Finite Group Databasebeta

← #11787635

Subgroups — #11787635

Order 1152.

665 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10594786C211yesmaximal, characteristic
2#10594776C211yesmaximal, characteristic, complemented
3C3×#5005C211yesmaximal, characteristic, complemented
4C3×#5002C211yesmaximal, characteristic, complemented
5#10591070C211yesmaximal, characteristic, complemented
6#10589399C211yesmaximal, characteristic, complemented
7#10589166C211yesmaximal, characteristic, complemented
8#70416—31nomaximal
9#70293—31nomaximal
10C2×C3×#235C2211yescharacteristic
11C2×C3×#235C2211yescharacteristic
12C32×#117C2211yescharacteristic
13C2×#3748C2211yescharacteristic
14C2×#3748C2211yescharacteristic
15C2×C3×#208C2211yescharacteristic
16C22×C3×C3⋊C8C2211yescharacteristic
17#63827—21no—
18#63827C412yescomplemented
19#63823—21no—
20#63823C412yescomplemented
21#63822—22no—
22#63815—22no—
23#63814—22no—
24D3×#208—22no—
25C3×#810—21no—
26C3×#809—21no—
27C3×#805—22no—
28(C2×D3)⋊(C3⋊C8)—22no—
29#5822—31no—
30#5822—31no—
31#5819—31no—
32#5745—31no—
33C3×#421—31no—
34C3×#416—31no—
35#5234—31no—
36#5110—31no—
37#5005D311yescharacteristic, complemented
38#5002D311yescharacteristic, complemented
39#4558—31no—
40#4552—31no—
41#4533—31no—
42#4533—31no—
43#4533—61no—
44C22×C32×C4C2311yescharacteristic
45C2×C3×#67—21no—
46C2×C3×#67—21no—
47C3×#235—21no—
48C3×#235—21no—
49C3×#235—21no—
50C3×#235—21no—
51C3×#235—21no—
52C3×#235—21no—
53C3×#235C2×C412yes—
54C3×#235D412yes—
55C2×C3×D12C2×C411yescharacteristic
56C2×C3×D12D411yescharacteristic
57C2×C3×C4×D3—21no—
58C2×C3×C4×D3—21no—
59C2×C3×Dic6C2×C411yescharacteristic
60C2×C3×Dic6D411yescharacteristic
61C32×C4⋊C4—21no—
62C32×C22⋊C4—21no—
63C32×C42—21no—
64#3748—21no—
65#3748—21no—
66#3748—21no—
67#3748C2×C412yes—
68C2×C3⋊(C3⋊C8)—21no—
69C2×C3⋊(C3⋊C8)D412yes—
70C3×#208—21no—
71C3×#208D412yes—
72C2×C3×C3⋊C8—21no—
73C2×C3×C3⋊C8—21no—
74C2×C3×C3⋊C8—21no—
75C2×C3×C3⋊C8—21no—
76#3718—22no—
77#3715—22no—
78#3715—22no—
79#3713—22no—
80#3711—24no—
81D3×C3⋊C8—24no—
82#1881—91noSylow
83C2×C3×#41—31no—
84C2×C3×#41—31no—
85C2×#235C2×D311yescharacteristic
86C2×#235C2×D311yescharacteristic
87C2×C3×M16—31no—
88C22×C3×C8—31no—
89C3×#117—21no—
90C3×#117C2×D311yescharacteristic
91C3×#117C2×D311yescharacteristic
92#951—32no—
93#951—61no—
94#950—62no—
95#950—62no—
96#949—32no—
97#949—61no—
98#948—62no—
99#948—62no—
100#942—32no—
101#942—61no—
102#932—32no—
103#932—61no—
104C2×#208—31no—
105C2×#208—31no—
106C2×#208—31no—
107C2×#208—31no—
108C2×#208—61no—
109C2×#208—61no—
110C2×#208—62no—
111C2×#208C2×D311yescharacteristic
112C22×C3⋊C8C2×D311yescharacteristic
113#907—62no—
114D3×M16—62no—
115#901—62no—
116C3×#104—62no—
117C3×#103—61no—
118C3×#102—61no—
119#830—62no—
120(C2×D3)⋊C8—62no—
121#810—21no—
122#809—21no—
123#805—22no—
124C23×C32#4111yescharacteristic
125C22×C3×D3—21no—
126C22×C3×D3—21no—
127C2×C32×C4—41no—
128C2×C32×C4C22×C411yescharacteristic
129C2×C32×C4C2×D411yescharacteristic
130C2×C32×C4C2×D411yescharacteristic
131C2×C32×C4C2×D411yescharacteristic
132C2×C32×C4C2×D411yescharacteristic
133C2×C32×C4#4111yescharacteristic
134C3×#67—22no—
135C3×#67—22no—
136C3×#67—41no—
137C3×#67—41no—
138C2×C3×Dic3—21no—
139C2×C3×Dic3—21no—
140C3×D12—21no—
141C3×D12—21no—
142C3×D12—21no—
143C3×D12C22⋊C412yes—
144C3×C4×D3—22no—
145C3×C4×D3—22no—
146C3×C4×D3—41no—
147C3×C4×D3—41no—
148C3×Dic6—21no—
149C3×Dic6—21no—
150C3×Dic6—21no—
151C3×Dic6C22⋊C412yes—
152C3⋊(C3⋊C8)—22no—
153C3⋊(C3⋊C8)—22no—
154C3×C3⋊C8—22no—
155C3×C3⋊C8—41no—
156#575—91no—
157#568—91no—
158#421—31no—
159#416—31no—
160#408—91no—
161#344—91no—
162#341—91no—
163C3×#41—32no—
164C3×#41—32no—
165C3×#41—61no—
166C3×#41—61no—
167C3×#41—61no—
168C3×#41—61no—
169C3×#41—61no—
170C3×#41—61no—
171C2×C3×Q8—31no—
172C2×C3×Q8—31no—
173C2×C3×D4—31no—
174C2×C3×D4—31no—
175C2×C3×D4—61no—
176C2×C3×D4—61no—
177C22×C3×C4—21no—
178C22×C3×C4—61no—
179C22×C3×C4—61no—
180C22×C3×C4C22×D311yescharacteristic
181C22×C3×C4C22×D311yescharacteristic
182C2×#67—21no—
183C2×#67—21no—
184#235—21no—
185#235—21no—
186#235—21no—
187#235—21no—
188#235—21no—
189#235—21no—
190#235C4×D312yes—
191#235D1212yes—
192C2×D12C4×D311yescharacteristic
193C2×D12#6711yescharacteristic
194C2×C4×D3—21no—
195C2×C4×D3—21no—
196C2×Dic6C4×D311yescharacteristic
197C2×Dic6#6711yescharacteristic
198C3×M16—32no—
199C3×M16—61no—
200C2×C3×C8—61no—
201C2×C3×C8—61no—
202C2×C3×C8—61no—
203C2×C3×C8—61no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—41no—
207C3×C22⋊C4—22no—
208C3×C22⋊C4—21no—
209C3×C22⋊C4—21no—
210C3×C42—21no—
211C3×C42—21no—
212C3×C42—41no—
213#216—62no—
214#216—62no—
215#215—62no—
216#215—62no—
217#214—62no—
218#214—62no—
219#213—62no—
220#213—62no—
221#208—21no—
222#208—32no—
223#208—32no—
224#208—62no—
225#208—64no—
226#208—61no—
227#208—61no—
228#208—61no—
229#208—61no—
230#208—61no—
231#208—61no—
232#208—121no—
233#208—121no—
234#208—121no—
235#208#6712yes—
236C2×C3⋊C8—21no—
237C2×C3⋊C8—21no—
238C2×C3⋊C8—21no—
239C2×C3⋊C8—21no—
240C2×C3⋊C8—32no—
241C2×C3⋊C8—32no—
242C2×C3⋊C8—62no—
243C2×C3⋊C8—64no—
244C2×C3⋊C8—61no—
245C2×C3⋊C8—61no—
246C2×C3⋊C8—121no—
247ΓL(1,𝔽25)—64no—
248D3×C8—64no—
249C22×C32—41no—
250C22×C32C4×D411yescharacteristic
251C22×C32#12111yescharacteristic
252C22×C32#12311yescharacteristic
253C2×C3×D3—22no—
254C2×C3×D3—42no—
255C2×C3×D3—41no—
256C2×C3×D3—81no—
257C32×C4—81no—
258C32×C4C2×C22⋊C411yescharacteristic
259C32×C4C4×D411yescharacteristic
260C32×C4#12011yescharacteristic
261C32×C4#12111yescharacteristic
262C3×Dic3—22no—
263C3×Dic3—41no—
264C2×#41—31no—
265C2×#41—31no—
266#137—92no—
267#137—181no—
268AGL(1,ℤ/8)—92no—
269AGL(1,ℤ/8)—181no—
270#135—182no—
271#131—182no—
272#131—182no—
273C2×M16—31no—
274C2×M16—91no—
275C2×M16—91no—
276C2×M16—182no—
277C22×C8—31no—
278#117D3211yescharacteristic
279#104—62no—
280#103—61no—
281#102—61no—
282C22⋊C8—182no—
283C23×C3—21no—
284C23×C3—61no—
285C23×C3—61no—
286C23×C3#23711yescharacteristic
287C23×C3#23711yescharacteristic
288C22×D3—21no—
289C22×D3—21no—
290C3×Q8—32no—
291C3×Q8—61no—
292C3×Q8—61no—
293C3×Q8—61no—
294C3×D4—32no—
295C3×D4—62no—
296C3×D4—62no—
297C3×D4—61no—
298C3×D4—61no—
299C3×D4—61no—
300C3×D4—121no—
301C3×D4—121no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—42no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—61no—
312C2×C3×C4—62no—
313C2×C3×C4—62no—
314C2×C3×C4—61no—
315C2×C3×C4—121no—
316C2×C3×C4—121no—
317C2×C3×C4C2×C4×D311yescharacteristic
318C2×C3×C4C2×C4×D311yescharacteristic
319C2×C3×C4C2×D1211yescharacteristic
320C2×C3×C4#23511yescharacteristic
321C2×C3×C4D3×D411yescharacteristic
322C2×C3×C4D3×D411yescharacteristic
323C2×C3×C4D3×D411yescharacteristic
324C2×C3×C4D3×D411yescharacteristic
325C2×C3×C4D3×D411yescharacteristic
326C2×C3×C4#23711yescharacteristic
327C2×C3×C4C2×#6711yescharacteristic
328C2×C3×C4C2×#6711yescharacteristic
329#67—22no—
330#67—22no—
331#67—41no—
332#67—41no—
333C2×Dic3—21no—
334C2×Dic3—21no—
335D12—21no—
336D12—21no—
337D12—21no—
338D12(C2×D3)⋊C412yes—
339C4×D3—22no—
340C4×D3—22no—
341C4×D3—41no—
342C4×D3—41no—
343Dic6—21no—
344Dic6—21no—
345Dic6—21no—
346Dic6(C2×D3)⋊C412yes—
347C3×C8—62no—
348C3×C8—121no—
349C3⋊C8—22no—
350C3⋊C8—41no—
351C3⋊C8—62no—
352C3⋊C8—62no—
353C3⋊C8—62no—
354C3⋊C8—62no—
355C3⋊C8—122no—
356C3⋊C8—122no—
357C2×C32—21no—
358C2×C32—21no—
359C2×C32—21no—
360C2×C32#38611yescharacteristic
361C3×D3—42no—
362C3×D3—81no—
363#41—32no—
364#41—32no—
365#41—61no—
366#41—61no—
367#41—61no—
368#41—61no—
369#41—61no—
370#41—61no—
371C2×Q8—31no—
372C2×Q8—31no—
373C2×D4—31no—
374C2×D4—31no—
375C2×D4—61no—
376C2×D4—61no—
377C22×C4—61no—
378C22×C4—61no—
379C22×C4C2×D3211yescharacteristic
380Q16—182no—
381SD16—182no—
382SD16—182no—
383D8—182no—
384M16—32no—
385M16—61no—
386M16—92no—
387M16—184no—
388M16—181no—
389M16—181no—
390M16—181no—
391C2×C8—61no—
392C2×C8—61no—
393C2×C8—61no—
394C2×C8—61no—
395C2×C8—92no—
396C2×C8—184no—
397C2×C8—181no—
398C4⋊C4—21no—
399C22⋊C4—21no—
400C42—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—41no—
405C22×C3—41no—
406C22×C3—62no—
407C22×C3—81no—
408C22×C3—122no—
409C22×C3—121no—
410C22×C3—241no—
411C22×C3#67⋊C411yescharacteristic
412C22×C3#67⋊C411yescharacteristic
413C22×C3#88311yescharacteristic
414C22×C3#88611yescharacteristic
415C22×C3#93811yescharacteristic
416C22×C3#93911yescharacteristic
417C2×D3—22no—
418C2×D3—42no—
419C2×D3—41no—
420C2×D3—81no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—62no—
426C3×C4—81no—
427C3×C4—81no—
428C3×C4—82no—
429C3×C4—121no—
430C3×C4D3×C22⋊C411yescharacteristic
431C3×C4#67⋊C411yescharacteristic
432C3×C4#88211yescharacteristic
433C3×C4#88411yescharacteristic
434C3×C4C2×(C2×D3)⋊C411yescharacteristic
435C3×C4C4×#6711yescharacteristic
436C3×C4#93711yescharacteristic
437C3×C4#93911yescharacteristic
438Dic3—22no—
439Dic3—41no—
440C32#188111yescharacteristic, complemented, Sylow
441C23—61no—
442C23—61no—
443C23#380511yescharacteristic
444Q8—32no—
445Q8—61no—
446Q8—61no—
447Q8—61no—
448D4—32no—
449D4—62no—
450D4—62no—
451D4—61no—
452D4—61no—
453D4—61no—
454D4—121no—
455D4—121no—
456C2×C4—41no—
457C2×C4—61no—
458C2×C4—62no—
459C2×C4—62no—
460C2×C4—61no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4#379511yescharacteristic
464C2×C4C4×D3211yescharacteristic
465C2×C4D3×D1211yescharacteristic
466C2×C4#380211yescharacteristic
467C2×C4D3×#6711yescharacteristic
468C2×C4D3×#6711yescharacteristic
469C8—62no—
470C8—121no—
471C8—182no—
472C8—182no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—41no—
481C2×C3—41no—
482C2×C3—41no—
483C2×C3—122no—
484C2×C3—241no—
485C2×C3#495611yescharacteristic
486C2×C3#495811yescharacteristic
487D3—42no—
488D3—81no—
489C22—41no—
490C22—62no—
491C22—122no—
492C22—121no—
493C22—241no—
494C22#6389111yescharacteristic
495C22#6390211yescharacteristic
496C22#6397011yescharacteristic
497C4—62no—
498C4—81no—
499C4—121no—
500C4#6387911yescharacteristic
501C4#6388511yescharacteristic
502C4D3×(C2×D3)⋊C411yescharacteristic
503C4#6392211yescharacteristic
504C3—21no—
505C3#7029311yescharacteristic, complemented
506C3#7041611yescharacteristic, complemented
507C2—21no—
508C2—21no—
509C2—21no—
510C2—122no—
511C2—241no—
512C2#1059136311yescharacteristic
subgroupG/H
centerC4#63922
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC32×#117C22
socleC2×C32#386
prime radical#10594786C2
cosocleC22×C32×C4C23
hypercenter#117D32
hypocenterC32#1881
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. #10594776C2semidirect
3. C3×#5005C2semidirect
4. C3×#5002C2semidirect
5. #10591070C2semidirect
6. #10589399C2semidirect
7. #10589166C2semidirect
18. #63827C4semidirect
20. #63823C4semidirect
37. #5005D3semidirect
38. #5002D3semidirect
440. C32#1881semidirect
505. C3#70293semidirect
506. C3#70416semidirect

Maximal subgroups

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

subgroupindex
#105947862
#105947762
C3×#50052
C3×#50022
#105910702
#105893992
#105891662
#704163
#702933