Finite Group Databasebeta

← #10697568

Subgroups — #10697568

Order 768.

590 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 105 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×#2563C211yesmaximal, characteristic, complemented
2#83216C211yesmaximal, characteristic, complemented
3#82307C211yesmaximal, characteristic
4#71775C211yesmaximal, characteristic, complemented
5#71597C211yesmaximal, characteristic, complemented
6#69469C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#10241—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C2×#932—21no—
12#5477C2211yescharacteristic
13C3×#393—22no—
14C3×#390—21no—
15C3×#382—21no—
16C3×#380—22no—
17C3×C4×C4⋊C4C2211yescharacteristic
18#5088—21no—
19#5084—22no—
20#5080—21no—
21#5019—21no—
22#5016—22no—
23#5015—21no—
24#4968—22no—
25#4966—21no—
26#4925C2211yescharacteristic
27#4925C2211yescharacteristic
28#4924C2211yescharacteristic
29#4547—22no—
30D4⋊(C3⋊C8)—22no—
31#4451—22no—
32#3227—31no—
33#3036—31no—
34#2563D311yescharacteristic, complemented
35#1549—31no—
36#1502—31no—
37#1478—31no—
38#1303—31no—
39C22×C3×Q8—21no—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4—21no—
42C2×#235—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C4⋊C4D412yes—
55C2×C3×C22⋊C4—21no—
56C2×C3×C22⋊C4—24no—
57C2×C3×C42—21no—
58C2×C3×C42C2311yescharacteristic
59C2×#214—22no—
60#932—42no—
61#932—42no—
62C2×#213—22no—
63#930—21no—
64#930—21no—
65C2×#208—21no—
66#875—21no—
67#874D411yescharacteristic
68C4×D12—21no—
69C4×D12—21no—
70#869D411yescharacteristic
71C3×#95—24no—
72C3×#95D412yes—
73#830—21no—
74#830—21no—
75#810—22no—
76#809—22no—
77#809—24no—
78#808—24no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)D412yes—
82C4⋊(C3⋊C8)D412yes—
83C4×C3⋊C8—21no—
84C4×C3⋊C8—21no—
85C2×AGL(1,ℤ/8)—61no—
86#532—31no—
87C2×C4⋊Q8C2×D311yescharacteristic
88C2×C4×D4C2×D311yescharacteristic
89#498—61no—
90#486—61no—
91#481—62no—
92#476—62no—
93#463—62no—
94#461—61no—
95#459—61no—
96#431—31no—
97#431—31no—
98#423—31no—
99#417—61no—
100#393—22no—
101#390—21no—
102#382—21no—
103#380—22no—
104C4×C4⋊C4C2×D311yescharacteristic
105#332—62no—
106#327—62no—
107D4⋊C8—62no—
108C24×C3—21no—
109C2×C3×Q8—41no—
110C2×C3×Q8—41no—
111C2×C3×Q8—82no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4SD1612yes—
118C22×C3×C4—21no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—44no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C2×#67—41no—
128#235—41no—
129#235—41no—
130#235—41no—
131#235—41no—
132C2×D12—21no—
133C2×D12—21no—
134C2×D12—21no—
135C2×C4×D3—41no—
136C2×Dic6—21no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—42no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C22⋊C4—41no—
146C3×C22⋊C4—84no—
147C3×C42—41no—
148C3×C42—41no—
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42#4111yescharacteristic
153#214—44no—
154#213—44no—
155(C2×D3)⋊C4—41no—
156(C2×D3)⋊C4—41no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159#208—41no—
160#208—41no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—41no—
164C2×#41—61no—
165C22×Q8—21no—
166C22×D4#6711yescharacteristic
167C23×C4—21no—
168AGL(1,ℤ/8)—122no—
169AGL(1,ℤ/8)—122no—
170C2×SD16—62no—
171C2×D8—62no—
172C2×M16—61no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—31no—
178#127—31no—
179#124—61no—
180#121—61no—
181#121—61no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—61no—
187C4×D4—61no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4#6711yescharacteristic
191C2×C4⋊C4#6712yes—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—24no—
194C2×C42—21no—
195C2×C42C22×D311yescharacteristic
196#106—64no—
197C4⋊C8—32no—
198C4⋊C8—32no—
199C4⋊C8—61no—
200C4⋊C8—61no—
201#103—62no—
202#102—62no—
203#102—64no—
204C22⋊C8—61no—
205C22⋊C8—61no—
206C4×C8—61no—
207C4×C8—61no—
208#95—24no—
209#95#6712yes—
210C23×C3—22no—
211C23×C3—42no—
212C23×C3—42no—
213C23×C3—41no—
214C23×C3—41no—
215C23×C3#12011yescharacteristic
216C22×D3—41no—
217C3×Q8—82no—
218C3×Q8—82no—
219C3×D4—22no—
220C3×D4—22no—
221C3×D4—41no—
222C3×D4—41no—
223C2×C3×C4—22no—
224C2×C3×C4—41no—
225C2×C3×C4—42no—
226C2×C3×C4—44no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—41no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—82no—
233C2×C3×C4—82no—
234C2×C3×C4#12011yescharacteristic
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12711yescharacteristic
240C2×C3×C4C2×SD1611yescharacteristic
241C2×C3×C4#13511yescharacteristic
242C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
243C2×C3×C4#13711yescharacteristic
244#67—81no—
245#67—81no—
246C2×Dic3—41no—
247D12—41no—
248D12—41no—
249D12—42no—
250D12—81no—
251C4×D3—81no—
252C4×D3—81no—
253Dic6—41no—
254Dic6—41no—
255C3⋊C8—42no—
256C3⋊C8—42no—
257C3⋊C8—81no—
258C24—21no—
259#41—121no—
260#41—121no—
261#41—121no—
262#41—121no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—61no—
266C2×Q8—82no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—61no—
273C2×D4—61no—
274C2×D4—61no—
275C2×D4—121no—
276C2×D4#21512yes—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—44no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—121no—
284C22×C4C2×#6711yescharacteristic
285C22×C4C2×#6711yescharacteristic
286C22×C4C2×#6711yescharacteristic
287SD16—124no—
288D8—124no—
289M16—121no—
290M16—121no—
291C2×C8—62no—
292C2×C8—62no—
293C2×C8—121no—
294C4⋊C4—21no—
295C4⋊C4—22no—
296C4⋊C4—21no—
297C4⋊C4—22no—
298C4⋊C4—42no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—84no—
306C22⋊C4—121no—
307C22⋊C4—121no—
308C42—41no—
309C42—41no—
310C42D3×D411yescharacteristic
311C42D3×D411yescharacteristic
312C42D3×D411yescharacteristic
313C42#23711yescharacteristic
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—44no—
317C22×C3—42no—
318C22×C3—82no—
319C22×C3—81no—
320C22×C3—81no—
321C22×C3#39211yescharacteristic
322C22×C3#44811yescharacteristic
323C22×C3#44911yescharacteristic
324C2×D3—81no—
325C2×D3—81no—
326C2×D3—81no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—82no—
331C3×C4—81no—
332C3×C4—82no—
333C3×C4#45011yescharacteristic
334C3×C4#45211yescharacteristic
335C3×C4#45412yes—
336C3×C4#46511yescharacteristic
337C3×C4#46811yescharacteristic
338Dic3—81no—
339C23—22no—
340C23—42no—
341C23—42no—
342C23—41no—
343C23—41no—
344C23—121no—
345C23#95311yescharacteristic
346Q8—82no—
347Q8—82no—
348Q8—121no—
349Q8—121no—
350D4—22no—
351D4—22no—
352D4—41no—
353D4—41no—
354D4—121no—
355D4—121no—
356D4—122no—
357D4—241no—
358D4—241no—
359D4—241no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—121no—
372C2×C4—241no—
373C2×C4—241no—
374C2×C4#93711yescharacteristic
375C2×C4#93711yescharacteristic
376C2×C4#93811yescharacteristic
377C2×C4#93911yescharacteristic
378C2×C4#93911yescharacteristic
379C2×C4#94011yescharacteristic
380C2×C4C2×#21511yescharacteristic
381C2×C4#94211yescharacteristic
382C2×C4#94911yescharacteristic
383C2×C4#95011yescharacteristic
384C8—122no—
385C8—122no—
386C8—241no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—82no—
390C2×C3#200011yescharacteristic
391C2×C3#200311yescharacteristic
392C2×C3#200811yescharacteristic
393D3—161no—
394C22—21no—
395C22—21no—
396C22—44no—
397C22—42no—
398C22—82no—
399C22—81no—
400C22—81no—
401C22—241no—
402C22—241no—
403C22—241no—
404C22#522311yescharacteristic
405C22#522711yescharacteristic
406C22#523811yescharacteristic
407C4—21no—
408C4—21no—
409C4—21no—
410C4—82no—
411C4—81no—
412C4—82no—
413C4—241no—
414C4#504611yescharacteristic
415C4#504711yescharacteristic
416C4#504811yescharacteristic
417C4#504911yescharacteristic
418C4#506212yes—
419C3#1024111yescharacteristic, complemented, Sylow
420C2—21no—
421C2—21no—
422C2—82no—
423C2—481no—
424C2#7169911yescharacteristic
425C2#7173011yescharacteristic
426C2#7185811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2563C2
socleC22×C3#392
prime radical#82307C2
hypercenter#2563D3
hypocenterC3#10241
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2563

Complements

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

NH ≅ G/Ntype
1. C3×#2563C2semidirect
2. #83216C2semidirect
4. #71775C2semidirect
5. #71597C2semidirect
6. #69469C2semidirect
7. #67620C2semidirect
34. #2563D3semidirect
419. C3#10241semidirect

Maximal subgroups

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

subgroupindex
C3×#25632
#832162
#823072
#717752
#715972
#694692
#676202
#102413