Finite Group Databasebeta

← #10692100

Subgroups — #10692100

Order 768.

574 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 97 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
1#83316C211yesmaximal, characteristic
2#73383C211yesmaximal, characteristic, complemented
3C3×#1505C211yesmaximal, characteristic
4#71818C211yesmaximal, characteristic
5#70802C211yesmaximal, characteristic, complemented
6#67749C211yesmaximal, characteristic, complemented
7#67615C211yesmaximal, characteristic, complemented
8#9326—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C4×Dic6—21no—
11C2×C4×Dic6C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#376C2211yescharacteristic
14C2×#837—22no—
15#5091C412yescomplemented
16#5075C412yescomplemented
17#5040—21no—
18#5033—21no—
19#5018—22no—
20C4×C4⋊Dic3—21no—
21C4×C4⋊Dic3C2211yescharacteristic
22#4926C2211yescharacteristic
23C4×#208C2211yescharacteristic
24#4649—22no—
25#4649—21no—
26#4648—21no—
27C3×#327—21no—
28C3×#327—21no—
29#4484—22no—
30#3112—31no—
31#2276—31no—
32#1980—31no—
33#1505D311yescharacteristic
34#1491—31no—
35#1300—31no—
36#1285—31no—
37C2×C3×#41—21no—
38C22×Dic6D411yescharacteristic
39C3×C4⋊Q8C2×C411yescharacteristic
40C3×#127C2×C411yescharacteristic
41C3×#124—21no—
42C3×#121—21no—
43C3×#121—21no—
44C3×C4×D4—21no—
45C3×C4×D4—21no—
46C2×C3×C42—21no—
47C2×C3×C42C2311yescharacteristic
48C2×#214—22no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3D411yescharacteristic
51C2×Dic3⋊C4—22no—
52C2×Dic3⋊C4—21no—
53C2×C4×Dic3—22no—
54C2×C4×Dic3—21no—
55C2×C4×Dic3—22no—
56C2×C4×Dic3D412yes—
57C2×#208—21no—
58C4×Dic6—28no—
59C4×Dic6—21no—
60C4×Dic6—21no—
61C4×Dic6C2×C412yes—
62C4×Dic6D412yes—
63C3×C4⋊C8—21no—
64C3×C22⋊C8—21no—
65C3×#97—21no—
66C3×#95—21no—
67C3×#95D411yescharacteristic
68C3×#95D411yescharacteristic
69#837—44no—
70#832—22no—
71C3⋊#95—22no—
72C3⋊#95—21no—
73#830—21no—
74#810—22no—
75#807—22no—
76C4⋊(C3⋊C8)C2×C412yes—
77#803—21no—
78#803—21no—
79C4×C3⋊C8—21no—
80#532C2×D311yescharacteristic
81C2×C4×Q8—31no—
82C2×C4×Q8—61no—
83#480—61no—
84#477—32no—
85#460—32no—
86#448—61no—
87#440—62no—
88#432—31no—
89#432C2×D311yescharacteristic
90C2×#104—62no—
91C4×M16—31no—
92C4×C4⋊C4—31no—
93C4×C4⋊C4—62no—
94C4×C4⋊C4—61no—
95C4×C4⋊C4—61no—
96#376—61no—
97#376C2×D311yescharacteristic
98#327—21no—
99#327—21no—
100Q8⋊C8—62no—
101C3×#41—42no—
102C3×#41—42no—
103C2×C3×Q8—21no—
104C2×C3×D4—21no—
105C2×C3×D4—21no—
106C2×C3×D4—21no—
107C2×C3×D4—41no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4—41no—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×Dic3—21no—
115C22×Dic3—21no—
116C22×Dic3—22no—
117C2×Dic6—21no—
118C2×Dic6—24no—
119C2×Dic6C22⋊C412yes—
120C2×C3×C8—41no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—21no—
123C3×C22⋊C4—41no—
124C3×C22⋊C4—41no—
125C3×C42—24no—
126C3×C42C22×C411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42#4111yescharacteristic
129C3×C42#4111yescharacteristic
130#214—44no—
131C4⋊Dic3—24no—
132C4⋊Dic3—21no—
133C4⋊Dic3C22⋊C412yes—
134Dic3⋊C4—28no—
135Dic3⋊C4—22no—
136Dic3⋊C4—22no—
137C4×Dic3—28no—
138C4×Dic3—22no—
139C4×Dic3—22no—
140C4×Dic3—42no—
141C4×Dic3—82no—
142#208—22no—
143#208—41no—
144C2×C3⋊C8—22no—
145C2×C3⋊C8—41no—
146C2×#41—21no—
147C22×Q8—31no—
148C2×SD16—62no—
149C2×M16—61no—
150C4⋊Q8C4×D311yescharacteristic
151#127C4×D311yescharacteristic
152#124—21no—
153#121—21no—
154#121—21no—
155C4×Q8—32no—
156C4×Q8—32no—
157C4×Q8—68no—
158C4×Q8—61no—
159C4×Q8—61no—
160C4×D4—21no—
161C4×D4—21no—
162C2×C4⋊C4—31no—
163C2×C4⋊C4—62no—
164C2×C4⋊C4—61no—
165C2×C4⋊C4—61no—
166C2×C42—21no—
167C2×C42—32no—
168C2×C42—62no—
169C2×C42—61no—
170C2×C42—62no—
171C2×C42C22×D311yescharacteristic
172#107—62no—
173C4⋊C8—21no—
174C4⋊C8—32no—
175#104—124no—
176#103—62no—
177#102—62no—
178C22⋊C8—21no—
179C22⋊C8—61no—
180#97—21no—
181#97—61no—
182#97—61no—
183C4×C8—61no—
184#95—21no—
185#95—62no—
186#95—61no—
187#95#6711yescharacteristic
188#95#6711yescharacteristic
189C23×C3—41no—
190C23×C3#12011yescharacteristic
191C3×Q8—42no—
192C3×D4—42no—
193C3×D4—42no—
194C3×D4—81no—
195C3×D4—81no—
196C3×D4—81no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—22no—
200C2×C3×C4—22no—
201C2×C3×C4—22no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—82no—
209C2×C3×C4C2×C22⋊C411yescharacteristic
210C2×C3×C4C4×D411yescharacteristic
211C2×C3×C4C4×D411yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12311yescharacteristic
215C2×Dic3—24no—
216C2×Dic3—22no—
217C2×Dic3—24no—
218C2×Dic3—22no—
219C2×Dic3—42no—
220C2×Dic3—42no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223Dic6—24no—
224Dic6—22no—
225Dic6#10414yescomplemented
226C3×C8—81no—
227C3⋊C8—42no—
228C3⋊C8—42no—
229#41—42no—
230#41—42no—
231C2×Q8—21no—
232C2×Q8—32no—
233C2×Q8—64no—
234C2×Q8—61no—
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4—41no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—41no—
242C22×C4—61no—
243C22×C4—62no—
244C22×C4—61no—
245C22×C4D3×D411yescharacteristic
246C22×C4D3×D411yescharacteristic
247C22×C4C2×#6711yescharacteristic
248SD16—124no—
249M16—62no—
250M16—121no—
251C2×C8—41no—
252C2×C8—62no—
253C2×C8—121no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—32no—
257C4⋊C4—68no—
258C4⋊C4—64no—
259C4⋊C4—62no—
260C4⋊C4—61no—
261C4⋊C4—62no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C42—24no—
265C42—68no—
266C42—62no—
267C42—62no—
268C42—122no—
269C42—242no—
270C42C2×C4×D311yescharacteristic
271C42#23511yescharacteristic
272C42D3×D411yescharacteristic
273C42#23711yescharacteristic
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—81no—
277C22×C3—81no—
278C22×C3—81no—
279C22×C3#38611yescharacteristic
280C22×C3#45311yescharacteristic
281C22×C3#45411yescharacteristic
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—42no—
288C3×C4—42no—
289C3×C4—81no—
290C3×C4C2×#10412yes—
291Dic3—24no—
292Dic3—44no—
293Dic3—42no—
294C23—41no—
295C23#93711yescharacteristic
296Q8—34no—
297Q8—42no—
298Q8—64no—
299Q8—62no—
300D4—42no—
301D4—42no—
302D4—81no—
303D4—81no—
304D4—81no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—64no—
317C2×C4—62no—
318C2×C4—64no—
319C2×C4—62no—
320C2×C4—82no—
321C2×C4—122no—
322C2×C4—122no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4D3×C22⋊C411yescharacteristic
326C2×C4#67⋊C411yescharacteristic
327C2×C4#88311yescharacteristic
328C2×C4#88411yescharacteristic
329C2×C4C4×#6711yescharacteristic
330C2×C4#93811yescharacteristic
331C8—81no—
332C8—122no—
333C8—122no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3—161no—
337C2×C3#188311yescharacteristic
338C2×C3#189611yescharacteristic
339C2×C3#190511yescharacteristic
340C22—21no—
341C22—21no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22#495611yescharacteristic
346C22#519811yescharacteristic
347C22#520111yescharacteristic
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—42no—
354C4—42no—
355C4—64no—
356C4—81no—
357C4—124no—
358C4—122no—
359C4D3×#10412yes—
360C3#932611yescharacteristic, complemented, Sylow
361C2—21no—
362C2—21no—
363C2—161no—
364C2#7005211yescharacteristic
365C2#7032811yescharacteristic
366C2#7038211yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1505C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#1505D3
hypocenterC3#9326
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1505

Complements

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

NH ≅ G/Ntype
2. #73383C2semidirect
5. #70802C2semidirect
6. #67749C2semidirect
7. #67615C2semidirect
15. #5091C4semidirect
16. #5075C4semidirect
225. Dic6#104semidirect
360. C3#9326semidirect

Maximal subgroups

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

subgroupindex
#833162
#733832
C3×#15052
#718182
#708022
#677492
#676152
#93263