Finite Group Databasebeta

← #10695591

Subgroups — #10695591

Order 768.

654 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2561C211yesmaximal, characteristic
2#82727C211yesmaximal, characteristic, complemented
3#82113C211yesmaximal, characteristic
4#71689C211yesmaximal, characteristic, complemented
5#71539C211yesmaximal, characteristic
6#69442C211yesmaximal, characteristic, complemented
7#67616C211yesmaximal, characteristic, complemented
8#9944—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×C3×C22⋊C4C2211yescharacteristic
11C2×#926C2211yescharacteristic
12C3×#397—21no—
13C3×#393—21no—
14C3×#388—22no—
15C3×#381—21no—
16C3×#379—21no—
17C3×C4×C22⋊C4C2211yescharacteristic
18#5247—21no—
19#5225—21no—
20#5211C2211yescharacteristic
21#5210—21no—
22#5207C2211yescharacteristic
23#5207C2211yescharacteristic
24#5045—22no—
25#5044—21no—
26#5043—21no—
27#4964—21no—
28#4953—21no—
29#4949—21no—
30#4944—22no—
31#4538—22no—
32#4525—22no—
33#4523—22no—
34#3095—31no—
35#2561D311yescharacteristic
36#2482—31no—
37#1826—31no—
38#1511—31no—
39#1462—31no—
40#1284—31no—
41C25×C3D411yescharacteristic
42C22×C3×Q8D411yescharacteristic
43C23×C3×C4D411yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C22⋊C4—22no—
52C2×C3×C22⋊C4—21no—
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4—21no—
55C2×C3×C22⋊C4—21no—
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—22no—
61C2×C3×C42D412yes—
62C2×#217—22no—
63C2×#217—21no—
64C2×#217—21no—
65C2×#217—22no—
66#926—21no—
67#926—21no—
68#926—21no—
69#926—21no—
70C2×C4⋊Dic3D411yescharacteristic
71C2×C4⋊Dic3D411yescharacteristic
72C2×Dic3⋊C4—22no—
73C2×C4×Dic3—21no—
74C2×#208—21no—
75C2×#208—21no—
76C3×#95—22no—
77C3×#95—22no—
78C3×#95—21no—
79#835—22no—
80#835—22no—
81C3⋊#95—22no—
82C3⋊#95—21no—
83C3⋊#95D412yes—
84#830—21no—
85#830—21no—
86#830—21no—
87#830D412yes—
88#808—22no—
89#807—22no—
90C2×C22⋊Q8C2×D311yescharacteristic
91C2×#117—31no—
92C22×C22⋊C4C2×D311yescharacteristic
93#485—62no—
94#484—61no—
95#483—61no—
96#428—61no—
97#419—61no—
98#407—31no—
99#407—31no—
100#399—61no—
101#397—21no—
102#397—61no—
103#393—21no—
104#388—22no—
105#388—62no—
106#385—61no—
107#381—21no—
108#381—31no—
109#379—21no—
110#379—61no—
111C4×C22⋊C4C2×D311yescharacteristic
112#339—62no—
113#328—62no—
114#323—62no—
115C24×C3—21no—
116C24×C3—21no—
117C24×C3—21no—
118C24×C3—42no—
119C24×C3—42no—
120C24×C3—42no—
121C24×C3C2×D411yescharacteristic
122C2×C3×Q8—21no—
123C2×C3×Q8—21no—
124C2×C3×Q8—42no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—22no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—42no—
132C22×C3×C4—42no—
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4#4111yescharacteristic
139C22×Dic3—22no—
140C22×Dic3—21no—
141C22×Dic3—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—41no—
146C3×C22⋊C4—42no—
147C3×C22⋊C4—44no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C22⋊C4—82no—
155C3×C42—42no—
156#217—42no—
157#217—44no—
158#217—41no—
159#217—41no—
160#217—82no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165Dic3⋊C4—82no—
166C4×Dic3—41no—
167C4×Dic3—41no—
168#208—42no—
169#208—42no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—41no—
172C25#6711yescharacteristic
173C22×Q8#6711yescharacteristic
174C23×C4#6711yescharacteristic
175C23×C4C22×D311yescharacteristic
176C2×M16—61no—
177C2×M16—61no—
178C22⋊Q8—21no—
179C22⋊Q8—21no—
180C22⋊Q8—21no—
181C22⋊Q8—21no—
182#117—61no—
183#117—61no—
184#117—61no—
185#117—61no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—31no—
189C2×C4⋊C4—62no—
190C2×C4⋊C4#6711yescharacteristic
191C2×C22⋊C4—22no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—62no—
202C2×C22⋊C4—61no—
203C2×C22⋊C4—61no—
204C2×C22⋊C4—62no—
205C2×C42—61no—
206C2×C42#6712yes—
207#107—62no—
208#106—62no—
209#103—62no—
210#103—62no—
211C22⋊C8—32no—
212C22⋊C8—61no—
213C22⋊C8—61no—
214C22⋊C8—61no—
215#95—22no—
216#95—22no—
217#95—21no—
218#95—32no—
219#95—62no—
220#95—61no—
221C23×C3—22no—
222C23×C3—22no—
223C23×C3—22no—
224C23×C3—22no—
225C23×C3—22no—
226C23×C3—41no—
227C23×C3—42no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—42no—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3—42no—
234C23×C3—42no—
235C23×C3—81no—
236C23×C3—81no—
237C23×C3—88no—
238C23×C3#12011yescharacteristic
239C23×C3#12011yescharacteristic
240C23×C3#12111yescharacteristic
241C3×Q8—42no—
242C3×Q8—81no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—81no—
260C2×C3×C4#12011yescharacteristic
261C2×C3×C4#12111yescharacteristic
262C2×C3×C4#12111yescharacteristic
263C2×C3×C4#12711yescharacteristic
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3—42no—
267C2×Dic3—41no—
268C2×Dic3—41no—
269C2×Dic3—42no—
270C2×Dic3—42no—
271C3⋊C8—42no—
272C3⋊C8—81no—
273C24—21no—
274C24—21no—
275C24—21no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24C2×#6711yescharacteristic
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—42no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—62no—
292C22×C4—61no—
293C22×C4—61no—
294C22×C4D3×D411yescharacteristic
295C22×C4D3×D411yescharacteristic
296C22×C4D3×D411yescharacteristic
297C22×C4#23711yescharacteristic
298C22×C4C2×#6711yescharacteristic
299C22×C4C2×#6711yescharacteristic
300M16—122no—
301M16—122no—
302C2×C8—62no—
303C2×C8—121no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—61no—
309C4⋊C4—61no—
310C4⋊C4—61no—
311C4⋊C4—61no—
312C4⋊C4—242no—
313C22⋊C4—42no—
314C22⋊C4—44no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—82no—
322C22⋊C4—122no—
323C22⋊C4—124no—
324C22⋊C4—121no—
325C22⋊C4—121no—
326C22⋊C4—242no—
327C42—42no—
328C42—121no—
329C42—121no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—44no—
338C22×C3—44no—
339C22×C3—44no—
340C22×C3—42no—
341C22×C3—82no—
342C22×C3—82no—
343C22×C3—82no—
344C22×C3—82no—
345C22×C3—81no—
346C22×C3—81no—
347C22×C3#39211yescharacteristic
348C22×C3#45512yes—
349C22×C3#45812yes—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—21no—
354C3×C4—82no—
355C3×C4—81no—
356C3×C4—81no—
357C3×C4—81no—
358Dic3—82no—
359Dic3—81no—
360Dic3—81no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—42no—
370C23—42no—
371C23—41no—
372C23—41no—
373C23—42no—
374C23—42no—
375C23—81no—
376C23—81no—
377C23—88no—
378C23#93711yescharacteristic
379C23#93911yescharacteristic
380C23#95311yescharacteristic
381Q8—42no—
382Q8—81no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—81no—
400C2×C4—122no—
401C2×C4—122no—
402C2×C4—122no—
403C2×C4—121no—
404C2×C4—121no—
405C2×C4—122no—
406C2×C4—122no—
407C2×C4#93711yescharacteristic
408C2×C4#93811yescharacteristic
409C2×C4#93911yescharacteristic
410C2×C4#94011yescharacteristic
411C8—122no—
412C8—241no—
413C2×C3—22no—
414C2×C3—22no—
415C2×C3—41no—
416C2×C3—82no—
417C2×C3#200411yescharacteristic
418C2×C3#201011yescharacteristic
419C2×C3#202211yescharacteristic
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—44no—
428C22—44no—
429C22—44no—
430C22—42no—
431C22—82no—
432C22—82no—
433C22—82no—
434C22—82no—
435C22—81no—
436C22—81no—
437C22#506112yes—
438C22#522311yescharacteristic
439C22#524312yes—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—82no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4—242no—
449C4—241no—
450C4—241no—
451C3#994411yescharacteristic, complemented, Sylow
452C2—22no—
453C2—22no—
454C2—41no—
455C2—82no—
456C2#7171511yescharacteristic
457C2#7173111yescharacteristic
458C2#7184911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2561C2
socleC22×C3#392
prime radicalC25×C3D4
hypercenter#2561D3
hypocenterC3#9944
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2561

Complements

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

NH ≅ G/Ntype
2. #82727C2semidirect
4. #71689C2semidirect
6. #69442C2semidirect
7. #67616C2semidirect
451. C3#9944semidirect

Maximal subgroups

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

subgroupindex
C3×#25612
#827272
#821132
#716892
#715392
#694422
#676162
#99443