Finite Group Databasebeta

← #11923057

Subgroups — #11923057

Order 1152.

898 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 898 conjugacy classes of subgroups — too many to draw. Only its 99 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, complemented
2C2×C4⋊(C3⋊S4)C211yesmaximal, characteristic
3#10593647C214yesmaximal, complemented
4C4×C3⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
5#85508—33nomaximal
6#84066—31nomaximal
7C22×C3⋊S4C2212yes—
8C3×Q8×A4—22no—
9C2×C3×C4×A4C2212yes—
10C2×C3×C4×A4C2211yescharacteristic
11C2×C3⋊SL(2,ℤ/4)C2212yes—
12C4⋊(C3⋊S4)—22no—
13#64156—41nomaximal
14C23×C3×Q8D311yescharacteristic, complemented
15C2×Q8×A4D313yescomplemented
16C2×C4⋊S4—33no—
17C2×#947—31no—
18C2×#947—31no—
19C2×#930—31no—
20(C2×S4)⋊C4—312no—
21C4×SL(2,ℤ/4)—33no—
22#5231—34no—
23#5229—31no—
24#5229—31no—
25#5213—34no—
26C4×#217—31no—
27C22×C3×A4C2311yescharacteristic
28C2×C3⋊S4—22no—
29C2×C3⋊S4—24no—
30C2×C32×Q8—41no—
31C2×C4⋊(C3⋊D3)—41no—
32C3×C4×A4—22no—
33C3×C4×A4D412yes—
34C3⋊SL(2,ℤ/4)—22no—
35#3752—44no—
36C4×C3⋊Dic3—41no—
37#2436—91noSylow
38C22×S4—36no—
39C22×C3×Q8—22no—
40C22×C3×Q8—31no—
41C22×C3×Q8—31no—
42C22×C3×Q8—32no—
43C22×C3×Q8—62no—
44C23×C3×C4C2×D312yes—
45C23×C3×C4C2×D311yescharacteristic
46C22×#67—32no—
47C22×D12—31no—
48Q8×A4—26no—
49C2×C4×A4C2×D316yes—
50C2×C4×A4C2×D313yes—
51C2×SL(2,ℤ/4)—36no—
52C4⋊S4—66no—
53C2×#217—32no—
54#947—64no—
55#947—62no—
56#947—61no—
57#947—61no—
58#947—123no—
59#930—62no—
60#930—62no—
61C2×(C2×D3)⋊C4—34no—
62C2×(C2×D3)⋊C4—34no—
63C2×(C2×D3)⋊C4—32no—
64C2×C4⋊Dic3—31no—
65C2×C4×Dic3—31no—
66C2×C4×Dic3—31no—
67C3⋊#95—32no—
68C3⋊#95—34no—
69C3⋊#95—34no—
70C22×C3⋊D3—42no—
71C2×C3×A4C2×D411yescharacteristic
72C2×C3×A4#4112yes—
73C3⋊S4—42no—
74C32×Q8—82no—
75C2×C32×C4—41no—
76C2×C32×C4—42no—
77C2×C3⋊Dic3—42no—
78C4⋊(C3⋊D3)—82no—
79C23×Q8C3⋊D311yescharacteristic, complemented
80C2×#124—91no—
81C2×#124—91no—
82C2×#121—91no—
83#393—94no—
84#391—91no—
85#391—91no—
86#386—94no—
87C4×C22⋊C4—91no—
88C24×C3C22×D311yescharacteristic
89C23×D3—32no—
90C22×A4C22×D313yes—
91C2×S4—612no—
92C2×S4—66no—
93C2×C3×Q8—34no—
94C2×C3×Q8—32no—
95C2×C3×Q8—31no—
96C2×C3×Q8—43no—
97C2×C3×Q8—61no—
98C2×C3×Q8—62no—
99C2×C3×Q8—62no—
100C2×C3×Q8—62no—
101C2×C3×Q8—64no—
102C2×C3×Q8—64no—
103C2×C3×Q8S411yescharacteristic, complemented
104C22×C3×C4—22no—
105C22×C3×C4—31no—
106C22×C3×C4—31no—
107C22×C3×C4—32no—
108C22×C3×C4—32no—
109C22×C3×C4—32no—
110C22×C3×C4—62no—
111C22×C3×C4#6712yes—
112C2×#67—62no—
113C2×#67—64no—
114C2×#67—64no—
115C2×#67—62no—
116C22×Dic3—32no—
117C22×Dic3—32no—
118C22×Dic3—32no—
119C2×D12—62no—
120C2×D12—61no—
121C2×D12—62no—
122C2×D12—61no—
123C2×D12—123no—
124C4×A4—26no—
125C4×A4#6716yes—
126SL(2,ℤ/4)—66no—
127#217—62no—
128#217—62no—
129(C2×D3)⋊C4—64no—
130(C2×D3)⋊C4—68no—
131(C2×D3)⋊C4—64no—
132(C2×D3)⋊C4—64no—
133(C2×D3)⋊C4—1212no—
134C4⋊Dic3—62no—
135C4×Dic3—62no—
136C4×Dic3—61no—
137C4×Dic3—61no—
138C4×Dic3—123no—
139C22×C32—41no—
140C2×C3⋊D3—82no—
141C2×C3⋊D3—84no—
142C3×A4#12411yescharacteristic, complemented
143C32×C4—42no—
144C32×C4—82no—
145C3⋊Dic3—82no—
146C22×Q8—22no—
147C22×Q8—31no—
148C22×Q8—31no—
149C22×Q8—32no—
150C22×Q8—62no—
151C22×D4—91no—
152C22×D4—92no—
153C23×C4C2×C3⋊D312yes—
154C23×C4C2×C3⋊D311yescharacteristic
155#124—184no—
156#124—182no—
157#124—181no—
158#124—181no—
159#121—182no—
160#121—182no—
161C2×C4⋊C4—91no—
162C2×C22⋊C4—92no—
163C2×C22⋊C4—94no—
164C2×C22⋊C4—94no—
165C2×C22⋊C4—92no—
166C2×C42—91no—
167C2×C42—91no—
168#95—92no—
169#95—94no—
170#95—94no—
171C23×C3—31no—
172C23×C3—32no—
173C23×C3—31no—
174C23×C3#23912yes—
175C23×C3C2×#6711yescharacteristic
176C22×D3—62no—
177C22×D3—62no—
178C22×D3—64no—
179C22×D3—64no—
180C22×D3—62no—
181C22×D3—126no—
182C2×A4#23916yes—
183C2×A4C2×#6713yes—
184S4—126no—
185C3×Q8—22no—
186C3×Q8—64no—
187C3×Q8—64no—
188C3×Q8—62no—
189C3×Q8—86no—
190C2×C3×C4—31no—
191C2×C3×C4—32no—
192C2×C3×C4—32no—
193C2×C3×C4—32no—
194C2×C3×C4—34no—
195C2×C3×C4—32no—
196C2×C3×C4—43no—
197C2×C3×C4—46no—
198C2×C3×C4—62no—
199C2×C3×C4—62no—
200C2×C3×C4—61no—
201C2×C3×C4—62no—
202C2×C3×C4C2×S412yes—
203C2×C3×C4C2×S411yescharacteristic
204#67—124no—
205#67—122no—
206#67—122no—
207C2×Dic3—62no—
208C2×Dic3—62no—
209C2×Dic3—62no—
210C2×Dic3—62no—
211C2×Dic3—62no—
212C2×Dic3—62no—
213C2×Dic3—64no—
214C2×Dic3—62no—
215C2×Dic3—126no—
216D12—122no—
217D12—122no—
218D12—246no—
219C2×C32—42no—
220C2×C32—41no—
221C3⋊D3—162no—
222C24—92no—
223C24C22×C3⋊D311yescharacteristic
224C2×Q8—34no—
225C2×Q8—32no—
226C2×Q8—31no—
227C2×Q8—61no—
228C2×Q8—62no—
229C2×Q8—62no—
230C2×Q8—62no—
231C2×Q8—64no—
232C2×Q8—64no—
233C2×Q8C3⋊S411yescharacteristic, complemented
234C2×D4—182no—
235C2×D4—181no—
236C2×D4—182no—
237C2×D4—182no—
238C2×D4—184no—
239C2×D4—184no—
240C2×D4—182no—
241C2×D4—181no—
242C22×C4—22no—
243C22×C4—31no—
244C22×C4—31no—
245C22×C4—32no—
246C22×C4—32no—
247C22×C4—32no—
248C22×C4—62no—
249C22×C4—92no—
250C22×C4—92no—
251C22×C4—92no—
252C22×C4#63812yes—
253C4⋊C4—182no—
254C22⋊C4—184no—
255C22⋊C4—188no—
256C22⋊C4—184no—
257C22⋊C4—182no—
258C22⋊C4—182no—
259C22⋊C4—184no—
260C42—181no—
261C42—182no—
262C42—181no—
263C22×C3—32no—
264C22×C3—31no—
265C22×C3—32no—
266C22×C3—32no—
267C22×C3—31no—
268C22×C3—31no—
269C22×C3—43no—
270C22×C3—61no—
271C22×C3#94711yescharacteristic, complemented
272C22×C3C22×S411yescharacteristic
273C2×D3—122no—
274C2×D3—124no—
275C2×D3—122no—
276C2×D3—124no—
277C2×D3—122no—
278C2×D3—246no—
279C2×D3—2412no—
280A4#94713yescomplemented
281C3×C4—22no—
282C3×C4—32no—
283C3×C4—46no—
284C3×C4—62no—
285C3×C4—86no—
286C3×C4GL(2,ℤ/4)12yes—
287Dic3—122no—
288Dic3—122no—
289Dic3—122no—
290Dic3—246no—
291C32—41noSylow
292C23—31no—
293C23—32no—
294C23—31no—
295C23—182no—
296C23—182no—
297C23—184no—
298C23—184no—
299C23—182no—
300C23#383212yes—
301C23C2×#63811yescharacteristic
302Q8—22no—
303Q8—64no—
304Q8—64no—
305Q8—62no—
306D4—362no—
307D4—364no—
308D4—362no—
309D4—362no—
310D4—362no—
311C2×C4—31no—
312C2×C4—32no—
313C2×C4—32no—
314C2×C4—32no—
315C2×C4—34no—
316C2×C4—32no—
317C2×C4—62no—
318C2×C4—62no—
319C2×C4—61no—
320C2×C4—62no—
321C2×C4—182no—
322C2×C4—182no—
323C2×C4—182no—
324C2×C4—182no—
325C2×C4—184no—
326C2×C4—182no—
327C2×C4—182no—
328C2×C4—182no—
329C2×C4C2×C3⋊S412yes—
330C2×C4C2×C3⋊S411yescharacteristic
331C2×C3—31no—
332C2×C3—32no—
333C2×C3—31no—
334C2×C3—46no—
335C2×C3—43no—
336C2×C3#591912yes—
337C2×C3C2×GL(2,ℤ/4)11yescharacteristic
338D3—242no—
339D3—486no—
340C22—32no—
341C22—31no—
342C22—32no—
343C22—32no—
344C22—31no—
345C22—31no—
346C22—61no—
347C22—362no—
348C22—364no—
349C22—362no—
350C22—364no—
351C22—362no—
352C22#6415611yescharacteristic, complemented
353C22C22×C3⋊S411yescharacteristic
354C4—22no—
355C4—32no—
356C4—62no—
357C4—362no—
358C4—362no—
359C4—362no—
360C4#6426912yes—
361C3—43no—
362C3#8550811yescharacteristic, complemented
363C2—31no—
364C2—32no—
365C2—31no—
366C2—722no—
367C2#1059658212yes—
368C2C2×#6426911yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC22×C3×A4C23
FittingC23×C3×Q8D3
socleC24×C3C22×D3
prime radicalC2×C4⋊(C3⋊S4)C2
hypercenterC2×Q8C3⋊S4
hypocenterC3×A4#124
LCSG ▷ C22×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×Q8×A4C2semidirect
3. #10593647C2semidirect
4. C4×C3⋊SL(2,ℤ/4)C2semidirect
14. C23×C3×Q8D3semidirect
15. C2×Q8×A4D3semidirect
79. C23×Q8C3⋊D3semidirect
103. C2×C3×Q8S4semidirect
142. C3×A4#124semidirect
233. C2×Q8C3⋊S4semidirect
271. C22×C3#947semidirect
280. A4#947semidirect
352. C22#64156semidirect
362. C3#85508semidirect

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×C4⋊(C3⋊S4)2
#105936472
C4×C3⋊SL(2,ℤ/4)2
#855083
#840663
#641564