Finite Group Databasebeta

← #11788759

Subgroups — #11788759

Order 1152.

783 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5805C211yesmaximal, characteristic
2#10594776C211yesmaximal, characteristic, complemented
3#10592914C211yesmaximal, characteristic, complemented
4C3×#5215C211yesmaximal, characteristic, complemented
5#10591231C211yesmaximal, characteristic, complemented
6#10589158C211yesmaximal, characteristic, complemented
7#10589157C211yesmaximal, characteristic, complemented
8#71854—31nomaximal
9#70330—31nomaximal
10C22×C32×D4C2211yescharacteristic
11C2×C3×#235C2211yescharacteristic
12#64135C2211yescharacteristic
13C2×#3748C2211yescharacteristic
14C3×#953—22no—
15C3×#940C412yescomplemented
16C3×#938—22no—
17C3×#936C412yescomplemented
18C3×#926C2211yescharacteristic
19C3×#926C2211yescharacteristic
20C2×C3×#208C2211yescharacteristic
21#63815—22no—
22#63814—22no—
23D3×#208—22no—
24#63659—22no—
25C3×#832—22no—
26#63580—22no—
27C3×#535—31no—
28#5819—31no—
29#5805D311yescharacteristic
30#5745—31no—
31C3×#418—31no—
32#5240—31no—
33#5215—31no—
34#5215—31no—
35#5215—61no—
36#5215D311yescharacteristic, complemented
37#5006—31no—
38#4557—31no—
39#4531—31no—
40#4531—31no—
41#4531—31no—
42C24×C32—21no—
43C2×C32×D4—21no—
44C2×C32×D4—21no—
45C2×C32×D4—21no—
46C2×C32×D4—21no—
47C2×C32×D4—21no—
48C2×C32×D4C2×C412yes—
49C22×C32×C4C2311yescharacteristic
50C2×C3×#67—24no—
51C2×C3×#67D412yes—
52C3×#235—21no—
53C3×#235—21no—
54C3×#235—21no—
55C3×#235—21no—
56C2×C3×D12C2×C411yescharacteristic
57C2×C3×C4×D3D412yes—
58C2×C3×Dic6C2×C411yescharacteristic
59#3757—21no—
60C4⋊(C3⋊Dic3)—21no—
61C4×C3⋊Dic3—21no—
62#3748—21no—
63#3748—21no—
64C2×C3⋊(C3⋊C8)D412yescomplemented
65C3×#217—21no—
66C3×#217—24no—
67C3×#217—21no—
68C3×C4⋊Dic3—21no—
69C3×C4⋊Dic3—21no—
70C3×C4×Dic3—21no—
71C3×C4×Dic3C2×C412yes—
72C3×#208—21no—
73C3×#208—21no—
74C2×C3×C3⋊C8D412yescomplemented
75#3711—24no—
76D3×C3⋊C8—24no—
77#1910—91noSylow
78C2×C3×#41—31no—
79C22×C3×D4—21no—
80C22×C3×D4C2×D311yescharacteristic
81C22×C3×D4C2×D311yescharacteristic
82C2×#235C2×D311yescharacteristic
83C2×C3×M16—31no—
84C3×#127—32no—
85C3×#124—32no—
86C3×#121—62no—
87C3×#120—62no—
88C3×#117—31no—
89C3×#117—31no—
90#953—22no—
91#948—62no—
92#948—62no—
93#940Dic312yescomplemented
94#938—22no—
95#936Dic312yescomplemented
96#926—31no—
97#926—31no—
98#926—61no—
99#926C2×D311yescharacteristic
100#926C2×D311yescharacteristic
101C2×#208—31no—
102C2×#208—31no—
103C2×#208—61no—
104C2×#208—62no—
105C2×#208C2×D311yescharacteristic
106#907—62no—
107D3×M16—62no—
108#901—62no—
109C3×#102—62no—
110#834—62no—
111#832—22no—
112#832—62no—
113#832—62no—
114#832—64no—
115#806—62no—
116C23×C32—22no—
117C23×C32—41no—
118C23×C32—41no—
119C23×C32—82no—
120C23×C32C2×D411yescharacteristic
121C22×C3×D3C22⋊C412yes—
122C32×D4—22no—
123C32×D4—22no—
124C32×D4—41no—
125C32×D4—41no—
126C2×C32×C4C22×C411yescharacteristic
127C2×C32×C4C2×D411yescharacteristic
128C2×C32×C4C2×D411yescharacteristic
129C2×C32×C4C2×D411yescharacteristic
130C2×C32×C4#4111yescharacteristic
131C2×C32×C4#4111yescharacteristic
132C2×C3⋊Dic3—41no—
133C3×#67—22no—
134C3×#67—22no—
135C3×#67—84no—
136C2×C3×Dic3—22no—
137C2×C3×Dic3—41no—
138C2×C3×Dic3C22⋊C412yes—
139C3×D12—21no—
140C3×D12—21no—
141C3×C4×D3—22no—
142C3×C4×D3—22no—
143C3×Dic6—21no—
144C3×Dic6—21no—
145C3⋊(C3⋊C8)—22no—
146C3×C3⋊C8—22no—
147#568—91no—
148#535—31no—
149#418—31no—
150#418—91no—
151#410—91no—
152H(3,ℤ/4)—91no—
153H(3,ℤ/4)—91no—
154C24×C3—21no—
155C24×C3—21no—
156C24×C3—41no—
157C3×#41—61no—
158C3×#41—61no—
159C3×#41—61no—
160C3×#41—61no—
161C2×C3×Q8—31no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—21no—
165C2×C3×D4—21no—
166C2×C3×D4—21no—
167C2×C3×D4—21no—
168C2×C3×D4—21no—
169C2×C3×D4—21no—
170C2×C3×D4—21no—
171C2×C3×D4—21no—
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4—22no—
175C2×C3×D4—32no—
176C2×C3×D4—31no—
177C2×C3×D4—41no—
178C2×C3×D4—41no—
179C2×C3×D4—41no—
180C2×C3×D4—64no—
181C2×C3×D4C4×D312yes—
182C2×C3×D4C2×Dic312yes—
183C22×C3×C4—21no—
184C22×C3×C4—32no—
185C22×C3×C4C22×D311yescharacteristic
186C22×C3×C4C22×D311yescharacteristic
187C2×#67—24no—
188C2×#67#6712yes—
189#235—21no—
190#235—21no—
191#235—21no—
192#235—21no—
193C2×D12C2×Dic311yescharacteristic
194C2×C4×D3#6712yes—
195C2×Dic6C2×Dic311yescharacteristic
196C3×M16—61no—
197C3×M16—61no—
198C2×C3×C8—32no—
199C3×C4⋊C4—61no—
200C3×C4⋊C4—61no—
201C3×C22⋊C4—64no—
202C3×C22⋊C4—61no—
203C3×C22⋊C4—61no—
204C3×C42—32no—
205C3×C42—61no—
206#217—21no—
207#217—24no—
208#217—21no—
209#217—61no—
210#217—61no—
211#217—121no—
212C4⋊Dic3—21no—
213C4⋊Dic3—21no—
214C4⋊Dic3—62no—
215C4⋊Dic3—61no—
216C4⋊Dic3—61no—
217C4×Dic3—21no—
218C4×Dic3—61no—
219C4×Dic3—61no—
220C4×Dic3—121no—
221C4×Dic3C2×Dic312yes—
222#208—21no—
223#208—21no—
224#208—64no—
225#208—61no—
226#208—61no—
227#208—61no—
228#208—61no—
229#208—121no—
230#208—121no—
231C2×C3⋊C8—32no—
232C2×C3⋊C8—32no—
233C2×C3⋊C8—62no—
234C2×C3⋊C8—64no—
235C2×C3⋊C8#6712yescomplemented
236ΓL(1,𝔽25)—64no—
237D3×C8—64no—
238C22×C32—41no—
239C22×C32—42no—
240C22×C32—44no—
241C22×C32—82no—
242C22×C32—82no—
243C22×C32C2×C22⋊C411yescharacteristic
244C22×C32#12111yescharacteristic
245C22×C32#12111yescharacteristic
246C2×C3×D3—22no—
247C2×C3×D3—42no—
248C32×C4C4×D411yescharacteristic
249C32×C4C4×D411yescharacteristic
250C32×C4#12011yescharacteristic
251C32×C4#12311yescharacteristic
252C3⋊Dic3—81no—
253C3×Dic3—22no—
254C3×Dic3—42no—
255C3×Dic3—81no—
256C2×#41—31no—
257C22×D4D3211yescharacteristic
258#131—182no—
259#131—182no—
260C2×M16—31no—
261C2×M16—91no—
262C2×M16—182no—
263#127—32no—
264#124—32no—
265#121—62no—
266#120—62no—
267#117—31no—
268#117—31no—
269#117—91no—
270#102—62no—
271#102—182no—
272#99—182no—
273C23×C3—22no—
274C23×C3—22no—
275C23×C3—21no—
276C23×C3—32no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—41no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3—41no—
284C23×C3—82no—
285C23×C3—82no—
286C23×C3—84no—
287C23×C3D3×D411yescharacteristic
288C23×C3C2×#6711yescharacteristic
289C22×D3#21712yes—
290C3×Q8—61no—
291C3×Q8—61no—
292C3×D4—24no—
293C3×D4—22no—
294C3×D4—22no—
295C3×D4—22no—
296C3×D4—22no—
297C3×D4—42no—
298C3×D4—42no—
299C3×D4—42no—
300C3×D4—41no—
301C3×D4—41no—
302C3×D4—41no—
303C3×D4—41no—
304C3×D4—62no—
305C3×D4—62no—
306C3×D4—61no—
307C3×D4—61no—
308C3×D4—244no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—32no—
316C2×C3×C4—62no—
317C2×C3×C4—62no—
318C2×C3×C4—62no—
319C2×C3×C4—121no—
320C2×C3×C4C2×C4×D311yescharacteristic
321C2×C3×C4#23511yescharacteristic
322C2×C3×C4D3×D411yescharacteristic
323C2×C3×C4D3×D411yescharacteristic
324C2×C3×C4D3×D411yescharacteristic
325C2×C3×C4D3×D411yescharacteristic
326C2×C3×C4#23711yescharacteristic
327C2×C3×C4#23711yescharacteristic
328C2×C3×C4#23711yescharacteristic
329C2×C3×C4C22×Dic311yescharacteristic
330C2×C3×C4C2×#6711yescharacteristic
331C2×C3×C4C2×#6711yescharacteristic
332#67—22no—
333#67—22no—
334#67—84no—
335C2×Dic3—22no—
336C2×Dic3—41no—
337C2×Dic3—122no—
338C2×Dic3—121no—
339C2×Dic3—121no—
340C2×Dic3#21712yes—
341D12—21no—
342D12—21no—
343C4×D3—22no—
344C4×D3—22no—
345Dic6—21no—
346Dic6—21no—
347C3×C8—62no—
348C3⋊C8—22no—
349C3⋊C8—62no—
350C3⋊C8—62no—
351C3⋊C8—122no—
352C2×C32—21no—
353C2×C32—21no—
354C2×C32—21no—
355C2×C32—82no—
356C2×C32#38611yescharacteristic
357C3×D3—42no—
358C24—21no—
359#41—61no—
360#41—61no—
361#41—61no—
362#41—61no—
363C2×Q8—31no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—32no—
370C2×D4—31no—
371C2×D4—64no—
372C2×D4D3×Dic312yes—
373C22×C4—32no—
374C22×C4C2×D3211yescharacteristic
375M16—61no—
376M16—61no—
377M16—184no—
378M16—181no—
379M16—181no—
380C2×C8—32no—
381C2×C8—92no—
382C2×C8—184no—
383C4⋊C4—61no—
384C4⋊C4—61no—
385C4⋊C4—181no—
386C22⋊C4—64no—
387C22⋊C4—61no—
388C22⋊C4—61no—
389C22⋊C4—181no—
390C42—32no—
391C42—61no—
392C42—181no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—41no—
397C22×C3—41no—
398C22×C3—42no—
399C22×C3—44no—
400C22×C3—44no—
401C22×C3—44no—
402C22×C3—42no—
403C22×C3—42no—
404C22×C3—62no—
405C22×C3—84no—
406C22×C3—84no—
407C22×C3—82no—
408C22×C3—82no—
409C22×C3—82no—
410C22×C3—82no—
411C22×C3—84no—
412C22×C3—122no—
413C22×C3D3×C22⋊C411yescharacteristic
414C22×C3#88411yescharacteristic
415C22×C3#93811yescharacteristic
416C22×C3#93911yescharacteristic
417C22×C3#93911yescharacteristic
418C22×C3C2×#21711yescharacteristic
419C2×D3—22no—
420C2×D3—42no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—62no—
426C3×C4—122no—
427C3×C4—241no—
428C3×C4#67⋊C411yescharacteristic
429C3×C4#88311yescharacteristic
430C3×C4C4×#6711yescharacteristic
431C3×C4D4×Dic311yescharacteristic
432C3×C4D4×Dic311yescharacteristic
433C3×C4#93511yescharacteristic
434C3×C4#93711yescharacteristic
435C3×C4#93711yescharacteristic
436Dic3—22no—
437Dic3—42no—
438Dic3—81no—
439Dic3—242no—
440Dic3—241no—
441Dic3—241no—
442C32#191011yescharacteristic, complemented, Sylow
443C23—22no—
444C23—32no—
445C23—41no—
446C23—41no—
447C23—82no—
448C23D3×#6711yescharacteristic
449Q8—61no—
450Q8—61no—
451D4—22no—
452D4—22no—
453D4—41no—
454D4—41no—
455D4—62no—
456D4—62no—
457D4—61no—
458D4—61no—
459D4—244no—
460C2×C4—32no—
461C2×C4—62no—
462C2×C4—62no—
463C2×C4—62no—
464C2×C4—121no—
465C2×C4—361no—
466C2×C4C2×D3×Dic311yescharacteristic
467C2×C4#380411yescharacteristic
468C2×C4#380511yescharacteristic
469C2×C4D3×#6711yescharacteristic
470C2×C4D3×#6711yescharacteristic
471C2×C4#381111yescharacteristic
472C8—62no—
473C8—182no—
474C2×C3—22no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—41no—
483C2×C3—41no—
484C2×C3—84no—
485C2×C3—82no—
486C2×C3—82no—
487C2×C3—122no—
488C2×C3#495611yescharacteristic
489C2×C3#522011yescharacteristic
490D3—42no—
491C22—41no—
492C22—44no—
493C22—42no—
494C22—62no—
495C22—82no—
496C22—82no—
497C22—122no—
498C22D3×#21711yescharacteristic
499C22#6397011yescharacteristic
500C22#6397111yescharacteristic
501C4—62no—
502C4—122no—
503C4—241no—
504C4—721no—
505C4#6396911yescharacteristic
506C4Dic3×#6711yescharacteristic
507C4#67⋊Dic311yescharacteristic
508C4#6398311yescharacteristic
509C3—21no—
510C3#7033011yescharacteristic, complemented
511C3#7185411yescharacteristic, complemented
512C2—21no—
513C2—21no—
514C2—21no—
515C2—82no—
516C2—122no—
517C2#1059172111yescharacteristic
subgroupG/H
centerC2#10591721
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC22×C32×D4C22
socleC2×C32#386
prime radicalC3×#5805C2
cosocleC22×C32×C4C23
hypercenterC22×D4D32
hypocenterC32#1910
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. #10594776C2semidirect
3. #10592914C2semidirect
4. C3×#5215C2semidirect
5. #10591231C2semidirect
6. #10589158C2semidirect
7. #10589157C2semidirect
15. C3×#940C4semidirect
17. C3×#936C4semidirect
36. #5215D3semidirect
64. C2×C3⋊(C3⋊C8)D4semidirect
74. C2×C3×C3⋊C8D4semidirect
93. #940Dic3semidirect
95. #936Dic3semidirect
235. C2×C3⋊C8#67semidirect
442. C32#1910semidirect
510. C3#70330semidirect
511. C3#71854semidirect

Maximal subgroups

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

subgroupindex
C3×#58052
#105947762
#105929142
C3×#52152
#105912312
#105891582
#105891572
#718543
#703303