Finite Group Databasebeta

← #11921097

Subgroups — #11921097

Order 1152.

634 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C22×C3×C4×A4C211yesmaximal, characteristic
2C2×C3⋊(A4⋊C8)C212yesmaximal, complemented
3#73141—33nomaximal
4#72126—31nomaximal
5C23×C3×A4C411yescharacteristic
6C2×C3×C4×A4—22no—
7C2×C3×C4×A4C411yescharacteristic
8C2×C3×C4×A4C2211yescharacteristic
9C3⋊(A4⋊C8)—22no—
10#63657—41nomaximal
11C24×C3×C4D311yescharacteristic
12C22×C4×A4D313yes—
13C2×A4⋊C8—36no—
14C2×#830—32no—
15C2×#830—31no—
16C2×#830—31no—
17C2×#830—32no—
18C22×C3×A4—21no—
19C22×C3×A4—21no—
20C22×C3×A4C812yescomplemented
21C22×C3×A4C2×C411yescharacteristic
22C22×C32×C4—41no—
23C3×C4×A4—21no—
24C3×C4×A4D412yes—
25C2×C3⋊(C3⋊C8)—42no—
26#1779—91noSylow
27C25×C3Dic311yescharacteristic
28C23×A4Dic313yes—
29C23×C3×C4—22no—
30C23×C3×C4—31no—
31C23×C3×C4—31no—
32C23×C3×C4—31no—
33C23×C3×C4—31no—
34C23×C3×C4—62no—
35C23×C3×C4Dic311yescharacteristic
36C23×C3×C4C2×D311yescharacteristic
37C2×C4×A4—26no—
38C2×C4×A4Dic313yes—
39C2×C4×A4C2×D313yes—
40C22×C3⋊C8—32no—
41C22×C3⋊C8—32no—
42A4⋊C8—66no—
43#830—62no—
44#830—64no—
45#830—61no—
46#830—61no—
47#830—61no—
48#830—62no—
49#830—61no—
50#830—123no—
51C23×C32—41no—
52C2×C3×A4—22no—
53C2×C3×A4C22⋊C411yescharacteristic
54C2×C3×A4C2×C811yescharacteristic
55C2×C3×A4M1611yescharacteristic
56C2×C32×C4—41no—
57C2×C32×C4—41no—
58C2×C32×C4—82no—
59C3⋊(C3⋊C8)—82no—
60C24×C4C3⋊D311yescharacteristic
61C2×C22⋊C8—92no—
62C2×C22⋊C8—91no—
63C2×C22⋊C8—91no—
64C2×C22⋊C8—92no—
65C24×C3—21no—
66C24×C3—21no—
67C24×C3—31no—
68C24×C3—31no—
69C24×C3—32no—
70C24×C3—61no—
71C24×C3—61no—
72C24×C3C3⋊C812yescomplemented
73C24×C3C2×Dic311yescharacteristic
74C22×A4—23no—
75C22×A4—23no—
76C22×A4C3⋊C816yescomplemented
77C22×A4C2×Dic313yes—
78C22×C3×C4—21no—
79C22×C3×C4—31no—
80C22×C3×C4—32no—
81C22×C3×C4—31no—
82C22×C3×C4—31no—
83C22×C3×C4—34no—
84C22×C3×C4—31no—
85C22×C3×C4—31no—
86C22×C3×C4—31no—
87C22×C3×C4—31no—
88C22×C3×C4—43no—
89C22×C3×C4—62no—
90C22×C3×C4—62no—
91C22×C3×C4—61no—
92C22×C3×C4—61no—
93C22×C3×C4—62no—
94C22×C3×C4—62no—
95C22×C3×C4—62no—
96C22×C3×C4—62no—
97C22×C3×C4—62no—
98C22×C3×C4#6712yes—
99C22×C3×C4S411yescharacteristic
100C4×A4—23no—
101C4×A4#6716yes—
102C2×C3⋊C8—62no—
103C2×C3⋊C8—62no—
104C2×C3⋊C8—62no—
105C2×C3⋊C8—62no—
106C2×C3⋊C8—62no—
107C2×C3⋊C8—62no—
108C2×C3⋊C8—126no—
109C22×C32—42no—
110C22×C32—41no—
111C22×C32—81no—
112C22×C32—81no—
113C3×A4C22⋊C811yescharacteristic, complemented
114C32×C4—42no—
115C32×C4—81no—
116C25C3⋊Dic311yescharacteristic
117C23×C4—22no—
118C23×C4—31no—
119C23×C4—31no—
120C23×C4—31no—
121C23×C4—31no—
122C23×C4—62no—
123C23×C4C3⋊Dic311yescharacteristic
124C23×C4C2×C3⋊D311yescharacteristic
125C22×C8—92no—
126C22×C8—92no—
127C22⋊C8—182no—
128C22⋊C8—184no—
129C22⋊C8—181no—
130C22⋊C8—181no—
131C22⋊C8—181no—
132C22⋊C8—182no—
133C22⋊C8—181no—
134C23×C3—22no—
135C23×C3—31no—
136C23×C3—31no—
137C23×C3—32no—
138C23×C3—31no—
139C23×C3—31no—
140C23×C3—32no—
141C23×C3—32no—
142C23×C3—32no—
143C23×C3—32no—
144C23×C3—31no—
145C23×C3—43no—
146C23×C3—61no—
147C23×C3—61no—
148C23×C3—61no—
149C23×C3—61no—
150C23×C3—62no—
151C23×C3—61no—
152C23×C3—61no—
153C23×C3—62no—
154C23×C3—62no—
155C23×C3—61no—
156C23×C3—64no—
157C23×C3C2×C3⋊C811yescharacteristic
158C23×C3#20811yescharacteristic
159C23×C3#21711yescharacteristic
160C23×C3SL(2,ℤ/4)11yescharacteristic
161C2×A4—26no—
162C2×A4C2×C3⋊C813yes—
163C2×A4#20813yes—
164C2×A4#21713yes—
165C2×C3×C4—22no—
166C2×C3×C4—31no—
167C2×C3×C4—32no—
168C2×C3×C4—32no—
169C2×C3×C4—32no—
170C2×C3×C4—32no—
171C2×C3×C4—31no—
172C2×C3×C4—43no—
173C2×C3×C4—43no—
174C2×C3×C4—62no—
175C2×C3×C4—61no—
176C2×C3×C4—61no—
177C2×C3×C4—61no—
178C2×C3×C4—62no—
179C2×C3×C4—64no—
180C2×C3×C4—62no—
181C2×C3×C4—61no—
182C2×C3×C4—86no—
183C2×C3×C4SL(2,ℤ/4)11yescharacteristic
184C2×C3×C4C2×S411yescharacteristic
185C3⋊C8—122no—
186C3⋊C8—122no—
187C3⋊C8—246no—
188C2×C32—41no—
189C2×C32—41no—
190C2×C32—41no—
191C2×C32—82no—
192C24—21no—
193C24—21no—
194C24—31no—
195C24—31no—
196C24—32no—
197C24—61no—
198C24—61no—
199C24C3⋊(C3⋊C8)12yescomplemented
200C24C2×C3⋊Dic311yescharacteristic
201C22×C4—21no—
202C22×C4—31no—
203C22×C4—32no—
204C22×C4—31no—
205C22×C4—31no—
206C22×C4—34no—
207C22×C4—31no—
208C22×C4—31no—
209C22×C4—31no—
210C22×C4—31no—
211C22×C4—62no—
212C22×C4—62no—
213C22×C4—61no—
214C22×C4—61no—
215C22×C4—62no—
216C22×C4—62no—
217C22×C4—62no—
218C22×C4—62no—
219C22×C4—62no—
220C22×C4#63812yes—
221C22×C4C3⋊S411yescharacteristic
222C2×C8—182no—
223C2×C8—182no—
224C2×C8—182no—
225C2×C8—182no—
226C2×C8—182no—
227C2×C8—182no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—31no—
231C22×C3—31no—
232C22×C3—31no—
233C22×C3—31no—
234C22×C3—31no—
235C22×C3—31no—
236C22×C3—31no—
237C22×C3—31no—
238C22×C3—31no—
239C22×C3—34no—
240C22×C3—32no—
241C22×C3—46no—
242C22×C3—43no—
243C22×C3—62no—
244C22×C3—62no—
245C22×C3—62no—
246C22×C3—62no—
247C22×C3—61no—
248C22×C3—62no—
249C22×C3—62no—
250C22×C3—62no—
251C22×C3—61no—
252C22×C3—61no—
253C22×C3—83no—
254C22×C3—83no—
255C22×C3#83011yescharacteristic, complemented
256C22×C3A4⋊C812yescomplemented
257C22×C3C2×SL(2,ℤ/4)11yescharacteristic
258A4#83013yescomplemented
259C3×C4—21no—
260C3×C4—32no—
261C3×C4—46no—
262C3×C4—61no—
263C3×C4—83no—
264C3×C4GL(2,ℤ/4)12yes—
265C32—41noSylow
266C23—22no—
267C23—31no—
268C23—31no—
269C23—32no—
270C23—31no—
271C23—31no—
272C23—32no—
273C23—32no—
274C23—32no—
275C23—32no—
276C23—31no—
277C23—61no—
278C23—61no—
279C23—61no—
280C23—61no—
281C23—62no—
282C23—61no—
283C23—61no—
284C23—62no—
285C23—62no—
286C23—61no—
287C23—64no—
288C23C2×C3⋊(C3⋊C8)11yescharacteristic
289C23#374811yescharacteristic
290C23#375711yescharacteristic
291C23C3⋊SL(2,ℤ/4)11yescharacteristic
292C2×C4—22no—
293C2×C4—31no—
294C2×C4—32no—
295C2×C4—32no—
296C2×C4—32no—
297C2×C4—32no—
298C2×C4—31no—
299C2×C4—62no—
300C2×C4—61no—
301C2×C4—61no—
302C2×C4—61no—
303C2×C4—62no—
304C2×C4—64no—
305C2×C4—62no—
306C2×C4—61no—
307C2×C4C3⋊SL(2,ℤ/4)11yescharacteristic
308C2×C4C2×C3⋊S411yescharacteristic
309C8—362no—
310C8—362no—
311C2×C3—22no—
312C2×C3—31no—
313C2×C3—31no—
314C2×C3—31no—
315C2×C3—31no—
316C2×C3—43no—
317C2×C3—43no—
318C2×C3—43no—
319C2×C3—62no—
320C2×C3—86no—
321C2×C3C2×A4⋊C811yescharacteristic
322C2×C3#540911yescharacteristic
323C2×C3#543211yescharacteristic
324C22—21no—
325C22—21no—
326C22—31no—
327C22—31no—
328C22—31no—
329C22—31no—
330C22—31no—
331C22—31no—
332C22—31no—
333C22—31no—
334C22—31no—
335C22—34no—
336C22—32no—
337C22—62no—
338C22—62no—
339C22—62no—
340C22—62no—
341C22—61no—
342C22—62no—
343C22—62no—
344C22—62no—
345C22—61no—
346C22—61no—
347C22#6365711yescharacteristic, complemented
348C22C3⋊(A4⋊C8)12yescomplemented
349C22C2×C3⋊SL(2,ℤ/4)11yescharacteristic
350C4—21no—
351C4—32no—
352C4—61no—
353C4#6426912yes—
354C3—43no—
355C3#7314111yescharacteristic, complemented
356C2—22no—
357C2—31no—
358C2—31no—
359C2—31no—
360C2—31no—
361C2—62no—
362C2C2×C3⋊(A4⋊C8)11yescharacteristic
363C2#1059364311yescharacteristic
364C2#1059366611yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×C3⋊S4
commutatorC2×C3×A4C2×C8
FittingC24×C3×C4D3
socleC24×C3C2×Dic3
prime radicalC23×C3×A4C4
cosocleC2×C3×C4×A4C22
hypercenterC22×C4C3⋊S4
hypocenterC3×A4C22⋊C8
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊(A4⋊C8)C2semidirect
20. C22×C3×A4C8semidirect
72. C24×C3C3⋊C8semidirect
76. C22×A4C3⋊C8semidirect
113. C3×A4C22⋊C8semidirect
199. C24C3⋊(C3⋊C8)semidirect
255. C22×C3#830semidirect
256. C22×C3A4⋊C8semidirect
258. A4#830semidirect
347. C22#63657semidirect
348. C22C3⋊(A4⋊C8)semidirect
355. C3#73141semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×A42
C2×C3⋊(A4⋊C8)2
#731413
#721263
#636574