Finite Group Databasebeta

← #11792612

Subgroups — #11792612

Order 1152.

881 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#63864C211yesmaximal, characteristic
2C22×#3718C211yesmaximal, characteristic, complemented
3C22×#3717C211yesmaximal, characteristic
4C2×#63636C211yesmaximal, characteristic
5C3×#4973C211yesmaximal, characteristic, complemented
6C2×C3×#810C211yesmaximal, characteristic, complemented
7#10589155C211yesmaximal, characteristic, complemented
8#70278—31nomaximal
9#69689—31nomaximal
10C22×C4⋊(C3⋊D3)C2211yescharacteristic
11C22×C3×Dic6C2211yescharacteristic
12C22×C3×Dic6C2211yescharacteristic
13C2×C32×C4⋊C4C2211yescharacteristic
14C22×C3⋊(C3⋊C8)C2211yescharacteristic
15C2×C3×C4×Dic3C2211yescharacteristic
16C22×C3×C3⋊C8C2211yescharacteristic
17C2×#3722—22no—
18#63864—22no—
19#63864—22no—
20C2×#3718—22no—
21C2×#3718—22no—
22C2×#3718—22no—
23C2×#3717—22no—
24C2×#3717—22no—
25C2×#3717—22no—
26#63636—22no—
27C3×C3⋊#95—22no—
28C3×#810—22no—
29C2×#947—31no—
30C22×#216—31no—
31C22×#216—31no—
32C22×#215—31no—
33C22×#204—31no—
34C2×#875—31no—
35C2×C3×#103—31no—
36C3×#391—31no—
37#4973D311yescharacteristic, complemented
38C2×#810D311yescharacteristic, complemented
39C2×#809—31no—
40C2×#809—31no—
41C2×#809—61no—
42#4550—31no—
43#4524—31no—
44C23×C3⋊D3—21no—
45C22×C32×C4—21no—
46C22×C32×C4C2311yescharacteristic
47C2×C4⋊(C3⋊D3)—21no—
48C2×C4⋊(C3⋊D3)—21no—
49C2×C4⋊(C3⋊D3)—21no—
50C2×C4⋊(C3⋊D3)—21no—
51C2×C4⋊(C3⋊D3)—21no—
52C2×C4⋊(C3⋊D3)D412yes—
53C22×C3×Dic3—21no—
54C22×C3×Dic3—21no—
55C22×C3×Dic3D412yes—
56C2×C3×Dic6—21no—
57C2×C3×Dic6—21no—
58C2×C3×Dic6—21no—
59C2×C3×Dic6—21no—
60C2×C3×Dic6—21no—
61C2×C3×Dic6—21no—
62C2×C3×Dic6—21no—
63C2×C3×Dic6—21no—
64C2×C3×Dic6—21no—
65C2×C3×Dic6—21no—
66C2×C3×Dic6D412yes—
67C2×C3×Dic6D412yes—
68C32×C4⋊C4—21no—
69C32×C4⋊C4—21no—
70C2×C3⋊(C3⋊C8)—21no—
71C2×C3⋊(C3⋊C8)—21no—
72C2×C3⋊(C3⋊C8)D412yes—
73C3×C4×Dic3—21no—
74C3×C4×Dic3—21no—
75C2×C3×C3⋊C8—21no—
76C2×C3×C3⋊C8—21no—
77C2×C3×C3⋊C8D412yes—
78#3722—28no—
79#3718—28no—
80#3717—28no—
81#2015—91noSylow
82C22×C3×Q8—31no—
83C22×C3×Q8—31no—
84C22×D12—31no—
85C22×D12—31no—
86C22×D12—61no—
87C22×Dic6C2×D311yescharacteristic
88C22×Dic6C2×D311yescharacteristic
89C22×C3×C8—31no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2×D311yescharacteristic
92C2×C3×C4⋊C4C2×D311yescharacteristic
93C2×C3×C42—31no—
94#947—62no—
95#947—62no—
96C2×#216—62no—
97C2×#216—62no—
98C2×#216—62no—
99C2×#216—62no—
100C2×#216—62no—
101C2×#216—62no—
102C2×#215—62no—
103C2×#215—62no—
104C2×#215—62no—
105C2×(C2×D3)⋊C4—62no—
106C2×(C2×D3)⋊C4—62no—
107C2×C4×Dic3C2×D311yescharacteristic
108C22×C3⋊C8—31no—
109C22×C3⋊C8—31no—
110C22×C3⋊C8—61no—
111C22×C3⋊C8C2×D311yescharacteristic
112C2×#204—62no—
113C2×#204—62no—
114C2×#204—62no—
115#875—62no—
116#875—62no—
117C3×#103—62no—
118C3×#95—62no—
119C3⋊#95—22no—
120#810—22no—
121#809—62no—
122#809—62no—
123#809—64no—
124C23×C32C2×D411yescharacteristic
125C22×C3⋊D3—22no—
126C22×C3⋊D3—41no—
127C22×C3⋊D3—41no—
128C22×C3⋊D3—42no—
129C22×C3⋊D3—42no—
130C2×C32×C4—41no—
131C2×C32×C4—42no—
132C2×C32×C4C2×D411yescharacteristic
133C2×C32×C4C2×D411yescharacteristic
134C2×C32×C4C2×D411yescharacteristic
135C2×C32×C4C2×D411yescharacteristic
136C2×C32×C4C2×D411yescharacteristic
137C2×C32×C4#4111yescharacteristic
138C4⋊(C3⋊D3)—22no—
139C4⋊(C3⋊D3)—22no—
140C4⋊(C3⋊D3)—41no—
141C4⋊(C3⋊D3)—41no—
142C2×C3×Dic3—22no—
143C2×C3×Dic3—22no—
144C2×C3×Dic3—22no—
145C2×C3×Dic3—22no—
146C2×C3×Dic3—22no—
147C2×C3×Dic3—41no—
148C2×C3×Dic3—41no—
149C2×C3×Dic3—41no—
150C2×C3×Dic3—41no—
151C3×Dic6—22no—
152C3×Dic6—22no—
153C3×Dic6—22no—
154C3×Dic6—22no—
155C3×Dic6—41no—
156C3×Dic6—41no—
157C3×Dic6—41no—
158C3×Dic6—41no—
159C3⋊(C3⋊C8)—22no—
160C3×C3⋊C8—22no—
161C22×Q16—91no—
162C22×SD16—91no—
163C2×#124—91no—
164C2×#103—31no—
165C2×#102—91no—
166#391—31no—
167#336—91no—
168C23×D3—61no—
169C23×D3—61no—
170C23×D3—121no—
171C2×C3×Q8—32no—
172C2×C3×Q8—32no—
173C2×C3×Q8—61no—
174C2×C3×Q8—61no—
175C2×C3×Q8—61no—
176C2×C3×Q8—61no—
177C2×C3×Q8—61no—
178C2×C3×Q8—61no—
179C2×C3×Q8—61no—
180C2×C3×Q8—61no—
181C2×C3×Q8—61no—
182C2×C3×Q8—61no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—32no—
187C22×C3×C4—41no—
188C22×C3×C4—61no—
189C22×C3×C4—61no—
190C22×C3×C4C22×D311yescharacteristic
191C22×C3×C4C22×D311yescharacteristic
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3#6712yes—
195C2×D12—32no—
196C2×D12—32no—
197C2×D12—62no—
198C2×D12—62no—
199C2×D12—62no—
200C2×D12—61no—
201C2×D12—61no—
202C2×D12—61no—
203C2×D12—61no—
204C2×D12—61no—
205C2×D12—61no—
206C2×D12—61no—
207C2×D12—61no—
208C2×D12—61no—
209C2×D12—61no—
210C2×D12—121no—
211C2×D12—121no—
212C2×D12—121no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6D1212yes—
224C2×Dic6D1212yes—
225C2×C3×C8—32no—
226C2×C3×C8—61no—
227C2×C3×C8—61no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—41no—
234C3×C42—61no—
235C3×C42—61no—
236#216—68no—
237#216—68no—
238#215—68no—
239(C2×D3)⋊C4—68no—
240(C2×D3)⋊C4—68no—
241C4×Dic3—21no—
242C4×Dic3—21no—
243C2×C3⋊C8—21no—
244C2×C3⋊C8—21no—
245C2×C3⋊C8—32no—
246C2×C3⋊C8—32no—
247C2×C3⋊C8—62no—
248C2×C3⋊C8—61no—
249C2×C3⋊C8—61no—
250C2×C3⋊C8—61no—
251C2×C3⋊C8—61no—
252C2×C3⋊C8—121no—
253C2×C3⋊C8—121no—
254C2×C3⋊C8#6712yes—
255#204—68no—
256C22×C32#12011yescharacteristic
257C22×C32#12111yescharacteristic
258C22×C32#12711yescharacteristic
259C22×C32#13512yes—
260C22×C32#13712yes—
261C2×C3⋊D3—42no—
262C2×C3⋊D3—44no—
263C2×C3⋊D3—82no—
264C2×C3⋊D3—81no—
265C2×C3⋊D3—81no—
266C32×C4—81no—
267C32×C4#12011yescharacteristic
268C32×C4#12011yescharacteristic
269C32×C4#12111yescharacteristic
270C32×C4#12111yescharacteristic
271C3×Dic3—42no—
272C3×Dic3—42no—
273C3×Dic3—42no—
274C22×Q8—31no—
275C22×Q8—31no—
276C22×D4—91no—
277C2×Q16—182no—
278C2×Q16—182no—
279C2×Q16—182no—
280C2×SD16—182no—
281C2×SD16—182no—
282C2×SD16—182no—
283C22×C8—31no—
284C22×C8—91no—
285#124—182no—
286#124—182no—
287C2×C4⋊C4D3211yescharacteristic
288C2×C22⋊C4—182no—
289C2×C42—31no—
290#103—62no—
291#102—182no—
292#95—62no—
293C23×C3—21no—
294C23×C3D3×D411yescharacteristic
295C23×C3C2×#6711yescharacteristic
296C22×D3—62no—
297C22×D3—62no—
298C22×D3—122no—
299C22×D3—122no—
300C22×D3—122no—
301C22×D3—121no—
302C22×D3—122no—
303C22×D3—122no—
304C22×D3—121no—
305C22×D3—121no—
306C22×D3—121no—
307C22×D3—122no—
308C22×D3—122no—
309C22×D3—242no—
310C22×D3—242no—
311C3×Q8—62no—
312C3×Q8—62no—
313C3×Q8—62no—
314C3×Q8—62no—
315C3×Q8—121no—
316C3×Q8—121no—
317C3×Q8—121no—
318C3×Q8—121no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—42no—
330C2×C3×C4—62no—
331C2×C3×C4—62no—
332C2×C3×C4—62no—
333C2×C3×C4—62no—
334C2×C3×C4—62no—
335C2×C3×C4—82no—
336C2×C3×C4—121no—
337C2×C3×C4—121no—
338C2×C3×C4—121no—
339C2×C3×C4—121no—
340C2×C3×C4C2×D1211yescharacteristic
341C2×C3×C4C2×D1211yescharacteristic
342C2×C3×C4#23511yescharacteristic
343C2×C3×C4#23511yescharacteristic
344C2×C3×C4D3×D411yescharacteristic
345C2×C3×C4D3×D411yescharacteristic
346C2×C3×C4D3×D411yescharacteristic
347C2×C3×C4D3×D411yescharacteristic
348C2×C3×C4D3×D411yescharacteristic
349C2×C3×C4D3×D411yescharacteristic
350C2×C3×C4D3×D411yescharacteristic
351C2×C3×C4C2×#6711yescharacteristic
352C2×Dic3—22no—
353C2×Dic3—22no—
354C2×Dic3—22no—
355C2×Dic3—22no—
356C2×Dic3—22no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361D12—64no—
362D12—62no—
363D12—62no—
364D12—62no—
365D12—62no—
366D12—122no—
367D12—122no—
368D12—122no—
369D12—121no—
370D12—121no—
371D12—121no—
372D12—121no—
373Dic6—22no—
374Dic6—22no—
375Dic6—22no—
376Dic6—22no—
377Dic6—41no—
378Dic6—41no—
379Dic6—41no—
380Dic6—41no—
381C3×C8—62no—
382C3⋊C8—22no—
383C3⋊C8—62no—
384C3⋊C8—62no—
385C3⋊C8—122no—
386C2×C32#39211yescharacteristic
387C2×C32#45212yes—
388C2×C32#46412yes—
389C2×C32#49511yescharacteristic
390C2×C32#49711yescharacteristic
391C3⋊D3—82no—
392C24—181no—
393C2×Q8—32no—
394C2×Q8—32no—
395C2×Q8—61no—
396C2×Q8—61no—
397C2×Q8—61no—
398C2×Q8—61no—
399C2×Q8—61no—
400C2×Q8—61no—
401C2×Q8—61no—
402C2×Q8—61no—
403C2×Q8—61no—
404C2×Q8—61no—
405C2×D4—92no—
406C2×D4—181no—
407C2×D4—181no—
408C2×D4—181no—
409C2×D4—181no—
410C2×D4—181no—
411C22×C4—21no—
412C22×C4—32no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4C2×D3211yescharacteristic
416Q16—188no—
417SD16—188no—
418C2×C8—32no—
419C2×C8—61no—
420C2×C8—61no—
421C2×C8—92no—
422C2×C8—181no—
423C2×C8—181no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C22⋊C4—188no—
427C42—61no—
428C42—61no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—21no—
434C22×C3#88211yescharacteristic
435C22×C3#88411yescharacteristic
436C22×C3#91612yes—
437C22×C3#91812yes—
438C22×C3#93011yescharacteristic
439C22×C3#93711yescharacteristic
440C22×C3#94011yescharacteristic
441C22×C3#94011yescharacteristic
442C22×C3#94212yes—
443C22×C3#95012yes—
444C2×D3—124no—
445C2×D3—122no—
446C2×D3—122no—
447C2×D3—124no—
448C2×D3—124no—
449C2×D3—244no—
450C2×D3—244no—
451C2×D3—242no—
452C2×D3—242no—
453C2×D3—241no—
454C2×D3—241no—
455C2×D3—242no—
456C2×D3—242no—
457C2×D3—241no—
458C2×D3—241no—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—21no—
463C3×C4—82no—
464C3×C4—81no—
465C3×C4—81no—
466C3×C4—122no—
467C3×C4—122no—
468C3×C4—122no—
469C3×C4#88211yescharacteristic
470C3×C4#88211yescharacteristic
471C3×C4#88211yescharacteristic
472C3×C4#88411yescharacteristic
473C3×C4#88411yescharacteristic
474C3×C4#88411yescharacteristic
475C3×C4#93011yescharacteristic
476C3×C4#93711yescharacteristic
477Dic3—42no—
478Dic3—42no—
479Dic3—42no—
480C32#201511yescharacteristic, complemented, Sylow
481C23—182no—
482C23—361no—
483C23—362no—
484C23—362no—
485C23—361no—
486C23D3×#6711yescharacteristic
487Q8—62no—
488Q8—62no—
489Q8—62no—
490Q8—62no—
491Q8—121no—
492Q8—121no—
493Q8—121no—
494Q8—121no—
495D4—182no—
496D4—182no—
497D4—361no—
498D4—361no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—62no—
502C2×C4—62no—
503C2×C4—62no—
504C2×C4—62no—
505C2×C4—62no—
506C2×C4—121no—
507C2×C4—121no—
508C2×C4—121no—
509C2×C4—121no—
510C2×C4#379811yescharacteristic
511C2×C4D3×D1211yescharacteristic
512C2×C4D3×D1211yescharacteristic
513C2×C4#380211yescharacteristic
514C2×C4D3×#6711yescharacteristic
515C2×C4#381111yescharacteristic
516C8—62no—
517C8—182no—
518C2×C3—22no—
519C2×C3—21no—
520C2×C3—22no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3#479812yes—
524C2×C3#480812yes—
525C2×C3#496011yescharacteristic
526C2×C3#496011yescharacteristic
527C2×C3#502812yes—
528C2×C3#504712yes—
529C2×C3#505911yescharacteristic
530C2×C3#508611yescharacteristic
531C2×C3#516811yescharacteristic
532C2×C3#519011yescharacteristic
533D3—244no—
534D3—242no—
535D3—242no—
536C22—364no—
537C22—362no—
538C22—722no—
539C22—721no—
540C22—721no—
541C22#6390611yescharacteristic
542C22#6392211yescharacteristic
543C22#6393512yes—
544C22#6394812yes—
545C22#6397911yescharacteristic
546C4—81no—
547C4—122no—
548C4—122no—
549C4—122no—
550C4#6390611yescharacteristic
551C4#6391211yescharacteristic
552C4#6392211yescharacteristic
553C4#6392311yescharacteristic
554C3—21no—
555C3#6968911yescharacteristic, complemented
556C3#7027811yescharacteristic, complemented
557C2—722no—
558C2#1059053612yes—
559C2#1059061612yes—
560C2#1059139311yescharacteristic
561C2#1059147311yescharacteristic
562C2#1059160811yescharacteristic
subgroupG/H
centerC23D3×#67
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC2×C32×C4⋊C4C22
socleC23×C32C2×D4
prime radicalC22×C4⋊(C3⋊D3)C22
hypercenterC2×C4⋊C4D32
hypocenterC32#2015
LCSG ▷ C22×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×#3718C2semidirect
5. C3×#4973C2semidirect
6. C2×C3×#810C2semidirect
7. #10589155C2semidirect
37. #4973D3semidirect
38. C2×#810D3semidirect
480. C32#2015semidirect
555. C3#69689semidirect
556. C3#70278semidirect

Maximal subgroups

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

subgroupindex
C2×#638642
C22×#37182
C22×#37172
C2×#636362
C3×#49732
C2×C3×#8102
#105891552
#702783
#696893