Finite Group Databasebeta

← #10696095

Subgroups — #10696095

Order 768.

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

normal subgroups only

This group has 518 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
2#83342C211yesmaximal, characteristic, complemented
3#82391C211yesmaximal, characteristic
4#71837C211yesmaximal, characteristic, complemented
5#71599C211yesmaximal, characteristic
6#70814C211yesmaximal, characteristic, complemented
7#67752C211yesmaximal, characteristic, complemented
8#10023—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#484—21no—
13C3×#474—21no—
14C3×#461—21no—
15C3×#450—21no—
16C3×#440—22no—
17C3×#432C2211yescharacteristic
18#5222—21no—
19#5210—22no—
20#5081—21no—
21#5081—21no—
22#5065—22no—
23#4995—21no—
24#4995—21no—
25#4983—21no—
26#4953—21no—
27#4949—21no—
28#4937C2211yescharacteristic
29#4926C2211yescharacteristic
30C4×#208C2211yescharacteristic
31#4538—22no—
32D4⋊(C3⋊C8)—22no—
33#4465—22no—
34#3166—31no—
35#3110D311yescharacteristic
36#2633—31no—
37#1534—31no—
38#1515—31no—
39#1490—31no—
40#1288—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C2×C3×SD16—22no—
44C3×C4⋊Q8D411yescharacteristic
45C3×C4⋊Q8D411yescharacteristic
46C3×#125—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×#117—21no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—21no—
58C2×C3×C42C2311yescharacteristic
59C2×#217—22no—
60C2×#217—22no—
61C2×C4⋊Dic3D411yescharacteristic
62C2×C4⋊Dic3D411yescharacteristic
63C2×Dic3⋊C4—21no—
64C2×Dic3⋊C4—21no—
65C2×Dic3⋊C4—22no—
66C2×#208—21no—
67#870—21no—
68#870—21no—
69#870—21no—
70#870—21no—
71C3×#107—22no—
72C3×C4⋊C8—21no—
73C3×#103—22no—
74C3×#102—22no—
75C3×C22⋊C8—21no—
76C3×#97D412yes—
77C3⋊#95—22no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95D412yes—
81#830—21no—
82#808—22no—
83#808—22no—
84#807—22no—
85#807—22no—
86C4⋊(C3⋊C8)—21no—
87#803—21no—
88#803—21no—
89C4×C3⋊C8D412yes—
90#533C2×D311yescharacteristic
91C2×#125—31no—
92C2×C4×D4C2×D311yescharacteristic
93#501—61no—
94#501—61no—
95#499—62no—
96#484—21no—
97#474—21no—
98#461—21no—
99#450—21no—
100#440—22no—
101#432—31no—
102#432C2×D311yescharacteristic
103#428—61no—
104#428—61no—
105C4×M16—31no—
106#399—61no—
107#398—61no—
108#397—61no—
109#394—61no—
110#385—62no—
111#383—31no—
112#330—62no—
113#328—62no—
114D4⋊C8—62no—
115C24×C3—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—42no—
121C22×C3×C4—21no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4—82no—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×Dic3—22no—
129C22×Dic3—21no—
130C22×Dic3—21no—
131C3×SD16—44no—
132C2×C3×C8—22no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C42—41no—
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42#4111yescharacteristic
149#217—44no—
150#217—82no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155Dic3⋊C4—41no—
156Dic3⋊C4—41no—
157Dic3⋊C4—41no—
158Dic3⋊C4—41no—
159Dic3⋊C4—82no—
160#208—24no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—41no—
163C22×D4#6711yescharacteristic
164C23×C4—21no—
165C2×SD16—22no—
166C2×M16—61no—
167C4⋊Q8#6711yescharacteristic
168C4⋊Q8#6711yescharacteristic
169#125—21no—
170#125—61no—
171#125—61no—
172#125—61no—
173#125—61no—
174C22⋊Q8—21no—
175C22⋊Q8—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182#117—21no—
183C2×C4⋊C4—31no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4—62no—
188C2×C4⋊C4#6711yescharacteristic
189C2×C22⋊C4—21no—
190C2×C22⋊C4—62no—
191C2×C22⋊C4—62no—
192C2×C42C22×D311yescharacteristic
193#107—22no—
194#107—62no—
195#107—62no—
196#106—62no—
197#106—62no—
198C4⋊C8—21no—
199C4⋊C8—61no—
200#103—22no—
201#102—22no—
202C22⋊C8—21no—
203C22⋊C8—61no—
204#97—61no—
205#97—61no—
206#97#6712yes—
207C4×C8—32no—
208#95—32no—
209#95—62no—
210#95—61no—
211#95—61no—
212C23×C3—22no—
213C23×C3—42no—
214C23×C3—41no—
215C23×C3—41no—
216C23×C3—42no—
217C23×C3#12011yescharacteristic
218C3×Q8—42no—
219C3×Q8—81no—
220C3×D4—22no—
221C3×D4—22no—
222C3×D4—41no—
223C3×D4—41no—
224C2×C3×C4—22no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—42no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—82no—
232C2×C3×C4—82no—
233C2×C3×C4#12011yescharacteristic
234C2×C3×C4#12011yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12711yescharacteristic
239C2×Dic3—41no—
240C2×Dic3—42no—
241C2×Dic3—42no—
242C2×Dic3—42no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—42no—
247C3×C8—42no—
248C3⋊C8—24no—
249C3⋊C8—81no—
250C24—21no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—42no—
256C22×C4—21no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—62no—
260C22×C4—61no—
261C22×C4—61no—
262C22×C4—82no—
263C22×C4D3×D411yescharacteristic
264C22×C4D3×D411yescharacteristic
265C22×C4C2×#6711yescharacteristic
266SD16—44no—
267M16—64no—
268C2×C8—22no—
269C2×C8—62no—
270C2×C8—121no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C4⋊C4—121no—
286C4⋊C4—121no—
287C4⋊C4—242no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—124no—
292C22⋊C4—242no—
293C42—41no—
294C42D3×D411yescharacteristic
295C42#23711yescharacteristic
296C42C2×#6711yescharacteristic
297C42C2×#6711yescharacteristic
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—44no—
301C22×C3—42no—
302C22×C3—82no—
303C22×C3—81no—
304C22×C3—81no—
305C22×C3#39211yescharacteristic
306C22×C3#45612yes—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—81no—
314C3×C4—81no—
315C3×C4—81no—
316Dic3—82no—
317Dic3—81no—
318Dic3—81no—
319C23—22no—
320C23—42no—
321C23—41no—
322C23—42no—
323C23—41no—
324C23#93711yescharacteristic
325Q8—42no—
326Q8—81no—
327D4—22no—
328D4—22no—
329D4—41no—
330D4—41no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—122no—
341C2×C4—122no—
342C2×C4—121no—
343C2×C4—122no—
344C2×C4—121no—
345C2×C4—122no—
346C2×C4—121no—
347C2×C4—121no—
348C2×C4#93711yescharacteristic
349C2×C4#93811yescharacteristic
350C2×C4#93911yescharacteristic
351C2×C4#93911yescharacteristic
352C2×C4#94011yescharacteristic
353C2×C4#95311yescharacteristic
354C8—42no—
355C8—64no—
356C8—241no—
357C2×C3—22no—
358C2×C3—82no—
359C2×C3#200511yescharacteristic
360C2×C3#200811yescharacteristic
361C2×C3#201011yescharacteristic
362C22—21no—
363C22—21no—
364C22—44no—
365C22—42no—
366C22—82no—
367C22—81no—
368C22—81no—
369C22#520512yes—
370C22#522311yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—81no—
378C4—81no—
379C4—81no—
380C4—242no—
381C4—241no—
382C4—241no—
383C3#1002311yescharacteristic, complemented, Sylow
384C2—22no—
385C2—82no—
386C2#7171511yescharacteristic
387C2#7209911yescharacteristic
388C2#7210811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3110C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenter#3110D3
hypocenterC3#10023
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
2. #83342C2semidirect
4. #71837C2semidirect
6. #70814C2semidirect
7. #67752C2semidirect
383. C3#10023semidirect

Maximal subgroups

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

subgroupindex
C3×#31102
#833422
#823912
#718372
#715992
#708142
#677522
#100233