Finite Group Databasebeta

← #11921009

Subgroups — #11921009

Order 1152.

534 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 77 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
2C3×A4×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×C3⋊(A4⋊C8)C211yesmaximal, characteristic, complemented
4#73053—33nomaximal
5#71560—31nomaximal
6C22×C3⋊S4—21no—
7C2×C3×C4×A4—21no—
8C2×C3×C4×A4C2211yescharacteristic
9C4⋊(C3⋊S4)—21no—
10C4⋊(C3⋊S4)C412yescomplemented
11C3⋊(A4⋊C8)—21no—
12#63636—41nomaximal
13C2×C4⋊S4—33no—
14C22×C3×C4⋊C4D311yescharacteristic, complemented
15C2×#930—31no—
16A4×C4⋊C4D313yescomplemented
17C2×A4⋊C8—33no—
18C2×#830—31no—
19C2×#809—31no—
20C2×#809—31no—
21#4529—31no—
22#4529—31no—
23C22×C3×A4D411yescharacteristic
24C2×C3⋊S4—22no—
25C2×C3⋊S4—41no—
26C2×C3⋊S4—41no—
27C2×C4⋊(C3⋊D3)—41no—
28C3×C4×A4—41no—
29C3×C4×A4C2×C411yescharacteristic
30C3×C4×A4D411yescharacteristic
31C32×C4⋊C4—41no—
32C2×C3⋊(C3⋊C8)—41no—
33#1874—91noSylow
34C22×S4—63no—
35C23×C3×C4—21no—
36C23×C3×C4C2×D311yescharacteristic
37C22×#67—61no—
38C22×D12—31no—
39C2×C4×A4—23no—
40C2×C4×A4C2×D313yes—
41C4⋊S4—36no—
42C4⋊S4—63no—
43C2×C3×C4⋊C4—31no—
44C2×C3×C4⋊C4—31no—
45C2×C3×C4⋊C4—61no—
46#930—32no—
47#930—32no—
48#930—61no—
49#930—61no—
50C2×(C2×D3)⋊C4—61no—
51C2×C4⋊Dic3—31no—
52C22×C3⋊C8—31no—
53C22×C3⋊C8—31no—
54A4⋊C8—63no—
55#830—61no—
56#830—61no—
57#809—62no—
58#809—61no—
59#809—61no—
60#809—123no—
61C22×C3⋊D3—81no—
62C2×C3×A4C22⋊C411yescharacteristic
63C2×C3×A4D811yescharacteristic
64C2×C3×A4SD1611yescharacteristic
65C3⋊S4—42no—
66C2×C32×C4—41no—
67C2×C32×C4—81no—
68C4⋊(C3⋊D3)—42no—
69C4⋊(C3⋊D3)—81no—
70C3⋊(C3⋊C8)—81no—
71C2×#121—91no—
72C22×C4⋊C4C3⋊D311yescharacteristic, complemented
73C2×#102—91no—
74C2×#102—91no—
75C2×C22⋊C8—91no—
76#340—91no—
77#340—91no—
78C24×C3#6711yescharacteristic
79C23×D3—61no—
80C22×A4#6713yes—
81C2×S4—66no—
82C2×S4—123no—
83C2×S4—123no—
84C22×C3×C4—31no—
85C22×C3×C4—31no—
86C22×C3×C4—31no—
87C22×C3×C4—31no—
88C22×C3×C4—41no—
89C22×C3×C4—61no—
90C22×C3×C4—61no—
91C22×C3×C4—121no—
92C22×C3×C4C4×D311yescharacteristic
93C22×C3×C4D1211yescharacteristic
94C2×#67—62no—
95C2×#67—62no—
96C2×#67—121no—
97C2×#67—121no—
98C2×#67—121no—
99C2×#67—121no—
100C22×Dic3—61no—
101C2×D12—32no—
102C2×D12—61no—
103C2×D12—61no—
104C2×D12—61no—
105C2×D12—61no—
106C2×D12—61no—
107C2×D12—123no—
108C4×A4—43no—
109C4×A4C4×D313yes—
110C4×A4D1213yes—
111C3×C4⋊C4—32no—
112C3×C4⋊C4—31no—
113C3×C4⋊C4—43no—
114C3×C4⋊C4—61no—
115C3×C4⋊C4S411yescharacteristic, complemented
116(C2×D3)⋊C4—62no—
117(C2×D3)⋊C4—62no—
118C4⋊Dic3—32no—
119C4⋊Dic3—61no—
120C2×C3⋊C8—61no—
121C2×C3⋊C8—61no—
122C2×C3⋊C8—61no—
123C2×C3⋊C8—61no—
124C2×C3⋊C8—61no—
125C2×C3⋊C8—61no—
126C2×C3⋊C8—123no—
127C22×C32—41no—
128C2×C3⋊D3—82no—
129C2×C3⋊D3—161no—
130C2×C3⋊D3—161no—
131C3×A4#10211yescharacteristic, complemented
132C32×C4—41no—
133C32×C4—41no—
134C32×C4—161no—
135C22×D4—91no—
136C22×D4—181no—
137C23×C4—21no—
138C23×C4C2×C3⋊D311yescharacteristic
139C22×C8—91no—
140C22×C8—91no—
141#121—92no—
142#121—92no—
143#121—181no—
144#121—181no—
145C2×C4⋊C4—31no—
146C2×C4⋊C4—31no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4—91no—
149C2×C22⋊C4—181no—
150#102—182no—
151#102—181no—
152#102—181no—
153C22⋊C8—181no—
154C22⋊C8—181no—
155C23×C3—31no—
156C23×C3—31no—
157C23×C3—31no—
158C23×C3—31no—
159C23×C3(C2×D3)⋊C411yescharacteristic
160C23×C3#21311yescharacteristic
161C23×C3#21511yescharacteristic
162C22×D3—62no—
163C22×D3—121no—
164C22×D3—121no—
165C22×D3—121no—
166C22×D3—121no—
167C22×D3—121no—
168C22×D3—121no—
169C22×D3—243no—
170C2×A4(C2×D3)⋊C413yes—
171C2×A4#21313yes—
172C2×A4#21513yes—
173S4—126no—
174C2×C3×C4—21no—
175C2×C3×C4—31no—
176C2×C3×C4—31no—
177C2×C3×C4—31no—
178C2×C3×C4—31no—
179C2×C3×C4—31no—
180C2×C3×C4—31no—
181C2×C3×C4—31no—
182C2×C3×C4—43no—
183C2×C3×C4—61no—
184C2×C3×C4—61no—
185C2×C3×C4—83no—
186C2×C3×C4—121no—
187C2×C3×C4—121no—
188C2×C3×C4—121no—
189C2×C3×C4—121no—
190C2×C3×C4C2×S411yescharacteristic
191#67—122no—
192#67—122no—
193#67—122no—
194#67—122no—
195C2×Dic3—62no—
196C2×Dic3—121no—
197C2×Dic3—121no—
198D12—62no—
199D12—62no—
200D12—126no—
201D12—121no—
202D12—121no—
203D12—243no—
204C3⋊C8—121no—
205C3⋊C8—121no—
206C3⋊C8—243no—
207C2×C32—41no—
208C2×C32—41no—
209C2×C32—41no—
210C3⋊D3—162no—
211C24—181no—
212C24#63811yescharacteristic
213C2×D4—92no—
214C2×D4—181no—
215C2×D4—181no—
216C2×D4—181no—
217C2×D4—181no—
218C2×D4—182no—
219C2×D4—182no—
220C2×D4—181no—
221C2×D4—361no—
222C2×D4—361no—
223C2×D4—361no—
224C2×D4—361no—
225C22×C4—31no—
226C22×C4—31no—
227C22×C4—31no—
228C22×C4—31no—
229C22×C4—41no—
230C22×C4—61no—
231C22×C4—61no—
232C22×C4—121no—
233C22×C4—181no—
234C22×C4C4×C3⋊D311yescharacteristic
235C22×C4C4⋊(C3⋊D3)11yescharacteristic
236C2×C8—181no—
237C2×C8—181no—
238C2×C8—181no—
239C2×C8—181no—
240C2×C8—181no—
241C2×C8—181no—
242C4⋊C4—32no—
243C4⋊C4—31no—
244C4⋊C4—61no—
245C4⋊C4—92no—
246C4⋊C4—181no—
247C4⋊C4C3⋊S411yescharacteristic, complemented
248C22⋊C4—182no—
249C22⋊C4—182no—
250C22×C3—31no—
251C22×C3—31no—
252C22×C3—31no—
253C22×C3—31no—
254C22×C3—31no—
255C22×C3—31no—
256C22×C3—31no—
257C22×C3—31no—
258C22×C3—31no—
259C22×C3—43no—
260C22×C3—61no—
261C22×C3#80911yescharacteristic, complemented
262C22×C3GL(2,ℤ/4)11yescharacteristic
263C2×D3—122no—
264C2×D3—122no—
265C2×D3—122no—
266C2×D3—246no—
267C2×D3—241no—
268C2×D3—241no—
269C2×D3—241no—
270C2×D3—241no—
271C2×D3—483no—
272C2×D3—483no—
273A4#80913yescomplemented
274C3×C4—31no—
275C3×C4—31no—
276C3×C4—41no—
277C3×C4—43no—
278C3×C4—43no—
279C3×C4—121no—
280C3×C4—163no—
281C3×C4C4×S411yescharacteristic
282C3×C4C4⋊S411yescharacteristic
283Dic3—122no—
284C32—41noSylow
285C23—31no—
286C23—31no—
287C23—31no—
288C23—31no—
289C23—182no—
290C23—361no—
291C23—361no—
292C23—361no—
293C23—361no—
294C23—361no—
295C23—361no—
296C23#375211yescharacteristic
297C23#375311yescharacteristic
298C23#375511yescharacteristic
299D4—182no—
300D4—182no—
301D4—361no—
302D4—362no—
303D4—362no—
304D4—362no—
305D4—362no—
306D4—361no—
307C2×C4—21no—
308C2×C4—31no—
309C2×C4—31no—
310C2×C4—31no—
311C2×C4—31no—
312C2×C4—31no—
313C2×C4—31no—
314C2×C4—31no—
315C2×C4—61no—
316C2×C4—61no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4—121no—
320C2×C4—121no—
321C2×C4—182no—
322C2×C4—361no—
323C2×C4—361no—
324C2×C4C2×C3⋊S411yescharacteristic
325C8—361no—
326C8—361no—
327C2×C3—31no—
328C2×C3—31no—
329C2×C3—31no—
330C2×C3—31no—
331C2×C3—43no—
332C2×C3—43no—
333C2×C3—43no—
334C2×C3(C2×S4)⋊C411yescharacteristic
335C2×C3#541511yescharacteristic
336C2×C3#541711yescharacteristic
337D3—242no—
338D3—486no—
339C22—31no—
340C22—31no—
341C22—31no—
342C22—31no—
343C22—31no—
344C22—31no—
345C22—31no—
346C22—31no—
347C22—31no—
348C22—61no—
349C22—362no—
350C22—362no—
351C22—362no—
352C22—721no—
353C22—721no—
354C22—721no—
355C22—721no—
356C22#6363611yescharacteristic, complemented
357C22#6426911yescharacteristic
358C4—31no—
359C4—31no—
360C4—41no—
361C4—121no—
362C4—362no—
363C4C4×C3⋊S411yescharacteristic
364C4C4⋊(C3⋊S4)11yescharacteristic
365C3—43no—
366C3#7305311yescharacteristic, complemented
367C2—31no—
368C2—31no—
369C2—31no—
370C2—31no—
371C2—722no—
372C2#1059364711yescharacteristic
373C2#1059364911yescharacteristic
374C2#1059365111yescharacteristic
subgroupG/H
centerC22#64269
commutatorC3×C4×A4C2×C4
FrattiniC2×C4C2×C3⋊S4
FittingC22×C3×C4⋊C4D3
socleC24×C3#67
prime radicalC2×C4⋊(C3⋊S4)C2
cosocleC2×C3×C4×A4C22
hypercenterC4⋊C4C3⋊S4
hypocenterC3×A4#102
LCSG ▷ C3×C4×A4 ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×A4×C4⋊C4C2semidirect
3. C2×C3⋊(A4⋊C8)C2semidirect
10. C4⋊(C3⋊S4)C4semidirect
14. C22×C3×C4⋊C4D3semidirect
16. A4×C4⋊C4D3semidirect
72. C22×C4⋊C4C3⋊D3semidirect
115. C3×C4⋊C4S4semidirect
131. C3×A4#102semidirect
247. C4⋊C4C3⋊S4semidirect
261. C22×C3#809semidirect
273. A4#809semidirect
356. C22#63636semidirect
366. C3#73053semidirect

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
C3×A4×C4⋊C42
C2×C3⋊(A4⋊C8)2
#730533
#715603
#636364