Finite Group Databasebeta

← #10694320

Subgroups — #10694320

Order 768.

624 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82811C211yesmaximal, characteristic
2C2×C3×#328C211yesmaximal, characteristic
3C2×#4538C211yesmaximal, characteristic
4C2×#4467C211yesmaximal, characteristic
5#69908C211yesmaximal, characteristic
6#69401C211yesmaximal, characteristic
7C2×#4452C211yesmaximal, characteristic
8#9714—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×#944C411yescharacteristic, complemented
11C2×#924C2211yescharacteristic
12C22×C4×Dic3C2211yescharacteristic
13C2×C3×C22⋊C8C2211yescharacteristic
14C2×C3×#95C2211yescharacteristic
15#5229C411yescharacteristic, complemented
16C2×C3⋊#95—21no—
17C2×C3⋊#95—21no—
18C2×C3⋊#95C2211yescharacteristic
19C2×#830C2211yescharacteristic
20C2×Dic3⋊C8—21no—
21#4976—21no—
22#4976—21no—
23#4975C411yescharacteristic, complemented
24#4973C411yescharacteristic, complemented
25#4942—21no—
26#4942—21no—
27C3×#328—22no—
28#4550—21no—
29#4538—22no—
30#4467—22no—
31#4452—22no—
32#2857—31no—
33#1797—31no—
34C2×#328—31no—
35C2×#328—31no—
36C2×#328—31no—
37C2×#328D311yescharacteristic
38#1434—31no—
39C22×C3×Q8C2×C411yescharacteristic
40C23×C3×C4—21no—
41C23×C3×C4C2311yescharacteristic
42C23×Dic3—21no—
43C23×Dic3D411yescharacteristic
44C23×Dic3D411yescharacteristic
45C22×Dic6C2×C411yescharacteristic
46C22×C3×C8—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4C2×C411yescharacteristic
53C2×C3×C22⋊C4—21no—
54C2×#217—21no—
55#944—42no—
56C2×C4⋊Dic3C2×C411yescharacteristic
57#924—21no—
58#924—21no—
59#924—21no—
60#924—21no—
61C2×Dic3⋊C4—21no—
62C2×Dic3⋊C4—21no—
63C2×Dic3⋊C4—21no—
64C2×Dic3⋊C4—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×C4⋊Q8—31no—
88C2×C22⋊Q8—31no—
89C2×C22⋊Q8C2×D311yescharacteristic
90C22×C42—31no—
91C2×C4⋊C8—61no—
92C2×C22⋊C8—31no—
93C2×C22⋊C8C2×D311yescharacteristic
94#391—31no—
95#391—31no—
96#389—61no—
97#389—61no—
98#384—31no—
99#380—61no—
100#380—61no—
101C2×#95—31no—
102C2×#95—61no—
103C2×#95—61no—
104C2×#95C2×D311yescharacteristic
105#336—61no—
106#328—22no—
107#328—62no—
108#328—62no—
109#328—62no—
110C24×C3C2×D411yescharacteristic
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—41no—
114C2×C3×Q8—41no—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—22no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4C22×C411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4#4111yescharacteristic
131C22×C3×C4#4111yescharacteristic
132C22×Dic3—21no—
133C22×Dic3—21no—
134C22×Dic3—22no—
135C22×Dic3—21no—
136C22×Dic3—21no—
137C22×Dic3—22no—
138C22×Dic3—41no—
139C22×Dic3—41no—
140C22×Dic3—41no—
141C22×Dic3—41no—
142C22×Dic3—41no—
143C22×Dic3—41no—
144C22×Dic3—41no—
145C22×Dic3—41no—
146C22×Dic3—41no—
147C22×Dic3C22⋊C411yescharacteristic
148C22×Dic3C22⋊C411yescharacteristic
149C22×Dic3C22⋊C411yescharacteristic
150C22×Dic3C22⋊C411yescharacteristic
151C22×Dic3SD1612yes—
152C22×Dic3Q1612yes—
153C2×Dic6—21no—
154C2×Dic6—21no—
155C2×Dic6—41no—
156C2×Dic6—41no—
157C2×C3×C8—41no—
158C2×C3×C8—41no—
159C2×C3×C8—41no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—21no—
162C3×C4⋊C4—41no—
163C3×C4⋊C4—41no—
164C3×C22⋊C4—41no—
165C3×C22⋊C4—41no—
166#217—41no—
167#217—41no—
168C4⋊Dic3—21no—
169C4⋊Dic3—21no—
170Dic3⋊C4—42no—
171Dic3⋊C4—42no—
172Dic3⋊C4—42no—
173Dic3⋊C4—41no—
174Dic3⋊C4—41no—
175Dic3⋊C4—42no—
176C4×Dic3—22no—
177C4×Dic3—22no—
178C4×Dic3—41no—
179C4×Dic3—41no—
180C2×C3⋊C8—41no—
181C2×C3⋊C8—41no—
182C2×C3⋊C8—41no—
183C22×Q8—31no—
184C22×Q8C4×D311yescharacteristic
185C23×C4—21no—
186C23×C4—31no—
187C23×C4—31no—
188C23×C4—61no—
189C23×C4C22×D311yescharacteristic
190C22×C8—21no—
191C22×C8—61no—
192C4⋊Q8—122no—
193C22⋊Q8—21no—
194C22⋊Q8—21no—
195C22⋊Q8—21no—
196C22⋊Q8—21no—
197C22⋊Q8—61no—
198C22⋊Q8—61no—
199C22⋊Q8—61no—
200C22⋊Q8—61no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—31no—
203C2×C4⋊C4—61no—
204C2×C4⋊C4—61no—
205C2×C4⋊C4—61no—
206C2×C4⋊C4—61no—
207C2×C4⋊C4—61no—
208C2×C4⋊C4C4×D311yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—61no—
211C2×C42—31no—
212C2×C42—31no—
213C2×C42—31no—
214C2×C42—31no—
215C2×C42—122no—
216C4⋊C8—122no—
217C22⋊C8—21no—
218C22⋊C8—21no—
219C22⋊C8—61no—
220C22⋊C8—61no—
221#95—21no—
222#95—32no—
223#95—61no—
224#95—61no—
225#95—61no—
226#95—61no—
227#95—61no—
228#95—122no—
229#95—122no—
230#95#6712yes—
231C23×C3—21no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—21no—
235C23×C3C2×C22⋊C411yescharacteristic
236C23×C3#12111yescharacteristic
237C23×C3#12111yescharacteristic
238C23×C3C2×SD1611yescharacteristic
239C23×C3C2×Q1611yescharacteristic
240C23×C3#13711yescharacteristic
241C23×C3#13711yescharacteristic
242C3×Q8—81no—
243C3×Q8—81no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—42no—
261C2×C3×C4—81no—
262C2×C3×C4—81no—
263C2×C3×C4—81no—
264C2×C3×C4—81no—
265C2×C3×C4C4×D411yescharacteristic
266C2×C3×C4C4×D411yescharacteristic
267C2×C3×C4#12411yescharacteristic
268C2×C3×C4#12711yescharacteristic
269C2×Dic3—21no—
270C2×Dic3—21no—
271C2×Dic3—21no—
272C2×Dic3—21no—
273C2×Dic3—21no—
274C2×Dic3—21no—
275C2×Dic3—21no—
276C2×Dic3—21no—
277C2×Dic3—21no—
278C2×Dic3—42no—
279C2×Dic3—42no—
280C2×Dic3—42no—
281C2×Dic3—42no—
282C2×Dic3—42no—
283C2×Dic3—42no—
284C2×Dic3—41no—
285C2×Dic3—41no—
286C2×Dic3—41no—
287C2×Dic3—41no—
288C2×Dic3—41no—
289C2×Dic3—41no—
290C2×Dic3—42no—
291C2×Dic3—81no—
292C2×Dic3—81no—
293C2×Dic3—81no—
294C2×Dic3—81no—
295C2×Dic3#9912yes—
296C2×Dic3#10312yes—
297C2×Dic3#10312yes—
298Dic6—81no—
299Dic6—81no—
300C3×C8—81no—
301C3⋊C8—81no—
302C24D3×D411yescharacteristic
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—61no—
308C2×Q8—61no—
309C2×Q8—121no—
310C2×Q8—121no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—31no—
315C22×C4—31no—
316C22×C4—31no—
317C22×C4—32no—
318C22×C4—32no—
319C22×C4—31no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—62no—
329C22×C4—61no—
330C22×C4—61no—
331C22×C4—61no—
332C22×C4—61no—
333C22×C4—62no—
334C22×C4—121no—
335C22×C4—121no—
336C22×C4—121no—
337C22×C4—121no—
338C22×C4—121no—
339C22×C4—121no—
340C22×C4—121no—
341C22×C4—121no—
342C22×C4—121no—
343C22×C4C2×C4×D311yescharacteristic
344C22×C4#23511yescharacteristic
345C22×C4D3×D411yescharacteristic
346C22×C4D3×D411yescharacteristic
347C22×C4#23711yescharacteristic
348C22×C4C2×#6711yescharacteristic
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—121no—
353C2×C8—121no—
354C2×C8—121no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—61no—
360C4⋊C4—61no—
361C4⋊C4—122no—
362C4⋊C4—122no—
363C4⋊C4—122no—
364C4⋊C4—122no—
365C4⋊C4—121no—
366C4⋊C4—121no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—121no—
370C22⋊C4—121no—
371C42—62no—
372C42—62no—
373C42—121no—
374C42—121no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3#39011yescharacteristic
388C22×C3C2×#9911yescharacteristic
389C22×C3#41011yescharacteristic
390C22×C3C2×#10311yescharacteristic
391C22×C3#41911yescharacteristic
392C22×C3#42111yescharacteristic
393C22×C3#42111yescharacteristic
394C22×C3#46111yescharacteristic
395C22×C3#46111yescharacteristic
396C22×C3#46211yescharacteristic
397C22×C3#46211yescharacteristic
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—82no—
403C3×C4—81no—
404C3×C4—81no—
405Dic3—22no—
406Dic3—22no—
407Dic3—22no—
408Dic3—41no—
409Dic3—82no—
410Dic3—81no—
411Dic3—81no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23D3×C22⋊C411yescharacteristic
417C23#88411yescharacteristic
418C23D3×SD1611yescharacteristic
419C23#91511yescharacteristic
420C23D3×Q1611yescharacteristic
421C23#91811yescharacteristic
422C23#93911yescharacteristic
423Q8—81no—
424Q8—81no—
425Q8—241no—
426Q8—241no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—32no—
432C2×C4—32no—
433C2×C4—32no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—61no—
448C2×C4—61no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—61no—
452C2×C4—61no—
453C2×C4—61no—
454C2×C4—61no—
455C2×C4—61no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—122no—
461C2×C4—122no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4—122no—
466C2×C4—121no—
467C2×C4—121no—
468C2×C4—121no—
469C2×C4—121no—
470C2×C4—122no—
471C2×C4—121no—
472C2×C4—121no—
473C2×C4—241no—
474C2×C4—241no—
475C2×C4—241no—
476C2×C4—241no—
477C2×C4#67⋊C411yescharacteristic
478C2×C4#88511yescharacteristic
479C2×C4C4×#6711yescharacteristic
480C2×C4#94011yescharacteristic
481C8—81no—
482C8—241no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3#151111yescharacteristic
488C2×C3#151511yescharacteristic
489C2×C3#151811yescharacteristic
490C2×C3#152011yescharacteristic
491C2×C3#196311yescharacteristic
492C2×C3#196411yescharacteristic
493C2×C3#198111yescharacteristic
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22#474011yescharacteristic
507C22D3×#9911yescharacteristic
508C22#479511yescharacteristic
509C22D3×#10311yescharacteristic
510C22#480211yescharacteristic
511C22#481811yescharacteristic
512C22#482111yescharacteristic
513C22#482111yescharacteristic
514C22#495711yescharacteristic
515C22#516211yescharacteristic
516C22#518211yescharacteristic
517C4—21no—
518C4—21no—
519C4—21no—
520C4—21no—
521C4—62no—
522C4—62no—
523C4—62no—
524C4—82no—
525C4—81no—
526C4—81no—
527C4—121no—
528C4—242no—
529C4—241no—
530C4—241no—
531C3#971411yescharacteristic, complemented, Sylow
532C2—21no—
533C2—21no—
534C2—21no—
535C2—21no—
536C2#6831311yescharacteristic
537C2#6831411yescharacteristic
538C2#6833611yescharacteristic
539C2#6833911yescharacteristic
540C2#7003011yescharacteristic
541C2#7022911yescharacteristic
542C2#7035711yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#328C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenterC2×#328D3
hypocenterC3#9714
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
10. C2×#944C4semidirect
15. #5229C4semidirect
23. #4975C4semidirect
24. #4973C4semidirect
531. C3#9714semidirect

Maximal subgroups

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

subgroupindex
#828112
C2×C3×#3282
C2×#45382
C2×#44672
#699082
#694012
C2×#44522
#97143