Finite Group Databasebeta

← #11895962

Subgroups — #11895962

Order 1152.

825 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10595351C211yesmaximal, characteristic, complemented
2#10594520C211yesmaximal, characteristic
3#10592487C211yesmaximal, characteristic, complemented
4C3×#4817C212yesmaximal, complemented
5#10591622C211yesmaximal, characteristic, complemented
6#10591617C212yesmaximal, complemented
7#10590885C211yesmaximal, characteristic, complemented
8#10590882C212yesmaximal, complemented
9#10590787C211yesmaximal, characteristic, complemented
10#10590542C212yesmaximal, complemented
11#10590430C211yesmaximal, characteristic, complemented
12#79752—32nomaximal
13C2×#3798—21no—
14Q8×C3⋊Dic3C2211yescharacteristic
15C2×#3755C2211yescharacteristic
16#64102C2211yescharacteristic
17C3×Q8×Dic3C2212yes—
18C2×C3×#215C2212yes—
19C3×#893C2212yes—
20#63919—21no—
21#63919C2211yescharacteristic
22C4×#627—21no—
23C4×#627C2211yescharacteristic
24#63914C2211yescharacteristic
25#63912—21no—
26#63911C2212yes—
27#63906—22no—
28#63869—22no—
29#63869C2212yes—
30C32×#103C2211yescharacteristic
31#63648C2211yescharacteristic
32C8×C3⋊Dic3C2211yescharacteristic
33#63635C2211yescharacteristic
34C3×#821C2212yes—
35C3×#815C2212yes—
36C3×#807C2212yes—
37#63572C2211yescharacteristic
38Dic3⋊(C3⋊C8)C2212yes—
39#63564C2212yes—
40#63563C2212yes—
41#63562C2212yes—
42#5569—32no—
43#5553—32no—
44C3×#440—32no—
45#5163—32no—
46#5163—32no—
47#5027—32no—
48#5009—32no—
49#4891—32no—
50#4890—32no—
51#4817D312yescomplemented
52#4788—32no—
53#4788—61no—
54#4764—32no—
55#4760—32no—
56#4713—32no—
57#4701—32no—
58C2×C32×Q8C2311yescharacteristic
59C2×C4⋊(C3⋊D3)C2311yescharacteristic
60C2×C4×C3⋊D3—21no—
61C2×C3×D12C2312yes—
62C2×C3×C4×D3—22no—
63C2×#627—21no—
64C2×#627D411yescharacteristic
65C2×#626—22no—
66C2×#626—22no—
67C2×#625—21no—
68C2×#625—21no—
69#3798—44no—
70C2×C32×C8C2311yescharacteristic
71C32×C4⋊C4C2311yescharacteristic
72#3755—22no—
73#3752—21no—
74C4⋊(C3⋊Dic3)—21no—
75C4⋊(C3⋊Dic3)C2311yescharacteristic
76C4×C3⋊Dic3—21no—
77C4×C3⋊Dic3C2311yescharacteristic
78C2×C3⋊(C3⋊C8)C2311yescharacteristic
79C3×#215—24no—
80C3×(C2×D3)⋊C4—22no—
81C3×C4⋊Dic3—22no—
82C3×C4⋊Dic3C2312yes—
83C3×C4×Dic3—22no—
84C3×C4×Dic3C2312yes—
85C2×C3×C3⋊C8C2312yes—
86#3724—21no—
87#3724—22no—
88#3724D411yescharacteristic
89Dic3⋊Dic3—22no—
90Dic3⋊Dic3D412yes—
91#3368—91noSylow
92C2×#235—62no—
93C2×C3×SD16—32no—
94C3×C4×Q8—32no—
95C3×C4×D4—32no—
96Q8×Dic3—32no—
97Q8×Dic3—61no—
98Q8×Dic3C2×D312yes—
99C2×#215—32no—
100C2×#215—61no—
101C2×#215C2×D312yes—
102#940—32no—
103#940—32no—
104#930—62no—
105#893—32no—
106#893—61no—
107#893C2×D312yes—
108#890—32no—
109#890—62no—
110#888—32no—
111#888—62no—
112#884—62no—
113#884—62no—
114#874—32no—
115#870—32no—
116#870—62no—
117C4×Dic6—32no—
118C4×Dic6—62no—
119C3×#107—32no—
120C3×#103—21no—
121C3×#103C2×D312yes—
122C3×#102—32no—
123C3×#97—32no—
124#832—32no—
125#832—32no—
126#821—32no—
127#821—32no—
128#821—32no—
129#821—61no—
130#821C2×D312yes—
131#815C2×D312yes—
132Dic3⋊C8—32no—
133Dic3⋊C8—32no—
134C8×Dic3—32no—
135C8×Dic3—61no—
136#809—32no—
137#809—32no—
138#808—32no—
139#808—61no—
140#807C2×D312yes—
141C4⋊(C3⋊C8)—32no—
142C22×C3⋊D3—21no—
143C22×C3×D3—22no—
144C32×Q8—21no—
145C32×Q8#4112yescomplemented
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4C2411yescharacteristic
149C2×C3⋊Dic3—21no—
150C2×C3⋊Dic3C2×D411yescharacteristic
151C2×C3⋊Dic3C2×D411yescharacteristic
152C4⋊(C3⋊D3)—21no—
153C4⋊(C3⋊D3)—21no—
154C4×C3⋊D3—42no—
155C2×C3×Dic3—22no—
156C2×C3×Dic3C2×D412yes—
157C2×C3×Dic3C2×D412yes—
158C3×D12—22no—
159C3×D12—22no—
160C3×C4×D3—44no—
161#627—22no—
162#627—42no—
163#626—44no—
164#626—82no—
165#625—42no—
166#625—81no—
167C32×C8—21no—
168C3⋊(C3⋊C8)—21no—
169C3×C3⋊C8—22no—
170#557—91no—
171#550—91no—
172#490—92no—
173#486—91no—
174#478—92no—
175#474—91no—
176#460—92no—
177#460—91no—
178#446—91no—
179#440—32no—
180C4×SD16—91no—
181C2×C3×Q8—21no—
182C2×C3×Q8C22×D312yes—
183C2×C3×D4—32no—
184C22×C3×C4—62no—
185C2×#67—62no—
186C2×#67—62no—
187C2×#67—62no—
188C2×#67—62no—
189#235—128no—
190C2×D12—32no—
191C2×D12—61no—
192C2×D12—62no—
193C2×D12—62no—
194C2×D12C22×D312yes—
195C2×C4×D3—22no—
196C2×C4×D3—62no—
197C2×C4×D3—121no—
198C2×Dic6—32no—
199C2×Dic6—62no—
200C3×SD16—64no—
201C2×C3×C8—21no—
202C2×C3×C8—32no—
203C2×C3×C8C22×D312yes—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—32no—
206C3×C4⋊C4—62no—
207C3×C4⋊C4C22×D312yes—
208C3×C22⋊C4—62no—
209C3×C42—32no—
210C3×C42—62no—
211#215—24no—
212#215—64no—
213#215—64no—
214(C2×D3)⋊C4—22no—
215(C2×D3)⋊C4—62no—
216(C2×D3)⋊C4—121no—
217C4⋊Dic3—22no—
218C4⋊Dic3—32no—
219C4⋊Dic3—32no—
220C4⋊Dic3—61no—
221C4⋊Dic3—62no—
222C4⋊Dic3—62no—
223C4⋊Dic3—121no—
224C4⋊Dic3C22×D312yes—
225Dic3⋊C4—32no—
226Dic3⋊C4—32no—
227Dic3⋊C4—62no—
228Dic3⋊C4—62no—
229Dic3⋊C4—62no—
230Dic3⋊C4—62no—
231C4×Dic3—22no—
232C4×Dic3—32no—
233C4×Dic3—61no—
234C4×Dic3—62no—
235C4×Dic3—121no—
236C4×Dic3C22×D312yes—
237C2×C3⋊C8—32no—
238C2×C3⋊C8—61no—
239C2×C3⋊C8C22×D312yes—
240C22×C32C22×D411yescharacteristic
241C2×C3⋊D3—41no—
242C2×C3⋊D3—41no—
243C2×C3⋊D3—41no—
244C2×C3×D3—42no—
245C2×C3×D3—42no—
246C2×C3×D3—42no—
247C32×C4—22no—
248C32×C4—41no—
249C32×C4C2×#4111yescharacteristic
250C32×C42−1+411yescharacteristic
251C3⋊Dic3—21no—
252C3⋊Dic3—41no—
253C3⋊Dic3AGL(1,ℤ/8)12yes—
254C3×Dic3—22no—
255C3×Dic3—22no—
256C3×Dic3—42no—
257C2×#41—181no—
258C2×SD16—32no—
259C2×SD16—91no—
260C4⋊Q8—91no—
261C4⋊Q8—181no—
262#127—92no—
263#127—91no—
264#125—92no—
265#125—182no—
266#121—181no—
267#121—182no—
268C4×Q8—32no—
269C4×Q8—91no—
270C4×Q8—91no—
271C4×Q8—181no—
272C4×D4—32no—
273C4×D4—91no—
274#107—32no—
275#106—91no—
276C4⋊C8—92no—
277C4⋊C8—91no—
278#103D3211yescharacteristic
279#102—32no—
280#102—92no—
281#102—92no—
282#102—92no—
283#102—91no—
284#97—32no—
285C4×C8—91no—
286C23×C3—62no—
287C22×D3—22no—
288C22×D3—62no—
289C22×D3—121no—
290C3×Q8—22no—
291C3×Q8—22no—
292C3×Q8—41no—
293C3×Q8#23514yescomplemented
294C3×D4—62no—
295C3×D4—62no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—21no—
299C2×C3×C4—32no—
300C2×C3×C4—32no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—62no—
304C2×C3×C4—124no—
305C2×C3×C4C23×D312yes—
306#67—124no—
307#67—124no—
308#67—242no—
309#67—242no—
310C2×Dic3—22no—
311C2×Dic3—32no—
312C2×Dic3—32no—
313C2×Dic3—61no—
314C2×Dic3—61no—
315C2×Dic3—62no—
316C2×Dic3—121no—
317C2×Dic3D3×D412yes—
318C2×Dic3D3×D412yes—
319D12—22no—
320D12—22no—
321D12—62no—
322D12—62no—
323D12—62no—
324D12—121no—
325D12—124no—
326D12—242no—
327C4×D3—44no—
328C4×D3—124no—
329C4×D3—242no—
330Dic6—64no—
331Dic6—124no—
332C3×C8—22no—
333C3×C8—41no—
334C3×C8—62no—
335C3⋊C8—22no—
336C3⋊C8—62no—
337C3⋊C8—62no—
338C2×C32#54811yescharacteristic
339C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
340C2×C32#57711yescharacteristic
341C3⋊D3—81no—
342C3×D3—82no—
343#41—364no—
344C2×Q8—91no—
345C2×Q8—181no—
346C2×Q8C2×D3211yescharacteristic
347C2×D4—32no—
348C2×D4—91no—
349C2×D4—182no—
350C2×D4—182no—
351C2×D4—181no—
352C2×D4—181no—
353C22×C4—62no—
354C22×C4—181no—
355SD16—64no—
356SD16—182no—
357C2×C8—32no—
358C2×C8—91no—
359C2×C8C2×D3211yescharacteristic
360C4⋊C4—32no—
361C4⋊C4—62no—
362C4⋊C4—91no—
363C4⋊C4—92no—
364C4⋊C4—91no—
365C4⋊C4—181no—
366C4⋊C4—182no—
367C4⋊C4—182no—
368C4⋊C4—181no—
369C4⋊C4C2×D3211yescharacteristic
370C22⋊C4—62no—
371C22⋊C4—181no—
372C42—32no—
373C42—62no—
374C42—91no—
375C42—181no—
376C22×C3—21no—
377C22×C3—122no—
378C22×C3—122no—
379C22×C3—122no—
380C22×C3C2×D3×D412yes—
381C2×D3—42no—
382C2×D3—42no—
383C2×D3—42no—
384C2×D3—122no—
385C2×D3—122no—
386C2×D3—122no—
387C2×D3—122no—
388C2×D3—241no—
389C2×D3—241no—
390C3×C4—24no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—42no—
394C3×C4—42no—
395C3×C4—62no—
396C3×C4—62no—
397C3×C4—81no—
398C3×C4—122no—
399C3×C4C2×#23512yes—
400C3×C4#101012yes—
401Dic3—22no—
402Dic3—22no—
403Dic3—34no—
404Dic3—42no—
405Dic3—62no—
406Dic3—62no—
407Dic3—121no—
408Dic3—122no—
409Dic3—241no—
410C32#336811yescharacteristic, complemented, Sylow
411C23—62no—
412C23—181no—
413Q8—21no—
414Q8—182no—
415Q8—362no—
416Q8#379812yescomplemented
417D4—62no—
418D4—62no—
419D4—181no—
420D4—181no—
421D4—364no—
422D4—362no—
423D4—722no—
424D4—721no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—32no—
428C2×C4—32no—
429C2×C4—62no—
430C2×C4—91no—
431C2×C4—91no—
432C2×C4—124no—
433C2×C4—181no—
434C2×C4—362no—
435C2×C4C22×D3211yescharacteristic
436C8—21no—
437C8—62no—
438C8—181no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—242no—
443C2×C3#555212yes—
444C2×C3D3×AGL(1,ℤ/8)12yes—
445C2×C3#577812yes—
446D3—82no—
447D3—242no—
448D3—242no—
449C22—122no—
450C22—122no—
451C22—122no—
452C22—361no—
453C22—361no—
454C22—361no—
455C22D32×D411yescharacteristic
456C4—22no—
457C4—41no—
458C4—62no—
459C4—62no—
460C4—92no—
461C4—122no—
462C4—181no—
463C4—361no—
464C4C2×#379811yescharacteristic
465C4#6430911yescharacteristic
466C3—21no—
467C3#7975212yescomplemented
468C2—242no—
469C2—721no—
470C2#1059418911yescharacteristic
471C2#1059474011yescharacteristic
472C2#1059476711yescharacteristic
subgroupG/H
centerC22D32×D4
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×#103C22
socleC22×C32C22×D4
prime radical#10594520C2
hypercenter#103D32
hypocenterC32#3368
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
1. #10595351C2semidirect
3. #10592487C2semidirect
4. C3×#4817C2semidirect
5. #10591622C2semidirect
6. #10591617C2semidirect
7. #10590885C2semidirect
8. #10590882C2semidirect
9. #10590787C2semidirect
10. #10590542C2semidirect
11. #10590430C2semidirect
51. #4817D3semidirect
145. C32×Q8#41semidirect
293. C3×Q8#235semidirect
410. C32#3368semidirect
416. Q8#3798semidirect
467. C3#79752semidirect

Maximal subgroups

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

subgroupindex
#105953512
#105945202
#105924872
C3×#48172
#105916222
#105916172
#105908852
#105908822
#105907872
#105905422
#105904302
#797523