Finite Group Databasebeta

← #11806922

Subgroups — #11806922

Order 1152.

1896 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×Q8C211yesmaximal, characteristic, complemented
2C2×#64156C211yesmaximal, characteristic
3C2×#64102C211yesmaximal, characteristic
4#10592928C211yesmaximal, characteristic, complemented
5#10592689C212yesmaximal, complemented
6#10592688C212yesmaximal, complemented
7C4⋊C4×C3⋊Dic3C211yesmaximal, characteristic, complemented
8#10592639C212yesmaximal, complemented
9#10592638C211yesmaximal, characteristic
10C4×#3752C212yesmaximal, complemented
11C4×#3750C211yesmaximal, characteristic, complemented
12#82466—34nomaximal
13C22×C32×Q8C2211yescharacteristic
14C22×C4⋊(C3⋊D3)C2211yescharacteristic
15C22×C4×C3⋊D3C2212yes—
16C32×C4×Q8—24no—
17C2×C32×C4⋊C4C2212yes—
18C2×C32×C4⋊C4C2211yescharacteristic
19C2×C32×C42C2211yescharacteristic
20C2×C32×C42C2212yes—
21#64156C418yescomplemented
22C2×#3752C2212yes—
23C2×#3752C2212yes—
24C2×#3752C2212yes—
25C2×#3752C2212yes—
26C2×#3752C2212yes—
27C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
28C2×#3750C2211yescharacteristic
29C2×#3750C2211yescharacteristic
30C2×C4×C3⋊Dic3C2211yescharacteristic
31C2×C4×C3⋊Dic3C2212yes—
32C2×C4×C3⋊Dic3C2211yescharacteristic
33#64102—24no—
34#63658C2211yescharacteristic
35#63658C2211yescharacteristic
36#63658C2212yes—
37#63658C2212yes—
38#63658C2212yes—
39C2×C3×C4×Q8D314yescomplemented
40C2×#947—34no—
41C2×#893—34no—
42#5229—34no—
43#4990—38no—
44#4989—38no—
45Dic3×C4⋊C4—34no—
46#4940—38no—
47#4939—34no—
48C4×(C2×D3)⋊C4—38no—
49C4×Dic3⋊C4—34no—
50C23×C3⋊D3C2312yes—
51C2×C32×Q8—22no—
52C2×C32×Q8—22no—
53C2×C32×Q8C2×C414yescomplemented
54C22×C32×C4C2311yescharacteristic
55C22×C32×C4C2312yes—
56C22×C32×C4C2312yes—
57C22×C32×C4C2311yescharacteristic
58C22×C32×C4C2311yescharacteristic
59C22×C3⋊Dic3C2311yescharacteristic
60C22×C3⋊Dic3C2312yes—
61C22×C3⋊Dic3C2311yescharacteristic
62C22×C3⋊Dic3C2311yescharacteristic
63C22×C3⋊Dic3C2311yescharacteristic
64C2×C4⋊(C3⋊D3)—22no—
65C2×C4⋊(C3⋊D3)—22no—
66C2×C4⋊(C3⋊D3)C2×C414yes—
67C2×C4×C3⋊D3—24no—
68C2×C4×C3⋊D3—24no—
69C2×C4×C3⋊D3—42no—
70C32×C4⋊C4—24no—
71C32×C4⋊C4D414yes—
72C32×C42—24no—
73C32×C42—22no—
74#3752—24no—
75#3752—24no—
76#3752—42no—
77#3752C2×C418yescomplemented
78#3752C2×C418yescomplemented
79C4⋊(C3⋊Dic3)—22no—
80#3750—22no—
81#3750—22no—
82C4×C3⋊Dic3—22no—
83C4×C3⋊Dic3—42no—
84C4×C3⋊Dic3C2×C414yescomplemented
85#2367—91noSylow
86C22×C3×Q8C2×D314yes—
87C22×D12—34no—
88C22×C4×D3—38no—
89C3×C4×Q8—216no—
90C2×C3×C4⋊C4C2×D318yes—
91C2×C3×C4⋊C4C2×D314yes—
92C2×C3×C42C2×D314yes—
93C2×C3×C42C2×D318yes—
94#947—332no—
95C2×(C2×D3)⋊C4—38no—
96C2×(C2×D3)⋊C4—38no—
97C2×(C2×D3)⋊C4—38no—
98C2×(C2×D3)⋊C4—38no—
99C2×(C2×D3)⋊C4—38no—
100C2×C4⋊Dic3—34no—
101C2×Dic3⋊C4—34no—
102C2×Dic3⋊C4—34no—
103C2×C4×Dic3—34no—
104C2×C4×Dic3—34no—
105C2×C4×Dic3—38no—
106#893—616no—
107C3⋊#95—34no—
108C3⋊#95—34no—
109C3⋊#95—38no—
110C3⋊#95—38no—
111C3⋊#95—38no—
112C23×C32C2411yescharacteristic
113C22×C3⋊D3—22no—
114C22×C3⋊D3—22no—
115C22×C3⋊D3—22no—
116C22×C3⋊D3—22no—
117C22×C3⋊D3—22no—
118C22×C3⋊D3—22no—
119C22×C3⋊D3C22×C414yes—
120C32×Q8—28no—
121C2×C32×C4—21no—
122C2×C32×C4—22no—
123C2×C32×C4—22no—
124C2×C32×C4—22no—
125C2×C32×C4—22no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4C22×C414yes—
131C2×C32×C4C22×C412yes—
132C2×C32×C4C2×D414yes—
133C2×C32×C4C2×D412yes—
134C2×C32×C4#4112yes—
135C2×C32×C4#4112yes—
136C2×C3⋊Dic3—21no—
137C2×C3⋊Dic3—21no—
138C2×C3⋊Dic3—21no—
139C2×C3⋊Dic3—21no—
140C2×C3⋊Dic3—21no—
141C2×C3⋊Dic3—21no—
142C2×C3⋊Dic3—21no—
143C2×C3⋊Dic3—21no—
144C2×C3⋊Dic3—22no—
145C2×C3⋊Dic3—22no—
146C2×C3⋊Dic3—22no—
147C2×C3⋊Dic3—21no—
148C2×C3⋊Dic3C22×C412yes—
149C2×C3⋊Dic3C22×C412yes—
150C2×C3⋊Dic3#4112yes—
151C4⋊(C3⋊D3)—28no—
152C4×C3⋊D3—48no—
153C2×#124—91no—
154C2×C4×Q8C3⋊D311yescharacteristic, complemented
155C2×C4×D4—91no—
156#391—91no—
157#390—91no—
158#388—92no—
159#388—92no—
160#385—92no—
161C4×C4⋊C4—91no—
162C4×C4⋊C4—91no—
163C4×C22⋊C4—92no—
164C23×D3—38no—
165C2×C3×Q8—28no—
166C2×C3×Q8—28no—
167C2×C3×Q8C4×D3116yescomplemented
168C22×C3×C4C22×D314yes—
169C22×C3×C4C22×D318yes—
170C22×C3×C4C22×D314yes—
171C22×C3×C4C22×D318yes—
172C22×C3×C4C22×D314yes—
173C22×Dic3—34no—
174C22×Dic3—34no—
175C22×Dic3—34no—
176C22×Dic3—34no—
177C22×Dic3—38no—
178C2×D12—316no—
179C2×D12—68no—
180C2×D12—68no—
181C2×C4×D3—616no—
182C2×C4×D3—616no—
183C2×C4×D3—128no—
184C3×C4⋊C4—216no—
185C3×C4⋊C4#67116yes—
186C3×C42—216no—
187C3×C42—28no—
188(C2×D3)⋊C4—332no—
189(C2×D3)⋊C4—332no—
190(C2×D3)⋊C4—616no—
191(C2×D3)⋊C4—616no—
192(C2×D3)⋊C4—128no—
193C4⋊Dic3—68no—
194Dic3⋊C4—68no—
195Dic3⋊C4—68no—
196C4×Dic3—316no—
197C4×Dic3—68no—
198C4×Dic3—128no—
199C22×C32C23×C411yescharacteristic
200C22×C32C22×D411yescharacteristic
201C22×C32C2×#4111yescharacteristic
202C22×C32C2×#4111yescharacteristic
203C22×C32C2×#4111yescharacteristic
204C22×C32H(5,𝔽2)11yescharacteristic
205C22×C322−1+411yescharacteristic
206C2×C3⋊D3—24no—
207C2×C3⋊D3—24no—
208C2×C3⋊D3—24no—
209C2×C3⋊D3—42no—
210C2×C3⋊D3—42no—
211C2×C3⋊D3—42no—
212C2×C3⋊D3—42no—
213C32×C4—24no—
214C32×C4—24no—
215C32×C4—22no—
216C32×C4—41no—
217C32×C4C4×D414yes—
218C3⋊Dic3—22no—
219C3⋊Dic3—22no—
220C3⋊Dic3—22no—
221C3⋊Dic3—41no—
222C3⋊Dic3—42no—
223C22×Q8C2×C3⋊D311yescharacteristic
224C22×D4—91no—
225C23×C4—92no—
226#124—98no—
227C4×Q8—24no—
228C4×D4—184no—
229C2×C4⋊C4—91no—
230C2×C4⋊C4—91no—
231C2×C4⋊C4—91no—
232C2×C4⋊C4C2×C3⋊D312yes—
233C2×C4⋊C4C2×C3⋊D311yescharacteristic
234C2×C22⋊C4—92no—
235C2×C22⋊C4—92no—
236C2×C22⋊C4—92no—
237C2×C22⋊C4—92no—
238C2×C22⋊C4—92no—
239C2×C42—91no—
240C2×C42—91no—
241C2×C42—92no—
242C2×C42C2×C3⋊D312yes—
243C2×C42C2×C3⋊D311yescharacteristic
244#95—91no—
245#95—92no—
246#95—92no—
247#95—91no—
248#95—92no—
249C23×C3C23×D314yes—
250C22×D3—316no—
251C22×D3—68no—
252C22×D3—68no—
253C22×D3—68no—
254C22×D3—68no—
255C22×D3—68no—
256C22×D3—68no—
257C3×Q8—232no—
258C2×C3×C4—28no—
259C2×C3×C4—24no—
260C2×C3×C4—28no—
261C2×C3×C4—28no—
262C2×C3×C4—24no—
263C2×C3×C4—28no—
264C2×C3×C4—24no—
265C2×C3×C4—24no—
266C2×C3×C4—24no—
267C2×C3×C4C2×C4×D3116yes—
268C2×C3×C4C2×C4×D318yes—
269C2×C3×C4#23518yes—
270C2×C3×C4#23918yes—
271C2×C3×C4C2×#67116yes—
272C2×C3×C4C2×#6718yes—
273C2×Dic3—38no—
274C2×Dic3—38no—
275C2×Dic3—38no—
276C2×Dic3—64no—
277C2×Dic3—64no—
278C2×Dic3—64no—
279C2×Dic3—64no—
280C2×Dic3—64no—
281C2×Dic3—64no—
282C2×Dic3—68no—
283C2×Dic3—64no—
284C2×Dic3—68no—
285C2×Dic3—64no—
286C2×Dic3—68no—
287C2×Dic3—64no—
288D12—632no—
289C4×D3—1232no—
290C2×C32C2×C4×D411yescharacteristic
291C2×C32C4×#4111yescharacteristic
292C2×C32#52011yescharacteristic
293C2×C32#54611yescharacteristic
294C2×C32#54811yescharacteristic
295C2×C32#55611yescharacteristic
296C2×C32#55911yescharacteristic
297C3⋊D3—44no—
298C24—92no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8C4×C3⋊D314yescomplemented
302C2×D4—94no—
303C2×D4—182no—
304C2×D4—182no—
305C22×C4—91no—
306C22×C4—91no—
307C22×C4—91no—
308C22×C4—92no—
309C22×C4—91no—
310C22×C4—184no—
311C22×C4—184no—
312C22×C4—362no—
313C22×C4C22×C3⋊D311yescharacteristic
314C22×C4C22×C3⋊D312yes—
315C22×C4C22×C3⋊D311yescharacteristic
316C22×C4C22×C3⋊D312yes—
317C22×C4C22×C3⋊D311yescharacteristic
318C4⋊C4—24no—
319C4⋊C4—182no—
320C4⋊C4—182no—
321C4⋊C4—182no—
322C4⋊C4#63814yes—
323C22⋊C4—98no—
324C22⋊C4—98no—
325C22⋊C4—184no—
326C22⋊C4—184no—
327C22⋊C4—362no—
328C42—24no—
329C42—22no—
330C42—94no—
331C42—182no—
332C42—362no—
333C22×C3C22×C4×D314yes—
334C22×C3C2×#23514yes—
335C22×C3C2×#23914yes—
336C22×C3#100714yes—
337C22×C3D3×#4114yes—
338C22×C3#100914yes—
339C22×C3C22×#6714yes—
340C2×D3—616no—
341C2×D3—616no—
342C2×D3—616no—
343C2×D3—128no—
344C2×D3—128no—
345C2×D3—128no—
346C2×D3—128no—
347C3×C4—216no—
348C3×C4—28no—
349C3×C4—216no—
350C3×C4—44no—
351C3×C4C4×#67116yes—
352Dic3—68no—
353Dic3—68no—
354Dic3—68no—
355Dic3—124no—
356Dic3—128no—
357C32#236711yescharacteristic, complemented, Sylow
358C23—94no—
359C23—182no—
360C23—182no—
361C23—182no—
362C23—182no—
363C23—182no—
364C23—182no—
365C23C23×C3⋊D311yescharacteristic
366Q8—28no—
367D4—188no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—92no—
378C2×C4—92no—
379C2×C4—92no—
380C2×C4—181no—
381C2×C4—181no—
382C2×C4—181no—
383C2×C4—181no—
384C2×C4—181no—
385C2×C4—181no—
386C2×C4—182no—
387C2×C4—182no—
388C2×C4—181no—
389C2×C4—181no—
390C2×C4—182no—
391C2×C4—181no—
392C2×C4—368no—
393C2×C4C2×C4×C3⋊D314yes—
394C2×C4C2×C4×C3⋊D312yes—
395C2×C4#382812yes—
396C2×C4#383212yes—
397C2×C4C2×#63814yes—
398C2×C4C2×#63812yes—
399C2×C3#556814yes—
400C2×C3C4×#23914yes—
401C2×C3#557414yes—
402C2×C3#558514yes—
403C2×C3C2×C4×#6714yes—
404C2×C3#581714yes—
405C2×C3#582914yes—
406D3—1216no—
407C22—184no—
408C22—184no—
409C22—184no—
410C22—362no—
411C22—362no—
412C22—362no—
413C22—362no—
414C22C22×C4×C3⋊D311yescharacteristic
415C22C2×#382811yescharacteristic
416C22C2×#383211yescharacteristic
417C22#6436411yescharacteristic
418C22#41×C3⋊D311yescharacteristic
419C22#6436611yescharacteristic
420C22C22×#63811yescharacteristic
421C4—24no—
422C4—22no—
423C4—24no—
424C4—41no—
425C4—182no—
426C4—182no—
427C4—182no—
428C4—361no—
429C4—362no—
430C4C4×#63814yes—
431C3#8246614yescomplemented
432C2—364no—
433C2#1059603911yescharacteristic
434C2C4×#383211yescharacteristic
435C2#1059604511yescharacteristic
436C2#1059605611yescharacteristic
437C2C2×C4×#63811yescharacteristic
438C2#1059628811yescharacteristic
439C2#1059630011yescharacteristic
subgroupG/H
center, FrattiniC23C23×C3⋊D3
commutatorC22×C32C23×C4
FittingC2×C32×C4×Q8C2
socle, cosocleC23×C32C24
prime radicalC22×C4⋊(C3⋊D3)C22
hypercenterC2×C4×Q8C3⋊D3
hypocenterC32#2367
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C32×C4×Q8C2semidirect
4. #10592928C2semidirect
5. #10592689C2semidirect
6. #10592688C2semidirect
7. C4⋊C4×C3⋊Dic3C2semidirect
8. #10592639C2semidirect
10. C4×#3752C2semidirect
11. C4×#3750C2semidirect
21. #64156C4semidirect
39. C2×C3×C4×Q8D3semidirect
53. C2×C32×Q8C2×C4semidirect
77. #3752C2×C4semidirect
78. #3752C2×C4semidirect
84. C4×C3⋊Dic3C2×C4semidirect
154. C2×C4×Q8C3⋊D3semidirect
167. C2×C3×Q8C4×D3semidirect
301. C2×Q8C4×C3⋊D3semidirect
357. C32#2367semidirect
431. C3#82466semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4×Q82
C2×#641562
C2×#641022
#105929282
#105926892
#105926882
C4⋊C4×C3⋊Dic32
#105926392
#105926382
C4×#37522
C4×#37502
#824663