Finite Group Databasebeta

← #11792601

Subgroups — #11792601

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×C3×#936C211yesmaximal, characteristic
2C22×C3×#214C211yesmaximal, characteristic
3C22×#3718C211yesmaximal, characteristic, complemented
4C2×#63659C211yesmaximal, characteristic
5#10591220C211yesmaximal, characteristic, complemented
6C2×#63566C211yesmaximal, characteristic, complemented
7#10589155C211yesmaximal, characteristic, complemented
8#72058—31nomaximal
9#71729—31nomaximal
10C22×C32×D4C2211yescharacteristic
11C22×C3×Dic6C2211yescharacteristic
12C22×C3×Dic6C2211yescharacteristic
13C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C22×C3⋊(C3⋊C8)C2211yescharacteristic
15C2×C3×#217—22no—
16C3×#936—22no—
17C3×#936—22no—
18C2×C3×#214—22no—
19C2×C3×#214—22no—
20C2×C3×#214—22no—
21C2×C3×C4×Dic3C2211yescharacteristic
22C22×C3×C3⋊C8C2211yescharacteristic
23C2×#3718—22no—
24C2×#3718—22no—
25C2×#3718—22no—
26#63659—22no—
27#63579—22no—
28#63566—22no—
29C22×C3×SD16—31no—
30C2×C3×#124—31no—
31C22×#216—31no—
32C22×#216—31no—
33C2×#936D311yescharacteristic
34C22×#214D311yescharacteristic
35#5229—31no—
36C2×#835—31no—
37C2×#832—31no—
38C2×#832—31no—
39C2×#832—61no—
40C2×#816—31no—
41#4929—31no—
42#4550—31no—
43#4524—31no—
44C24×C32—21no—
45C2×C32×D4—21no—
46C2×C32×D4—21no—
47C2×C32×D4—21no—
48C2×C32×D4—21no—
49C2×C32×D4—21no—
50C2×C32×D4D412yes—
51C22×C32×C4C2311yescharacteristic
52C22×C3⋊Dic3—21no—
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—
68C4⋊(C3⋊Dic3)—21no—
69C4⋊(C3⋊Dic3)—21no—
70C2×C3⋊(C3⋊C8)—21no—
71C2×C3⋊(C3⋊C8)—21no—
72C2×C3⋊(C3⋊C8)D412yes—
73C3×#217—28no—
74C3×#214—28no—
75C3×C4×Dic3—21no—
76C3×C4×Dic3—21no—
77C2×C3×C3⋊C8—21no—
78C2×C3×C3⋊C8—21no—
79C2×C3×C3⋊C8D412yes—
80#3718—28no—
81#2015—91noSylow
82C22×C3×Q8—31no—
83C22×C3×Q8—31no—
84C22×C3×D4—21no—
85C22×C3×D4C2×D311yescharacteristic
86C22×C3×D4C2×D311yescharacteristic
87C22×Dic6C2×D311yescharacteristic
88C22×Dic6C2×D311yescharacteristic
89C2×C3×SD16—62no—
90C2×C3×SD16—62no—
91C2×C3×SD16—62no—
92C22×C3×C8—31no—
93C3×#124—62no—
94C3×#124—62no—
95C2×C3×C22⋊C4—62no—
96C2×C3×C42—31no—
97C2×#217—22no—
98C2×#216—62no—
99C2×#216—62no—
100C2×#216—62no—
101C2×#216—62no—
102C2×#216—62no—
103C2×#216—62no—
104#936—22no—
105#936—22no—
106C2×#214—22no—
107C2×#214—22no—
108C2×#214—22no—
109C2×C4⋊Dic3—31no—
110C2×C4⋊Dic3—31no—
111C2×C4⋊Dic3—61no—
112C2×C4×Dic3C2×D311yescharacteristic
113C22×C3⋊C8—31no—
114C22×C3⋊C8—31no—
115C22×C3⋊C8—61no—
116C22×C3⋊C8C2×D311yescharacteristic
117#835—62no—
118#832—62no—
119#832—62no—
120#832—64no—
121C3⋊#95—62no—
122C3⋊#95—62no—
123#816—62no—
124C23×C32—22no—
125C23×C32—41no—
126C23×C32—41no—
127C23×C32—42no—
128C23×C32—42no—
129C23×C32C2×D411yescharacteristic
130C32×D4—22no—
131C32×D4—22no—
132C32×D4—41no—
133C32×D4—41no—
134C2×C32×C4C2×D411yescharacteristic
135C2×C32×C4C2×D411yescharacteristic
136C2×C32×C4C2×D411yescharacteristic
137C2×C32×C4C2×D411yescharacteristic
138C2×C32×C4C2×D411yescharacteristic
139C2×C32×C4#4111yescharacteristic
140C2×C3⋊Dic3—41no—
141C2×C3⋊Dic3—42no—
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—31no—
163C2×#124—31no—
164C2×#103—91no—
165C2×#102—91no—
166#391—91no—
167#336—91no—
168C24×C3—21no—
169C24×C3—21no—
170C24×C3—41no—
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—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4—22no—
196C2×C3×D4—41no—
197C2×C3×D4—41no—
198C2×C3×D4—41no—
199C2×C3×D4#6712yes—
200C2×C3×D4#6712yes—
201C22×C3×C4—21no—
202C22×C3×C4—32no—
203C22×C3×C4—61no—
204C22×C3×C4—61no—
205C22×C3×C4C22×D311yescharacteristic
206C22×C3×C4C22×D311yescharacteristic
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—61no—
210C22×Dic3—61no—
211C22×Dic3—121no—
212C22×Dic3#6712yes—
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×Dic6#6712yes—
224C2×Dic6#6712yes—
225C3×SD16—68no—
226C2×C3×C8—32no—
227C2×C3×C8—61no—
228C2×C3×C8—61no—
229C3×C22⋊C4—68no—
230C3×C42—61no—
231C3×C42—61no—
232#217—28no—
233#216—68no—
234#216—68no—
235#214—28no—
236C4⋊Dic3—62no—
237C4⋊Dic3—61no—
238C4⋊Dic3—61no—
239C4⋊Dic3—61no—
240C4⋊Dic3—61no—
241C4⋊Dic3—121no—
242C4×Dic3—21no—
243C4×Dic3—21no—
244C2×C3⋊C8—21no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—32no—
247C2×C3⋊C8—32no—
248C2×C3⋊C8—62no—
249C2×C3⋊C8—61no—
250C2×C3⋊C8—61no—
251C2×C3⋊C8—61no—
252C2×C3⋊C8—61no—
253C2×C3⋊C8—121no—
254C2×C3⋊C8—121no—
255C2×C3⋊C8#6712yes—
256C22×C32—42no—
257C22×C32—44no—
258C22×C32—82no—
259C22×C32—81no—
260C22×C32—81no—
261C22×C32#12011yescharacteristic
262C22×C32#12111yescharacteristic
263C22×C32#12711yescharacteristic
264C22×C32#13512yes—
265C22×C32#13712yes—
266C32×C4#12011yescharacteristic
267C32×C4#12011yescharacteristic
268C32×C4#12111yescharacteristic
269C32×C4#12111yescharacteristic
270C3⋊Dic3—81no—
271C3×Dic3—42no—
272C3×Dic3—42no—
273C3×Dic3—42no—
274C22×Q8—31no—
275C22×Q8—31no—
276C22×D4D3211yescharacteristic
277C2×Q16—182no—
278C2×Q16—182no—
279C2×Q16—182no—
280C2×SD16—62no—
281C2×SD16—62no—
282C2×SD16—62no—
283C22×C8—31no—
284C22×C8—91no—
285#124—62no—
286#124—62no—
287C2×C4⋊C4—91no—
288C2×C22⋊C4—62no—
289C2×C42—31no—
290#103—182no—
291#102—182no—
292#95—182no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—21no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3—41no—
302C23×C3—41no—
303C23×C3—41no—
304C23×C3—41no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3—82no—
308C23×C3—82no—
309C23×C3D3×D411yescharacteristic
310C23×C3C2×#6711yescharacteristic
311C3×Q8—62no—
312C3×Q8—62no—
313C3×Q8—62no—
314C3×Q8—62no—
315C3×Q8—121no—
316C3×Q8—121no—
317C3×Q8—121no—
318C3×Q8—121no—
319C3×D4—24no—
320C3×D4—22no—
321C3×D4—22no—
322C3×D4—22no—
323C3×D4—22no—
324C3×D4—42no—
325C3×D4—42no—
326C3×D4—42no—
327C3×D4—41no—
328C3×D4—41no—
329C3×D4—41no—
330C3×D4—41no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—62no—
338C2×C3×C4—62no—
339C2×C3×C4—62no—
340C2×C3×C4—62no—
341C2×C3×C4—62no—
342C2×C3×C4—121no—
343C2×C3×C4—121no—
344C2×C3×C4—121no—
345C2×C3×C4—121no—
346C2×C3×C4D3×D411yescharacteristic
347C2×C3×C4D3×D411yescharacteristic
348C2×C3×C4D3×D411yescharacteristic
349C2×C3×C4D3×D411yescharacteristic
350C2×C3×C4D3×D411yescharacteristic
351C2×C3×C4#23711yescharacteristic
352C2×C3×C4#23711yescharacteristic
353C2×C3×C4C2×#6711yescharacteristic
354C2×C3×C4C2×#6711yescharacteristic
355C2×C3×C4C2×#6711yescharacteristic
356C2×C3×C4C2×#6711yescharacteristic
357C2×C3×C4C2×#6711yescharacteristic
358C2×Dic3—22no—
359C2×Dic3—22no—
360C2×Dic3—22no—
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—122no—
368C2×Dic3—122no—
369C2×Dic3—121no—
370C2×Dic3—121no—
371C2×Dic3—122no—
372C2×Dic3—242no—
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—82no—
387C2×C32#39211yescharacteristic
388C2×C32#45212yes—
389C2×C32#46412yes—
390C2×C32#49511yescharacteristic
391C2×C32#49711yescharacteristic
392C24—21no—
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—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4#62512yes—
411C22×C4—32no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—181no—
415C22×C4C2×D3211yescharacteristic
416Q16—188no—
417SD16—68no—
418C2×C8—32no—
419C2×C8—61no—
420C2×C8—61no—
421C2×C8—92no—
422C2×C8—181no—
423C2×C8—181no—
424C4⋊C4—181no—
425C4⋊C4—181no—
426C22⋊C4—68no—
427C42—61no—
428C42—61no—
429C22×C3—22no—
430C22×C3—22no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—21no—
434C22×C3—44no—
435C22×C3—44no—
436C22×C3—42no—
437C22×C3—42no—
438C22×C3—44no—
439C22×C3—84no—
440C22×C3—82no—
441C22×C3—84no—
442C22×C3—82no—
443C22×C3—82no—
444C22×C3—81no—
445C22×C3—81no—
446C22×C3—82no—
447C22×C3—81no—
448C22×C3—81no—
449C22×C3#91512yes—
450C22×C3#91612yes—
451C22×C3#93711yescharacteristic
452C22×C3#93811yescharacteristic
453C22×C3#93911yescharacteristic
454C22×C3#94011yescharacteristic
455C22×C3#94011yescharacteristic
456C22×C3#95012yes—
457C22×C3#95112yes—
458C22×C3#95311yescharacteristic
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—21no—
463C3×C4—122no—
464C3×C4—122no—
465C3×C4—122no—
466C3×C4#93711yescharacteristic
467C3×C4#93711yescharacteristic
468C3×C4#93711yescharacteristic
469C3×C4#93811yescharacteristic
470C3×C4#93911yescharacteristic
471C3×C4#93911yescharacteristic
472C3×C4#93911yescharacteristic
473C3×C4#95311yescharacteristic
474Dic3—42no—
475Dic3—42no—
476Dic3—42no—
477Dic3—242no—
478Dic3—241no—
479Dic3—241no—
480C32#201511yescharacteristic, complemented, Sylow
481C23—22no—
482C23—42no—
483C23—42no—
484C23—41no—
485C23—41no—
486C23D3×#6711yescharacteristic
487Q8—62no—
488Q8—62no—
489Q8—62no—
490Q8—62no—
491Q8—121no—
492Q8—121no—
493Q8—121no—
494Q8—121no—
495D4—22no—
496D4—22no—
497D4—41no—
498D4—41no—
499C2×C4—62no—
500C2×C4—62no—
501C2×C4—62no—
502C2×C4—62no—
503C2×C4—62no—
504C2×C4—121no—
505C2×C4—121no—
506C2×C4—121no—
507C2×C4—121no—
508C2×C4—362no—
509C2×C4—361no—
510C2×C4#380511yescharacteristic
511C2×C4C2×#62511yescharacteristic
512C2×C4D3×#6711yescharacteristic
513C2×C4D3×#6711yescharacteristic
514C2×C4D3×#6711yescharacteristic
515C2×C4#381111yescharacteristic
516C8—62no—
517C8—182no—
518C2×C3—21no—
519C2×C3—22no—
520C2×C3—22no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—84no—
524C2×C3—82no—
525C2×C3—82no—
526C2×C3#505412yes—
527C2×C3#505911yescharacteristic
528C2×C3#506011yescharacteristic
529C2×C3#516612yes—
530C2×C3#516711yescharacteristic
531C2×C3#516811yescharacteristic
532C2×C3#517312yes—
533C2×C3#522311yescharacteristic
534C2×C3#522311yescharacteristic
535C2×C3#523912yes—
536C22—44no—
537C22—42no—
538C22—81no—
539C22—82no—
540C22—81no—
541C22#6392912yes—
542C22#6393512yes—
543C22#6397011yescharacteristic
544C22#6397611yescharacteristic
545C22#6397911yescharacteristic
546C4—122no—
547C4—122no—
548C4—122no—
549C4—721no—
550C4#6397011yescharacteristic
551C4#6397611yescharacteristic
552C4#6398011yescharacteristic
553C4#6398311yescharacteristic
554C3—21no—
555C3#7172911yescharacteristic, complemented
556C3#7205811yescharacteristic, complemented
557C2—82no—
558C2#1059144212yes—
559C2#1059146112yes—
560C2#1059147311yescharacteristic
561C2#1059149111yescharacteristic
562C2#1059172311yescharacteristic
subgroupG/H
centerC23D3×#67
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
Fitting, prime radicalC22×C32×D4C22
socleC23×C32C2×D4
hypercenterC22×D4D32
hypocenterC32#2015
LCSG ▷ C22×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
3. C22×#3718C2semidirect
5. #10591220C2semidirect
6. C2×#63566C2semidirect
7. #10589155C2semidirect
480. C32#2015semidirect
555. C3#71729semidirect
556. C3#72058semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9362
C22×C3×#2142
C22×#37182
C2×#636592
#105912202
C2×#635662
#105891552
#720583
#717293