Finite Group Databasebeta

← #11882239

Subgroups — #11882239

Order 1152.

1176 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10596199C211yesmaximal, characteristic, complemented
2#10596019C211yesmaximal, characteristic
3C32×#476C211yesmaximal, characteristic, complemented
4#10592783C211yesmaximal, characteristic, complemented
5#10592718C211yesmaximal, characteristic, complemented
6#10592563C212yesmaximal, complemented
7#10592553C212yesmaximal, complemented
8#10592540C211yesmaximal, characteristic, complemented
9C3⋊D3×C4⋊C8C211yesmaximal, characteristic, complemented
10#10592477C212yesmaximal, complemented
11#10592461C212yesmaximal, complemented
12#81070—34nomaximal
13C2×#3828—21no—
14C32×C4⋊Q8C2211yescharacteristic
15C32×C4×D4C2211yescharacteristic
16#64155—21no—
17Q8×C3⋊Dic3—21no—
18#64149C2211yescharacteristic
19#64147C2211yescharacteristic
20C2×#3754C2212yes—
21C4×#638—21no—
22C2×#3744C2212yes—
23C2×C8×C3⋊D3C2212yes—
24#64106C2212yes—
25#64104C2211yescharacteristic
26#64101—21no—
27#64099—21no—
28#64097C2211yescharacteristic
29#64096C2212yes—
30#64094—21no—
31#64093—21no—
32C4×C4⋊(C3⋊D3)C2211yescharacteristic
33C42×C3⋊D3C2211yescharacteristic
34#64078C2211yescharacteristic
35C32×#106C2212yes—
36C32×C4⋊C8C2211yescharacteristic
37C32×#102C2212yes—
38#63643C2212yes—
39#63641C2212yes—
40#63637C2212yes—
41#63636C2212yes—
42#63635C2212yes—
43C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
44#5728—34no—
45#5548—34no—
46C3×#476D314yescomplemented
47#5084—34no—
48#5019—34no—
49#4864—38no—
50#4854—38no—
51#4841—34no—
52D3×C4⋊C8—34no—
53#4778—38no—
54#4762—38no—
55C2×C32×Q8—21no—
56C2×C32×D4C2311yescharacteristic
57C22×C32×C4—21no—
58C2×#638—21no—
59C2×#638—21no—
60C2×#638—21no—
61#3828—44no—
62C2×C4⋊(C3⋊D3)—21no—
63C2×C4⋊(C3⋊D3)C2311yescharacteristic
64C2×C4×C3⋊D3—21no—
65C2×C4×C3⋊D3D411yescharacteristic
66C2×C4×C3⋊D3D411yescharacteristic
67C2×C4×C3⋊D3C2311yescharacteristic
68C2×#634—21no—
69C2×#634C2312yes—
70C2×C32×C8C2312yes—
71C32×C4⋊C4—21no—
72C32×C4⋊C4C2312yes—
73C32×C4⋊C4C2311yescharacteristic
74C32×C22⋊C4—21no—
75C32×C42C2311yescharacteristic
76#3757—21no—
77#3754—42no—
78#3752—22no—
79#3752—21no—
80#3752—21no—
81C4⋊(C3⋊Dic3)—21no—
82C4⋊(C3⋊Dic3)—21no—
83C4⋊(C3⋊Dic3)—22no—
84C4⋊(C3⋊Dic3)C2311yescharacteristic
85#3750—21no—
86#3750—22no—
87C4×C3⋊Dic3—22no—
88C4×C3⋊Dic3D411yescharacteristic
89C4×C3⋊Dic3D411yescharacteristic
90C4×C3⋊Dic3C2311yescharacteristic
91C2×C3⋊(C3⋊C8)C2312yes—
92#3744—42no—
93C8×C3⋊D3—42no—
94#3246—91noSylow
95C2×#235—64no—
96C3×C4⋊Q8C2×D314yes—
97C3×C4×D4C2×D314yes—
98#946—64no—
99Q8×Dic3—64no—
100#940—34no—
101#938—34no—
102C2×#214—38no—
103C4×#67—64no—
104C2×#204—38no—
105C2×D3×C8—38no—
106#897—38no—
107#895—34no—
108#892—64no—
109#890—64no—
110#888—34no—
111#887—38no—
112#885—64no—
113#884—64no—
114C4×D12—34no—
115C42×D3—34no—
116#869—34no—
117C3×#106C2×D318yes—
118C3×C4⋊C8C2×D314yes—
119C3×#102C2×D318yes—
120#816—38no—
121Dic3⋊C8—38no—
122#810—38no—
123#809—38no—
124#808—38no—
125C4⋊(C3⋊C8)—34no—
126C23×C32—21no—
127C22×C3⋊D3—21no—
128C22×C3⋊D3C2×D411yescharacteristic
129C32×Q8—42no—
130C32×D4—21no—
131C32×D4—21no—
132C2×C32×C4—22no—
133C2×C32×C4—21no—
134C2×C32×C4—42no—
135C2×C32×C4C2×D411yescharacteristic
136C2×C32×C4C2×D411yescharacteristic
137C2×C32×C4C2411yescharacteristic
138#638—42no—
139#638—42no—
140#638—42no—
141C2×C3⋊Dic3—22no—
142C2×C3⋊Dic3—21no—
143C2×C3⋊Dic3C2×D411yescharacteristic
144C2×C3⋊Dic3C2×D411yescharacteristic
145C2×C3⋊Dic3C2×D411yescharacteristic
146C4⋊(C3⋊D3)—21no—
147C4⋊(C3⋊D3)—21no—
148C4⋊(C3⋊D3)—42no—
149C4×C3⋊D3—21no—
150C4×C3⋊D3—21no—
151C4×C3⋊D3—41no—
152C4×C3⋊D3—42no—
153C4×C3⋊D3SD1614yescomplemented
154#634—22no—
155#634—22no—
156#634—42no—
157C32×C8—22no—
158C3⋊(C3⋊C8)—22no—
159#533—91no—
160#532—91no—
161#476—91no—
162#476C3⋊D311yescharacteristic, complemented
163#475—92no—
164#465—92no—
165#465—92no—
166#461—91no—
167#461—91no—
168#460—92no—
169C2×C4⋊C8—91no—
170C2×C3×Q8—24no—
171C2×C3×D4C22×D314yes—
172C22×C3×C4—24no—
173C2×#67—64no—
174C2×#67—64no—
175C2×#67—64no—
176#235—1216no—
177C2×D12—34no—
178C2×D12—64no—
179C2×C4×D3—34no—
180C2×C4×D3—34no—
181C2×C4×D3—34no—
182C2×C4×D3—64no—
183C2×Dic6—38no—
184C2×Dic6—64no—
185C2×C3×C8C22×D318yes—
186C3×C4⋊C4—24no—
187C3×C4⋊C4C22×D318yes—
188C3×C4⋊C4C22×D314yes—
189C3×C22⋊C4—24no—
190C3×C42C22×D314yes—
191#217—64no—
192#214—128no—
193(C2×D3)⋊C4—64no—
194(C2×D3)⋊C4—68no—
195(C2×D3)⋊C4—64no—
196C4⋊Dic3—34no—
197C4⋊Dic3—68no—
198C4⋊Dic3—64no—
199C4⋊Dic3—64no—
200Dic3⋊C4—64no—
201Dic3⋊C4—68no—
202C4×Dic3—34no—
203C4×Dic3—34no—
204C4×Dic3—34no—
205C4×Dic3—68no—
206C2×C3⋊C8—38no—
207#204—128no—
208D3×C8—128no—
209C22×C32—41no—
210C22×C32—41no—
211C22×C32—41no—
212C22×C32C22×D411yescharacteristic
213C2×C3⋊D3—21no—
214C2×C3⋊D3—21no—
215C2×C3⋊D3—41no—
216C2×C3⋊D3—41no—
217C2×C3⋊D3—41no—
218C2×C3⋊D3C2×SD1612yes—
219C32×C4—21no—
220C32×C4—41no—
221C32×C4—42no—
222C32×C4C2×SD1612yes—
223C32×C4H(5,𝔽2)11yescharacteristic
224C32×C42−1+411yescharacteristic
225C3⋊Dic3—21no—
226C3⋊Dic3—21no—
227C3⋊Dic3—42no—
228C3⋊Dic3—41no—
229C3⋊Dic3C2×SD1612yes—
230C2×#41—181no—
231C2×SD16—92no—
232C2×SD16—92no—
233C22×C8—92no—
234C4⋊Q8—91no—
235C4⋊Q8—91no—
236C4⋊Q8C2×C3⋊D311yescharacteristic
237#127—91no—
238#125—181no—
239#124—181no—
240C22⋊Q8—92no—
241C22⋊Q8—181no—
242#121—91no—
243#121—91no—
244#121—181no—
245C4×Q8—92no—
246C4×Q8—181no—
247C4×D4—91no—
248C4×D4—181no—
249C4×D4C2×C3⋊D311yescharacteristic
250#117—181no—
251C2×C42—91no—
252#106—92no—
253#106C2×C3⋊D312yes—
254C4⋊C8—92no—
255C4⋊C8—91no—
256C4⋊C8C2×C3⋊D311yescharacteristic
257#103—92no—
258#103—92no—
259#102—92no—
260#102C2×C3⋊D312yes—
261C23×C3—24no—
262C22×D3—34no—
263C22×D3—64no—
264C3×Q8—48no—
265C3×D4—24no—
266C3×D4—24no—
267C2×C3×C4—28no—
268C2×C3×C4—24no—
269C2×C3×C4—48no—
270C2×C3×C4D3×D414yes—
271C2×C3×C4D3×D414yes—
272C2×C3×C4C23×D314yes—
273#67—128no—
274#67—128no—
275#67—128no—
276C2×Dic3—34no—
277C2×Dic3—34no—
278C2×Dic3—34no—
279C2×Dic3—68no—
280C2×Dic3—64no—
281D12—64no—
282D12—64no—
283D12—128no—
284C4×D3—316no—
285C4×D3—64no—
286C4×D3—64no—
287C4×D3—124no—
288C4×D3—128no—
289Dic6—68no—
290Dic6—68no—
291Dic6—128no—
292C3×C8—28no—
293C3⋊C8—68no—
294C2×C32—81no—
295C2×C32#53711yescharacteristic
296C2×C32C22×SD1611yescharacteristic
297C2×C32#57511yescharacteristic
298C3⋊D3—22no—
299C3⋊D3—81no—
300#41—364no—
301C2×Q8—21no—
302C2×Q8—92no—
303C2×Q8—181no—
304C2×D4—91no—
305C2×D4—181no—
306C2×D4—181no—
307C2×D4—181no—
308C2×D4—181no—
309C2×D4C22×C3⋊D311yescharacteristic
310C22×C4—21no—
311C22×C4—91no—
312C22×C4—91no—
313C22×C4—91no—
314C22×C4—181no—
315SD16—362no—
316SD16—362no—
317C2×C8—92no—
318C2×C8—362no—
319C2×C8C22×C3⋊D312yes—
320C4⋊C4—21no—
321C4⋊C4—91no—
322C4⋊C4—181no—
323C4⋊C4—182no—
324C4⋊C4—182no—
325C4⋊C4—181no—
326C4⋊C4—181no—
327C4⋊C4C22×C3⋊D312yes—
328C4⋊C4C22×C3⋊D311yescharacteristic
329C22⋊C4—21no—
330C22⋊C4—181no—
331C22⋊C4—182no—
332C22⋊C4—181no—
333C22⋊C4—181no—
334C42—91no—
335C42—91no—
336C42—91no—
337C42—182no—
338C42C22×C3⋊D311yescharacteristic
339C22×C3—44no—
340C22×C3—44no—
341C22×C3—44no—
342C22×C3C2×D3×D414yes—
343C2×D3—38no—
344C2×D3—64no—
345C2×D3—64no—
346C2×D3—124no—
347C2×D3—124no—
348C2×D3—124no—
349C3×C4—24no—
350C3×C4—44no—
351C3×C4—48no—
352C3×C4D3×SD1618yes—
353C3×C4#100714yes—
354C3×C4#100914yes—
355Dic3—38no—
356Dic3—64no—
357Dic3—64no—
358Dic3—124no—
359Dic3—128no—
360C32#324611yescharacteristic, complemented, Sylow
361C23—21no—
362C23—91no—
363C23—181no—
364Q8—42no—
365Q8—182no—
366Q8—182no—
367Q8—362no—
368D4—21no—
369D4—21no—
370D4—181no—
371D4—181no—
372D4—362no—
373D4—362no—
374D4—362no—
375D4—362no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—42no—
379C2×C4—91no—
380C2×C4—94no—
381C2×C4—91no—
382C2×C4—91no—
383C2×C4—181no—
384C2×C4—181no—
385C2×C4—182no—
386C2×C4—181no—
387C2×C4—361no—
388C2×C4—362no—
389C2×C4D4×C3⋊D311yescharacteristic
390C2×C4D4×C3⋊D311yescharacteristic
391C2×C4C23×C3⋊D311yescharacteristic
392C8—22no—
393C8—182no—
394C2×C3—84no—
395C2×C3#562914yes—
396C2×C3C2×D3×SD1614yes—
397C2×C3#577314yes—
398D3—68no—
399D3—244no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—92no—
404C22—181no—
405C22—181no—
406C22—361no—
407C22—361no—
408C22—361no—
409C22C2×D4×C3⋊D311yescharacteristic
410C4—21no—
411C4—41no—
412C4—42no—
413C4—92no—
414C4—181no—
415C4—181no—
416C4—362no—
417C4—361no—
418C4SD16×C3⋊D312yes—
419C4#6436411yescharacteristic
420C4#6436611yescharacteristic
421C3#8107014yescomplemented
422C2—81no—
423C2—182no—
424C2—721no—
425C2#1059610011yescharacteristic
426C2C2×SD16×C3⋊D311yescharacteristic
427C2#1059624411yescharacteristic
subgroupG/H
centerC22C2×D4×C3⋊D3
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#476C2
socleC22×C32C22×D4
prime radical#10596019C2
hypercenter#476C3⋊D3
hypocenterC32#3246
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #10596199C2semidirect
3. C32×#476C2semidirect
4. #10592783C2semidirect
5. #10592718C2semidirect
6. #10592563C2semidirect
7. #10592553C2semidirect
8. #10592540C2semidirect
9. C3⋊D3×C4⋊C8C2semidirect
10. #10592477C2semidirect
11. #10592461C2semidirect
46. C3×#476D3semidirect
153. C4×C3⋊D3SD16semidirect
162. #476C3⋊D3semidirect
360. C32#3246semidirect
421. C3#81070semidirect

Maximal subgroups

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

subgroupindex
#105961992
#105960192
C32×#4762
#105927832
#105927182
#105925632
#105925532
#105925402
C3⋊D3×C4⋊C82
#105924772
#105924612
#810703