Finite Group Databasebeta

← #11788684

Subgroups — #11788684

Order 1152.

705 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#942C211yesmaximal, characteristic
2C2×#63815C211yesmaximal, characteristic
3C2×#63662C211yesmaximal, characteristic, complemented
4C2×C3×#836C211yesmaximal, characteristic, complemented
5#10591088C211yesmaximal, characteristic
6#10589168C211yesmaximal, characteristic, complemented
7#10589165C211yesmaximal, characteristic, complemented
8#72103—31nomaximal
9#70123—31nomaximal
10C22×C32×Q8C2211yescharacteristic
11C2×C3×#235C2211yescharacteristic
12C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
13C22×C3⋊(C3⋊C8)C2211yescharacteristic
14C2×C3×#216—21no—
15C3×#942—22no—
16C3×#942C414yescomplemented
17C2×C3×#215—21no—
18C2×C3×#208C2211yescharacteristic
19C2×C3×#208C2211yescharacteristic
20C2×C3×#208C2211yescharacteristic
21C2×#3711—21no—
22#63815—22no—
23#63815—22no—
24C2×D3×C3⋊C8—21no—
25#63662—21no—
26#63662—21no—
27C3×#836—22no—
28#63566—21no—
29#63563—21no—
30C2×C3×#137—31no—
31C2×#948—31no—
32C2×#942D311yescharacteristic
33C2×#907—31no—
34C2×C3×#101—31no—
35#5232—31no—
36C2×#836D311yescharacteristic, complemented
37C2×#835—31no—
38C2×#835—31no—
39C2×#835—61no—
40#5133—31no—
41#4558—31no—
42#4558—31no—
43#4554—31no—
44#4554—31no—
45C2×C32×Q8—21no—
46C2×C32×Q8—21no—
47C2×C32×Q8—21no—
48C2×C32×Q8—21no—
49C2×C32×Q8—21no—
50C2×C32×Q8C2×C412yes—
51C22×C32×C4—21no—
52C22×C32×C4C2311yescharacteristic
53C22×C3⋊Dic3—21no—
54C2×C3×#67—21no—
55C3×#235—21no—
56C3×#235—21no—
57C3×#235C2×C412yes—
58C3×#235D412yes—
59C2×C3×D12D411yescharacteristic
60C2×C3×C4×D3—21no—
61C2×C3×Dic6D411yescharacteristic
62C4⋊(C3⋊Dic3)—21no—
63C4⋊(C3⋊Dic3)—21no—
64C2×C3⋊(C3⋊C8)—21no—
65C2×C3⋊(C3⋊C8)—21no—
66C2×C3⋊(C3⋊C8)D411yescharacteristic
67C2×C3⋊(C3⋊C8)D411yescharacteristic
68C3×#216—24no—
69C3×#215—24no—
70C3×#208—21no—
71C3×#208—21no—
72C3×#208—21no—
73C3×#208—21no—
74C3×#208C2×C412yes—
75C3×#208D412yes—
76C2×C3×C3⋊C8—21no—
77C2×C3×C3⋊C8—21no—
78C2×C3×C3⋊C8—21no—
79#3711—24no—
80D3×C3⋊C8—24no—
81#1908—91noSylow
82C2×C3×#41—31no—
83C22×C3×Q8—21no—
84C22×C3×Q8C2×D311yescharacteristic
85C22×C3×Q8C2×D311yescharacteristic
86C2×#235C2×D311yescharacteristic
87C3×#137—34no—
88C3×#137—62no—
89C2×C3×Q16—61no—
90C2×C3×SD16—61no—
91C2×C3×M16—31no—
92C2×C3×M16—31no—
93C2×C3×M16—31no—
94#948—62no—
95#948—62no—
96C2×#216—21no—
97#942—22no—
98#942Dic314yescomplemented
99C2×#215—21no—
100C2×C4⋊Dic3—31no—
101C2×C4⋊Dic3—31no—
102C2×C4⋊Dic3—61no—
103C2×#208—61no—
104C2×#208C2×D311yescharacteristic
105C2×#208C2×D311yescharacteristic
106C2×#208C2×D311yescharacteristic
107C22×C3⋊C8—31no—
108C22×C3⋊C8—31no—
109C22×C3⋊C8—61no—
110C22×C3⋊C8—61no—
111#907—62no—
112#907—62no—
113C2×ΓL(1,𝔽25)—61no—
114C2×D3×C8—61no—
115C3×#101—62no—
116#836—22no—
117#835—61no—
118#835—61no—
119#835—61no—
120#835—61no—
121#835—61no—
122#835—62no—
123#835—62no—
124#832—61no—
125#821—61no—
126#816—61no—
127C23×C32#4111yescharacteristic
128C22×C3×D3—21no—
129C32×Q8—22no—
130C32×Q8—22no—
131C32×Q8—41no—
132C32×Q8—41no—
133C2×C32×C4—22no—
134C2×C32×C4—41no—
135C2×C32×C4—41no—
136C2×C32×C4C22×C411yescharacteristic
137C2×C32×C4C2×D411yescharacteristic
138C2×C32×C4C2×D411yescharacteristic
139C2×C32×C4C2×D411yescharacteristic
140C2×C32×C4C2×D411yescharacteristic
141C2×C32×C4#4111yescharacteristic
142C2×C3⋊Dic3—41no—
143C2×C3⋊Dic3—81no—
144C3×#67—22no—
145C3×#67—22no—
146C2×C3×Dic3—21no—
147C3×D12—21no—
148C3×D12C22⋊C412yes—
149C3×C4×D3—22no—
150C3×C4×D3—22no—
151C3×Dic6—21no—
152C3×Dic6C22⋊C412yes—
153C3⋊(C3⋊C8)—21no—
154C3⋊(C3⋊C8)—21no—
155C3×C3⋊C8—22no—
156C3×C3⋊C8—22no—
157C3×C3⋊C8—41no—
158C2×#137—31no—
159C2×#131—91no—
160#417—91no—
161C2×#103—91no—
162C2×#101—31no—
163#343—91no—
164#341—91no—
165C3×#41—32no—
166C3×#41—32no—
167C3×#41—61no—
168C3×#41—61no—
169C2×C3×Q8—21no—
170C2×C3×Q8—21no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—21no—
174C2×C3×Q8—21no—
175C2×C3×Q8—21no—
176C2×C3×Q8—21no—
177C2×C3×Q8—21no—
178C2×C3×Q8—21no—
179C2×C3×Q8—21no—
180C2×C3×Q8—21no—
181C2×C3×Q8—22no—
182C2×C3×Q8—31no—
183C2×C3×Q8—41no—
184C2×C3×Q8—41no—
185C2×C3×Q8—41no—
186C2×C3×Q8C4×D312yes—
187C2×C3×Q8C2×Dic312yes—
188C2×C3×D4—31no—
189C2×C3×D4—61no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—41no—
194C22×C3×C4—61no—
195C22×C3×C4C22×D311yescharacteristic
196C22×C3×C4C22×D311yescharacteristic
197C2×#67—21no—
198C22×Dic3—61no—
199C22×Dic3—61no—
200C22×Dic3—121no—
201#235—21no—
202#235—21no—
203#235C2×Dic312yes—
204#235#6712yes—
205C2×D12#6711yescharacteristic
206C2×C4×D3—21no—
207C2×Dic6#6711yescharacteristic
208C3×Q16—64no—
209C3×SD16—64no—
210C3×M16—32no—
211C3×M16—32no—
212C3×M16—61no—
213C3×M16—61no—
214C3×M16—61no—
215C3×M16—61no—
216C2×C3×C8—61no—
217C2×C3×C8—61no—
218C2×C3×C8—61no—
219#216—24no—
220#215—24no—
221C4⋊Dic3—62no—
222C4⋊Dic3—61no—
223C4⋊Dic3—61no—
224C4⋊Dic3—61no—
225C4⋊Dic3—61no—
226C4⋊Dic3—121no—
227#208—21no—
228#208—21no—
229#208—21no—
230#208—21no—
231#208—64no—
232#208C2×Dic312yes—
233#208#6712yes—
234C2×C3⋊C8—21no—
235C2×C3⋊C8—21no—
236C2×C3⋊C8—21no—
237C2×C3⋊C8—31no—
238C2×C3⋊C8—31no—
239C2×C3⋊C8—31no—
240C2×C3⋊C8—31no—
241C2×C3⋊C8—61no—
242C2×C3⋊C8—61no—
243C2×C3⋊C8—64no—
244C2×C3⋊C8—61no—
245C2×C3⋊C8—61no—
246C2×C3⋊C8—61no—
247C2×C3⋊C8—61no—
248C2×C3⋊C8—121no—
249C2×C3⋊C8—121no—
250ΓL(1,𝔽25)—64no—
251D3×C8—64no—
252C22×C32—21no—
253C22×C32—21no—
254C22×C32C4×D411yescharacteristic
255C22×C32#12111yescharacteristic
256C22×C32#12311yescharacteristic
257C2×C3×D3—22no—
258C2×C3×D3—41no—
259C2×C3×D3—41no—
260C32×C4—42no—
261C32×C4C2×C22⋊C411yescharacteristic
262C32×C4C4×D411yescharacteristic
263C32×C4#12011yescharacteristic
264C32×C4#12111yescharacteristic
265C3⋊Dic3—81no—
266C3×Dic3—22no—
267C2×#41—31no—
268C22×Q8D3211yescharacteristic
269#137—34no—
270#137—62no—
271C2×Q16—61no—
272C2×SD16—61no—
273#131—182no—
274#131—182no—
275C2×M16—31no—
276C2×M16—31no—
277C2×M16—31no—
278C2×M16—181no—
279C22×C8—91no—
280C22×C8—181no—
281C2×C4⋊C4—91no—
282#103—181no—
283#103—181no—
284#103—181no—
285#102—181no—
286#101—62no—
287C23×C3—21no—
288C23×C3—61no—
289C23×C3#23511yescharacteristic
290C23×C3#23711yescharacteristic
291C22×D3—21no—
292C3×Q8—24no—
293C3×Q8—22no—
294C3×Q8—22no—
295C3×Q8—22no—
296C3×Q8—22no—
297C3×Q8—32no—
298C3×Q8—42no—
299C3×Q8—42no—
300C3×Q8—42no—
301C3×Q8—41no—
302C3×Q8—41no—
303C3×Q8—41no—
304C3×Q8—41no—
305C3×Q8—61no—
306C3×D4—32no—
307C3×D4—62no—
308C3×D4—62no—
309C3×D4—61no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—61no—
326C2×C3×C4—62no—
327C2×C3×C4—62no—
328C2×C3×C4C2×C4×D311yescharacteristic
329C2×C3×C4D3×D411yescharacteristic
330C2×C3×C4D3×D411yescharacteristic
331C2×C3×C4D3×D411yescharacteristic
332C2×C3×C4D3×D411yescharacteristic
333C2×C3×C4D3×D411yescharacteristic
334C2×C3×C4#23711yescharacteristic
335C2×C3×C4#23711yescharacteristic
336C2×C3×C4C22×Dic311yescharacteristic
337C2×C3×C4C2×#6711yescharacteristic
338C2×C3×C4C2×#6711yescharacteristic
339C2×C3×C4C2×#6711yescharacteristic
340#67—22no—
341#67—22no—
342C2×Dic3—21no—
343C2×Dic3—122no—
344C2×Dic3—121no—
345C2×Dic3—121no—
346C2×Dic3—242no—
347C2×Dic3—241no—
348C2×Dic3—241no—
349D12—21no—
350D12#21712yes—
351C4×D3—22no—
352C4×D3—22no—
353Dic6—21no—
354Dic6#21712yes—
355C3×C8—62no—
356C3×C8—62no—
357C3×C8—121no—
358C3⋊C8—22no—
359C3⋊C8—22no—
360C3⋊C8—41no—
361C3⋊C8—61no—
362C3⋊C8—61no—
363C3⋊C8—61no—
364C3⋊C8—61no—
365C3⋊C8—121no—
366C3⋊C8—121no—
367C2×C32—21no—
368C2×C32—21no—
369C2×C32#38611yescharacteristic
370C2×C32#47111yescharacteristic
371C2×C32#47311yescharacteristic
372C3×D3—42no—
373#41—32no—
374#41—32no—
375#41—61no—
376#41—61no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—31no—
383C2×Q8D3×Dic312yes—
384C2×D4—31no—
385C2×D4—61no—
386C22×C4—21no—
387C22×C4—61no—
388C22×C4—181no—
389C22×C4C2×D3211yescharacteristic
390Q16—64no—
391SD16—64no—
392M16—32no—
393M16—32no—
394M16—61no—
395M16—61no—
396M16—61no—
397M16—61no—
398M16—184no—
399C2×C8—61no—
400C2×C8—61no—
401C2×C8—61no—
402C2×C8—91no—
403C2×C8—91no—
404C2×C8—184no—
405C2×C8—181no—
406C2×C8—181no—
407C4⋊C4—181no—
408C4⋊C4—181no—
409C22×C3—22no—
410C22×C3—22no—
411C22×C3—21no—
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—62no—
419C22×C3—121no—
420C22×C3—121no—
421C22×C3#88311yescharacteristic
422C22×C3#88411yescharacteristic
423C22×C3C4×#6711yescharacteristic
424C22×C3D4×Dic311yescharacteristic
425C22×C3#93511yescharacteristic
426C22×C3#93911yescharacteristic
427C2×D3—22no—
428C2×D3—41no—
429C2×D3—41no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—21no—
434C3×C4—44no—
435C3×C4—42no—
436C3×C4—42no—
437C3×C4—62no—
438C3×C4D3×C22⋊C411yescharacteristic
439C3×C4#67⋊C411yescharacteristic
440C3×C4D4×Dic311yescharacteristic
441C3×C4#93711yescharacteristic
442C3×C4#93711yescharacteristic
443C3×C4#93811yescharacteristic
444C3×C4#93911yescharacteristic
445C3×C4C2×#21711yescharacteristic
446Dic3—22no—
447Dic3—242no—
448Dic3—241no—
449Dic3—241no—
450C32#190811yescharacteristic, complemented, Sylow
451C23—61no—
452C23#380411yescharacteristic
453Q8—22no—
454Q8—22no—
455Q8—32no—
456Q8—41no—
457Q8—41no—
458Q8—61no—
459D4—32no—
460D4—62no—
461D4—62no—
462D4—61no—
463C2×C4—22no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—61no—
467C2×C4—62no—
468C2×C4—62no—
469C2×C4—361no—
470C2×C4—721no—
471C2×C4C2×D3×Dic311yescharacteristic
472C2×C4#380511yescharacteristic
473C2×C4D3×#6711yescharacteristic
474C2×C4D3×#6711yescharacteristic
475C2×C4D3×#6711yescharacteristic
476C2×C4#381111yescharacteristic
477C8—62no—
478C8—62no—
479C8—121no—
480C8—181no—
481C8—181no—
482C2×C3—22no—
483C2×C3—22no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—122no—
492C2×C3#475411yescharacteristic
493C2×C3#475511yescharacteristic
494C2×C3#495611yescharacteristic
495C2×C3#520311yescharacteristic
496C2×C3#520411yescharacteristic
497C2×C3#522011yescharacteristic
498D3—42no—
499C22—21no—
500C22—21no—
501C22—62no—
502C22—121no—
503C22—121no—
504C22#6396911yescharacteristic
505C22#6397111yescharacteristic
506C22Dic3×#6711yescharacteristic
507C4—42no—
508C4—62no—
509C4—721no—
510C4D3×#21711yescharacteristic
511C4#6397011yescharacteristic
512C4#67⋊Dic311yescharacteristic
513C4#6398311yescharacteristic
514C3—21no—
515C3#7012311yescharacteristic, complemented
516C3#7210311yescharacteristic, complemented
517C2—21no—
518C2—21no—
519C2—122no—
520C2#1059164311yescharacteristic
521C2#1059164411yescharacteristic
522C2#1059172111yescharacteristic
subgroupG/H
centerC22#63971
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC22×C32×Q8C22
socleC22×C32#121
prime radicalC2×C3×#235C22
cosocleC22×C32×C4C23
hypercenterC22×Q8D32
hypocenterC32#1908
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
3. C2×#63662C2semidirect
4. C2×C3×#836C2semidirect
6. #10589168C2semidirect
7. #10589165C2semidirect
16. C3×#942C4semidirect
36. C2×#836D3semidirect
98. #942Dic3semidirect
450. C32#1908semidirect
515. C3#70123semidirect
516. C3#72103semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9422
C2×#638152
C2×#636622
C2×C3×#8362
#105910882
#105891682
#105891652
#721033
#701233