Finite Group Databasebeta

← #10696088

Subgroups — #10696088

Order 768.

576 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 81 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×#3110C211yesmaximal, characteristic, complemented
2#83338C211yesmaximal, characteristic, complemented
3#82421C211yesmaximal, characteristic
4#71837C211yesmaximal, characteristic, complemented
5#71599C211yesmaximal, characteristic, complemented
6#70808C211yesmaximal, characteristic, complemented
7#67752C211yesmaximal, characteristic, complemented
8#10022—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C2×#875C2211yescharacteristic
12C3×#484—21no—
13C3×#474—21no—
14C3×#461—21no—
15C3×#450—21no—
16C3×#440—22no—
17C3×#432C2211yescharacteristic
18#5223—21no—
19#5213—22no—
20#5089—21no—
21#5089—21no—
22#5068—22no—
23#5004—21no—
24#5001—21no—
25#4992—21no—
26#4960—21no—
27#4947—21no—
28#4937C2211yescharacteristic
29#4926C2211yescharacteristic
30C4×#208C2211yescharacteristic
31#4538—22no—
32D4⋊(C3⋊C8)—22no—
33#4464—22no—
34#3140—31no—
35#3110D311yescharacteristic, complemented
36#2631—31no—
37#1533—31no—
38#1515—31no—
39#1490—31no—
40#1288—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C22×#67—22no—
44C22×D12D411yescharacteristic
45C22×Dic6D411yescharacteristic
46C2×C3×SD16—22no—
47C3×C4⋊Q8D411yescharacteristic, complemented
48C3×C4⋊Q8D411yescharacteristic, complemented
49C3×#125—21no—
50C3×C22⋊Q8—21no—
51C3×C22⋊Q8—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C3×#117—21no—
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C22⋊C4—21no—
61C2×C3×C42C2311yescharacteristic
62C2×#217—22no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—21no—
65C2×(C2×D3)⋊C4—22no—
66C2×#208—21no—
67#875—21no—
68#875—21no—
69#875—21no—
70#875—21no—
71C3×#107—22no—
72C3×C4⋊C8—21no—
73C3×#103—22no—
74C3×#102—22no—
75C3×C22⋊C8—21no—
76C3×#97D412yescomplemented
77C3⋊#95—22no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95D412yes—
81#830—21no—
82#810—22no—
83#810—22no—
84#809—22no—
85#809—22no—
86C4⋊(C3⋊C8)—21no—
87#803—21no—
88#803—21no—
89C4×C3⋊C8D412yescomplemented
90#533C2×D311yescharacteristic
91C2×#124—31no—
92C2×C4×D4C2×D311yescharacteristic
93#489—61no—
94#489—61no—
95#488—62no—
96#484—21no—
97#474—21no—
98#461—21no—
99#450—21no—
100#440—22no—
101#432—31no—
102#432C2×D311yescharacteristic
103#419—61no—
104#418—61no—
105C4×M16—31no—
106#397—61no—
107#393—61no—
108#392—61no—
109#392—61no—
110#386—62no—
111#383—31no—
112#329—62no—
113#328—62no—
114D4⋊C8—62no—
115C24×C3—21no—
116C23×D3—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—42no—
122C22×C3×C4—21no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4—82no—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C2×#67—44no—
130C2×#67—82no—
131C2×#67—82no—
132C22×Dic3—22no—
133C22×Dic3—21no—
134C2×D12—21no—
135C2×D12—21no—
136C2×D12—42no—
137C2×Dic6—21no—
138C2×Dic6—21no—
139C2×Dic6—42no—
140C3×SD16—44no—
141C2×C3×C8—22no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—41no—
149C3×C4⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—41no—
153C3×C42—41no—
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157C3×C42#4111yescharacteristic
158#217—82no—
159(C2×D3)⋊C4—41no—
160(C2×D3)⋊C4—41no—
161(C2×D3)⋊C4—41no—
162(C2×D3)⋊C4—41no—
163(C2×D3)⋊C4—82no—
164#208—42no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—41no—
167C22×Q8—31no—
168C22×D4—31no—
169C22×D4—62no—
170C22×D4#6711yescharacteristic
171C23×C4—21no—
172C2×SD16—22no—
173C2×M16—61no—
174C4⋊Q8#6711yescharacteristic, complemented
175C4⋊Q8#6711yescharacteristic, complemented
176#125—21no—
177#124—61no—
178#124—61no—
179#124—61no—
180#124—61no—
181C22⋊Q8—21no—
182C22⋊Q8—21no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189#117—21no—
190C2×C4⋊C4#6711yescharacteristic
191C2×C22⋊C4—21no—
192C2×C22⋊C4—61no—
193C2×C22⋊C4—61no—
194C2×C22⋊C4—62no—
195C2×C22⋊C4—62no—
196C2×C42C22×D311yescharacteristic
197#107—22no—
198C4⋊C8—21no—
199C4⋊C8—61no—
200#103—22no—
201#103—62no—
202#103—62no—
203#102—22no—
204#102—62no—
205#102—62no—
206C22⋊C8—21no—
207C22⋊C8—61no—
208#97—61no—
209#97—61no—
210#97#6712yescomplemented
211C4×C8—32no—
212#95—32no—
213#95—62no—
214#95—61no—
215#95—61no—
216C23×C3—22no—
217C23×C3—42no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—42no—
221C23×C3#12011yescharacteristic
222C22×D3—41no—
223C22×D3—42no—
224C22×D3—41no—
225C22×D3—41no—
226C22×D3—42no—
227C3×Q8—42no—
228C3×Q8—81no—
229C3×D4—22no—
230C3×D4—22no—
231C3×D4—41no—
232C3×D4—41no—
233C2×C3×C4—22no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—42no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—82no—
241C2×C3×C4—82no—
242C2×C3×C4#12011yescharacteristic
243C2×C3×C4#12011yescharacteristic
244C2×C3×C4#12111yescharacteristic
245C2×C3×C4#12111yescharacteristic
246C2×C3×C4#12111yescharacteristic
247C2×C3×C4#12711yescharacteristic
248#67—88no—
249C2×Dic3—42no—
250C2×Dic3—42no—
251C2×Dic3—41no—
252C2×Dic3—42no—
253C2×Dic3—42no—
254D12—42no—
255D12—81no—
256Dic6—42no—
257Dic6—81no—
258C3×C8—42no—
259C3⋊C8—42no—
260C3⋊C8—81no—
261C24—21no—
262C24—61no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—61no—
266C2×Q8—61no—
267C2×Q8—122no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—42no—
271C2×D4—61no—
272C2×D4—61no—
273C2×D4—122no—
274C2×D4—124no—
275C2×D4—242no—
276C2×D4—242no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—62no—
281C22×C4—61no—
282C22×C4—82no—
283C22×C4D3×D411yescharacteristic
284C22×C4D3×D411yescharacteristic
285C22×C4C2×#6711yescharacteristic
286SD16—44no—
287M16—122no—
288C2×C8—22no—
289C2×C8—62no—
290C2×C8—121no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—121no—
303C22⋊C4—121no—
304C22⋊C4—121no—
305C22⋊C4—121no—
306C22⋊C4—242no—
307C22⋊C4—242no—
308C42—41no—
309C42D3×D411yescharacteristic
310C42#23711yescharacteristic
311C42C2×#6711yescharacteristic
312C42C2×#6711yescharacteristic
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—44no—
316C22×C3—42no—
317C22×C3—82no—
318C22×C3—81no—
319C22×C3—81no—
320C22×C3#39211yescharacteristic
321C22×C3#45512yes—
322C2×D3—82no—
323C2×D3—82no—
324C2×D3—81no—
325C2×D3—81no—
326C2×D3—81no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—81no—
334C3×C4—81no—
335C3×C4—81no—
336Dic3—82no—
337Dic3—81no—
338C23—22no—
339C23—42no—
340C23—41no—
341C23—42no—
342C23—41no—
343C23—121no—
344C23—122no—
345C23—121no—
346C23—122no—
347C23—121no—
348C23#93711yescharacteristic
349Q8—42no—
350Q8—81no—
351Q8—122no—
352Q8—241no—
353D4—22no—
354D4—22no—
355D4—41no—
356D4—41no—
357D4—122no—
358D4—241no—
359D4—248no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—122no—
370C2×C4—122no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4—121no—
374C2×C4#93711yescharacteristic
375C2×C4#93811yescharacteristic
376C2×C4#93911yescharacteristic
377C2×C4#93911yescharacteristic
378C2×C4#94011yescharacteristic
379C2×C4#95311yescharacteristic
380C8—42no—
381C8—122no—
382C8—241no—
383C2×C3—22no—
384C2×C3—82no—
385C2×C3#200411yescharacteristic
386C2×C3#200811yescharacteristic
387C2×C3#201011yescharacteristic
388D3—161no—
389C22—21no—
390C22—21no—
391C22—44no—
392C22—42no—
393C22—82no—
394C22—81no—
395C22—81no—
396C22—242no—
397C22—242no—
398C22—241no—
399C22—241no—
400C22—241no—
401C22#520212yes—
402C22#522311yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—81no—
410C4—81no—
411C4—81no—
412C4—242no—
413C4—241no—
414C3#1002211yescharacteristic, complemented, Sylow
415C2—22no—
416C2—82no—
417C2—481no—
418C2#7171511yescharacteristic
419C2#7209911yescharacteristic
420C2#7210611yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3110C2
socleC22×C3#392
prime radical#82421C2
hypercenter#3110D3
hypocenterC3#10022
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
1. C3×#3110C2semidirect
2. #83338C2semidirect
4. #71837C2semidirect
5. #71599C2semidirect
6. #70808C2semidirect
7. #67752C2semidirect
35. #3110D3semidirect
47. C3×C4⋊Q8D4semidirect
48. C3×C4⋊Q8D4semidirect
76. C3×#97D4semidirect
89. C4×C3⋊C8D4semidirect
174. C4⋊Q8#67semidirect
175. C4⋊Q8#67semidirect
210. #97#67semidirect
414. C3#10022semidirect

Maximal subgroups

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

subgroupindex
C3×#31102
#833382
#824212
#718372
#715992
#708082
#677522
#100223