Finite Group Databasebeta

← #11806037

Subgroups — #11806037

Order 1152.

1680 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#638C211yesmaximal, characteristic
2C32×C4×C22⋊C4C211yesmaximal, characteristic, complemented
3#10592910C211yesmaximal, characteristic, complemented
4#10592656C212yesmaximal
5#10592644C212yesmaximal, complemented
6#10592639C212yesmaximal, complemented
7C4×#3750C212yesmaximal, complemented
8#10592365C211yesmaximal, characteristic, complemented
9#10592363C211yesmaximal, characteristic, complemented
10#10592349C211yesmaximal, characteristic, complemented
11#10592348C211yesmaximal, characteristic, complemented
12#73463—34nomaximal
13C23×C32×C4C2211yescharacteristic
14C22×#638C2211yescharacteristic
15C22×C4×C3⋊D3C2211yescharacteristic
16C2×C32×C22⋊C4C2212yes—
17C2×C32×C42C2212yes—
18C2×#3757C2212yes—
19C2×#3757C2211yescharacteristic
20C4×#638C418yescomplemented
21C2×#3752C2212yes—
22C2×#3752C2212yes—
23C2×#3752C2211yescharacteristic
24C2×#3750C2212yes—
25C2×#3750C2212yes—
26C2×#3750C2211yescharacteristic
27C2×C4×C3⋊Dic3C2212yes—
28C2×C4×C3⋊Dic3C2212yes—
29C2×C4×C3⋊Dic3C2211yescharacteristic
30C32×#95C2211yescharacteristic
31C32×#95C2211yescharacteristic
32#63658C2211yescharacteristic
33#63658C2211yescharacteristic
34#63658C2212yes—
35#63658C2212yes—
36#63658C2212yes—
37C2×C4×#67—34no—
38C3×C4×C22⋊C4D314yescomplemented
39#5211—34no—
40#4957—38no—
41#4945—38no—
42#4940—38no—
43C4×Dic3⋊C4—38no—
44#4666—34no—
45#4664—34no—
46#4650—34no—
47#4649—34no—
48C24×C32C2311yescharacteristic
49C23×C3⋊D3C2311yescharacteristic
50C22×C32×C4—42no—
51C22×C32×C4C2×C414yes—
52C22×C32×C4C2312yes—
53C22×C32×C4C2311yescharacteristic
54C22×C32×C4C2311yescharacteristic
55C22×C32×C4C2312yes—
56C2×#638—22no—
57C2×#638—22no—
58C2×#638C2×C414yes—
59C22×C3⋊Dic3C2312yes—
60C22×C3⋊Dic3C2312yes—
61C22×C3⋊Dic3C2311yescharacteristic
62C22×C3⋊Dic3C2311yescharacteristic
63C22×C3⋊Dic3C2311yescharacteristic
64C2×C4×C3⋊D3—42no—
65C2×C4×C3⋊D3C2×C414yes—
66C32×C22⋊C4—24no—
67C32×C42—42no—
68#3757—42no—
69#3757C2×C414yes—
70#3752—24no—
71#3752—42no—
72#3752C2×C414yes—
73#3750—24no—
74#3750—24no—
75#3750C2×C414yes—
76C4×C3⋊Dic3—24no—
77C4×C3⋊Dic3—42no—
78C4×C3⋊Dic3C2×C414yes—
79#2357—91noSylow
80C23×C3×C4C2×D314yes—
81C22×#67—34no—
82C22×C4×D3—34no—
83C2×C3×C22⋊C4C2×D318yes—
84C2×C3×C42C2×D318yes—
85C2×#217—34no—
86C2×#217—38no—
87C4×#67—332no—
88C2×(C2×D3)⋊C4—34no—
89C2×(C2×D3)⋊C4—38no—
90C2×(C2×D3)⋊C4—38no—
91C2×Dic3⋊C4—34no—
92C2×Dic3⋊C4—38no—
93C2×Dic3⋊C4—38no—
94C2×C4×Dic3—34no—
95C2×C4×Dic3—38no—
96C2×C4×Dic3—38no—
97C3×#95C2×D314yes—
98C3×#95C2×D314yes—
99C3⋊#95—34no—
100C3⋊#95—34no—
101C3⋊#95—38no—
102C3⋊#95—38no—
103C3⋊#95—38no—
104C23×C32—22no—
105C23×C32—21no—
106C23×C32—21no—
107C23×C32—22no—
108C23×C32C22×C412yes—
109C23×C32C2411yescharacteristic
110C22×C3⋊D3—22no—
111C22×C3⋊D3—21no—
112C22×C3⋊D3—21no—
113C22×C3⋊D3—22no—
114C22×C3⋊D3C22×C412yes—
115C2×C32×C4—22no—
116C2×C32×C4—22no—
117C2×C32×C4—22no—
118C2×C32×C4—22no—
119C2×C32×C4—22no—
120C2×C32×C4—22no—
121C2×C32×C4—22no—
122C2×C32×C4—44no—
123C2×C32×C4C22×C412yes—
124C2×C32×C4C22×C412yes—
125C2×C32×C4#4114yes—
126#638—28no—
127C2×C3⋊Dic3—22no—
128C2×C3⋊Dic3—22no—
129C2×C3⋊Dic3—22no—
130C2×C3⋊Dic3—22no—
131C2×C3⋊Dic3—22no—
132C2×C3⋊Dic3—22no—
133C2×C3⋊Dic3—22no—
134C2×C3⋊Dic3—22no—
135C2×C3⋊Dic3C22×C412yes—
136C2×C3⋊Dic3C22×C412yes—
137C2×C3⋊Dic3C22×C412yes—
138C2×C3⋊Dic3#4112yes—
139C2×C3⋊Dic3#4112yes—
140C4×C3⋊D3—44no—
141C2×C4×D4—91no—
142#390—92no—
143#388—92no—
144#388—92no—
145#382—91no—
146#381—91no—
147#381—91no—
148C4×C4⋊C4—92no—
149C4×C4⋊C4—91no—
150C4×C22⋊C4—91no—
151C4×C22⋊C4C3⋊D311yescharacteristic, complemented
152C24×C3C22×D314yes—
153C23×D3—34no—
154C22×C3×C4—48no—
155C22×C3×C4C4×D3116yes—
156C22×C3×C4C22×D318yes—
157C22×C3×C4C22×D314yes—
158C22×C3×C4C22×D314yes—
159C22×C3×C4C22×D318yes—
160C2×#67—316no—
161C2×#67—68no—
162C2×#67—68no—
163C22×Dic3—34no—
164C22×Dic3—34no—
165C22×Dic3—34no—
166C22×Dic3—38no—
167C22×Dic3—38no—
168C2×C4×D3—316no—
169C2×C4×D3—128no—
170C3×C22⋊C4—216no—
171C3×C42—48no—
172#217—316no—
173#217—128no—
174(C2×D3)⋊C4—316no—
175(C2×D3)⋊C4—616no—
176(C2×D3)⋊C4—128no—
177Dic3⋊C4—316no—
178Dic3⋊C4—616no—
179Dic3⋊C4—616no—
180C4×Dic3—316no—
181C4×Dic3—616no—
182C4×Dic3—128no—
183C22×C32—22no—
184C22×C32—22no—
185C22×C32—22no—
186C22×C32—42no—
187C22×C32—42no—
188C22×C32C23×C411yescharacteristic
189C22×C32C2×#4112yes—
190C22×C32C2×#4111yescharacteristic
191C22×C32C2×#4111yescharacteristic
192C22×C32H(5,𝔽2)12yes—
193C2×C3⋊D3—22no—
194C2×C3⋊D3—22no—
195C2×C3⋊D3—22no—
196C2×C3⋊D3—42no—
197C2×C3⋊D3—42no—
198C32×C4—22no—
199C32×C4—22no—
200C32×C4—24no—
201C32×C4—42no—
202C3⋊Dic3—22no—
203C3⋊Dic3—22no—
204C3⋊Dic3—42no—
205C3⋊Dic3—42no—
206C3⋊Dic3#11714yes—
207C22×D4—91no—
208C23×C4—91no—
209C23×C4C2×C3⋊D311yescharacteristic
210C4×D4—98no—
211C2×C4⋊C4—91no—
212C2×C4⋊C4—92no—
213C2×C4⋊C4—92no—
214C2×C22⋊C4—91no—
215C2×C22⋊C4—91no—
216C2×C22⋊C4—92no—
217C2×C22⋊C4—92no—
218C2×C22⋊C4—92no—
219C2×C22⋊C4C2×C3⋊D312yes—
220C2×C42—91no—
221C2×C42—92no—
222C2×C42—92no—
223C2×C42C2×C3⋊D312yes—
224#95—91no—
225#95—91no—
226#95—92no—
227#95—92no—
228#95—92no—
229#95C2×C3⋊D311yescharacteristic
230#95C2×C3⋊D311yescharacteristic
231C23×C3—28no—
232C23×C3—28no—
233C23×C3—24no—
234C23×C3—24no—
235C23×C3C2×C4×D318yes—
236C23×C3C23×D314yes—
237C22×D3—38no—
238C22×D3—68no—
239C22×D3—68no—
240C22×D3—64no—
241C22×D3—64no—
242C2×C3×C4—28no—
243C2×C3×C4—28no—
244C2×C3×C4—28no—
245C2×C3×C4—28no—
246C2×C3×C4—28no—
247C2×C3×C4—28no—
248C2×C3×C4—28no—
249C2×C3×C4—416no—
250C2×C3×C4C2×C4×D318yes—
251C2×C3×C4C2×C4×D318yes—
252C2×C3×C4#235116yes—
253#67—632no—
254C2×Dic3—38no—
255C2×Dic3—38no—
256C2×Dic3—38no—
257C2×Dic3—38no—
258C2×Dic3—38no—
259C2×Dic3—68no—
260C2×Dic3—68no—
261C2×Dic3—68no—
262C2×Dic3—68no—
263C2×Dic3—68no—
264C2×Dic3—68no—
265C2×Dic3—68no—
266C2×Dic3—68no—
267C4×D3—1216no—
268C2×C32—42no—
269C2×C32C2×#11711yescharacteristic
270C2×C32C4×#4111yescharacteristic
271C2×C32#51811yescharacteristic
272C2×C32#55312yes—
273C2×C32#55912yes—
274C3⋊D3—42no—
275C24—91no—
276C24C22×C3⋊D311yescharacteristic
277C2×D4—94no—
278C2×D4—182no—
279C2×D4—182no—
280C22×C4—42no—
281C22×C4—91no—
282C22×C4—91no—
283C22×C4—91no—
284C22×C4—94no—
285C22×C4—92no—
286C22×C4—92no—
287C22×C4—362no—
288C22×C4C4×C3⋊D314yes—
289C22×C4C22×C3⋊D312yes—
290C22×C4C22×C3⋊D311yescharacteristic
291C22×C4C22×C3⋊D311yescharacteristic
292C22×C4C22×C3⋊D312yes—
293C4⋊C4—94no—
294C4⋊C4—184no—
295C4⋊C4—184no—
296C22⋊C4—24no—
297C22⋊C4—94no—
298C22⋊C4—94no—
299C22⋊C4—184no—
300C22⋊C4—362no—
301C22⋊C4—362no—
302C42—42no—
303C42—94no—
304C42—184no—
305C42—362no—
306C22×C3—28no—
307C22×C3—28no—
308C22×C3—28no—
309C22×C3—48no—
310C22×C3—48no—
311C22×C3C22×C4×D314yes—
312C22×C3C2×#23518yes—
313C22×C3#100418yes—
314C22×C3D3×#4114yes—
315C22×C3D3×#4114yes—
316C2×D3—68no—
317C2×D3—68no—
318C2×D3—68no—
319C2×D3—128no—
320C2×D3—128no—
321C3×C4—28no—
322C3×C4—28no—
323C3×C4—216no—
324C3×C4—48no—
325Dic3—316no—
326Dic3—68no—
327Dic3—68no—
328Dic3—128no—
329Dic3—128no—
330C32#235711yescharacteristic, complemented, Sylow
331C23—22no—
332C23—22no—
333C23—21no—
334C23—21no—
335C23—92no—
336C23—182no—
337C23—182no—
338C23—181no—
339C23—181no—
340C23C2×C4×C3⋊D312yes—
341C23C23×C3⋊D311yescharacteristic
342D4—188no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—44no—
351C2×C4—92no—
352C2×C4—92no—
353C2×C4—92no—
354C2×C4—92no—
355C2×C4—92no—
356C2×C4—182no—
357C2×C4—182no—
358C2×C4—182no—
359C2×C4—182no—
360C2×C4—182no—
361C2×C4—182no—
362C2×C4—182no—
363C2×C4—182no—
364C2×C4—364no—
365C2×C4C2×C4×C3⋊D312yes—
366C2×C4C2×C4×C3⋊D312yes—
367C2×C4#382814yes—
368C2×C3—48no—
369C2×C3C4×#23514yes—
370C2×C3#548614yes—
371C2×C3D3×#11714yes—
372C2×C3#554018yes—
373C2×C3#555918yes—
374D3—128no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—42no—
379C22—42no—
380C22—182no—
381C22—182no—
382C22—182no—
383C22—362no—
384C22—362no—
385C22C22×C4×C3⋊D311yescharacteristic
386C22C2×#382812yes—
387C22#6436112yes—
388C22#41×C3⋊D311yescharacteristic
389C22#41×C3⋊D311yescharacteristic
390C4—22no—
391C4—22no—
392C4—24no—
393C4—42no—
394C4—94no—
395C4—182no—
396C4—182no—
397C4—362no—
398C4—362no—
399C3#7346314yescomplemented
400C2—42no—
401C2—362no—
402C2C4×#382811yescharacteristic
403C2#1059595711yescharacteristic
404C2C3⋊D3×#11711yescharacteristic
405C2#1059601112yes—
406C2#1059603012yes—
subgroupG/H
center, FrattiniC23C23×C3⋊D3
commutatorC22×C32C23×C4
FittingC32×C4×C22⋊C4C2
socle, cosocleC23×C32C24
prime radicalC22×#638C22
hypercenterC4×C22⋊C4C3⋊D3
hypocenterC32#2357
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C32×C4×C22⋊C4C2semidirect
3. #10592910C2semidirect
5. #10592644C2semidirect
6. #10592639C2semidirect
7. C4×#3750C2semidirect
8. #10592365C2semidirect
9. #10592363C2semidirect
10. #10592349C2semidirect
11. #10592348C2semidirect
20. C4×#638C4semidirect
38. C3×C4×C22⋊C4D3semidirect
151. C4×C22⋊C4C3⋊D3semidirect
330. C32#2357semidirect
399. C3#73463semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#6382
C32×C4×C22⋊C42
#105929102
#105926562
#105926442
#105926392
C4×#37502
#105923652
#105923632
#105923492
#105923482
#734633