Finite Group Databasebeta

← #11805744

Subgroups — #11805744

Order 1152.

1764 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2C32×#376C211yesmaximal, characteristic, complemented
3#10592639C212yesmaximal, complemented
4C4×#3752C212yesmaximal, complemented
5#10592636C211yesmaximal, characteristic, complemented
6#10592366C211yesmaximal, characteristic, complemented
7#10592365C211yesmaximal, characteristic, complemented
8#10592364C212yesmaximal
9#10592363C211yesmaximal, characteristic, complemented
10C3⋊D3×#95C211yesmaximal, characteristic, complemented
11#10592353C212yesmaximal, complemented
12#73417—34nomaximal
13C22×C4⋊(C3⋊D3)C2211yescharacteristic
14C22×C4×C3⋊D3C2212yes—
15C22×C4×C3⋊D3C2211yescharacteristic
16C22×C4×C3⋊D3C2211yescharacteristic
17C2×C32×C42C2212yes—
18C2×C32×C42C2211yescharacteristic
19C2×#3752C2212yes—
20C2×#3752C2212yes—
21C2×#3752C2212yes—
22C2×#3752C2211yescharacteristic
23C2×#3752C2211yescharacteristic
24C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
25C2×#3750C2212yes—
26C2×C4×C3⋊Dic3C2212yes—
27C4×C4⋊(C3⋊D3)C418yescomplemented
28C32×#95C2212yes—
29C32×#95C2211yescharacteristic
30C32×#95C2211yescharacteristic
31#63658C2212yes—
32#63658C2212yes—
33#63658C2212yes—
34#63658C2211yescharacteristic
35#63658C2211yescharacteristic
36#63658C2212yes—
37C2×C4×D12—34no—
38C3×#376D314yescomplemented
39#4940—38no—
40C4×(C2×D3)⋊C4—38no—
41#4937—34no—
42#4667—34no—
43#4666—34no—
44#4665—38no—
45#4664—34no—
46D3×#95—34no—
47#4654—38no—
48C23×C3⋊D3C2311yescharacteristic
49C23×C3⋊D3C2311yescharacteristic
50C22×C32×C4C2312yes—
51C22×C32×C4C2311yescharacteristic
52C22×C32×C4C2311yescharacteristic
53C22×C32×C4C2311yescharacteristic
54C22×C32×C4C2312yes—
55C22×C3⋊Dic3C2312yes—
56C22×C3⋊Dic3C2312yes—
57C22×C3⋊Dic3C2311yescharacteristic
58C22×C3⋊Dic3C2311yescharacteristic
59C2×C4⋊(C3⋊D3)—22no—
60C2×C4⋊(C3⋊D3)—22no—
61C2×C4⋊(C3⋊D3)C2×C414yes—
62C2×C4×C3⋊D3—24no—
63C2×C4×C3⋊D3—24no—
64C2×C4×C3⋊D3—42no—
65C2×C4×C3⋊D3—42no—
66C2×C4×C3⋊D3—42no—
67C2×C4×C3⋊D3C2×C414yes—
68C2×C4×C3⋊D3C2×C414yescomplemented
69C32×C42—24no—
70C32×C42C2×C414yescomplemented
71#3752—24no—
72#3752—24no—
73#3752—42no—
74#3752C2×C414yes—
75#3752C2×C414yescomplemented
76C4⋊(C3⋊Dic3)C2×C414yescomplemented
77#3750—42no—
78C4×C3⋊Dic3—42no—
79#2353—91noSylow
80C22×D12—34no—
81C22×C4×D3—34no—
82C22×C4×D3—34no—
83C22×C4×D3—38no—
84C2×C3×C42C2×D318yes—
85C2×C3×C42C2×D314yes—
86C2×(C2×D3)⋊C4—34no—
87C2×(C2×D3)⋊C4—34no—
88C2×(C2×D3)⋊C4—38no—
89C2×(C2×D3)⋊C4—38no—
90C2×(C2×D3)⋊C4—38no—
91C2×C4⋊Dic3—34no—
92C2×Dic3⋊C4—38no—
93C2×C4×Dic3—38no—
94C4×D12—332no—
95C3×#95C2×D318yes—
96C3×#95C2×D314yes—
97C3×#95C2×D314yes—
98C3⋊#95—38no—
99C3⋊#95—38no—
100C3⋊#95—38no—
101C3⋊#95—34no—
102C3⋊#95—34no—
103C3⋊#95—38no—
104C23×C32C2411yescharacteristic
105C22×C3⋊D3—22no—
106C22×C3⋊D3—22no—
107C22×C3⋊D3—22no—
108C22×C3⋊D3—21no—
109C22×C3⋊D3—21no—
110C22×C3⋊D3—22no—
111C22×C3⋊D3C22×C412yes—
112C22×C3⋊D3C22×C412yes—
113C22×C3⋊D3#4112yes—
114C22×C3⋊D3#4112yes—
115C2×C32×C4—22no—
116C2×C32×C4—22no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—22no—
120C2×C32×C4—22no—
121C2×C32×C4—22no—
122C2×C32×C4—22no—
123C2×C32×C4—22no—
124C2×C32×C4C22×C412yes—
125C2×C32×C4C22×C412yes—
126C2×C32×C4C22×C412yes—
127C2×C32×C4#4114yes—
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⋊Dic3—22no—
136C2×C3⋊Dic3C22×C412yes—
137C2×C3⋊Dic3C22×C412yes—
138C4⋊(C3⋊D3)—28no—
139C4×C3⋊D3—44no—
140C4×C3⋊D3—44no—
141C4×C3⋊D3—48no—
142C2×C4×D4—91no—
143#388—92no—
144#387—92no—
145#386—92no—
146#383—91no—
147#381—91no—
148#380—91no—
149C4×C22⋊C4—92no—
150C4×C22⋊C4—91no—
151#376C3⋊D311yescharacteristic, complemented
152C2×#95—91no—
153C23×D3—34no—
154C23×D3—34no—
155C22×C3×C4C22×D318yes—
156C22×C3×C4C22×D314yes—
157C22×C3×C4C22×D314yes—
158C22×C3×C4C22×D314yes—
159C22×C3×C4C22×D318yes—
160C22×Dic3—34no—
161C22×Dic3—34no—
162C22×Dic3—38no—
163C22×Dic3—38no—
164C2×D12—316no—
165C2×D12—68no—
166C2×D12—68no—
167C2×C4×D3—316no—
168C2×C4×D3—316no—
169C2×C4×D3—616no—
170C2×C4×D3—616no—
171C2×C4×D3—128no—
172C2×C4×D3—128no—
173C2×C4×D3—128no—
174C3×C42—216no—
175C3×C42C4×D3116yescomplemented
176(C2×D3)⋊C4—316no—
177(C2×D3)⋊C4—316no—
178(C2×D3)⋊C4—616no—
179(C2×D3)⋊C4—616no—
180(C2×D3)⋊C4—128no—
181C4⋊Dic3—316no—
182Dic3⋊C4—128no—
183C4×Dic3—128no—
184C22×C32C23×C411yescharacteristic
185C22×C32C2×#4112yes—
186C22×C32C2×#4111yescharacteristic
187C22×C32C2×#4111yescharacteristic
188C22×C32H(5,𝔽2)12yes—
189C2×C3⋊D3—22no—
190C2×C3⋊D3—22no—
191C2×C3⋊D3—24no—
192C2×C3⋊D3—24no—
193C2×C3⋊D3—22no—
194C2×C3⋊D3—22no—
195C2×C3⋊D3—22no—
196C2×C3⋊D3—42no—
197C2×C3⋊D3—42no—
198C2×C3⋊D3#11714yes—
199C32×C4—22no—
200C32×C4—22no—
201C32×C4—22no—
202C32×C4—24no—
203C32×C4—42no—
204C3⋊Dic3—22no—
205C3⋊Dic3—22no—
206C3⋊Dic3—42no—
207C3⋊Dic3—42no—
208C22×D4—91no—
209C23×C4—91no—
210C23×C4—91no—
211C23×C4—92no—
212C4×D4—98no—
213C2×C4⋊C4—91no—
214C2×C4⋊C4—92no—
215C2×C22⋊C4—91no—
216C2×C22⋊C4—91no—
217C2×C22⋊C4—92no—
218C2×C22⋊C4—92no—
219C2×C22⋊C4—92no—
220C2×C42—92no—
221C2×C42C2×C3⋊D312yes—
222C2×C42C2×C3⋊D311yescharacteristic
223#95—91no—
224#95—92no—
225#95—91no—
226#95—92no—
227#95—92no—
228#95—92no—
229#95C2×C3⋊D311yescharacteristic
230#95C2×C3⋊D311yescharacteristic
231#95C2×C3⋊D312yes—
232C23×C3C23×D314yes—
233C22×D3—38no—
234C22×D3—38no—
235C22×D3—38no—
236C22×D3—38no—
237C22×D3—68no—
238C22×D3—68no—
239C22×D3—68no—
240C22×D3—64no—
241C22×D3—64no—
242C22×D3—68no—
243C2×C3×C4—28no—
244C2×C3×C4—28no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—28no—
248C2×C3×C4—28no—
249C2×C3×C4—28no—
250C2×C3×C4—28no—
251C2×C3×C4—28no—
252C2×C3×C4C2×C4×D318yes—
253C2×C3×C4C2×C4×D318yes—
254C2×C3×C4C2×C4×D318yes—
255C2×C3×C4#235116yes—
256C2×Dic3—38no—
257C2×Dic3—38no—
258C2×Dic3—68no—
259C2×Dic3—68no—
260C2×Dic3—68no—
261C2×Dic3—68no—
262C2×Dic3—68no—
263C2×Dic3—68no—
264C2×Dic3—68no—
265C2×Dic3—68no—
266D12—632no—
267C4×D3—1216no—
268C4×D3—1216no—
269C4×D3—1232no—
270C2×C32C2×#11711yescharacteristic
271C2×C32C4×#4111yescharacteristic
272C2×C32#51811yescharacteristic
273C2×C32#55112yes—
274C2×C32#55312yes—
275C3⋊D3—24no—
276C3⋊D3—42no—
277C24—91no—
278C24—91no—
279C2×D4—94no—
280C2×D4—182no—
281C2×D4—182no—
282C22×C4—91no—
283C22×C4—91no—
284C22×C4—94no—
285C22×C4—94no—
286C22×C4—92no—
287C22×C4—92no—
288C22×C4—184no—
289C22×C4—184no—
290C22×C4—362no—
291C22×C4—362no—
292C22×C4—362no—
293C22×C4C22×C3⋊D312yes—
294C22×C4C22×C3⋊D311yescharacteristic
295C22×C4C22×C3⋊D311yescharacteristic
296C22×C4C22×C3⋊D311yescharacteristic
297C22×C4C22×C3⋊D312yes—
298C4⋊C4—94no—
299C4⋊C4—362no—
300C22⋊C4—94no—
301C22⋊C4—94no—
302C22⋊C4—184no—
303C22⋊C4—184no—
304C22⋊C4—362no—
305C42—24no—
306C42—362no—
307C42C4×C3⋊D314yescomplemented
308C22×C3C22×C4×D314yes—
309C22×C3C2×#23518yes—
310C22×C3D3×#4114yes—
311C22×C3D3×#4114yes—
312C22×C3#100918yes—
313C2×D3—316no—
314C2×D3—68no—
315C2×D3—68no—
316C2×D3—616no—
317C2×D3—616no—
318C2×D3—68no—
319C2×D3—68no—
320C2×D3—68no—
321C2×D3—128no—
322C2×D3—128no—
323C3×C4—28no—
324C3×C4—28no—
325C3×C4—28no—
326C3×C4—216no—
327C3×C4—48no—
328Dic3—68no—
329Dic3—68no—
330Dic3—128no—
331Dic3—128no—
332C32#235311yescharacteristic, complemented, Sylow
333C23—92no—
334C23—92no—
335C23—92no—
336C23—92no—
337C23—182no—
338C23—182no—
339C23—182no—
340C23—181no—
341C23—181no—
342C23—182no—
343C23C23×C3⋊D311yescharacteristic
344D4—188no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
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×C4—364no—
366C2×C4—368no—
367C2×C4C2×C4×C3⋊D312yes—
368C2×C4C2×C4×C3⋊D312yes—
369C2×C4C2×C4×C3⋊D312yes—
370C2×C4#382814yes—
371C2×C3C4×#23514yes—
372C2×C3D3×#11714yes—
373C2×C3#552114yes—
374C2×C3#552718yes—
375C2×C3#553018yes—
376D3—616no—
377D3—128no—
378C22—94no—
379C22—182no—
380C22—182no—
381C22—184no—
382C22—184no—
383C22—182no—
384C22—182no—
385C22—182no—
386C22—362no—
387C22—362no—
388C22C22×C4×C3⋊D311yescharacteristic
389C22C2×#382812yes—
390C22#41×C3⋊D311yescharacteristic
391C22#41×C3⋊D311yescharacteristic
392C22#6436612yes—
393C4—22no—
394C4—22no—
395C4—22no—
396C4—24no—
397C4—42no—
398C4—182no—
399C4—182no—
400C4—362no—
401C4—362no—
402C3#7341714yescomplemented
403C2—184no—
404C2—362no—
405C2C4×#382811yescharacteristic
406C2C3⋊D3×#11711yescharacteristic
407C2#1059599211yescharacteristic
408C2#1059599812yes—
409C2#1059600112yes—
subgroupG/H
center, FrattiniC23C23×C3⋊D3
commutatorC22×C32C23×C4
FittingC32×#376C2
socle, cosocleC23×C32C24
prime radicalC22×C4⋊(C3⋊D3)C22
hypercenter#376C3⋊D3
hypocenterC32#2353
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ #376

Complements

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

NH ≅ G/Ntype
2. C32×#376C2semidirect
3. #10592639C2semidirect
4. C4×#3752C2semidirect
5. #10592636C2semidirect
6. #10592366C2semidirect
7. #10592365C2semidirect
9. #10592363C2semidirect
10. C3⋊D3×#95C2semidirect
11. #10592353C2semidirect
27. C4×C4⋊(C3⋊D3)C4semidirect
38. C3×#376D3semidirect
68. C2×C4×C3⋊D3C2×C4semidirect
70. C32×C42C2×C4semidirect
75. #3752C2×C4semidirect
76. C4⋊(C3⋊Dic3)C2×C4semidirect
151. #376C3⋊D3semidirect
175. C3×C42C4×D3semidirect
307. C42C4×C3⋊D3semidirect
332. C32#2353semidirect
402. C3#73417semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊(C3⋊D3)2
C32×#3762
#105926392
C4×#37522
#105926362
#105923662
#105923652
#105923642
#105923632
C3⋊D3×#952
#105923532
#734173