Finite Group Databasebeta

← #10691748

Subgroups — #10691748

Order 768.

606 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3187C211yesmaximal, characteristic
2#82638C211yesmaximal, characteristic
3#71701C211yesmaximal, characteristic, complemented
4C2×#4538C211yesmaximal, characteristic, complemented
5#71675C211yesmaximal, characteristic
6#67678C211yesmaximal, characteristic, complemented
7#67675C211yesmaximal, characteristic, complemented
8#9275—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C2×D4×Dic3—21no—
12C3×#483—21no—
13C3×#481C412yescomplemented
14C3×#465—22no—
15C3×#450—21no—
16C3×#448C412yescomplemented
17C2×C3×C22⋊C8C2211yescharacteristic
18#5220—21no—
19#5220—21no—
20#5212C2211yescharacteristic
21C4×#217C2211yescharacteristic
22C2×C3⋊#95C2211yescharacteristic
23C2×#830C2211yescharacteristic
24#4980—21no—
25#4951—21no—
26Dic3×C22⋊C4—21no—
27#4538—21no—
28#4538—21no—
29#4537—22no—
30#4526—21no—
31#4526—21no—
32#3187D311yescharacteristic
33#2364—31no—
34#1996—31no—
35#1519—31no—
36C2×#328—31no—
37#1294—31no—
38#1282—31no—
39C22×C3×Q8C2×C411yescharacteristic, complemented
40C22×C3×D4—21no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C23×Dic3—21no—
44C23×Dic3—21no—
45C2×C3×SD16—22no—
46C22×C3×C8—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C3×#121—21no—
52C3×#121—21no—
53C3×#121C2×C412yes—
54C3×#121D412yes—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—21no—
60C2×#217—21no—
61C2×#217—21no—
62C2×#217—21no—
63C2×#217—21no—
64C2×#217D411yescharacteristic
65C2×#217D411yescharacteristic
66D4×Dic3—42no—
67D4×Dic3—42no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3—21no—
70C2×C4×Dic3—21no—
71C2×C4×Dic3—21no—
72C22×C3⋊C8—21no—
73C3×#106—22no—
74C3×#103—22no—
75C3×#102—22no—
76C3×C22⋊C8—21no—
77C3×C22⋊C8C2×C412yescomplemented
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82C3⋊#95—21no—
83C3⋊#95D411yescharacteristic
84C3⋊#95D411yescharacteristic
85#830—21no—
86#830—21no—
87C2×C22⋊Q8C2×D311yescharacteristic
88C2×#121C2×D311yescharacteristic
89C2×C4×D4—61no—
90#483—21no—
91#481Dic312yescomplemented
92#465—22no—
93#450—21no—
94#448Dic312yescomplemented
95C2×C22⋊C8—31no—
96C2×C22⋊C8C2×D311yescharacteristic
97#388—61no—
98#387—61no—
99#386—61no—
100#386—61no—
101#380—31no—
102C4×C22⋊C4—31no—
103C4×C22⋊C4—61no—
104C2×#95—31no—
105#342—61no—
106#342—61no—
107#328—61no—
108#328—61no—
109#327—62no—
110C24×C3—21no—
111C24×C3C2×D411yescharacteristic
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8—81no—
115C2×C3×D4—22no—
116C2×C3×D4—22no—
117C2×C3×D4—21no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—81no—
123C2×C3×D4C22⋊C412yes—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—81no—
128C22×C3×C4C22×C411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4#4111yescharacteristic
133C22×C3×C4#4111yescharacteristic
134C22×Dic3—21no—
135C22×Dic3—21no—
136C22×Dic3—21no—
137C22×Dic3—21no—
138C22×Dic3—42no—
139C22×Dic3—42no—
140C22×Dic3—42no—
141C22×Dic3—41no—
142C22×Dic3—41no—
143C22×Dic3—41no—
144C22×Dic3—81no—
145C22×Dic3—81no—
146C3×SD16—82no—
147C2×C3×C8—22no—
148C2×C3×C8—81no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—41no—
153C3×C4⋊C4—41no—
154C3×C4⋊C4C22⋊C412yes—
155C3×C22⋊C4—22no—
156C3×C22⋊C4—22no—
157C3×C22⋊C4—41no—
158C3×C22⋊C4—41no—
159#217—21no—
160#217—21no—
161#217—21no—
162#217—42no—
163#217—42no—
164#217—41no—
165#217—41no—
166#217—81no—
167#217SD1612yescomplemented
168C4⋊Dic3—42no—
169C4⋊Dic3—81no—
170C4×Dic3—42no—
171C4×Dic3—81no—
172C2×C3⋊C8—41no—
173C2×C3⋊C8—81no—
174C22×Q8C2×Dic311yescharacteristic, complemented
175C22×D4—21no—
176C22×D4#6711yescharacteristic
177C23×C4—61no—
178C23×C4—61no—
179C23×C4C22×D311yescharacteristic
180C2×SD16—22no—
181C22×C8—21no—
182C22×C8—61no—
183C22⋊Q8—21no—
184C22⋊Q8—21no—
185C22⋊Q8—21no—
186C22⋊Q8—21no—
187#121—21no—
188#121—21no—
189#121C2×Dic312yes—
190#121#6712yes—
191C4×D4—122no—
192C4×D4—122no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—61no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4C2×Dic311yescharacteristic, complemented
197C2×C4⋊C4#6711yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—31no—
201C2×C22⋊C4—31no—
202C2×C22⋊C4—61no—
203C2×C22⋊C4—61no—
204C2×C22⋊C4—61no—
205C2×C22⋊C4—61no—
206C2×C42—61no—
207C2×C42—61no—
208#106—22no—
209#103—22no—
210#102—22no—
211C22⋊C8—21no—
212C22⋊C8—61no—
213C22⋊C8—61no—
214C22⋊C8C2×Dic312yescomplemented
215#95—31no—
216#95—31no—
217#95—61no—
218#95—61no—
219#95—61no—
220#95—61no—
221#95—61no—
222C23×C3—22no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3—41no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—41no—
231C23×C3C4×D411yescharacteristic
232C23×C3#12011yescharacteristic
233C23×C3#12111yescharacteristic
234C23×C3C2×SD1611yescharacteristic
235C23×C3#13511yescharacteristic
236C23×C3AGL(1,ℤ/8)11yescharacteristic
237C23×C3#13711yescharacteristic
238C3×Q8—42no—
239C3×Q8—81no—
240C3×D4—42no—
241C3×D4—42no—
242C3×D4—42no—
243C3×D4—42no—
244C3×D4—42no—
245C3×D4—81no—
246C2×C3×C4—22no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—81no—
258C2×C3×C4—81no—
259C2×C3×C4C2×C22⋊C411yescharacteristic
260C2×C3×C4C4×D411yescharacteristic
261C2×C3×C4#12111yescharacteristic
262C2×C3×C4#12311yescharacteristic
263C2×Dic3—22no—
264C2×Dic3—22no—
265C2×Dic3—41no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268C2×Dic3—41no—
269C2×Dic3—41no—
270C2×Dic3—41no—
271C2×Dic3—41no—
272C2×Dic3—41no—
273C2×Dic3—41no—
274C2×Dic3—41no—
275C2×Dic3—84no—
276C2×Dic3—82no—
277C2×Dic3—82no—
278C2×Dic3—81no—
279C2×Dic3—81no—
280C2×Dic3—81no—
281C3×C8—42no—
282C3⋊C8—81no—
283C24—21no—
284C24D3×D411yescharacteristic
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—81no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—81no—
296C2×D4#21712yes—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—61no—
301C22×C4—61no—
302C22×C4—61no—
303C22×C4—61no—
304C22×C4—81no—
305C22×C4—122no—
306C22×C4—122no—
307C22×C4—122no—
308C22×C4—121no—
309C22×C4—121no—
310C22×C4—121no—
311C22×C4—241no—
312C22×C4—241no—
313C22×C4D3×D411yescharacteristic
314C22×C4#23711yescharacteristic
315C22×C4#23711yescharacteristic
316C22×C4C22×Dic311yescharacteristic
317C22×C4C2×#6711yescharacteristic
318C22×C4C2×#6711yescharacteristic
319SD16—82no—
320C2×C8—22no—
321C2×C8—81no—
322C2×C8—121no—
323C2×C8—241no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—122no—
330C4⋊C4—241no—
331C4⋊C4#21712yes—
332C22⋊C4—22no—
333C22⋊C4—22no—
334C22⋊C4—32no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—61no—
338C22⋊C4—61no—
339C22⋊C4—61no—
340C22⋊C4—122no—
341C22⋊C4—122no—
342C22⋊C4—121no—
343C22⋊C4—121no—
344C22⋊C4—241no—
345C42—122no—
346C42—241no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3—41no—
357C22×C3—42no—
358C22×C3—42no—
359C22×C3—42no—
360C22×C3—81no—
361C22×C3—81no—
362C22×C3—81no—
363C22×C3—81no—
364C22×C3#38611yescharacteristic
365C22×C3C4×SD1611yescharacteristic
366C22×C3#44011yescharacteristic
367C22×C3#45011yescharacteristic
368C22×C3#45211yescharacteristic
369C22×C3#46111yescharacteristic
370C22×C3#46311yescharacteristic
371C3×C4—21no—
372C3×C4—21no—
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—42no—
376C3×C4—81no—
377C3×C4—81no—
378Dic3—42no—
379Dic3—42no—
380Dic3—81no—
381Dic3—81no—
382Dic3—161no—
383C23—22no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23D3×SD1611yescharacteristic
393C23#91411yescharacteristic
394C23#91511yescharacteristic
395C23#91611yescharacteristic
396C23D4×Dic311yescharacteristic
397C23#93711yescharacteristic
398C23#93911yescharacteristic
399Q8—42no—
400Q8—81no—
401D4—42no—
402D4—42no—
403D4—42no—
404D4—42no—
405D4—42no—
406D4—81no—
407C2×C4—22no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—62no—
419C2×C4—62no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—121no—
423C2×C4—121no—
424C2×C4—121no—
425C2×C4—121no—
426C2×C4—121no—
427C2×C4—121no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—244no—
433C2×C4—242no—
434C2×C4—242no—
435C2×C4—241no—
436C2×C4—241no—
437C2×C4—241no—
438C2×C4D4×Dic311yescharacteristic
439C2×C4#93511yescharacteristic
440C2×C4#93911yescharacteristic
441C2×C4C2×#21711yescharacteristic
442C8—42no—
443C8—241no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—82no—
451C2×C3#187711yescharacteristic
452C2×C3#190611yescharacteristic
453C2×C3#191311yescharacteristic
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—42no—
465C22—42no—
466C22—42no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22Dic3×SD1611yescharacteristic
472C22#516211yescharacteristic
473C22#516411yescharacteristic
474C22#516511yescharacteristic
475C22#516911yescharacteristic
476C22#517311yescharacteristic
477C22#522011yescharacteristic
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—42no—
483C4—81no—
484C4—81no—
485C4—122no—
486C4—122no—
487C4—241no—
488C4—241no—
489C4—481no—
490C3#927511yescharacteristic, complemented, Sylow
491C2—21no—
492C2—21no—
493C2—21no—
494C2—21no—
495C2—21no—
496C2—21no—
497C2—82no—
498C2#7184811yescharacteristic
499C2#7204711yescharacteristic
500C2#7211411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#3187C2
socleC22×C3#386
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenter#3187D3
hypocenterC3#9275
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3187

Complements

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

NH ≅ G/Ntype
3. #71701C2semidirect
4. C2×#4538C2semidirect
6. #67678C2semidirect
7. #67675C2semidirect
13. C3×#481C4semidirect
16. C3×#448C4semidirect
39. C22×C3×Q8C2×C4semidirect
56. C2×C3×C4⋊C4C2×C4semidirect
77. C3×C22⋊C8C2×C4semidirect
91. #481Dic3semidirect
94. #448Dic3semidirect
167. #217SD16semidirect
174. C22×Q8C2×Dic3semidirect
196. C2×C4⋊C4C2×Dic3semidirect
214. C22⋊C8C2×Dic3semidirect
490. C3#9275semidirect

Maximal subgroups

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

subgroupindex
C3×#31872
#826382
#717012
C2×#45382
#716752
#676782
#676752
#92753