Finite Group Databasebeta

← #11808371

Subgroups — #11808371

Order 1152.

2088 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#117C211yesmaximal, characteristic, complemented
2C2×C4×#638C212yesmaximal, complemented
3C2×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
4#10592693C212yesmaximal, complemented
5#10592690C211yesmaximal, characteristic, complemented
6#10592680C211yesmaximal, characteristic, complemented
7#10592660C212yesmaximal, complemented
8#10592659C211yesmaximal, characteristic
9#10592652C211yesmaximal, characteristic, complemented
10C4×#3752C212yesmaximal, complemented
11C4×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
12#82273—34nomaximal
13C23×C32×C4C2211yescharacteristic
14C22×#638C2212yes—
15C22×C4⋊(C3⋊D3)C2211yescharacteristic
16C22×C4×C3⋊D3C2212yes—
17C32×#117—24no—
18C2×C32×C4⋊C4C2211yescharacteristic
19C2×C32×C4⋊C4C2211yescharacteristic
20C2×C32×C22⋊C4C2211yescharacteristic
21C2×C32×C22⋊C4C2211yescharacteristic
22C2×C32×C42C2211yescharacteristic
23C2×C32×C42C2211yescharacteristic
24C2×#3757C2212yes—
25C4×#638—28no—
26C2×#3752C2212yes—
27C2×#3752C2212yes—
28C2×#3752C2212yes—
29C2×#3752C2212yes—
30C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
31C2×C4⋊(C3⋊Dic3)C2212yes—
32C2×#3750C2212yes—
33C2×C4×C3⋊Dic3C2212yes—
34C4×C4⋊(C3⋊D3)—24no—
35#63658C2211yescharacteristic
36#63658C2211yescharacteristic
37#63658C2212yes—
38#63658C2212yes—
39C2×C3×#117D314yescomplemented
40C2×C4×#67—38no—
41C2×C4×D12—34no—
42#4994—38no—
43#4991—34no—
44#4981—34no—
45#4961—38no—
46#4960—34no—
47#4953—34no—
48C4×(C2×D3)⋊C4—38no—
49C4×C4⋊Dic3—34no—
50C24×C32C2311yescharacteristic
51C23×C3⋊D3C2312yes—
52C22×C32×C4—22no—
53C22×C32×C4—22no—
54C22×C32×C4D414yescomplemented
55C22×C32×C4C2311yescharacteristic
56C22×C32×C4C2311yescharacteristic
57C22×C32×C4C2311yescharacteristic
58C22×C32×C4C2311yescharacteristic
59C22×C32×C4C2311yescharacteristic
60C22×C32×C4C2311yescharacteristic
61C2×#638—24no—
62C2×#638—24no—
63C2×#638—24no—
64C22×C3⋊Dic3C2312yes—
65C22×C3⋊Dic3C2312yes—
66C22×C3⋊Dic3C2312yes—
67C2×C4⋊(C3⋊D3)—24no—
68C2×C4⋊(C3⋊D3)—22no—
69C2×C4×C3⋊D3—24no—
70C2×C4×C3⋊D3—24no—
71C2×C4×C3⋊D3—24no—
72C32×C4⋊C4—22no—
73C32×C4⋊C4—22no—
74C32×C22⋊C4—22no—
75C32×C22⋊C4—22no—
76C32×C42—22no—
77C32×C42D414yescomplemented
78#3757—24no—
79#3752—24no—
80#3752—24no—
81#3752—24no—
82#3752—24no—
83C4⋊(C3⋊Dic3)—22no—
84C4⋊(C3⋊Dic3)—24no—
85#3750—24no—
86C4×C3⋊Dic3—24no—
87#2392—91noSylow
88C23×C3×C4C2×D314yes—
89C22×#67—38no—
90C22×D12—34no—
91C22×C4×D3—38no—
92C3×#117—216no—
93C2×C3×C4⋊C4C2×D314yes—
94C2×C3×C4⋊C4C2×D314yes—
95C2×C3×C22⋊C4C2×D314yes—
96C2×C3×C22⋊C4C2×D314yes—
97C2×C3×C42C2×D314yes—
98C2×C3×C42C2×D314yes—
99C2×#217—38no—
100C4×#67—632no—
101C2×(C2×D3)⋊C4—38no—
102C2×(C2×D3)⋊C4—38no—
103C2×(C2×D3)⋊C4—38no—
104C2×(C2×D3)⋊C4—38no—
105C2×C4⋊Dic3—38no—
106C2×C4⋊Dic3—34no—
107C2×Dic3⋊C4—38no—
108C2×C4×Dic3—38no—
109C4×D12—616no—
110C3⋊#95—34no—
111C3⋊#95—34no—
112C3⋊#95—38no—
113C3⋊#95—38no—
114C23×C32—21no—
115C23×C32—22no—
116C23×C32—22no—
117C23×C32—21no—
118C23×C32C2×D412yes—
119C23×C32C2411yescharacteristic
120C22×C3⋊D3—22no—
121C22×C3⋊D3—22no—
122C22×C3⋊D3—22no—
123C22×C3⋊D3—22no—
124C22×C3⋊D3—22no—
125C22×C3⋊D3—22no—
126C22×C3⋊D3#4114yes—
127C2×C32×C4—21no—
128C2×C32×C4—22no—
129C2×C32×C4—22no—
130C2×C32×C4—21no—
131C2×C32×C4—28no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—22no—
135C2×C32×C4—22no—
136C2×C32×C4C2×D412yes—
137C2×C32×C4C2×D412yes—
138C2×C32×C4C2×D412yes—
139C2×C32×C4C2×D412yes—
140C2×C32×C4C2×D412yes—
141C2×C32×C4#4112yes—
142#638—216no—
143C2×C3⋊Dic3—22no—
144C2×C3⋊Dic3—22no—
145C2×C3⋊Dic3—22no—
146C2×C3⋊Dic3—22no—
147C2×C3⋊Dic3—22no—
148C2×C3⋊Dic3—22no—
149C2×C3⋊Dic3—22no—
150C2×C3⋊Dic3—22no—
151C2×C3⋊Dic3#4114yes—
152C4⋊(C3⋊D3)—28no—
153C4×C3⋊D3—48no—
154C2×C4×D4—92no—
155C2×C4×D4—91no—
156C2×#117C3⋊D311yescharacteristic, complemented
157#398—91no—
158#397—91no—
159#396—92no—
160#394—91no—
161#394—92no—
162#392—91no—
163C4×C4⋊C4—91no—
164C4×C22⋊C4—92no—
165C24×C3C22×D314yes—
166C23×D3—38no—
167C22×C3×C4—28no—
168C22×C3×C4—28no—
169C22×C3×C4D12116yescomplemented
170C22×C3×C4C22×D314yes—
171C22×C3×C4C22×D314yes—
172C22×C3×C4C22×D314yes—
173C22×C3×C4C22×D314yes—
174C22×C3×C4C22×D314yes—
175C22×C3×C4C22×D314yes—
176C2×#67—616no—
177C2×#67—616no—
178C2×#67—616no—
179C22×Dic3—38no—
180C22×Dic3—38no—
181C22×Dic3—38no—
182C2×D12—616no—
183C2×D12—68no—
184C2×C4×D3—616no—
185C2×C4×D3—616no—
186C2×C4×D3—616no—
187C3×C4⋊C4—28no—
188C3×C4⋊C4—28no—
189C3×C22⋊C4—28no—
190C3×C22⋊C4—28no—
191C3×C42—28no—
192C3×C42#67116yescomplemented
193#217—616no—
194(C2×D3)⋊C4—616no—
195(C2×D3)⋊C4—616no—
196(C2×D3)⋊C4—616no—
197(C2×D3)⋊C4—616no—
198C4⋊Dic3—616no—
199C4⋊Dic3—68no—
200Dic3⋊C4—616no—
201C4×Dic3—616no—
202C22×C32—22no—
203C22×C32—24no—
204C22×C32—41no—
205C22×C32—42no—
206C22×C32—41no—
207C22×C32C22×D411yescharacteristic
208C22×C32C22×D411yescharacteristic
209C22×C32C2×#4111yescharacteristic
210C22×C32C2×#4112yes—
211C22×C32C2×#4112yes—
212C2×C3⋊D3—24no—
213C2×C3⋊D3—24no—
214C2×C3⋊D3—24no—
215C2×C3⋊D3—42no—
216C2×C3⋊D3—42no—
217C2×C3⋊D3—42no—
218C2×C3⋊D3—42no—
219C32×C4—22no—
220C32×C4—22no—
221C32×C4—22no—
222C32×C4—41no—
223C32×C4—41no—
224C32×C4#12114yes—
225C3⋊Dic3—24no—
226C3⋊Dic3—42no—
227C3⋊Dic3—42no—
228C22×D4—92no—
229C22×D4—91no—
230C23×C4—92no—
231C23×C4C2×C3⋊D311yescharacteristic
232C4×D4—188no—
233C4×D4—184no—
234#117—24no—
235C2×C4⋊C4—92no—
236C2×C4⋊C4—91no—
237C2×C4⋊C4—92no—
238C2×C4⋊C4C2×C3⋊D311yescharacteristic
239C2×C4⋊C4C2×C3⋊D311yescharacteristic
240C2×C22⋊C4—92no—
241C2×C22⋊C4—92no—
242C2×C22⋊C4—92no—
243C2×C22⋊C4—92no—
244C2×C22⋊C4—92no—
245C2×C22⋊C4C2×C3⋊D311yescharacteristic
246C2×C22⋊C4C2×C3⋊D311yescharacteristic
247C2×C42—92no—
248C2×C42C2×C3⋊D311yescharacteristic
249C2×C42C2×C3⋊D311yescharacteristic
250#95—92no—
251#95—92no—
252#95—91no—
253#95—91no—
254C23×C3—24no—
255C23×C3—28no—
256C23×C3—28no—
257C23×C3—24no—
258C23×C3C2×D1218yes—
259C23×C3C23×D314yes—
260C22×D3—316no—
261C22×D3—68no—
262C22×D3—68no—
263C22×D3—68no—
264C22×D3—68no—
265C22×D3—68no—
266C22×D3—68no—
267C2×C3×C4—24no—
268C2×C3×C4—28no—
269C2×C3×C4—28no—
270C2×C3×C4—24no—
271C2×C3×C4—232no—
272C2×C3×C4—28no—
273C2×C3×C4—28no—
274C2×C3×C4—24no—
275C2×C3×C4—24no—
276C2×C3×C4C2×D1218yes—
277C2×C3×C4C2×D1218yes—
278C2×C3×C4#23518yes—
279C2×C3×C4C2×#6718yes—
280C2×C3×C4C2×#6718yes—
281C2×C3×C4C2×#6718yes—
282#67—664no—
283C2×Dic3—316no—
284C2×Dic3—68no—
285C2×Dic3—68no—
286C2×Dic3—68no—
287C2×Dic3—68no—
288C2×Dic3—68no—
289C2×Dic3—68no—
290C2×Dic3—68no—
291C2×Dic3—68no—
292D12—632no—
293C4×D3—1232no—
294C2×C32—42no—
295C2×C32C2×#12111yescharacteristic
296C2×C32#52512yes—
297C2×C32#52511yescharacteristic
298C2×C32#52912yes—
299C2×C32#53211yescharacteristic
300C3⋊D3—44no—
301C24—92no—
302C24C22×C3⋊D311yescharacteristic
303C2×D4—184no—
304C2×D4—184no—
305C2×D4—184no—
306C2×D4—184no—
307C2×D4—182no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—92no—
311C22×C4—92no—
312C22×C4—92no—
313C22×C4—184no—
314C22×C4—184no—
315C22×C4—184no—
316C22×C4C4⋊(C3⋊D3)14yescomplemented
317C22×C4C22×C3⋊D311yescharacteristic
318C22×C4C22×C3⋊D311yescharacteristic
319C22×C4C22×C3⋊D311yescharacteristic
320C22×C4C22×C3⋊D311yescharacteristic
321C22×C4C22×C3⋊D311yescharacteristic
322C22×C4C22×C3⋊D311yescharacteristic
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—184no—
326C4⋊C4—184no—
327C4⋊C4—182no—
328C22⋊C4—22no—
329C22⋊C4—22no—
330C22⋊C4—184no—
331C22⋊C4—184no—
332C22⋊C4—184no—
333C22⋊C4—184no—
334C22⋊C4—184no—
335C42—22no—
336C42—184no—
337C42#63814yescomplemented
338C22×C3—216no—
339C22×C3—28no—
340C22×C3—48no—
341C22×C3—44no—
342C22×C3—44no—
343C22×C3C22×D1214yes—
344C22×C3C2×#23514yes—
345C22×C3D3×#4118yes—
346C22×C3D3×#4118yes—
347C22×C3C22×#6714yes—
348C2×D3—616no—
349C2×D3—616no—
350C2×D3—616no—
351C2×D3—128no—
352C2×D3—128no—
353C2×D3—128no—
354C2×D3—128no—
355C3×C4—28no—
356C3×C4—28no—
357C3×C4—28no—
358C3×C4—44no—
359C3×C4—44no—
360C3×C4#930116yes—
361Dic3—616no—
362Dic3—128no—
363Dic3—128no—
364C32#239211yescharacteristic, complemented, Sylow
365C23—21no—
366C23—22no—
367C23—22no—
368C23—21no—
369C23—94no—
370C23—182no—
371C23—182no—
372C23—182no—
373C23—182no—
374C23—182no—
375C23—182no—
376C23C2×C4⋊(C3⋊D3)12yes—
377C23C23×C3⋊D311yescharacteristic
378D4—188no—
379D4—1816no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—28no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—94no—
390C2×C4—182no—
391C2×C4—182no—
392C2×C4—182no—
393C2×C4—182no—
394C2×C4—182no—
395C2×C4—182no—
396C2×C4—182no—
397C2×C4—182no—
398C2×C4—368no—
399C2×C4C2×C4⋊(C3⋊D3)12yes—
400C2×C4C2×C4⋊(C3⋊D3)12yes—
401C2×C4#382812yes—
402C2×C4C2×#63812yes—
403C2×C4C2×#63812yes—
404C2×C4C2×#63812yes—
405C2×C3—48no—
406C2×C3#552418yes—
407C2×C3#552818yes—
408C2×C3C2×#93014yes—
409C2×C3#582814yes—
410C2×C3#583214yes—
411D3—1216no—
412C22—24no—
413C22—22no—
414C22—42no—
415C22—41no—
416C22—41no—
417C22—184no—
418C22—184no—
419C22—184no—
420C22—362no—
421C22—362no—
422C22—362no—
423C22—362no—
424C22C22×C4⋊(C3⋊D3)11yescharacteristic
425C22C2×#382811yescharacteristic
426C22#41×C3⋊D312yes—
427C22#41×C3⋊D312yes—
428C22C22×#63811yescharacteristic
429C4—22no—
430C4—22no—
431C4—22no—
432C4—41no—
433C4—41no—
434C4—184no—
435C4—362no—
436C4—362no—
437C4#6413914yes—
438C3#8227314yescomplemented
439C2—42no—
440C2—364no—
441C2#1059599512yes—
442C2#1059599912yes—
443C2C2×#6413911yescharacteristic
444C2#1059629911yescharacteristic
445C2#1059630311yescharacteristic
subgroupG/H
centerC22×C4C22×C3⋊D3
commutator, socle, cosocleC23×C32C24
FrattiniC23C23×C3⋊D3
FittingC2×C32×#117C2
prime radical#10592659C2
hypercenterC2×#117C3⋊D3
hypocenterC32#2392
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C22×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×C32×#117C2semidirect
2. C2×C4×#638C2semidirect
3. C2×C4×C4⋊(C3⋊D3)C2semidirect
4. #10592693C2semidirect
5. #10592690C2semidirect
6. #10592680C2semidirect
7. #10592660C2semidirect
9. #10592652C2semidirect
10. C4×#3752C2semidirect
11. C4×C4⋊(C3⋊Dic3)C2semidirect
39. C2×C3×#117D3semidirect
54. C22×C32×C4D4semidirect
77. C32×C42D4semidirect
156. C2×#117C3⋊D3semidirect
169. C22×C3×C4D12semidirect
192. C3×C42#67semidirect
316. C22×C4C4⋊(C3⋊D3)semidirect
337. C42#638semidirect
364. C32#2392semidirect
438. C3#82273semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#1172
C2×C4×#6382
C2×C4×C4⋊(C3⋊D3)2
#105926932
#105926902
#105926802
#105926602
#105926592
#105926522
C4×#37522
C4×C4⋊(C3⋊Dic3)2
#822733