Finite Group Databasebeta

← #11882291

Subgroups — #11882291

Order 1152.

1194 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#532C211yesmaximal, characteristic, complemented
2#10595950C211yesmaximal, characteristic, complemented
3#10592790C211yesmaximal, characteristic, complemented
4#10592783C211yesmaximal, characteristic, complemented
5#10592774C211yesmaximal, characteristic, complemented
6#10592771C211yesmaximal, characteristic, complemented
7#10592740C212yesmaximal, complemented
8#10592737C212yesmaximal, complemented
9#10592715C212yesmaximal, complemented
10#10592708C212yesmaximal, complemented
11C2×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
12#83311—34nomaximal
13C2×C32×#41—21no—
14C32×C4⋊Q8C2211yescharacteristic
15C32×#127C2211yescharacteristic
16C32×#124—21no—
17C32×#121—21no—
18C32×#121C2212yes—
19C32×C4×D4—21no—
20C32×C4×D4C2212yes—
21C2×C32×C42C2211yescharacteristic
22C2×#3754C2212yes—
23C2×#3753C2212yes—
24#64139C2211yescharacteristic
25#64138—21no—
26C4×#638—21no—
27#64133C2211yescharacteristic
28C22×C3⋊(C3⋊C8)C2212yes—
29#64085—21no—
30#64084C2211yescharacteristic
31C4×C4⋊(C3⋊D3)C2211yescharacteristic
32#64081—21no—
33#64079C2211yescharacteristic
34C4×#634C2211yescharacteristic
35#63659C2212yes—
36#63659C2212yes—
37#63637C2212yes—
38#63636C2212yes—
39#63635C2212yes—
40#63634C2212yes—
41C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
42C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
43C4⋊(C3⋊(C3⋊C8))C2212yes—
44C3×#532D314yescomplemented
45#5479—34no—
46#5091—34no—
47#5084—34no—
48#5075—34no—
49#5072—34no—
50#5041—38no—
51#5038—38no—
52#5016—38no—
53#5009—38no—
54C2×C4⋊(C3⋊C8)—34no—
55C32×#41—44no—
56C2×C32×Q8—21no—
57C2×C32×D4—22no—
58C2×C32×D4—21no—
59C2×C32×D4—21no—
60C2×C32×D4C2312yes—
61C22×C32×C4—22no—
62C22×C32×C4D411yescharacteristic
63C22×C32×C4D411yescharacteristic
64C22×C32×C4C2311yescharacteristic
65C2×#638—21no—
66C2×C4⋊(C3⋊D3)C2311yescharacteristic
67C2×C4×C3⋊D3—21no—
68C2×#634C2311yescharacteristic
69C32×C4⋊C4—21no—
70C32×C4⋊C4C2312yes—
71C32×C22⋊C4—22no—
72C32×C22⋊C4—22no—
73C32×C42D411yescharacteristic
74C32×C42D411yescharacteristic
75C32×C42C2311yescharacteristic
76C32×C42C2311yescharacteristic
77#3757—21no—
78#3754—42no—
79#3753—42no—
80#3752—21no—
81#3752—21no—
82#3752—21no—
83#3752—21no—
84C4⋊(C3⋊Dic3)C2311yescharacteristic
85C4⋊(C3⋊Dic3)C2311yescharacteristic
86#3750—21no—
87#3750—21no—
88#3750—21no—
89#3750—21no—
90C4×C3⋊Dic3—21no—
91C2×C3⋊(C3⋊C8)—42no—
92C2×C3⋊(C3⋊C8)C2312yes—
93C2×C3⋊(C3⋊C8)C2312yes—
94#3247—91noSylow
95C2×C3×#41—24no—
96C3×C4⋊Q8C2×D314yes—
97C3×#127C2×D314yes—
98C3×#124—24no—
99C3×#121—24no—
100C3×#121C2×D318yes—
101C3×C4×D4—24no—
102C3×C4×D4C2×D318yes—
103C2×C3×C42C2×D314yes—
104C2×#214—38no—
105C2×#213—38no—
106#930—34no—
107#929—64no—
108C4×#67—64no—
109#924—34no—
110C22×C3⋊C8—38no—
111#876—64no—
112#875—34no—
113C4×D12—34no—
114#872—64no—
115#870—34no—
116C4×Dic6—34no—
117#832—38no—
118#832—38no—
119#810—38no—
120#809—38no—
121#808—38no—
122#807—38no—
123C4⋊(C3⋊C8)—38no—
124C4⋊(C3⋊C8)—34no—
125C4⋊(C3⋊C8)—34no—
126C23×C32—22no—
127C23×C32C2×D411yescharacteristic
128C22×C3⋊D3—21no—
129C32×Q8—42no—
130C32×D4—22no—
131C32×D4—22no—
132C32×D4—42no—
133C32×D4—44no—
134C32×D4—42no—
135C2×C32×C4—22no—
136C2×C32×C4—22no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—22no—
140C2×C32×C4—44no—
141C2×C32×C4C2×D411yescharacteristic
142C2×C32×C4C2×D411yescharacteristic
143C2×C32×C4C2×D411yescharacteristic
144C2×C32×C4C2×D411yescharacteristic
145C2×C32×C4C2×D411yescharacteristic
146C2×C32×C4C2411yescharacteristic
147#638—42no—
148C2×C3⋊Dic3—21no—
149C2×C3⋊Dic3—21no—
150C2×C3⋊Dic3—21no—
151C4⋊(C3⋊D3)—21no—
152C4⋊(C3⋊D3)—21no—
153C4×C3⋊D3—42no—
154#634—21no—
155#634—21no—
156C3⋊(C3⋊C8)—22no—
157C3⋊(C3⋊C8)—22no—
158#532C3⋊D311yescharacteristic, complemented
159#529—91no—
160#478—92no—
161#477—91no—
162#476—91no—
163#466—92no—
164#465—92no—
165#463—92no—
166#460—91no—
167#459—91no—
168C2×C4⋊C8—91no—
169C3×#41—416no—
170C2×C3×Q8—24no—
171C2×C3×D4—28no—
172C2×C3×D4—24no—
173C2×C3×D4—24no—
174C2×C3×D4C22×D318yes—
175C22×C3×C4—28no—
176C22×C3×C4#6714yes—
177C22×C3×C4#6714yes—
178C22×C3×C4C22×D314yes—
179C2×#67—64no—
180C2×D12—34no—
181C2×C4×D3—64no—
182C2×Dic6—34no—
183C3×C4⋊C4—24no—
184C3×C4⋊C4C22×D318yes—
185C3×C22⋊C4—28no—
186C3×C22⋊C4—28no—
187C3×C42#6714yes—
188C3×C42#6714yes—
189C3×C42C22×D314yes—
190C3×C42C22×D314yes—
191#217—64no—
192#214—128no—
193#213—128no—
194(C2×D3)⋊C4—64no—
195(C2×D3)⋊C4—64no—
196(C2×D3)⋊C4—64no—
197(C2×D3)⋊C4—64no—
198C4⋊Dic3—34no—
199C4⋊Dic3—34no—
200Dic3⋊C4—64no—
201Dic3⋊C4—64no—
202Dic3⋊C4—64no—
203Dic3⋊C4—64no—
204C4×Dic3—64no—
205C2×C3⋊C8—38no—
206C2×C3⋊C8—38no—
207C2×C3⋊C8—128no—
208C22×C32—21no—
209C22×C32—21no—
210C22×C32—42no—
211C22×C32—42no—
212C22×C32—42no—
213C22×C32AGL(1,ℤ/8)12yes—
214C22×C32C22×D411yescharacteristic
215C2×C3⋊D3—41no—
216C2×C3⋊D3—41no—
217C2×C3⋊D3—41no—
218C32×C4—21no—
219C32×C4—21no—
220C32×C4—21no—
221C32×C4—21no—
222C32×C4—42no—
223C32×C4#13512yes—
224C32×C4H(5,𝔽2)11yescharacteristic
225C32×C42−1+411yescharacteristic
226C3⋊Dic3—41no—
227C3⋊Dic3—41no—
228C3⋊Dic3—41no—
229C2×#41—21no—
230C2×SD16—92no—
231C2×D8—92no—
232C22×C8—92no—
233C4⋊Q8C2×C3⋊D311yescharacteristic
234#127C2×C3⋊D311yescharacteristic
235#126—181no—
236#125—91no—
237#124—21no—
238#124—91no—
239#123—181no—
240C22⋊Q8—91no—
241#121—21no—
242#121—91no—
243#121C2×C3⋊D312yes—
244C4×Q8—91no—
245C4×D4—21no—
246C4×D4—91no—
247C4×D4—181no—
248C4×D4C2×C3⋊D312yes—
249#117—181no—
250C2×C42C2×C3⋊D311yescharacteristic
251#107—92no—
252#106—92no—
253C4⋊C8—91no—
254C4⋊C8—91no—
255C4⋊C8—92no—
256#103—92no—
257#102—92no—
258#102—92no—
259#102—92no—
260C23×C3—28no—
261C23×C3C2×#6714yes—
262C22×D3—64no—
263C3×Q8—48no—
264C3×D4—28no—
265C3×D4—28no—
266C3×D4—48no—
267C3×D4—416no—
268C3×D4—48no—
269C2×C3×C4—28no—
270C2×C3×C4—28no—
271C2×C3×C4—24no—
272C2×C3×C4—24no—
273C2×C3×C4—28no—
274C2×C3×C4—416no—
275C2×C3×C4C2×#6714yes—
276C2×C3×C4C2×#6714yes—
277C2×C3×C4C2×#6714yes—
278C2×C3×C4C2×#6714yes—
279C2×C3×C4C2×#6714yes—
280C2×C3×C4C23×D314yes—
281#67—128no—
282C2×Dic3—64no—
283C2×Dic3—64no—
284C2×Dic3—64no—
285D12—64no—
286D12—64no—
287C4×D3—128no—
288Dic6—64no—
289Dic6—64no—
290C3⋊C8—68no—
291C3⋊C8—68no—
292C2×C32—22no—
293C2×C32—82no—
294C2×C32#53711yescharacteristic
295C2×C32C2×#13511yescharacteristic
296C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
297C3⋊D3—81no—
298#41—44no—
299C2×Q8—21no—
300C2×Q8—91no—
301C2×D4—22no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—91no—
305C2×D4—181no—
306C2×D4C22×C3⋊D312yes—
307C22×C4—22no—
308C22×C4—181no—
309C22×C4#63811yescharacteristic
310C22×C4#63811yescharacteristic
311C22×C4C22×C3⋊D311yescharacteristic
312SD16—362no—
313D8—362no—
314C2×C8—92no—
315C2×C8—92no—
316C2×C8—362no—
317C4⋊C4—21no—
318C4⋊C4—91no—
319C4⋊C4—91no—
320C4⋊C4—181no—
321C4⋊C4—181no—
322C4⋊C4—181no—
323C4⋊C4—181no—
324C4⋊C4C22×C3⋊D312yes—
325C22⋊C4—22no—
326C22⋊C4—22no—
327C22⋊C4—181no—
328C22⋊C4—181no—
329C22⋊C4—181no—
330C22⋊C4—181no—
331C22⋊C4—181no—
332C42—181no—
333C42#63811yescharacteristic
334C42#63811yescharacteristic
335C42C22×C3⋊D311yescharacteristic
336C42C22×C3⋊D311yescharacteristic
337C22×C3—24no—
338C22×C3—24no—
339C22×C3—48no—
340C22×C3—48no—
341C22×C3—48no—
342C22×C3#93218yes—
343C22×C3C22×#6714yes—
344C2×D3—124no—
345C2×D3—124no—
346C2×D3—124no—
347C3×C4—24no—
348C3×C4—24no—
349C3×C4—24no—
350C3×C4—24no—
351C3×C4—48no—
352C3×C4#95018yes—
353C3×C4#100914yes—
354C3×C4#101014yes—
355Dic3—124no—
356Dic3—124no—
357Dic3—124no—
358C32#324711yescharacteristic, complemented, Sylow
359C23—22no—
360C23—181no—
361C23C2×#63811yescharacteristic
362Q8—42no—
363Q8—181no—
364Q8—181no—
365D4—22no—
366D4—22no—
367D4—42no—
368D4—44no—
369D4—42no—
370D4—181no—
371D4—181no—
372D4—362no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—44no—
379C2×C4—181no—
380C2×C4—181no—
381C2×C4—181no—
382C2×C4—362no—
383C2×C4C2×#63811yescharacteristic
384C2×C4C2×#63811yescharacteristic
385C2×C4C2×#63811yescharacteristic
386C2×C4C2×#63811yescharacteristic
387C2×C4C2×#63811yescharacteristic
388C2×C4C23×C3⋊D311yescharacteristic
389C8—182no—
390C8—182no—
391C2×C3—28no—
392C2×C3—88no—
393C2×C3C2×#93214yes—
394C2×C3C2×#95014yes—
395C2×C3#583314yes—
396D3—244no—
397C22—21no—
398C22—21no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—361no—
403C22—361no—
404C22—361no—
405C22#6414112yes—
406C22C22×#63811yescharacteristic
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—42no—
412C4—361no—
413C4—361no—
414C4—361no—
415C4#6415912yes—
416C4#6436611yescharacteristic
417C4#6436711yescharacteristic
418C3#8331114yescomplemented
419C2—22no—
420C2—82no—
421C2—721no—
422C2C2×#6414111yescharacteristic
423C2C2×#6415911yescharacteristic
424C2#1059630411yescharacteristic
subgroupG/H
centerC22C22×#638
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#532C2
socleC22×C32C22×D4
hypercenter#532C3⋊D3
hypocenterC32#3247
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C32×#532C2semidirect
2. #10595950C2semidirect
3. #10592790C2semidirect
4. #10592783C2semidirect
5. #10592774C2semidirect
6. #10592771C2semidirect
7. #10592740C2semidirect
8. #10592737C2semidirect
9. #10592715C2semidirect
10. #10592708C2semidirect
11. C2×C4⋊(C3⋊(C3⋊C8))C2semidirect
44. C3×#532D3semidirect
158. #532C3⋊D3semidirect
358. C32#3247semidirect
418. C3#83311semidirect

Maximal subgroups

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

subgroupindex
C32×#5322
#105959502
#105927902
#105927832
#105927742
#105927712
#105927402
#105927372
#105927152
#105927082
C2×C4⋊(C3⋊(C3⋊C8))2
#833113