Finite Group Databasebeta

← #11778217

Subgroups — #11778217

Order 1152.

695 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 695 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10595072C211yesmaximal, characteristic
2#10592926C211yesmaximal, characteristic, complemented
3C3×#5231C211yesmaximal, characteristic, complemented
4C3×#5227C211yesmaximal, characteristic, complemented
5#10589151C211yesmaximal, characteristic, complemented
6#10589149C211yesmaximal, characteristic, complemented
7#10589145C211yesmaximal, characteristic, complemented
8#70972—31nomaximal
9#69699—31nomaximal
10C22×C32×Q8C2211yescharacteristic, complemented
11C2×D3×#67—21no—
12C2×#3755—21no—
13#64139C2211yescharacteristic
14C2×C3×#215—21no—
15C3×#930C2211yescharacteristic
16C2×C3×(C2×D3)⋊C4—21no—
17C2×C3×(C2×D3)⋊C4C2211yescharacteristic
18#63978—21no—
19#63923—21no—
20#63922—21no—
21#63891—21no—
22#63890—21no—
23#63662—21no—
24#63657C2211yescharacteristic, complemented
25C3×#835—21no—
26C3×C3⋊#95—21no—
27C3×C3⋊#95C2211yescharacteristic, complemented
28C3×#830C2211yescharacteristic, complemented
29#5605—31no—
30#5493—31no—
31C3×#448—31no—
32C3×#393—31no—
33#5231D311yescharacteristic, complemented
34#5227—31no—
35#5227—31no—
36#5227—61no—
37#5227D311yescharacteristic, complemented
38#4537—31no—
39#4525—31no—
40#4470—31no—
41#4468—31no—
42#4451—31no—
43#4451—31no—
44C23×C3×D3D411yescharacteristic
45C22×D32—21no—
46C2×C32×Q8—21no—
47C2×C32×Q8—21no—
48C2×C32×Q8—42no—
49C22×C32×C4—21no—
50C22×C32×C4C2311yescharacteristic
51C2×#638—21no—
52C2×C4⋊(C3⋊D3)D411yescharacteristic
53C2×C3×#67—21no—
54C22×C3×Dic3—21no—
55C22×C3×Dic3D411yescharacteristic
56C2×C3×D12D411yescharacteristic
57D3×#67—42no—
58D3×#67—42no—
59C2×#626—21no—
60C2×#626—21no—
61C2×#625—21no—
62C2×D3×Dic3—21no—
63#3755—42no—
64#3752—21no—
65C4⋊(C3⋊Dic3)D411yescharacteristic
66C2×C3⋊(C3⋊C8)—21no—
67C3×#215—42no—
68C3×(C2×D3)⋊C4—21no—
69C3×(C2×D3)⋊C4—21no—
70C3×(C2×D3)⋊C4—21no—
71C3×(C2×D3)⋊C4—42no—
72C3×C4⋊Dic3D411yescharacteristic
73C2×C3×C3⋊C8—21no—
74Dic3⋊Dic3—21no—
75#3722—21no—
76(C2×D3)⋊Dic3—21no—
77#1601—91noSylow
78C22×C3×Q8—21no—
79C22×C3×Q8C2×D311yescharacteristic, complemented
80C22×C3×Q8C2×D311yescharacteristic, complemented
81C22×#67—61no—
82C2×D3×D4—61no—
83C2×C3×SD16—61no—
84C3×#121—31no—
85C2×C3×C22⋊C4—31no—
86C2×C3×C22⋊C4—61no—
87C2×#215—21no—
88C2×#215—61no—
89C2×#215—61no—
90C2×#215—62no—
91#939—61no—
92#937—61no—
93#930—31no—
94#930—31no—
95#930—61no—
96#930—61no—
97#930C2×D311yescharacteristic
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4C2×D311yescharacteristic
100#894—61no—
101#886—61no—
102#886—61no—
103#883—61no—
104#882—61no—
105#882—61no—
106#882—61no—
107C3×#103—61no—
108C3×C22⋊C8—31no—
109C3×#95—31no—
110C3×#95—61no—
111#835—21no—
112#835—61no—
113#835—61no—
114#835—62no—
115C3⋊#95—21no—
116C3⋊#95C2×D311yescharacteristic, complemented
117#830—31no—
118#830—31no—
119#830—61no—
120#830C2×D311yescharacteristic, complemented
121C23×C32C2×D411yescharacteristic
122C22×C3⋊D3—21no—
123C22×C3×D3—21no—
124C22×C3×D3—21no—
125C22×C3×D3—21no—
126C22×C3×D3—41no—
127C22×C3×D3—41no—
128C22×C3×D3SD1612yescomplemented
129C2×D32—42no—
130C2×D32—81no—
131C2×D32—81no—
132C32×Q8—22no—
133C32×Q8—22no—
134C32×Q8—41no—
135C32×Q8—41no—
136C2×C32×C4—22no—
137C2×C32×C4—41no—
138C2×C32×C4—41no—
139C2×C32×C4—41no—
140C2×C32×C4C2×D411yescharacteristic
141C2×C32×C4C2×D411yescharacteristic
142#638—41no—
143#638—41no—
144C2×C3⋊Dic3—21no—
145C4⋊(C3⋊D3)—41no—
146C3×#67—41no—
147C3×#67—41no—
148C2×C3×Dic3—21no—
149C2×C3×Dic3—21no—
150C2×C3×Dic3—41no—
151C2×C3×Dic3—41no—
152C2×C3×Dic3—41no—
153C2×C3×Dic3—41no—
154C3×D12—41no—
155#626—42no—
156#626—81no—
157#625—42no—
158D3×Dic3—42no—
159C3⋊(C3⋊C8)—41no—
160C3×C3⋊C8—41no—
161#534—91no—
162#448—31no—
163#448—91no—
164#393—31no—
165#327—91no—
166#327—91no—
167#323—91no—
168C24×C3—31no—
169C23×D3—61no—
170C23×D3—61no—
171C23×D3#6711yescharacteristic
172C2×C3×Q8—22no—
173C2×C3×Q8—21no—
174C2×C3×Q8—21no—
175C2×C3×Q8—21no—
176C2×C3×Q8—21no—
177C2×C3×Q8—41no—
178C2×C3×Q8—42no—
179C2×C3×Q8—42no—
180C2×C3×Q8—44no—
181C2×C3×D4—31no—
182C2×C3×D4—61no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—31no—
187C22×C3×C4—41no—
188C22×C3×C4—61no—
189C22×C3×C4C22×D311yescharacteristic
190C22×C3×C4C22×D311yescharacteristic
191C2×#67—21no—
192C2×#67—61no—
193C2×#67—61no—
194C2×#67—61no—
195C2×#67—61no—
196C2×#67—61no—
197C2×#67—61no—
198C2×#67—121no—
199C2×#67—122no—
200C2×#67—122no—
201C22×Dic3—21no—
202C22×Dic3—61no—
203C22×Dic3#6711yescharacteristic
204D3×D4—122no—
205D3×D4—122no—
206C2×D12—31no—
207C2×D12—31no—
208C2×D12—61no—
209C2×D12—61no—
210C2×D12—61no—
211C2×D12#6711yescharacteristic
212C2×C4×D3—61no—
213C3×SD16—122no—
214C2×C3×C8—61no—
215C3×C4⋊C4—31no—
216C3×C22⋊C4—61no—
217C3×C22⋊C4—61no—
218C3×C22⋊C4—61no—
219C3×C22⋊C4—122no—
220#217—61no—
221#215—42no—
222#215—122no—
223#215—122no—
224#215—124no—
225(C2×D3)⋊C4—21no—
226(C2×D3)⋊C4—21no—
227(C2×D3)⋊C4—21no—
228(C2×D3)⋊C4—42no—
229(C2×D3)⋊C4—61no—
230(C2×D3)⋊C4—61no—
231(C2×D3)⋊C4—61no—
232(C2×D3)⋊C4—61no—
233(C2×D3)⋊C4—61no—
234(C2×D3)⋊C4—121no—
235C4⋊Dic3—31no—
236C4⋊Dic3—31no—
237C4⋊Dic3—61no—
238C4⋊Dic3—61no—
239C4⋊Dic3#6711yescharacteristic
240Dic3⋊C4—61no—
241C2×C3⋊C8—21no—
242C2×C3⋊C8—62no—
243C2×C3⋊C8—61no—
244C2×C3⋊C8—61no—
245C22×C32—21no—
246C22×C32—21no—
247C22×C32#12011yescharacteristic
248C22×C32C2×SD1611yescharacteristic
249C22×C32AGL(1,ℤ/8)11yescharacteristic
250C2×C3⋊D3—41no—
251C2×C3⋊D3—41no—
252C2×C3⋊D3—41no—
253C2×C3×D3—22no—
254C2×C3×D3—41no—
255C2×C3×D3—41no—
256C2×C3×D3—42no—
257C2×C3×D3—41no—
258C2×C3×D3—41no—
259C2×C3×D3—41no—
260C2×C3×D3—81no—
261D32—84no—
262C32×C4—21no—
263C32×C4—21no—
264C32×C4—42no—
265C3⋊Dic3—41no—
266C3×Dic3—41no—
267C3×Dic3—41no—
268C3×Dic3—81no—
269C22×Q8D3211yescharacteristic, complemented
270C22×D4—181no—
271C2×SD16—61no—
272C2×SD16—181no—
273#123—181no—
274#123—181no—
275#121—31no—
276#121—91no—
277#121—181no—
278#120—181no—
279#120—181no—
280C2×C22⋊C4—31no—
281C2×C22⋊C4—61no—
282#103—61no—
283#103—181no—
284C22⋊C8—31no—
285C22⋊C8—91no—
286#95—31no—
287#95—61no—
288C23×C3—21no—
289C23×C3—32no—
290C23×C3—61no—
291C23×C3—61no—
292C23×C3—61no—
293C23×C3—121no—
294C23×C3—121no—
295C23×C3D3×D411yescharacteristic
296C23×C3C2×#6711yescharacteristic
297C22×D3—21no—
298C22×D3—21no—
299C22×D3—21no—
300C22×D3—41no—
301C22×D3—41no—
302C22×D3—61no—
303C22×D3—61no—
304C22×D3—121no—
305C22×D3—122no—
306C22×D3—122no—
307C22×D3—241no—
308C22×D3—241no—
309C22×D3—241no—
310C22×D3—241no—
311C22×D3#21512yescomplemented
312C3×Q8—22no—
313C3×Q8—24no—
314C3×Q8—22no—
315C3×Q8—22no—
316C3×Q8—22no—
317C3×Q8—42no—
318C3×Q8—42no—
319C3×Q8—41no—
320C3×Q8—41no—
321C3×Q8—41no—
322C3×Q8—41no—
323C3×Q8—42no—
324C3×D4—121no—
325C3×D4—121no—
326C3×D4—121no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—42no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—42no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—61no—
342C2×C3×C4—61no—
343C2×C3×C4—121no—
344C2×C3×C4—121no—
345C2×C3×C4—121no—
346C2×C3×C4—121no—
347C2×C3×C4D3×D411yescharacteristic
348C2×C3×C4D3×D411yescharacteristic
349C2×C3×C4D3×D411yescharacteristic
350C2×C3×C4C2×#6711yescharacteristic
351#67—41no—
352#67—41no—
353#67—122no—
354#67—122no—
355#67—122no—
356#67—121no—
357#67—121no—
358#67—121no—
359#67—121no—
360#67—241no—
361#67—241no—
362#67—241no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—61no—
370C2×Dic3—61no—
371C2×Dic3—121no—
372C2×Dic3—122no—
373D12—41no—
374D12—121no—
375D12—121no—
376D12—122no—
377D12—122no—
378D12—241no—
379C4×D3—122no—
380C3×C8—121no—
381C3⋊C8—41no—
382C3⋊C8—122no—
383C3⋊C8—121no—
384C3⋊C8—121no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32#45011yescharacteristic
388C2×C32#45511yescharacteristic
389C2×C32B(4,𝔽2)11yescharacteristic
390C3⋊D3—81no—
391C3×D3—42no—
392C3×D3—81no—
393C24—31no—
394C24—181no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—42no—
398C2×D4—31no—
399C2×D4—61no—
400C2×D4—91no—
401C2×D4—181no—
402C2×D4—181no—
403C2×D4—181no—
404C2×D4—181no—
405C2×D4—362no—
406C2×D4—362no—
407C22×C4—21no—
408C22×C4—31no—
409C22×C4—61no—
410C22×C4—181no—
411C22×C4C2×D3211yescharacteristic
412SD16—122no—
413SD16—362no—
414C2×C8—61no—
415C2×C8—181no—
416C4⋊C4—31no—
417C4⋊C4—91no—
418C4⋊C4—181no—
419C22⋊C4—61no—
420C22⋊C4—61no—
421C22⋊C4—61no—
422C22⋊C4—122no—
423C22⋊C4—181no—
424C22⋊C4—181no—
425C22⋊C4—181no—
426C22×C3—21no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—41no—
434C22×C3—41no—
435C22×C3—62no—
436C22×C3—122no—
437C22×C3—121no—
438C22×C3—121no—
439C22×C3—121no—
440C22×C3—121no—
441C22×C3—121no—
442C22×C3—241no—
443C22×C3D3×SD1611yescharacteristic
444C22×C3#91411yescharacteristic
445C22×C3#93711yescharacteristic
446C22×C3#93711yescharacteristic
447C22×C3C2×#21511yescharacteristic
448C22×C3#94911yescharacteristic
449C2×D3—22no—
450C2×D3—41no—
451C2×D3—41no—
452C2×D3—42no—
453C2×D3—41no—
454C2×D3—41no—
455C2×D3—41no—
456C2×D3—81no—
457C2×D3—121no—
458C2×D3—121no—
459C2×D3—121no—
460C2×D3—121no—
461C2×D3—121no—
462C2×D3—121no—
463C2×D3—241no—
464C2×D3—241no—
465C2×D3—241no—
466C2×D3—244no—
467C2×D3—244no—
468C3×C4—22no—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—41no—
474C3×C4—44no—
475C3×C4—42no—
476C3×C4—42no—
477C3×C4—121no—
478C3×C4—121no—
479C3×C4—241no—
480Dic3—41no—
481Dic3—41no—
482Dic3—81no—
483Dic3—121no—
484Dic3—121no—
485Dic3—241no—
486C32#160111yescharacteristic, complemented, Sylow
487C23—32no—
488C23—61no—
489C23—61no—
490C23—61no—
491C23—121no—
492C23—121no—
493C23—181no—
494C23—362no—
495C23—721no—
496C23—721no—
497C23D3×#6711yescharacteristic
498Q8—22no—
499Q8—22no—
500Q8—41no—
501Q8—41no—
502D4—121no—
503D4—121no—
504D4—121no—
505D4—361no—
506D4—362no—
507D4—362no—
508D4—361no—
509D4—361no—
510D4—721no—
511C2×C4—22no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—121no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4—181no—
522C2×C4—362no—
523C2×C4D3×#6711yescharacteristic
524C2×C4#381111yescharacteristic
525C8—121no—
526C8—361no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—41no—
535C2×C3—41no—
536C2×C3—122no—
537C2×C3—241no—
538C2×C3#503211yescharacteristic
539C2×C3#503211yescharacteristic
540C2×C3#504611yescharacteristic
541C2×C3#506111yescharacteristic
542C2×C3#516911yescharacteristic
543C2×C3#520211yescharacteristic
544D3—42no—
545D3—81no—
546D3—241no—
547D3—241no—
548D3—481no—
549C22—21no—
550C22—21no—
551C22—62no—
552C22—122no—
553C22—121no—
554C22—121no—
555C22—121no—
556C22—121no—
557C22—121no—
558C22—241no—
559C22—361no—
560C22—361no—
561C22—361no—
562C22—724no—
563C22D3×#21511yescharacteristic
564C22#6393911yescharacteristic
565C22#6398311yescharacteristic
566C4—21no—
567C4—21no—
568C4—42no—
569C4—121no—
570C4—121no—
571C4—241no—
572C4—361no—
573C3—21no—
574C3#6969911yescharacteristic, complemented
575C3#7097211yescharacteristic, complemented
576C2—21no—
577C2—21no—
578C2—122no—
579C2—241no—
580C2—721no—
581C2#1059165111yescharacteristic
582C2#1059165911yescharacteristic
583C2#1059172511yescharacteristic
subgroupG/H
centerC22#63983
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC22×C32×Q8C22
socleC22×C32#120
prime radical#10595072C2
hypercenterC22×Q8D32
hypocenterC32#1601
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. #10592926C2semidirect
3. C3×#5231C2semidirect
4. C3×#5227C2semidirect
5. #10589151C2semidirect
6. #10589149C2semidirect
7. #10589145C2semidirect
10. C22×C32×Q8C22semidirect
24. #63657C22semidirect
27. C3×C3⋊#95C22semidirect
28. C3×#830C22semidirect
33. #5231D3semidirect
37. #5227D3semidirect
79. C22×C3×Q8C2×D3semidirect
80. C22×C3×Q8C2×D3semidirect
116. C3⋊#95C2×D3semidirect
120. #830C2×D3semidirect
128. C22×C3×D3SD16semidirect
269. C22×Q8D32semidirect
311. C22×D3#215semidirect
486. C32#1601semidirect
574. C3#69699semidirect
575. C3#70972semidirect

Maximal subgroups

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

subgroupindex
#105950722
#105929262
C3×#52312
C3×#52272
#105891512
#105891492
#105891452
#709723
#696993