Finite Group Databasebeta

← #11865729

Subgroups — #11865729

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
1C32×#532C211yesmaximal, characteristic, complemented
2C2×#64159C212yesmaximal, complemented
3C4×#3828C211yesmaximal, characteristic, complemented
4#10592932C212yesmaximal, complemented
5#10592789C211yesmaximal, characteristic, complemented
6#10592782C211yesmaximal, characteristic, complemented
7#10592774C211yesmaximal, characteristic, complemented
8#10592771C211yesmaximal, characteristic, complemented
9#10592756C212yesmaximal, complemented
10#10592753C212yesmaximal, complemented
11C2×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
12#83296—34nomaximal
13C2×C32×#41C2212yes—
14C2×#3828C2211yescharacteristic
15C32×C4⋊Q8C2211yescharacteristic
16C32×#127C2211yescharacteristic
17C32×#124C2212yes—
18C32×#121—21no—
19C32×#121—21no—
20C32×C4×D4—21no—
21C32×C4×D4—21no—
22C2×C32×C42C2211yescharacteristic
23#64159—28no—
24C2×#3756C2212yes—
25C2×#3755C2212yes—
26C2×#3754C2212yes—
27C2×#3753C2212yes—
28C4×#638—21no—
29C4×#638—21no—
30#64135C2211yescharacteristic
31C22×C3⋊(C3⋊C8)C2212yes—
32C4×C4⋊(C3⋊D3)C2211yescharacteristic
33C4×C4⋊(C3⋊D3)C2211yescharacteristic
34#64081—21no—
35C42×C3⋊D3—21no—
36C4×#634C2211yescharacteristic
37C4×#634C2211yescharacteristic
38#63662C2212yes—
39#63662C2212yes—
40#63659C2212yes—
41#63659C2212yes—
42C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
43C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
44C4⋊(C3⋊(C3⋊C8))C2212yes—
45C3×#532D314yescomplemented
46C2×#950—38no—
47C4×#235—34no—
48#5233—38no—
49#5090—34no—
50#5083—34no—
51#5075—34no—
52#5072—34no—
53#5057—38no—
54#5054—38no—
55C2×C4⋊(C3⋊C8)—34no—
56C32×#41—24no—
57C32×#41—24no—
58C2×C32×Q8C2312yes—
59C2×C32×D4—22no—
60C2×C32×D4—21no—
61C2×C32×D4—21no—
62C2×C32×D4C2312yes—
63C22×C32×C4—22no—
64C22×C32×C4D411yescharacteristic
65C22×C32×C4D411yescharacteristic
66C22×C32×C4C2311yescharacteristic
67C2×#638—21no—
68#3828—22no—
69#3828D414yescomplemented
70C2×C4⋊(C3⋊D3)C2311yescharacteristic
71C2×C4×C3⋊D3—21no—
72C2×C4×C3⋊D3—21no—
73C2×C4×C3⋊D3—21no—
74C2×#634C2311yescharacteristic
75C32×C4⋊C4—21no—
76C32×C4⋊C4—21no—
77C32×C22⋊C4—22no—
78C32×C22⋊C4—22no—
79C32×C42D411yescharacteristic
80C32×C42D411yescharacteristic
81C32×C42C2311yescharacteristic
82C32×C42C2311yescharacteristic
83#3757—21no—
84#3756—24no—
85#3755—24no—
86#3754—24no—
87#3753—24no—
88#3752—21no—
89#3752—21no—
90C4⋊(C3⋊Dic3)C2311yescharacteristic
91C4⋊(C3⋊Dic3)C2311yescharacteristic
92#3750—21no—
93#3750—21no—
94C4×C3⋊Dic3—21no—
95C4×C3⋊Dic3—21no—
96C4×C3⋊Dic3—21no—
97C2×C3⋊(C3⋊C8)—24no—
98C2×C3⋊(C3⋊C8)C2312yes—
99C2×C3⋊(C3⋊C8)C2312yes—
100#3035—91noSylow
101C2×C3×#41C2×D318yes—
102C2×#235—34no—
103C3×C4⋊Q8C2×D314yes—
104C3×#127C2×D314yes—
105C3×#124C2×D318yes—
106C3×#121—24no—
107C3×#121—24no—
108C3×C4×D4—24no—
109C3×C4×D4—24no—
110C2×C3×C42C2×D314yes—
111#950—632no—
112C2×#216—38no—
113C2×#215—38no—
114C2×#214—38no—
115C2×#213—38no—
116C4×#67—64no—
117C4×#67—64no—
118#926—34no—
119C22×C3⋊C8—38no—
120C4×D12—34no—
121C4×D12—34no—
122#872—64no—
123C42×D3—64no—
124C4×Dic6—34no—
125C4×Dic6—34no—
126#835—38no—
127#835—38no—
128#832—38no—
129#832—38no—
130C4⋊(C3⋊C8)—38no—
131C4⋊(C3⋊C8)—34no—
132C4⋊(C3⋊C8)—34no—
133C23×C32—22no—
134C23×C32C2×D411yescharacteristic
135C22×C3⋊D3—21no—
136C32×Q8—22no—
137C32×Q8—22no—
138C32×D4—22no—
139C32×D4—22no—
140C32×D4—42no—
141C32×D4—44no—
142C32×D4—42no—
143C2×C32×C4—22no—
144C2×C32×C4—22no—
145C2×C32×C4—41no—
146C2×C32×C4—44no—
147C2×C32×C4C2×D412yes—
148C2×C32×C4C2×D411yescharacteristic
149C2×C32×C4C2×D411yescharacteristic
150C2×C32×C4C2×D411yescharacteristic
151C2×C32×C4C2×D411yescharacteristic
152C2×C32×C4C2×D411yescharacteristic
153C2×C32×C4#4112yes—
154C2×C32×C4C2411yescharacteristic
155#638—24no—
156C2×C3⋊Dic3—21no—
157C2×C3⋊Dic3—21no—
158C2×C3⋊Dic3—21no—
159C4⋊(C3⋊D3)—21no—
160C4⋊(C3⋊D3)C2×D412yes—
161C4×C3⋊D3—24no—
162C4×C3⋊D3—42no—
163C4×C3⋊D3—42no—
164#634—21no—
165#634C2×D412yes—
166C3⋊(C3⋊C8)—22no—
167C3⋊(C3⋊C8)—22no—
168C2×#135—92no—
169#532C3⋊D311yescharacteristic, complemented
170C4×#41—91no—
171#464—92no—
172#463—92no—
173#462—91no—
174#461—91no—
175#460—91no—
176#459—91no—
177C2×C4⋊C8—91no—
178#416—92no—
179C3×#41—216no—
180C3×#41—216no—
181C2×C3×Q8C22×D318yes—
182C2×C3×D4—28no—
183C2×C3×D4—24no—
184C2×C3×D4—24no—
185C2×C3×D4C22×D318yes—
186C22×C3×C4—28no—
187C22×C3×C4#6714yes—
188C22×C3×C4#6714yes—
189C22×C3×C4C22×D314yes—
190C2×#67—64no—
191#235—316no—
192#235—68no—
193C2×D12—34no—
194C2×C4×D3—64no—
195C2×C4×D3—64no—
196C2×C4×D3—64no—
197C2×Dic6—34no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—24no—
200C3×C22⋊C4—28no—
201C3×C22⋊C4—28no—
202C3×C42#6714yes—
203C3×C42#6714yes—
204C3×C42C22×D314yes—
205C3×C42C22×D314yes—
206#217—64no—
207#216—616no—
208#215—616no—
209#214—616no—
210#213—616no—
211(C2×D3)⋊C4—64no—
212(C2×D3)⋊C4—64no—
213C4⋊Dic3—34no—
214C4⋊Dic3—34no—
215Dic3⋊C4—64no—
216Dic3⋊C4—64no—
217C4×Dic3—64no—
218C4×Dic3—64no—
219C4×Dic3—64no—
220C2×C3⋊C8—38no—
221C2×C3⋊C8—38no—
222C2×C3⋊C8—616no—
223C22×C32—21no—
224C22×C32—21no—
225C22×C32—42no—
226C22×C32—42no—
227C22×C32—42no—
228C22×C32#12112yes—
229C22×C32C22×D411yescharacteristic
230C2×C3⋊D3—22no—
231C2×C3⋊D3—41no—
232C2×C3⋊D3—41no—
233C32×C4—21no—
234C32×C4—21no—
235C32×C4—21no—
236C32×C4—42no—
237C32×C4#12112yes—
238C32×C4#13512yescomplemented
239C32×C4C22×D411yescharacteristic
240C32×C4C2×#4111yescharacteristic
241C3⋊Dic3—22no—
242C3⋊Dic3—41no—
243C3⋊Dic3—41no—
244C2×#41—91no—
245C2×#41C2×C3⋊D312yes—
246#135—188no—
247C2×Q16—92no—
248C2×SD16—92no—
249C2×SD16—92no—
250C2×D8—92no—
251C22×C8—92no—
252C4⋊Q8C2×C3⋊D311yescharacteristic
253#127C2×C3⋊D311yescharacteristic
254#124C2×C3⋊D312yes—
255#121—21no—
256#121—21no—
257C4×Q8—91no—
258C4×Q8—91no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—91no—
262C4×D4—91no—
263C4×D4—181no—
264C4×D4—181no—
265#117—91no—
266#117—181no—
267C2×C42—181no—
268C2×C42C2×C3⋊D311yescharacteristic
269C4⋊C8—91no—
270C4⋊C8—91no—
271C4⋊C8—92no—
272#103—92no—
273#103—92no—
274#102—92no—
275#102—92no—
276C23×C3—28no—
277C23×C3C2×#6714yes—
278C22×D3—64no—
279C3×Q8—28no—
280C3×Q8—28no—
281C3×D4—28no—
282C3×D4—28no—
283C3×D4—48no—
284C3×D4—416no—
285C3×D4—48no—
286C2×C3×C4—28no—
287C2×C3×C4—28no—
288C2×C3×C4—44no—
289C2×C3×C4—416no—
290C2×C3×C4D3×D418yes—
291C2×C3×C4#23718yes—
292C2×C3×C4C2×#6714yes—
293C2×C3×C4C2×#6714yes—
294C2×C3×C4C2×#6714yes—
295C2×C3×C4C2×#6714yes—
296C2×C3×C4C2×#6714yes—
297C2×C3×C4C23×D314yes—
298#67—616no—
299C2×Dic3—64no—
300C2×Dic3—64no—
301C2×Dic3—64no—
302D12—38no—
303D12—64no—
304C4×D3—616no—
305C4×D3—128no—
306C4×D3—128no—
307Dic6—38no—
308Dic6—64no—
309C3⋊C8—68no—
310C3⋊C8—68no—
311C2×C32—22no—
312C2×C32—82no—
313C2×C32C2×#12111yescharacteristic
314C2×C32C2×#13511yescharacteristic
315C2×C32#57511yescharacteristic
316C3⋊D3—42no—
317#41—24no—
318#41—24no—
319#41—94no—
320#41—182no—
321C2×Q8—91no—
322C2×Q8C22×C3⋊D312yes—
323C2×D4—22no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—91no—
327C2×D4—181no—
328C2×D4C22×C3⋊D312yes—
329C22×C4—22no—
330C22×C4—181no—
331C22×C4—181no—
332C22×C4—181no—
333C22×C4#63811yescharacteristic
334C22×C4#63811yescharacteristic
335C22×C4C22×C3⋊D311yescharacteristic
336Q16—184no—
337SD16—184no—
338SD16—184no—
339D8—184no—
340C2×C8—92no—
341C2×C8—92no—
342C2×C8—184no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—91no—
346C4⋊C4—91no—
347C4⋊C4—181no—
348C4⋊C4—181no—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—181no—
352C22⋊C4—181no—
353C22⋊C4—181no—
354C42—181no—
355C42—181no—
356C42—181no—
357C42#63811yescharacteristic
358C42#63811yescharacteristic
359C42C22×C3⋊D311yescharacteristic
360C42C22×C3⋊D311yescharacteristic
361C22×C3—24no—
362C22×C3—24no—
363C22×C3—48no—
364C22×C3—48no—
365C22×C3—48no—
366C22×C3#93818yes—
367C22×C3C22×#6714yes—
368C2×D3—68no—
369C2×D3—124no—
370C2×D3—124no—
371C3×C4—24no—
372C3×C4—24no—
373C3×C4—24no—
374C3×C4—48no—
375C3×C4#93818yes—
376C3×C4#95018yescomplemented
377C3×C4C2×D3×D414yes—
378C3×C4C2×#23714yes—
379Dic3—68no—
380Dic3—124no—
381Dic3—124no—
382C32#303511yescharacteristic, complemented, Sylow
383C23—22no—
384C23—181no—
385C23C2×#63811yescharacteristic
386Q8—22no—
387Q8—22no—
388Q8—92no—
389Q8—181no—
390D4—22no—
391D4—22no—
392D4—42no—
393D4—44no—
394D4—42no—
395D4—92no—
396D4—184no—
397D4—181no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—41no—
401C2×C4—44no—
402C2×C4—181no—
403C2×C4—184no—
404C2×C4—181no—
405C2×C4—181no—
406C2×C4—362no—
407C2×C4—362no—
408C2×C4D4×C3⋊D312yes—
409C2×C4#383012yes—
410C2×C4C2×#63811yescharacteristic
411C2×C4C2×#63811yescharacteristic
412C2×C4C2×#63811yescharacteristic
413C2×C4C2×#63811yescharacteristic
414C2×C4C2×#63811yescharacteristic
415C2×C4C23×C3⋊D311yescharacteristic
416C8—182no—
417C8—182no—
418C2×C3—28no—
419C2×C3—88no—
420C2×C3C2×#93814yes—
421C2×C3C2×#95014yes—
422C2×C3#582214yes—
423D3—128no—
424C22—21no—
425C22—21no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—182no—
430C22—361no—
431C22—361no—
432C22#6414712yes—
433C22C22×#63811yescharacteristic
434C4—21no—
435C4—21no—
436C4—21no—
437C4—42no—
438C4—182no—
439C4—361no—
440C4—361no—
441C4#6414712yes—
442C4#6415912yescomplemented
443C4C2×D4×C3⋊D311yescharacteristic
444C4C2×#383011yescharacteristic
445C3#8329614yescomplemented
446C2—22no—
447C2—82no—
448C2—362no—
449C2C2×#6414711yescharacteristic
450C2C2×#6415911yescharacteristic
451C2#1059629311yescharacteristic
subgroupG/H
centerC2×C4C2×#638
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#532C2
socleC22×C32C22×D4
hypercenter#532C3⋊D3
hypocenterC32#3035
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C32×#532C2semidirect
2. C2×#64159C2semidirect
3. C4×#3828C2semidirect
4. #10592932C2semidirect
5. #10592789C2semidirect
6. #10592782C2semidirect
7. #10592774C2semidirect
8. #10592771C2semidirect
9. #10592756C2semidirect
10. #10592753C2semidirect
11. C2×C4⋊(C3⋊(C3⋊C8))C2semidirect
45. C3×#532D3semidirect
69. #3828D4semidirect
169. #532C3⋊D3semidirect
238. C32×C4#135semidirect
376. C3×C4#950semidirect
382. C32#3035semidirect
442. C4#64159semidirect
445. C3#83296semidirect

Maximal subgroups

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

subgroupindex
C32×#5322
C2×#641592
C4×#38282
#105929322
#105927892
#105927822
#105927742
#105927712
#105927562
#105927532
C2×C4⋊(C3⋊(C3⋊C8))2
#832963