Finite Group Databasebeta

← #10698328

Subgroups — #10698328

Order 768.

364 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 91 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#73671C211yesmaximal, characteristic
2C3×#1503C211yesmaximal, characteristic
3#71840C211yesmaximal, characteristic
4#71835C211yesmaximal, characteristic
5#67746C211yesmaximal, characteristic
6#67737C211yesmaximal, characteristic
7#67620C211yesmaximal, characteristic
8#10373—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C3×#431C2211yescharacteristic
11C3×C4×C4⋊C4C2211yescharacteristic
12#4986—21no—
13#4982—21no—
14C4×C4⋊Dic3—21no—
15C4×C4⋊Dic3C2211yescharacteristic
16#4932C2211yescharacteristic
17C4×Dic3⋊C4—21no—
18#4925C2211yescharacteristic
19#4924C2211yescharacteristic
20#4661—21no—
21#4660—21no—
22C3×#328—21no—
23C3×#328—21no—
24#4538—21no—
25#4538—21no—
26#4532—21no—
27#4532—21no—
28#2402—31no—
29#1995—31no—
30#1503—31no—
31#1503D311yescharacteristic
32#1303—31no—
33#1303—31no—
34#1274—31no—
35C3×C4⋊Q8Q811yescharacteristic
36C3×C4⋊Q8Q811yescharacteristic
37C3×#125—21no—
38C3×C22⋊Q8—21no—
39C3×C22⋊Q8—21no—
40C3×C4×Q8—21no—
41C3×C4×Q8—21no—
42C3×#117—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C42—21no—
45C2×C3×C42C2311yescharacteristic
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3—21no—
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×C4×Dic3—21no—
52C2×C4×Dic3D411yescharacteristic
53C2×C4×Dic3D411yescharacteristic
54C2×#208—21no—
55C3×C4⋊C8—21no—
56C3×C22⋊C8—21no—
57C3×C4×C8—21no—
58C3×#95D411yescharacteristic
59C3×#95D411yescharacteristic
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95D411yescharacteristic
67C3⋊#95D411yescharacteristic
68#830—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4×C3⋊C8—21no—
73#533C2×D311yescharacteristic
74#431—31no—
75#431C2×D311yescharacteristic
76#423—31no—
77#400—61no—
78#400—61no—
79#398—61no—
80#398—61no—
81C4×C4⋊C4—31no—
82C4×C4⋊C4—61no—
83C4×C4⋊C4—61no—
84C4×C4⋊C4C2×D311yescharacteristic
85#376—31no—
86#339—61no—
87#339—61no—
88#328—21no—
89#328—21no—
90#328—61no—
91#328—61no—
92C2×C3×Q8—21no—
93C2×C3×Q8—21no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—21no—
102C22×Dic3—21no—
103C2×C3×C8—41no—
104C3×C4⋊C4—21no—
105C3×C4⋊C4—21no—
106C3×C4⋊C4—21no—
107C3×C4⋊C4—21no—
108C3×C4⋊C4—42no—
109C3×C4⋊C4—41no—
110C3×C4⋊C4—41no—
111C3×C22⋊C4—41no—
112C3×C42—41no—
113C3×C42—41no—
114C3×C42—41no—
115C3×C42C2×Q811yescharacteristic
116C3×C42#4111yescharacteristic
117C3×C42#4111yescharacteristic
118C3×C42#4111yescharacteristic
119C4⋊Dic3—42no—
120C4⋊Dic3—42no—
121C4⋊Dic3—42no—
122Dic3⋊C4—42no—
123Dic3⋊C4—42no—
124C4×Dic3—21no—
125C4×Dic3—21no—
126C4×Dic3—21no—
127C4×Dic3—21no—
128C4×Dic3—41no—
129C4×Dic3—41no—
130#208—41no—
131#208—41no—
132C2×C3⋊C8—41no—
133C2×C3⋊C8—41no—
134C2×M16—61no—
135C4⋊Q8Dic611yescharacteristic
136C4⋊Q8Dic611yescharacteristic
137#125—21no—
138C22⋊Q8—21no—
139C22⋊Q8—21no—
140C4×Q8—21no—
141C4×Q8—21no—
142#117—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—61no—
145C2×C4⋊C4—61no—
146C2×C4⋊C4—61no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4—61no—
149C2×C42—21no—
150C2×C42—31no—
151C2×C42—31no—
152C2×C42—61no—
153C2×C42C22×D311yescharacteristic
154C4⋊C8—21no—
155C4⋊C8—61no—
156C4⋊C8—61no—
157C4⋊C8—61no—
158C22⋊C8—21no—
159C22⋊C8—61no—
160C4×C8—21no—
161C4×C8—61no—
162#95—31no—
163#95—31no—
164#95—61no—
165#95—61no—
166#95—61no—
167#95—61no—
168#95—61no—
169#95—61no—
170#95#6711yescharacteristic
171#95#6711yescharacteristic
172C23×C3#12011yescharacteristic
173C3×Q8—81no—
174C3×Q8—81no—
175C2×C3×C4—22no—
176C2×C3×C4—41no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4C22⋊Q811yescharacteristic
186C2×C3×C4C22⋊Q811yescharacteristic
187C2×C3×C4C22⋊Q811yescharacteristic
188C2×C3×C4#12411yescharacteristic
189C2×C3×C4#12411yescharacteristic
190C2×C3×C4#12411yescharacteristic
191C2×C3×C4#13511yescharacteristic
192C2×C3×C4#13511yescharacteristic
193C2×C3×C4#13711yescharacteristic
194C2×C3×C4#13711yescharacteristic
195C2×Dic3—22no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C3×C8—81no—
208C3⋊C8—81no—
209C3⋊C8—81no—
210C2×Q8—21no—
211C2×Q8—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—61no—
215C22×C4—61no—
216C22×C4—61no—
217C22×C4—61no—
218C22×C4D3×D411yescharacteristic
219C22×C4D3×D411yescharacteristic
220C22×C4C2×#6711yescharacteristic
221M16—121no—
222M16—121no—
223C2×C8—41no—
224C2×C8—121no—
225C2×C8—121no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—42no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—122no—
234C4⋊C4—122no—
235C4⋊C4—122no—
236C4⋊C4—122no—
237C4⋊C4—122no—
238C22⋊C4—41no—
239C42—41no—
240C42—41no—
241C42—41no—
242C42—61no—
243C42—61no—
244C42—61no—
245C42—61no—
246C42—121no—
247C42—121no—
248C42C2×Dic611yescharacteristic
249C42#23511yescharacteristic
250C42#23711yescharacteristic
251C42#23711yescharacteristic
252C22×C3—21no—
253C22×C3—21no—
254C22×C3#39311yescharacteristic
255C22×C3#45211yescharacteristic
256C22×C3#45211yescharacteristic
257C3×C4—21no—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—42no—
262C3×C4—81no—
263C3×C4—81no—
264C3×C4—81no—
265C3×C4#47911yescharacteristic
266C3×C4#47911yescharacteristic
267C3×C4#48811yescharacteristic
268C3×C4#49111yescharacteristic
269Dic3—42no—
270Dic3—81no—
271Dic3—81no—
272Dic3—81no—
273C23#93711yescharacteristic
274Q8—81no—
275Q8—81no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—62no—
287C2×C4—121no—
288C2×C4—121no—
289C2×C4—121no—
290C2×C4—121no—
291C2×C4—121no—
292C2×C4—121no—
293C2×C4—121no—
294C2×C4—121no—
295C2×C4—121no—
296C2×C4—121no—
297C2×C4—121no—
298C2×C4#87811yescharacteristic
299C2×C4#87811yescharacteristic
300C2×C4#88511yescharacteristic
301C2×C4#88511yescharacteristic
302C2×C4#91511yescharacteristic
303C2×C4#91611yescharacteristic
304C2×C4#91811yescharacteristic
305C2×C4#91911yescharacteristic
306C2×C4#92411yescharacteristic
307C2×C4#93611yescharacteristic
308C8—81no—
309C8—241no—
310C8—241no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3#202611yescharacteristic
314C2×C3#203111yescharacteristic
315C2×C3#203111yescharacteristic
316C22—21no—
317C22—21no—
318C22#494711yescharacteristic
319C22#517311yescharacteristic
320C22#518711yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—42no—
326C4—81no—
327C4—81no—
328C4—81no—
329C4—122no—
330C4—241no—
331C4—241no—
332C4—241no—
333C4#479611yescharacteristic
334C4#479711yescharacteristic
335C4#479911yescharacteristic
336C4#480011yescharacteristic
337C3#1037311yescharacteristic, complemented, Sylow
338C2—21no—
339C2—21no—
340C2#7003811yescharacteristic
341C2#7023211yescharacteristic
342C2#7036811yescharacteristic
subgroupG/H
centerC2×C4#885
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1503C2
socleC22×C3#393
prime radicalC23×C3#120
hypercenter#1503D3
hypocenterC3#10373
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
337. C3#10373semidirect

Maximal subgroups

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

subgroupindex
#736712
C3×#15032
#718402
#718352
#677462
#677372
#676202
#103733