34no—
193C2×Dic3—62no—
194C2×Dic3—62no—
195C2×Dic3—62no—
196C2×Dic3—62no—
197C2×Dic3—62no—
198C2×Dic3—62no—
199C2×Dic3—62no—
200C2×Dic3—64no—
201C2×Dic3—62no—
202C2×Dic3—126no—
203C2×Dic3—126no—
204Dic6—122no—
205Dic6—122no—
206Dic6—246no—
207C2×C32—42no—
208C2×C32—41no—
209C24C22×C3⋊D311yescharacteristic
210C2×Q8—34no—
211C2×Q8—32no—
212C2×Q8—31no—
213C2×Q8—61no—
214C2×Q8—62no—
215C2×Q8—62no—
216C2×Q8—62no—
217C2×Q8—64no—
218C2×Q8—64no—
219C2×Q8—182no—
220C2×Q8—181no—
221C2×Q8—182no—
222C2×Q8—181no—
223C2×Q8C3⋊S411yescharacteristic
224C22×C4—22no—
225C22×C4—31no—
226C22×C4—31no—
227C22×C4—32no—
228C22×C4—32no—
229C22×C4—32no—
230C22×C4—62no—
231C22×C4—92no—
232C22×C4—92no—
233C22×C4—92no—
234C22×C4—92no—
235C22×C4#63812yes—
236C4⋊C4—184no—
237C4⋊C4—182no—
238C4⋊C4—188no—
239C4⋊C4—184no—
240C4⋊C4—184no—
241C22⋊C4—94no—
242C22⋊C4—94no—
243C22⋊C4—182no—
244C22⋊C4—182no—
245C42—181no—
246C42—182no—
247C42—181no—
248C22×C3—32no—
249C22×C3—31no—
250C22×C3—32no—
251C22×C3—32no—
252C22×C3—31no—
253C22×C3—31no—
254C22×C3—43no—
255C22×C3—61no—
256C22×C3#94411yescharacteristic, complemented
257C22×C3C22×S411yescharacteristic
258A4#94413yescomplemented
259C3×C4—22no—
260C3×C4—32no—
261C3×C4—46no—
262C3×C4—62no—
263C3×C4—86no—
264C3×C4GL(2,ℤ/4)12yes—
265Dic3—64no—
266Dic3—64no—
267Dic3—1212no—
268Dic3—122no—
269Dic3—122no—
270Dic3—246no—
271C32—41noSylow
272C23—31no—
273C23—32no—
274C23—31no—
275C23Q8×C3⋊D312yes—
276C23C2×#63811yescharacteristic
277Q8—22no—
278Q8—64no—
279Q8—64no—
280Q8—62no—
281Q8—362no—
282Q8—362no—
283C2×C4—31no—
284C2×C4—32no—
285C2×C4—32no—
286C2×C4—32no—
287C2×C4—34no—
288C2×C4—32no—
289C2×C4—62no—
290C2×C4—62no—
291C2×C4—61no—
292C2×C4—62no—
293C2×C4—94no—
294C2×C4—94no—
295C2×C4—182no—
296C2×C4—182no—
297C2×C4—182no—
298C2×C4—182no—
299C2×C4—182no—
300C2×C4—182no—
301C2×C4—182no—
302C2×C4—184no—
303C2×C4—182no—
304C2×C4C2×C3⋊S412yes—
305C2×C4C2×C3⋊S411yescharacteristic
306C2×C3—32no—
307C2×C3—31no—
308C2×C3—31no—
309C2×C3—46no—
310C2×C3—43no—
311C2×C3Q8×S412yes—
312C2×C3C2×GL(2,ℤ/4)11yescharacteristic
313C22—32no—
314C22—31no—
315C22—32no—
316C22—32no—
317C22—31no—
318C22—31no—
319C22—61no—
320C22#6415311yescharacteristic, complemented
321C22C22×C3⋊S411yescharacteristic
322C4—22no—
323C4—32no—
324C4—62no—
325C4—184no—
326C4—184no—
327C4—362no—
328C4—362no—
329C4#6426912yes—
330C3—43no—
331C3#8550211yescharacteristic, complemented
332C2—32no—
333C2—31no—
334C2—31no—
335C2Q8×C3⋊S412yes—
336C2C2×#6426911yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, prime radical, cosocleC22×C3×A4C23
FittingC23×C3×Q8D3
socleC24×C3C22×D3
hypercenterC2×Q8C3⋊S4
hypocenterC3×A4C4⋊Q8
LCSG ▷ C22×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
33. C3⋊SL(2,ℤ/4)Q8semidirect
135. C3×A4C4⋊Q8semidirect
256. C22×C3#944semidirect
258. A4#944semidirect
320. C22#64153semidirect
331. C3#85502semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×A42
C2×#642592
#105936452
C4×C3⋊SL(2,ℤ/4)2
#855023
#840653
#641534