Finite Group Databasebeta

← #11922839

Subgroups — #11922839

Order 1152.

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

normal subgroups only

This group has 854 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×C4⋊(C3⋊S4)C211yesmaximal, characteristic
2C2×#64259C211yesmaximal, characteristic, complemented
3C3×C42×A4C211yesmaximal, characteristic, complemented
4#10593647C214yesmaximal, complemented
5#85290—33nomaximal
6#82077—31nomaximal
7C22×C3⋊S4C2212yes—
8C2×C3×C4×A4C2212yes—
9C2×C3×C4×A4C2211yescharacteristic
10C2×C3⋊SL(2,ℤ/4)C2212yes—
11C4⋊(C3⋊S4)—22no—
12#64259—22no—
13#64084—41nomaximal
14C2×C4⋊S4—33no—
15C2×A4⋊Q8—33no—
16C22×C3×C42D311yescharacteristic, complemented
17C2×#930—31no—
18C2×#924—31no—
19C2×#875—31no—
20C42×A4D313yescomplemented
21(C2×S4)⋊C4—312no—
22#5213—34no—
23#4940—34no—
24#4939—31no—
25#4936—31no—
26#4929—31no—
27C22×C3×A4C2311yescharacteristic
28C2×C3⋊S4—22no—
29C2×C3⋊S4—24no—
30C2×C4⋊(C3⋊D3)—41no—
31C2×#634—41no—
32C3×C4×A4—22no—
33C3×C4×A4D412yes—
34C3⋊SL(2,ℤ/4)—22no—
35C32×C42—41no—
36#3752—44no—
37#2850—91noSylow
38C22×S4—36no—
39C23×C3×C4C2×D312yes—
40C23×C3×C4C2×D311yescharacteristic
41C22×#67—32no—
42C22×D12—31no—
43C22×Dic6—31no—
44C2×C4×A4C2×D316yes—
45C2×C4×A4C2×D313yes—
46C2×SL(2,ℤ/4)—36no—
47C4⋊S4—66no—
48A4⋊Q8—66no—
49C2×C3×C42—31no—
50C2×C3×C42—31no—
51C2×C3×C42—32no—
52C2×#217—32no—
53#930—62no—
54#930—62no—
55C2×(C2×D3)⋊C4—34no—
56C2×(C2×D3)⋊C4—32no—
57C2×(C2×D3)⋊C4—34no—
58C2×C4⋊Dic3—31no—
59C2×C4⋊Dic3—31no—
60#924—62no—
61#924—62no—
62C2×Dic3⋊C4—32no—
63#875—61no—
64#875—62no—
65#875—61no—
66#875—123no—
67C3⋊#95—34no—
68C3⋊#95—34no—
69C22×C3⋊D3—42no—
70C2×C3×A4C2×D411yescharacteristic
71C2×C3×A4#4112yes—
72C3⋊S4—42no—
73C2×C32×C4—41no—
74C2×C32×C4—42no—
75C2×C3⋊Dic3—42no—
76C4⋊(C3⋊D3)—82no—
77#634—82no—
78C2×#124—91no—
79C2×C22⋊Q8—91no—
80C2×#121—91no—
81C22×C42C3⋊D311yescharacteristic, complemented
82#391—91no—
83#390—91no—
84#388—94no—
85#386—94no—
86#382—91no—
87C24×C3C22×D311yescharacteristic
88C23×D3—32no—
89C22×A4C22×D313yes—
90C2×S4—66no—
91C2×S4—612no—
92C22×C3×C4—22no—
93C22×C3×C4—31no—
94C22×C3×C4—32no—
95C22×C3×C4—31no—
96C22×C3×C4—32no—
97C22×C3×C4—32no—
98C22×C3×C4—62no—
99C22×C3×C4D1212yes—
100C2×#67—62no—
101C2×#67—64no—
102C2×#67—62no—
103C2×#67—64no—
104C22×Dic3—32no—
105C22×Dic3—32no—
106C22×Dic3—32no—
107C2×D12—61no—
108C2×D12—62no—
109C2×D12—62no—
110C2×D12—61no—
111C2×D12—123no—
112C2×Dic6—61no—
113C2×Dic6—62no—
114C2×Dic6—62no—
115C2×Dic6—61no—
116C2×Dic6—123no—
117C4×A4—26no—
118C4×A4D1216yes—
119SL(2,ℤ/4)—66no—
120C3×C42—32no—
121C3×C42—31no—
122C3×C42—43no—
123C3×C42—61no—
124C3×C42S411yescharacteristic, complemented
125#217—62no—
126#217—62no—
127(C2×D3)⋊C4—64no—
128(C2×D3)⋊C4—68no—
129(C2×D3)⋊C4—64no—
130(C2×D3)⋊C4—64no—
131(C2×D3)⋊C4—1212no—
132C4⋊Dic3—62no—
133C4⋊Dic3—62no—
134Dic3⋊C4—64no—
135C22×C32—41no—
136C2×C3⋊D3—82no—
137C2×C3⋊D3—84no—
138C3×A4#12411yescharacteristic, complemented
139C32×C4—42no—
140C32×C4—82no—
141C3⋊Dic3—82no—
142C22×Q8—91no—
143C22×D4—91no—
144C22×D4—92no—
145C23×C4C2×C3⋊D312yes—
146C23×C4C2×C3⋊D311yescharacteristic
147#124—181no—
148#124—182no—
149#124—181no—
150C22⋊Q8—182no—
151C22⋊Q8—182no—
152#121—182no—
153#121—182no—
154C2×C4⋊C4—91no—
155C2×C4⋊C4—91no—
156C2×C4⋊C4—92no—
157C2×C22⋊C4—92no—
158C2×C22⋊C4—94no—
159C2×C22⋊C4—94no—
160C2×C22⋊C4—92no—
161C2×C42—31no—
162C2×C42—32no—
163C2×C42—31no—
164#95—94no—
165#95—94no—
166C23×C3—31no—
167C23×C3—31no—
168C23×C3—32no—
169C23×C3C2×D1211yescharacteristic
170C23×C3#23512yes—
171C22×D3—62no—
172C22×D3—64no—
173C22×D3—62no—
174C22×D3—64no—
175C22×D3—62no—
176C22×D3—126no—
177C2×A4C2×D1213yes—
178C2×A4#23516yes—
179S4—126no—
180C2×C3×C4—31no—
181C2×C3×C4—32no—
182C2×C3×C4—32no—
183C2×C3×C4—32no—
184C2×C3×C4—34no—
185C2×C3×C4—32no—
186C2×C3×C4—43no—
187C2×C3×C4—46no—
188C2×C3×C4—62no—
189C2×C3×C4—62no—
190C2×C3×C4—61no—
191C2×C3×C4—62no—
192C2×C3×C4C2×S412yes—
193C2×C3×C4C2×S411yescharacteristic
194#67—122no—
195#67—124no—
196#67—122no—
197C2×Dic3—62no—
198C2×Dic3—62no—
199C2×Dic3—62no—
200C2×Dic3—64no—
201C2×Dic3—62no—
202C2×Dic3—64no—
203C2×Dic3—62no—
204C2×Dic3—126no—
205D12—122no—
206D12—122no—
207D12—246no—
208Dic6—122no—
209Dic6—122no—
210Dic6—246no—
211C2×C32—41no—
212C2×C32—42no—
213C3⋊D3—162no—
214C24—92no—
215C24C22×C3⋊D311yescharacteristic
216C2×Q8—182no—
217C2×Q8—181no—
218C2×Q8—182no—
219C2×Q8—181no—
220C2×D4—182no—
221C2×D4—181no—
222C2×D4—182no—
223C2×D4—182no—
224C2×D4—184no—
225C2×D4—182no—
226C2×D4—184no—
227C2×D4—181no—
228C22×C4—22no—
229C22×C4—31no—
230C22×C4—32no—
231C22×C4—31no—
232C22×C4—32no—
233C22×C4—32no—
234C22×C4—62no—
235C22×C4—92no—
236C22×C4—92no—
237C22×C4—92no—
238C22×C4C4⋊(C3⋊D3)12yes—
239C4⋊C4—182no—
240C4⋊C4—184no—
241C4⋊C4—182no—
242C22⋊C4—184no—
243C22⋊C4—184no—
244C22⋊C4—188no—
245C22⋊C4—182no—
246C22⋊C4—182no—
247C22⋊C4—184no—
248C42—32no—
249C42—31no—
250C42—61no—
251C42C3⋊S411yescharacteristic, complemented
252C22×C3—31no—
253C22×C3—32no—
254C22×C3—31no—
255C22×C3—31no—
256C22×C3—32no—
257C22×C3—32no—
258C22×C3—43no—
259C22×C3—61no—
260C22×C3#87511yescharacteristic, complemented
261C22×C3C22×S411yescharacteristic
262C2×D3—122no—
263C2×D3—122no—
264C2×D3—124no—
265C2×D3—122no—
266C2×D3—124no—
267C2×D3—246no—
268C2×D3—2412no—
269A4#87513yescomplemented
270C3×C4—22no—
271C3×C4—32no—
272C3×C4—46no—
273C3×C4—62no—
274C3×C4—86no—
275C3×C4C4⋊S412yes—
276Dic3—122no—
277Dic3—122no—
278Dic3—122no—
279Dic3—246no—
280C32—41noSylow
281C23—31no—
282C23—31no—
283C23—32no—
284C23—182no—
285C23—184no—
286C23—182no—
287C23—184no—
288C23—182no—
289C23C2×C4⋊(C3⋊D3)11yescharacteristic
290C23#382812yes—
291Q8—362no—
292Q8—362no—
293D4—362no—
294D4—362no—
295D4—364no—
296D4—362no—
297D4—362no—
298C2×C4—31no—
299C2×C4—32no—
300C2×C4—32no—
301C2×C4—32no—
302C2×C4—34no—
303C2×C4—32no—
304C2×C4—62no—
305C2×C4—62no—
306C2×C4—61no—
307C2×C4—62no—
308C2×C4—182no—
309C2×C4—182no—
310C2×C4—182no—
311C2×C4—184no—
312C2×C4—184no—
313C2×C4—182no—
314C2×C4—182no—
315C2×C4C2×C3⋊S412yes—
316C2×C4C2×C3⋊S411yescharacteristic
317C2×C3—31no—
318C2×C3—31no—
319C2×C3—32no—
320C2×C3—43no—
321C2×C3—46no—
322C2×C3C2×C4⋊S411yescharacteristic
323C2×C3#591212yes—
324D3—242no—
325D3—486no—
326C22—31no—
327C22—32no—
328C22—31no—
329C22—31no—
330C22—32no—
331C22—32no—
332C22—61no—
333C22—362no—
334C22—362no—
335C22—364no—
336C22—362no—
337C22—364no—
338C22#6408411yescharacteristic, complemented
339C22C22×C3⋊S411yescharacteristic
340C4—22no—
341C4—32no—
342C4—62no—
343C4—362no—
344C4—362no—
345C4—362no—
346C4C4⋊(C3⋊S4)12yes—
347C3—43no—
348C3#8529011yescharacteristic, complemented
349C2—31no—
350C2—31no—
351C2—32no—
352C2—722no—
353C2C2×C4⋊(C3⋊S4)11yescharacteristic
354C2#1059657512yes—
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC22×C3×A4C23
FittingC22×C3×C42D3
socleC24×C3C22×D3
prime radicalC2×C4⋊(C3⋊S4)C2
hypercenterC42C3⋊S4
hypocenterC3×A4#124
LCSG ▷ C22×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C2×#64259C2semidirect
3. C3×C42×A4C2semidirect
4. #10593647C2semidirect
16. C22×C3×C42D3semidirect
20. C42×A4D3semidirect
81. C22×C42C3⋊D3semidirect
124. C3×C42S4semidirect
138. C3×A4#124semidirect
251. C42C3⋊S4semidirect
260. C22×C3#875semidirect
269. A4#875semidirect
338. C22#64084semidirect
348. C3#85290semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(C3⋊S4)2
C2×#642592
C3×C42×A42
#105936472
#852903
#820773
#640844