Finite Group Databasebeta

← #11770845

Subgroups — #11770845

Order 1152.

585 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#3757C211yesmaximal, characteristic
2C2×C3×#830C211yesmaximal, characteristic
3C2×C3×#830C211yesmaximal, characteristic
4#67678—31nomaximal
5#67677—31nomaximal
6C23×C32×C4C2211yescharacteristic
7C2×#3757C412yes—
8C2×C4×C3⋊Dic3—21no—
9C22×C3×C3⋊C8—21no—
10C22×C3×C3⋊C8C412yescomplemented
11#63658—21no—
12C3×#830—21no—
13C3×#830—21no—
14C3×#830—21no—
15C3×#830C412yescomplemented
16C2×C3×C22⋊C8—31no—
17C2×C3×C22⋊C8—31no—
18C4×#217—31no—
19C4×#217—31no—
20C4×#217—61no—
21C2×#830D311yescharacteristic
22C2×#830D311yescharacteristic
23C24×C32C2×C411yescharacteristic
24C22×C32×C4—42no—
25C22×C32×C4C2×C411yescharacteristic
26C22×C32×C4C2×C411yescharacteristic
27C22×C32×C4D411yescharacteristic
28C22×C32×C4D411yescharacteristic
29C22×C32×C4D411yescharacteristic
30C22×C32×C4Q811yescharacteristic
31C22×C3⋊Dic3—22no—
32#3757—22no—
33#3757C814yescomplemented
34C4×C3⋊Dic3—42no—
35C2×C3×C3⋊C8—22no—
36C2×C3×C3⋊C8—22no—
37C2×C3×C3⋊C8—42no—
38C2×C3×C3⋊C8—42no—
39#1282—91noSylow
40C23×C3×C4—21no—
41C23×C3×C4C2×D311yescharacteristic
42C23×C3×C4C2×D311yescharacteristic
43C22×C3×C8—32no—
44C22×C3×C8—61no—
45C2×#217—32no—
46C2×#217—32no—
47C2×#217—62no—
48C2×C4×Dic3—62no—
49C2×C4×Dic3—61no—
50C2×C4×Dic3—61no—
51C22×C3⋊C8—21no—
52C22×C3⋊C8Dic312yescomplemented
53C3×C22⋊C8—32no—
54C3×C22⋊C8—61no—
55C3×C22⋊C8—61no—
56C3×C22⋊C8—61no—
57C3⋊#95—62no—
58C3⋊#95—61no—
59C3⋊#95—61no—
60#830—21no—
61#830—21no—
62#830—21no—
63#830Dic312yescomplemented
64C23×C32—41no—
65C23×C32—41no—
66C23×C32C4211yescharacteristic
67C23×C32C22⋊C411yescharacteristic
68C23×C32C4⋊C411yescharacteristic
69C23×C32C2×C812yes—
70C23×C32M1612yes—
71C2×C32×C4—22no—
72C2×C32×C4—22no—
73C2×C32×C4—22no—
74C2×C32×C4—41no—
75C2×C32×C4—41no—
76C2×C32×C4—41no—
77C2×C32×C4C22⋊C411yescharacteristic
78C2×C32×C4C22⋊C411yescharacteristic
79C2×C32×C4C4⋊C411yescharacteristic
80C2×C32×C4C4⋊C411yescharacteristic
81C2×C3⋊Dic3—24no—
82C2×C3⋊Dic3—42no—
83C2×C3⋊Dic3—42no—
84C3×C3⋊C8—42no—
85C3×C3⋊C8—42no—
86C2×C22⋊C8—31no—
87C2×C22⋊C8—31no—
88C4×C22⋊C4—91no—
89C24×C3—21no—
90C24×C3C4×D311yescharacteristic
91C24×C3C4×D311yescharacteristic
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—42no—
99C22×C3×C4—42no—
100C22×C3×C4—44no—
101C22×C3×C4Dic611yescharacteristic
102C22×C3×C4Dic611yescharacteristic
103C22×C3×C4C4×D311yescharacteristic
104C22×C3×C4C4×D311yescharacteristic
105C22×C3×C4D1211yescharacteristic
106C22×C3×C4D1211yescharacteristic
107C22×C3×C4C2×Dic311yescharacteristic
108C22×C3×C4C2×Dic311yescharacteristic
109C22×C3×C4#6711yescharacteristic
110C22×C3×C4#6711yescharacteristic
111C22×C3×C4#6711yescharacteristic
112C22×C3×C4#6711yescharacteristic
113C22×Dic3—64no—
114C22×Dic3—62no—
115C22×Dic3—62no—
116C2×C3×C8—62no—
117C2×C3×C8—62no—
118C2×C3×C8—122no—
119C2×C3×C8—122no—
120#217—34no—
121#217—34no—
122#217—64no—
123#217—64no—
124#217—62no—
125#217—62no—
126C4×Dic3—124no—
127C4×Dic3—122no—
128C4×Dic3—122no—
129C2×C3⋊C8—22no—
130C2×C3⋊C8—22no—
131C2×C3⋊C8—42no—
132C2×C3⋊C8—42no—
133C22×C32—21no—
134C22×C32—21no—
135C22×C32—21no—
136C22×C32—21no—
137C22×C32—21no—
138C22×C32—21no—
139C22×C32—42no—
140C22×C32—42no—
141C22×C32#9511yescharacteristic
142C22×C32C4×C811yescharacteristic
143C22×C32#9711yescharacteristic
144C22×C32C22⋊C812yes—
145C22×C32C4⋊C812yes—
146C32×C4—21no—
147C32×C4—21no—
148C32×C4—21no—
149C32×C4#9912yes—
150C3⋊Dic3—44no—
151C23×C4D3211yescharacteristic
152C22×C8—32no—
153C22×C8—61no—
154C2×C22⋊C4—92no—
155C2×C42—181no—
156C22⋊C8—32no—
157C22⋊C8—61no—
158C22⋊C8—61no—
159C22⋊C8—61no—
160#95—181no—
161C23×C3—22no—
162C23×C3—22no—
163C23×C3—21no—
164C23×C3—21no—
165C23×C3—21no—
166C23×C3—41no—
167C23×C3—41no—
168C23×C3—41no—
169C23×C3—41no—
170C23×C3—42no—
171C23×C3—42no—
172C23×C3D3×C812yes—
173C23×C3D3×C812yes—
174C23×C3ΓL(1,𝔽25)12yes—
175C23×C3ΓL(1,𝔽25)12yes—
176C23×C3C4×Dic311yescharacteristic
177C23×C3C4×Dic311yescharacteristic
178C23×C3Dic3⋊C411yescharacteristic
179C23×C3Dic3⋊C411yescharacteristic
180C23×C3(C2×D3)⋊C411yescharacteristic
181C23×C3(C2×D3)⋊C411yescharacteristic
182C2×C3×C4—24no—
183C2×C3×C4—22no—
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×C3×C4—22no—
187C2×C3×C4—22no—
188C2×C3×C4—22no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—42no—
194C2×C3×C4—42no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—42no—
202C2×C3×C4—42no—
203C2×C3×C4—42no—
204C2×C3×C4Dic3⋊C411yescharacteristic
205C2×C3×C4Dic3⋊C411yescharacteristic
206C2×C3×C4C4⋊Dic311yescharacteristic
207C2×C3×C4C4⋊Dic311yescharacteristic
208C2×C3×C4(C2×D3)⋊C411yescharacteristic
209C2×C3×C4(C2×D3)⋊C411yescharacteristic
210C2×C3×C4#21711yescharacteristic
211C2×C3×C4#21711yescharacteristic
212C2×Dic3—68no—
213C2×Dic3—64no—
214C2×Dic3—64no—
215C2×Dic3—124no—
216C2×Dic3—122no—
217C2×Dic3—122no—
218C2×Dic3—124no—
219C2×Dic3—122no—
220C2×Dic3—122no—
221C3×C8—122no—
222C3×C8—122no—
223C3⋊C8—42no—
224C3⋊C8—42no—
225C2×C32—22no—
226C2×C32—21no—
227C2×C32—21no—
228C2×C32—22no—
229C2×C32#33611yescharacteristic
230C2×C32#34211yescharacteristic
231C2×C32#34411yescharacteristic
232C24#62411yescharacteristic
233C22×C4—42no—
234C22×C4—182no—
235C22×C4D3×Dic311yescharacteristic
236C22×C4D3×Dic311yescharacteristic
237C22×C4#62511yescharacteristic
238C22×C4#62611yescharacteristic
239C22×C4#62611yescharacteristic
240C22×C4#62711yescharacteristic
241C2×C8—62no—
242C2×C8—62no—
243C2×C8—122no—
244C2×C8—122no—
245C22⋊C4—94no—
246C22⋊C4—182no—
247C42—362no—
248C22×C3—22no—
249C22×C3—22no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—21no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—22no—
265C22×C3—22no—
266C22×C3—21no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3—42no—
272C22×C3—42no—
273C22×C3—42no—
274C22×C3—42no—
275C22×C3—44no—
276C22×C3—44no—
277C22×C3C8×Dic311yescharacteristic
278C22×C3C8×Dic311yescharacteristic
279C22×C3Dic3⋊C812yes—
280C22×C3Dic3⋊C812yes—
281C22×C3#81511yescharacteristic
282C22×C3#81511yescharacteristic
283C22×C3(C2×D3)⋊C812yes—
284C22×C3(C2×D3)⋊C812yes—
285C22×C3C3⋊#9511yescharacteristic
286C22×C3C3⋊#9511yescharacteristic
287C3×C4—22no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—22no—
295C3×C4—41no—
296C3×C4—41no—
297C3×C4#80612yes—
298C3×C4#83412yes—
299Dic3—128no—
300Dic3—124no—
301Dic3—124no—
302C32#128211yescharacteristic, complemented, Sylow
303C23—41no—
304C23—41no—
305C23#371012yes—
306C23#371212yes—
307C23Dic3211yescharacteristic
308C23#372211yescharacteristic
309C23#372411yescharacteristic
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—184no—
317C2×C4—362no—
318C2×C4—362no—
319C2×C4(C2×D3)⋊Dic311yescharacteristic
320C2×C4(C2×D3)⋊Dic311yescharacteristic
321C2×C4Dic3⋊Dic311yescharacteristic
322C2×C4Dic3⋊Dic311yescharacteristic
323C8—122no—
324C8—122no—
325C2×C3—24no—
326C2×C3—22no—
327C2×C3—21no—
328C2×C3—22no—
329C2×C3—21no—
330C2×C3—22no—
331C2×C3—21no—
332C2×C3—22no—
333C2×C3—21no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—41no—
338C2×C3—41no—
339C2×C3—42no—
340C2×C3#452611yescharacteristic
341C2×C3#452711yescharacteristic
342C2×C3#455011yescharacteristic
343C2×C3#455011yescharacteristic
344C2×C3#455611yescharacteristic
345C2×C3#455911yescharacteristic
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—42no—
353C22—42no—
354C22#6355311yescharacteristic
355C22#6355511yescharacteristic
356C22#6355712yes—
357C22#6357212yes—
358C22#6357911yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—364no—
363C4#6358012yes—
364C3—21no—
365C3#6767711yescharacteristic, complemented
366C3#6767811yescharacteristic, complemented
367C2—22no—
368C2—21no—
369C2—21no—
370C2—22no—
371C2#1058915611yescharacteristic
372C2#1058917111yescharacteristic
373C2#1058918611yescharacteristic
subgroupG/H
centerC2×C4(C2×D3)⋊Dic3
commutatorC22×C32C4×C8
Frattini, hypercenterC23×C4D32
Fitting, cosocleC23×C32×C4C22
socleC22×C32#95
prime radicalC24×C32C2×C4
hypocenterC32#1282
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
10. C22×C3×C3⋊C8C4semidirect
15. C3×#830C4semidirect
33. #3757C8semidirect
52. C22×C3⋊C8Dic3semidirect
63. #830Dic3semidirect
302. C32#1282semidirect
365. C3#67677semidirect
366. C3#67678semidirect

Maximal subgroups

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

subgroupindex
C4×#37572
C2×C3×#8302
#676783
#676773