Finite Group Databasebeta

← #11865753

Subgroups — #11865753

Order 1152.

1488 conjugacy classes of subgroups.

The subgroup lattice has 453 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64113C212yesmaximal, complemented
2#10595948C211yesmaximal, characteristic, complemented
3C4×#3828C211yesmaximal, characteristic, complemented
4C2×C32×C4⋊C8C211yesmaximal, characteristic, complemented
5#10592812C212yesmaximal, complemented
6#10592548C211yesmaximal, characteristic, complemented
7#10592547C211yesmaximal, characteristic, complemented
8#10592544C212yesmaximal, complemented
9#10592543C212yesmaximal, complemented
10#10592542C211yesmaximal, characteristic, complemented
11#10592540C211yesmaximal, characteristic, complemented
12#79021—34nomaximal
13C2×#3828C2212yes—
14C2×#3828C2211yescharacteristic
15C22×C32×C8C2212yes—
16C2×C32×C42C2211yescharacteristic
17#64139—21no—
18#64139—21no—
19C4×#638—21no—
20C4×#638—21no—
21#64135C2211yescharacteristic
22C2×#3746C2212yes—
23#64113—28no—
24C2×#3745C2212yes—
25C2×#3744C2212yes—
26C2×#3744C2212yes—
27#64084C2212yes—
28#64083C2211yescharacteristic
29C4×C4⋊(C3⋊D3)—21no—
30C4×C4⋊(C3⋊D3)—21no—
31C4×C4⋊(C3⋊D3)C2211yescharacteristic
32C4×C4⋊(C3⋊D3)C2211yescharacteristic
33#64081—21no—
34C42×C3⋊D3—21no—
35#64078C2211yescharacteristic
36C4×#634C2211yescharacteristic
37C4×#634C2211yescharacteristic
38C32×C4⋊C8C2211yescharacteristic
39C32×C4⋊C8C2211yescharacteristic
40C32×C4⋊C8C2212yes—
41#63648C2212yes—
42#63648C2212yes—
43#63643C2212yes—
44#63643C2212yes—
45C2×#904—38no—
46#5477—34no—
47C4×#235—34no—
48C2×C3×C4⋊C8D314yescomplemented
49#5113—38no—
50#4849—34no—
51#4848—34no—
52#4845—38no—
53#4844—38no—
54#4843—34no—
55#4841—34no—
56C22×C32×C4D411yescharacteristic
57C22×C32×C4D411yescharacteristic
58C22×C32×C4C2311yescharacteristic
59C2×#638—21no—
60C2×#638—22no—
61#3828—22no—
62#3828—24no—
63#3828—24no—
64#3828D414yescomplemented
65C2×C4⋊(C3⋊D3)—21no—
66C2×C4⋊(C3⋊D3)—21no—
67C2×C4⋊(C3⋊D3)C2312yes—
68C2×C4⋊(C3⋊D3)C2311yescharacteristic
69C2×C4×C3⋊D3—21no—
70C2×C4×C3⋊D3—21no—
71C2×C4×C3⋊D3—21no—
72C2×C4×C3⋊D3—22no—
73C2×#634C2312yes—
74C2×#634C2311yescharacteristic
75C2×C32×C8—24no—
76C2×C32×C8C2312yes—
77C2×C32×C8C2312yes—
78C32×C42D411yescharacteristic
79C32×C42D411yescharacteristic
80C32×C42C2311yescharacteristic
81C32×C42C2311yescharacteristic
82#3757—21no—
83#3752—21no—
84#3752—21no—
85#3752—22no—
86#3752—22no—
87C4⋊(C3⋊Dic3)—21no—
88C4⋊(C3⋊Dic3)—21no—
89C4⋊(C3⋊Dic3)C2311yescharacteristic
90C4⋊(C3⋊Dic3)C2311yescharacteristic
91#3750—21no—
92#3750—21no—
93C4×C3⋊Dic3—21no—
94C4×C3⋊Dic3—21no—
95C4×C3⋊Dic3—21no—
96#3746—24no—
97#3745—24no—
98#3744—24no—
99#3744—24no—
100#3035—91noSylow
101C2×#235—34no—
102C2×#235—38no—
103C22×C3×C8C2×D318yes—
104C2×C3×C42C2×D314yes—
105#930—64no—
106#930—64no—
107C4×#67—64no—
108C4×#67—64no—
109#926—34no—
110C2×Dic12—38no—
111#904—632no—
112C2×D24—38no—
113C2×#204—38no—
114C2×#204—38no—
115#875—38no—
116#874—34no—
117C4×D12—34no—
118C4×D12—34no—
119C4×D12—64no—
120C4×D12—64no—
121#872—64no—
122C42×D3—64no—
123#869—34no—
124C4×Dic6—34no—
125C4×Dic6—34no—
126C3×C4⋊C8C2×D314yes—
127C3×C4⋊C8C2×D314yes—
128C3×C4⋊C8C2×D318yes—
129#821—38no—
130#821—38no—
131#816—38no—
132#816—38no—
133C23×C32C2×D411yescharacteristic
134C22×C3⋊D3—21no—
135C22×C3⋊D3—22no—
136C2×C32×C4—22no—
137C2×C32×C4—41no—
138C2×C32×C4C2×D411yescharacteristic
139C2×C32×C4C2×D411yescharacteristic
140C2×C32×C4C2×D411yescharacteristic
141C2×C32×C4C2×D411yescharacteristic
142C2×C32×C4C2×D411yescharacteristic
143C2×C32×C4C2×D412yes—
144C2×C32×C4#4112yes—
145C2×C32×C4C2411yescharacteristic
146#638—24no—
147#638—44no—
148C2×C3⋊Dic3—21no—
149C2×C3⋊Dic3—21no—
150C2×C3⋊Dic3—21no—
151C2×C3⋊Dic3—22no—
152C4⋊(C3⋊D3)—22no—
153C4⋊(C3⋊D3)—21no—
154C4⋊(C3⋊D3)—22no—
155C4⋊(C3⋊D3)—42no—
156C4⋊(C3⋊D3)—42no—
157C4⋊(C3⋊D3)C2×D412yes—
158C4×C3⋊D3—24no—
159C4×C3⋊D3—42no—
160C4×C3⋊D3—44no—
161C4×C3⋊D3—42no—
162#634—21no—
163#634—22no—
164#634—22no—
165#634C2×D412yes—
166C32×C8—22no—
167C32×C8—22no—
168C2×#135—92no—
169#532—91no—
170C4×#41—91no—
171#464—92no—
172#463—92no—
173#462—91no—
174#461—91no—
175#460—91no—
176#459—91no—
177C2×C4⋊C8C3⋊D311yescharacteristic, complemented
178#416—92no—
179C22×C3×C4D1214yes—
180C22×C3×C4D1214yes—
181C22×C3×C4C22×D314yes—
182C2×#67—64no—
183C2×#67—68no—
184#235—316no—
185#235—68no—
186#235—616no—
187#235—616no—
188C2×D12—34no—
189C2×D12—38no—
190C2×D12—64no—
191C2×D12—64no—
192C2×C4×D3—64no—
193C2×C4×D3—64no—
194C2×C4×D3—64no—
195C2×C4×D3—68no—
196C2×Dic6—34no—
197C2×Dic6—38no—
198C2×C3×C8—216no—
199C2×C3×C8C22×D318yes—
200C2×C3×C8C22×D318yes—
201C3×C42D1214yes—
202C3×C42D1214yes—
203C3×C42C22×D314yes—
204C3×C42C22×D314yes—
205#217—64no—
206(C2×D3)⋊C4—64no—
207(C2×D3)⋊C4—64no—
208(C2×D3)⋊C4—68no—
209(C2×D3)⋊C4—68no—
210C4⋊Dic3—34no—
211C4⋊Dic3—34no—
212C4⋊Dic3—64no—
213C4⋊Dic3—64no—
214Dic3⋊C4—64no—
215Dic3⋊C4—64no—
216C4×Dic3—64no—
217C4×Dic3—64no—
218C4×Dic3—64no—
219Dic12—616no—
220D24—616no—
221#204—616no—
222#204—616no—
223C22×C32—21no—
224C22×C32—21no—
225C22×C32#12112yes—
226C22×C32C22×D411yescharacteristic
227C2×C3⋊D3—22no—
228C2×C3⋊D3—41no—
229C2×C3⋊D3—41no—
230C2×C3⋊D3—42no—
231C2×C3⋊D3—42no—
232C2×C3⋊D3—42no—
233C32×C4—21no—
234C32×C4—21no—
235C32×C4—21no—
236C32×C4#12112yes—
237C32×C4#13512yescomplemented
238C32×C4C22×D411yescharacteristic
239C32×C4C2×#4111yescharacteristic
240C3⋊Dic3—22no—
241C3⋊Dic3—42no—
242C3⋊Dic3—41no—
243C3⋊Dic3—41no—
244C2×#41—91no—
245C2×#41—92no—
246#135—188no—
247C2×Q16—92no—
248C2×SD16—92no—
249C2×SD16—92no—
250C2×D8—92no—
251C22×C8C2×C3⋊D312yes—
252C4⋊Q8—91no—
253#127—91no—
254#124—92no—
255#121—181no—
256#121—181no—
257C4×Q8—91no—
258C4×Q8—91no—
259C4×D4—91no—
260C4×D4—91no—
261C4×D4—181no—
262C4×D4—181no—
263C4×D4—181no—
264C4×D4—181no—
265#117—91no—
266#117—181no—
267C2×C42—181no—
268C2×C42C2×C3⋊D311yescharacteristic
269C4⋊C8C2×C3⋊D311yescharacteristic
270C4⋊C8C2×C3⋊D311yescharacteristic
271C4⋊C8C2×C3⋊D312yes—
272#103—92no—
273#103—92no—
274#102—92no—
275#102—92no—
276C23×C3C2×D1214yes—
277C22×D3—64no—
278C22×D3—68no—
279C2×C3×C4—28no—
280C2×C3×C4—44no—
281C2×C3×C4C2×D1214yes—
282C2×C3×C4C2×D1214yes—
283C2×C3×C4C2×D1214yes—
284C2×C3×C4C2×D1214yes—
285C2×C3×C4C2×D1214yes—
286C2×C3×C4D3×D418yes—
287C2×C3×C4#23918yes—
288C2×C3×C4C23×D314yes—
289#67—616no—
290#67—1216no—
291C2×Dic3—64no—
292C2×Dic3—64no—
293C2×Dic3—64no—
294C2×Dic3—68no—
295D12—38no—
296D12—68no—
297D12—64no—
298D12—68no—
299D12—128no—
300D12—128no—
301C4×D3—616no—
302C4×D3—128no—
303C4×D3—1216no—
304C4×D3—128no—
305Dic6—38no—
306Dic6—64no—
307Dic6—68no—
308Dic6—68no—
309C3×C8—28no—
310C3×C8—28no—
311C2×C32—22no—
312C2×C32C2×#12111yescharacteristic
313C2×C32C2×#13511yescharacteristic
314C2×C32#57511yescharacteristic
315C3⋊D3—42no—
316C3⋊D3—82no—
317#41—94no—
318#41—182no—
319#41—184no—
320#41—184no—
321C2×Q8—91no—
322C2×Q8—92no—
323C2×D4—91no—
324C2×D4—92no—
325C2×D4—181no—
326C2×D4—182no—
327C2×D4—181no—
328C2×D4—181no—
329C22×C4—181no—
330C22×C4—181no—
331C22×C4—181no—
332C22×C4—182no—
333C22×C4C4⋊(C3⋊D3)11yescharacteristic
334C22×C4C4⋊(C3⋊D3)11yescharacteristic
335C22×C4C22×C3⋊D311yescharacteristic
336Q16—184no—
337SD16—184no—
338SD16—184no—
339D8—184no—
340C2×C8—24no—
341C2×C8C22×C3⋊D312yes—
342C2×C8C22×C3⋊D312yes—
343C4⋊C4—91no—
344C4⋊C4—91no—
345C4⋊C4—181no—
346C4⋊C4—181no—
347C4⋊C4—181no—
348C4⋊C4—181no—
349C22⋊C4—181no—
350C22⋊C4—181no—
351C22⋊C4—181no—
352C22⋊C4—182no—
353C22⋊C4—182no—
354C42—181no—
355C42—181no—
356C42—181no—
357C42C4⋊(C3⋊D3)11yescharacteristic
358C42C4⋊(C3⋊D3)11yescharacteristic
359C42C22×C3⋊D311yescharacteristic
360C42C22×C3⋊D311yescharacteristic
361C22×C3—24no—
362C22×C3—24no—
363C22×C3#89518yes—
364C22×C3C22×D1214yes—
365C2×D3—68no—
366C2×D3—124no—
367C2×D3—124no—
368C2×D3—128no—
369C2×D3—128no—
370C2×D3—128no—
371C3×C4—24no—
372C3×C4—24no—
373C3×C4—24no—
374C3×C4#89518yes—
375C3×C4#90418yescomplemented
376C3×C4C2×D3×D414yes—
377C3×C4C2×#23914yes—
378Dic3—68no—
379Dic3—124no—
380Dic3—124no—
381Dic3—128no—
382C32#303511yescharacteristic, complemented, Sylow
383C23—181no—
384C23—182no—
385C23C2×C4⋊(C3⋊D3)11yescharacteristic
386Q8—92no—
387Q8—181no—
388Q8—182no—
389Q8—182no—
390D4—92no—
391D4—184no—
392D4—182no—
393D4—181no—
394D4—182no—
395D4—362no—
396D4—364no—
397D4—362no—
398C2×C4—22no—
399C2×C4—41no—
400C2×C4—181no—
401C2×C4—184no—
402C2×C4—181no—
403C2×C4—181no—
404C2×C4—182no—
405C2×C4—362no—
406C2×C4—364no—
407C2×C4—362no—
408C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
409C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
410C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
411C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
412C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
413C2×C4D4×C3⋊D312yes—
414C2×C4#383212yes—
415C2×C4C23×C3⋊D311yescharacteristic
416C8—22no—
417C8—22no—
418C2×C3—28no—
419C2×C3C2×#89514yes—
420C2×C3C2×#90414yes—
421C2×C3#574814yes—
422D3—128no—
423D3—248no—
424C22—21no—
425C22—21no—
426C22—182no—
427C22—361no—
428C22—361no—
429C22—362no—
430C22—362no—
431C22—362no—
432C22#6410412yes—
433C22C22×C4⋊(C3⋊D3)11yescharacteristic
434C4—21no—
435C4—21no—
436C4—21no—
437C4—182no—
438C4—361no—
439C4—361no—
440C4—362no—
441C4#6410412yes—
442C4#6411312yescomplemented
443C4C2×D4×C3⋊D311yescharacteristic
444C4C2×#383211yescharacteristic
445C3#7902114yescomplemented
446C2—22no—
447C2—362no—
448C2—722no—
449C2C2×#6410411yescharacteristic
450C2C2×#6411311yescharacteristic
451C2#1059621911yescharacteristic
subgroupG/H
centerC2×C4C2×C4⋊(C3⋊D3)
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC2×C32×C4⋊C8C2
socleC22×C32C22×D4
hypercenterC2×C4⋊C8C3⋊D3
hypocenterC32#3035
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4⋊C8

Complements

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

NH ≅ G/Ntype
1. C2×#64113C2semidirect
2. #10595948C2semidirect
3. C4×#3828C2semidirect
4. C2×C32×C4⋊C8C2semidirect
5. #10592812C2semidirect
6. #10592548C2semidirect
7. #10592547C2semidirect
8. #10592544C2semidirect
9. #10592543C2semidirect
10. #10592542C2semidirect
11. #10592540C2semidirect
48. C2×C3×C4⋊C8D3semidirect
64. #3828D4semidirect
177. C2×C4⋊C8C3⋊D3semidirect
237. C32×C4#135semidirect
375. C3×C4#904semidirect
382. C32#3035semidirect
442. C4#64113semidirect
445. C3#79021semidirect

Maximal subgroups

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

subgroupindex
C2×#641132
#105959482
C4×#38282
C2×C32×C4⋊C82
#105928122
#105925482
#105925472
#105925442
#105925432
#105925422
#105925402
#790213