Finite Group Databasebeta

← #10694330

Subgroups — #10694330

Order 768.

650 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82698C211yesmaximal, characteristic
2C2×C3×#327C211yesmaximal, characteristic
3C2×#4537C211yesmaximal, characteristic
4C2×#4467C211yesmaximal, characteristic, complemented
5#69908C211yesmaximal, characteristic, complemented
6#69401C211yesmaximal, characteristic, complemented
7C2×#4452C211yesmaximal, characteristic, complemented
8#9716—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#936C411yescharacteristic, complemented
11C2×#924C2211yescharacteristic
12C22×C4×Dic3C2211yescharacteristic
13C2×C3×C22⋊C8C2211yescharacteristic
14C2×C3×#95C2211yescharacteristic
15#5221C411yescharacteristic, complemented
16#5220—21no—
17#5220—21no—
18C2×C3⋊#95—21no—
19C2×C3⋊#95—21no—
20C2×C3⋊#95C2211yescharacteristic
21C2×#830C2211yescharacteristic
22C2×Dic3⋊C8—21no—
23#4977C411yescharacteristic, complemented
24#4973C411yescharacteristic, complemented
25#4945—21no—
26#4945—21no—
27C3×#327—22no—
28#4550—21no—
29#4537—22no—
30#4467—22no—
31#4452—22no—
32#2850—31no—
33#1797—31no—
34C2×#328—31no—
35C2×#328—31no—
36C2×#327—31no—
37C2×#327D311yescharacteristic
38#1434—31no—
39C22×C3×D4—21no—
40C22×C3×D4C2×C411yescharacteristic
41C23×C3×C4—21no—
42C23×C3×C4C2311yescharacteristic
43C23×Dic3—21no—
44C23×Dic3D411yescharacteristic
45C23×Dic3D411yescharacteristic
46C22×Dic6C2×C411yescharacteristic
47C22×C3×C8—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×#121—21no—
52C2×C3×C4⋊C4C2×C411yescharacteristic
53C2×C3×C22⋊C4—21no—
54C2×#217—21no—
55C2×#217—21no—
56C2×#217—21no—
57C2×#217—21no—
58C2×#217—21no—
59#936—42no—
60C2×C4⋊Dic3C2×C411yescharacteristic
61#924—21no—
62#924—21no—
63#924—21no—
64#924—21no—
65C2×Dic3⋊C4—21no—
66C2×C4×Dic3—42no—
67C2×C4×Dic3C2×C411yescharacteristic
68C2×C4×Dic3C2×C411yescharacteristic
69C2×C4×Dic3D411yescharacteristic
70C2×C4×Dic3D411yescharacteristic
71C22×C3⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×#95—21no—
75C3×#95D412yes—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81C3⋊#95—42no—
82C3⋊#95—42no—
83C3⋊#95D412yes—
84#830—21no—
85#830—21no—
86Dic3⋊C8—42no—
87C2×#124—31no—
88C2×C22⋊Q8—31no—
89C2×#121C2×D311yescharacteristic
90C22×C42—31no—
91C2×C4⋊C8—61no—
92C2×C22⋊C8—31no—
93C2×C22⋊C8C2×D311yescharacteristic
94#391—31no—
95#390—31no—
96#388—61no—
97#388—61no—
98#386—61no—
99#386—61no—
100#382—31no—
101C2×#95—31no—
102C2×#95—61no—
103C2×#95—61no—
104C2×#95C2×D311yescharacteristic
105#336—61no—
106#328—62no—
107#328—62no—
108#327—22no—
109#327—62no—
110C24×C3—21no—
111C24×C3C2×D411yescharacteristic
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C22×C3×C4—21no—
123C22×C3×C4—22no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4—41no—
129C22×C3×C4—41no—
130C22×C3×C4—41no—
131C22×C3×C4—41no—
132C22×C3×C4C22×C411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4#4111yescharacteristic
137C22×C3×C4#4111yescharacteristic
138C22×Dic3—21no—
139C22×Dic3—21no—
140C22×Dic3—21no—
141C22×Dic3—21no—
142C22×Dic3—22no—
143C22×Dic3—21no—
144C22×Dic3—21no—
145C22×Dic3—22no—
146C22×Dic3—41no—
147C22×Dic3—41no—
148C22×Dic3—41no—
149C22×Dic3—41no—
150C22×Dic3—41no—
151C22×Dic3—41no—
152C22×Dic3—41no—
153C22×Dic3—41no—
154C22×Dic3—41no—
155C22×Dic3C22⋊C411yescharacteristic
156C22×Dic3C22⋊C411yescharacteristic
157C22×Dic3C22⋊C411yescharacteristic
158C22×Dic3C22⋊C411yescharacteristic
159C2×Dic6—21no—
160C2×Dic6—21no—
161C2×Dic6—41no—
162C2×Dic6—41no—
163C2×C3×C8—41no—
164C2×C3×C8—41no—
165C2×C3×C8—41no—
166C3×C4⋊C4—21no—
167C3×C4⋊C4—21no—
168C3×C22⋊C4—41no—
169C3×C22⋊C4—41no—
170#217—42no—
171#217—42no—
172#217—42no—
173#217—41no—
174#217—41no—
175#217—42no—
176C4⋊Dic3—21no—
177C4⋊Dic3—21no—
178Dic3⋊C4—41no—
179Dic3⋊C4—41no—
180C4×Dic3—22no—
181C4×Dic3—22no—
182C4×Dic3—41no—
183C4×Dic3—41no—
184C2×C3⋊C8—41no—
185C2×C3⋊C8—41no—
186C2×C3⋊C8—41no—
187C22×Q8—31no—
188C22×D4—21no—
189C22×D4C4×D311yescharacteristic
190C23×C4—21no—
191C23×C4—31no—
192C23×C4—31no—
193C23×C4—61no—
194C23×C4C22×D311yescharacteristic
195C22×C8—21no—
196C22×C8—61no—
197#124—122no—
198C22⋊Q8—61no—
199C22⋊Q8—61no—
200C22⋊Q8—61no—
201C22⋊Q8—61no—
202#121—21no—
203#121—21no—
204#121—21no—
205#121—21no—
206C2×C4⋊C4—31no—
207C2×C4⋊C4—61no—
208C2×C4⋊C4C4×D311yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—61no—
211C2×C22⋊C4—61no—
212C2×C22⋊C4—61no—
213C2×C22⋊C4—61no—
214C2×C22⋊C4—61no—
215C2×C42—31no—
216C2×C42—31no—
217C2×C42—31no—
218C2×C42—31no—
219C2×C42—122no—
220C4⋊C8—122no—
221C22⋊C8—21no—
222C22⋊C8—21no—
223C22⋊C8—61no—
224C22⋊C8—61no—
225#95—21no—
226#95—32no—
227#95—61no—
228#95—61no—
229#95—61no—
230#95—61no—
231#95—61no—
232#95—122no—
233#95—122no—
234#95#6712yes—
235C23×C3—21no—
236C23×C3—21no—
237C23×C3—21no—
238C23×C3—21no—
239C23×C3—41no—
240C23×C3—41no—
241C23×C3—41no—
242C23×C3—41no—
243C23×C3—41no—
244C23×C3—41no—
245C23×C3—41no—
246C23×C3C2×C22⋊C411yescharacteristic
247C23×C3#12111yescharacteristic
248C23×C3#12111yescharacteristic
249C23×C3#13511yescharacteristic
250C23×C3#13511yescharacteristic
251C23×C3AGL(1,ℤ/8)11yescharacteristic
252C23×C3#13711yescharacteristic
253C3×D4—81no—
254C3×D4—81no—
255C3×D4—81no—
256C3×D4—81no—
257C3×D4—81no—
258C3×D4—81no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—42no—
273C2×C3×C4—81no—
274C2×C3×C4—81no—
275C2×C3×C4—81no—
276C2×C3×C4—81no—
277C2×C3×C4C4×D411yescharacteristic
278C2×C3×C4C4×D411yescharacteristic
279C2×C3×C4#12411yescharacteristic
280C2×C3×C4#12711yescharacteristic
281C2×Dic3—21no—
282C2×Dic3—21no—
283C2×Dic3—21no—
284C2×Dic3—21no—
285C2×Dic3—21no—
286C2×Dic3—21no—
287C2×Dic3—21no—
288C2×Dic3—21no—
289C2×Dic3—21no—
290C2×Dic3—21no—
291C2×Dic3—21no—
292C2×Dic3—42no—
293C2×Dic3—42no—
294C2×Dic3—42no—
295C2×Dic3—42no—
296C2×Dic3—42no—
297C2×Dic3—42no—
298C2×Dic3—41no—
299C2×Dic3—41no—
300C2×Dic3—41no—
301C2×Dic3—41no—
302C2×Dic3—41no—
303C2×Dic3—41no—
304C2×Dic3—42no—
305C2×Dic3—81no—
306C2×Dic3—81no—
307C2×Dic3—81no—
308C2×Dic3—81no—
309C2×Dic3#9912yes—
310Dic6—81no—
311Dic6—81no—
312C3×C8—81no—
313C3⋊C8—81no—
314C24—21no—
315C24D3×D411yescharacteristic
316C2×Q8—61no—
317C2×Q8—61no—
318C2×Q8—121no—
319C2×Q8—121no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—31no—
333C22×C4—31no—
334C22×C4—31no—
335C22×C4—31no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—62no—
345C22×C4—61no—
346C22×C4—61no—
347C22×C4—61no—
348C22×C4—61no—
349C22×C4—61no—
350C22×C4—61no—
351C22×C4—62no—
352C22×C4—121no—
353C22×C4—121no—
354C22×C4—121no—
355C22×C4—121no—
356C22×C4—121no—
357C22×C4—121no—
358C22×C4—121no—
359C22×C4—121no—
360C22×C4—121no—
361C22×C4C2×C4×D311yescharacteristic
362C22×C4#23511yescharacteristic
363C22×C4D3×D411yescharacteristic
364C22×C4D3×D411yescharacteristic
365C22×C4#23711yescharacteristic
366C22×C4C2×#6711yescharacteristic
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—121no—
371C2×C8—121no—
372C2×C8—121no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—61no—
376C4⋊C4—61no—
377C4⋊C4—121no—
378C4⋊C4—121no—
379C22⋊C4—41no—
380C22⋊C4—41no—
381C22⋊C4—122no—
382C22⋊C4—122no—
383C22⋊C4—122no—
384C22⋊C4—122no—
385C22⋊C4—121no—
386C22⋊C4—121no—
387C42—62no—
388C42—62no—
389C42—121no—
390C42—121no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—81no—
404C22×C3—81no—
405C22×C3—81no—
406C22×C3—81no—
407C22×C3—81no—
408C22×C3—81no—
409C22×C3—81no—
410C22×C3#39011yescharacteristic
411C22×C3C2×#9911yescharacteristic
412C22×C3#41011yescharacteristic
413C22×C3#41611yescharacteristic
414C22×C3#41711yescharacteristic
415C22×C3#42111yescharacteristic
416C22×C3#42111yescharacteristic
417C22×C3#46311yescharacteristic
418C22×C3#46311yescharacteristic
419C22×C3#46411yescharacteristic
420C22×C3#46411yescharacteristic
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—82no—
426C3×C4—81no—
427Dic3—22no—
428Dic3—41no—
429Dic3—41no—
430Dic3—41no—
431Dic3—82no—
432Dic3—81no—
433Dic3—81no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23D3×C22⋊C411yescharacteristic
446C23#88411yescharacteristic
447C23#91111yescharacteristic
448C23#91211yescharacteristic
449C23#91511yescharacteristic
450C23#91611yescharacteristic
451C23#93911yescharacteristic
452Q8—241no—
453Q8—241no—
454D4—81no—
455D4—81no—
456D4—81no—
457D4—81no—
458D4—81no—
459D4—81no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—32no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—61no—
476C2×C4—61no—
477C2×C4—61no—
478C2×C4—61no—
479C2×C4—61no—
480C2×C4—61no—
481C2×C4—61no—
482C2×C4—61no—
483C2×C4—61no—
484C2×C4—61no—
485C2×C4—61no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—122no—
491C2×C4—122no—
492C2×C4—122no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4—122no—
496C2×C4—121no—
497C2×C4—121no—
498C2×C4—121no—
499C2×C4—121no—
500C2×C4—122no—
501C2×C4—121no—
502C2×C4—121no—
503C2×C4—241no—
504C2×C4—241no—
505C2×C4—241no—
506C2×C4—241no—
507C2×C4#67⋊C411yescharacteristic
508C2×C4#88511yescharacteristic
509C2×C4C4×#6711yescharacteristic
510C2×C4#94011yescharacteristic
511C8—81no—
512C8—241no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—161no—
518C2×C3#151211yescharacteristic
519C2×C3#151311yescharacteristic
520C2×C3#151611yescharacteristic
521C2×C3#152111yescharacteristic
522C2×C3#196311yescharacteristic
523C2×C3#196411yescharacteristic
524C2×C3#198311yescharacteristic
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22#474011yescharacteristic
545C22D3×#9911yescharacteristic
546C22#476711yescharacteristic
547C22#477111yescharacteristic
548C22#477211yescharacteristic
549C22#478711yescharacteristic
550C22#482111yescharacteristic
551C22#482111yescharacteristic
552C22#495711yescharacteristic
553C22#515311yescharacteristic
554C22#516611yescharacteristic
555C4—21no—
556C4—21no—
557C4—21no—
558C4—21no—
559C4—62no—
560C4—82no—
561C4—81no—
562C4—121no—
563C4—121no—
564C4—121no—
565C4—242no—
566C4—241no—
567C4—241no—
568C3#971611yescharacteristic, complemented, Sylow
569C2—21no—
570C2—21no—
571C2—21no—
572C2—21no—
573C2—161no—
574C2#6828011yescharacteristic
575C2#6828311yescharacteristic
576C2#6830511yescharacteristic
577C2#6830611yescharacteristic
578C2#7003011yescharacteristic
579C2#7016211yescharacteristic
580C2#7035711yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#327C2
socleC23×C3C2×C22⋊C4
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenterC2×#327D3
hypocenterC3#9716
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
4. C2×#4467C2semidirect
5. #69908C2semidirect
6. #69401C2semidirect
7. C2×#4452C2semidirect
10. C2×#936C4semidirect
15. #5221C4semidirect
23. #4977C4semidirect
24. #4973C4semidirect
568. C3#9716semidirect

Maximal subgroups

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

subgroupindex
#826982
C2×C3×#3272
C2×#45372
C2×#44672
#699082
#694012
C2×#44522
#97163