Finite Group Databasebeta

← #10708207

Subgroups — #10708207

Order 768.

342 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 57 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#77927C211yesmaximal, characteristic
2C3×#1611C211yesmaximal, characteristic
3#71901C211yesmaximal, characteristic
4#71898C211yesmaximal, characteristic
5#67650C211yesmaximal, characteristic
6#67458C211yesmaximal, characteristic
7#67452C211yesmaximal, characteristic
8#12251—31nomaximal, Sylow
9C3×#544C2211yescharacteristic
10C2×#889—21no—
11C3×#483—21no—
12C3×#398C2211yescharacteristic
13#5043—21no—
14#4995—21no—
15#4982—21no—
16#4982—21no—
17#4981C2211yescharacteristic
18#4977C2211yescharacteristic
19#4660—21no—
20#4658—21no—
21#4657—21no—
22C3×#328—21no—
23C3×#324C2211yescharacteristic
24#4538—21no—
25#4532—21no—
26#4530C2211yescharacteristic
27#4528C2211yescharacteristic
28#2822—31no—
29#2061—31no—
30#1611—31no—
31#1611D311yescharacteristic
32#1346—31no—
33#1346—31no—
34#1324—31no—
35C3×C4⋊Q8—41no—
36C3×#125—41no—
37C3×C22⋊Q8—21no—
38C3×C22⋊Q8—21no—
39C3×#117—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4D411yescharacteristic
42C2×C3×C4⋊C4C2311yescharacteristic
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3D411yescharacteristic
45C2×C4⋊Dic3D411yescharacteristic
46C2×Dic3⋊C4—21no—
47C2×Dic3⋊C4—21no—
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×C4×Dic3D411yescharacteristic
52C2×#208—21no—
53#889—42no—
54#889—42no—
55C3×#106—41no—
56C3×#103—41no—
57C3×C22⋊C8—21no—
58C3×#95—21no—
59C3×#95D411yescharacteristic
60#835—41no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95D411yescharacteristic
68#830—21no—
69#808—41no—
70#808—41no—
71#807—41no—
72#544C2×D311yescharacteristic
73C2×#125—61no—
74#483—21no—
75#483—61no—
76#428—61no—
77#400—61no—
78#400—61no—
79#398—31no—
80#398—61no—
81#398—61no—
82#398—61no—
83#398C2×D311yescharacteristic
84#382—31no—
85#343—31no—
86#339—61no—
87#328—21no—
88#328—61no—
89#324—31no—
90#324C2×D311yescharacteristic
91C2×C3×Q8—21no—
92C2×C3×Q8—21no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×Dic3—21no—
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—21no—
102C2×C3×C8—41no—
103C3×C4⋊C4—21no—
104C3×C4⋊C4—21no—
105C3×C4⋊C4—21no—
106C3×C4⋊C4—21no—
107C3×C4⋊C4—41no—
108C3×C4⋊C4—41no—
109C3×C4⋊C4—41no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—41no—
112C3×C4⋊C4—41no—
113C3×C22⋊C4—41no—
114C3×C42—41no—
115C4⋊Dic3—42no—
116C4⋊Dic3—41no—
117C4⋊Dic3—41no—
118Dic3⋊C4—42no—
119Dic3⋊C4—42no—
120Dic3⋊C4—42no—
121Dic3⋊C4—41no—
122Dic3⋊C4—41no—
123Dic3⋊C4—81no—
124C4×Dic3—21no—
125C4×Dic3SD1612yes—
126#208—22no—
127#208—41no—
128C2×C3⋊C8—41no—
129C2×C3⋊C8—41no—
130C2×M16—61no—
131C4⋊Q8—41no—
132#125—41no—
133#125—122no—
134#125—122no—
135C22⋊Q8—21no—
136C22⋊Q8—21no—
137#117—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—31no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4—61no—
145C2×C4⋊C4—61no—
146C2×C4⋊C4—61no—
147C2×C4⋊C4#6711yescharacteristic
148C2×C4⋊C4C22×D311yescharacteristic
149C2×C42—31no—
150#107—121no—
151#106—41no—
152#106—121no—
153#106—121no—
154#103—41no—
155#103—121no—
156C22⋊C8—21no—
157C22⋊C8—61no—
158#95—21no—
159#95—31no—
160#95—61no—
161#95—61no—
162#95—61no—
163#95—61no—
164#95—61no—
165#95—61no—
166#95#6711yescharacteristic
167C23×C3#12011yescharacteristic
168C3×Q8—81no—
169C3×Q8—81no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—41no—
175C2×C3×C4—41no—
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×C4C2×SD1611yescharacteristic
185C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
186C2×Dic3—22no—
187C2×Dic3—41no—
188C2×Dic3—41no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C2×Dic3—42no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C3×C8—81no—
198C3⋊C8—42no—
199C3⋊C8—81no—
200C2×Q8—21no—
201C2×Q8—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—61no—
205C22×C4—61no—
206C22×C4—61no—
207C22×C4—61no—
208C22×C4D3×D411yescharacteristic
209C22×C4D3×D411yescharacteristic
210C22×C4C2×#6711yescharacteristic
211M16—62no—
212M16—121no—
213C2×C8—41no—
214C2×C8—121no—
215C2×C8—121no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—121no—
227C4⋊C4—121no—
228C4⋊C4—122no—
229C4⋊C4—122no—
230C4⋊C4—122no—
231C4⋊C4—122no—
232C4⋊C4—121no—
233C4⋊C4—121no—
234C4⋊C4—241no—
235C22⋊C4—41no—
236C42—32no—
237C42—41no—
238C42—61no—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3#45011yescharacteristic
242C22×C3#45611yescharacteristic
243C22×C3B(4,𝔽2)11yescharacteristic
244C3×C4—21no—
245C3×C4—21no—
246C3×C4—41no—
247C3×C4—41no—
248C3×C4—81no—
249C3×C4—81no—
250C3×C4—81no—
251C3×C4—81no—
252Dic3—42no—
253Dic3—81no—
254Dic3—81no—
255Dic3—81no—
256C23#93711yescharacteristic
257Q8—81no—
258Q8—81no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—62no—
274C2×C4—121no—
275C2×C4—121no—
276C2×C4—121no—
277C2×C4—121no—
278C2×C4—122no—
279C2×C4—121no—
280C2×C4—121no—
281C2×C4—121no—
282C2×C4—121no—
283C2×C4—121no—
284C2×C4D3×SD1611yescharacteristic
285C2×C4#91411yescharacteristic
286C8—81no—
287C8—122no—
288C8—241no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3#160411yescharacteristic
292C2×C3#220111yescharacteristic
293C2×C3#220311yescharacteristic
294C22—21no—
295C22—21no—
296C22#503211yescharacteristic
297C22#516911yescharacteristic
298C22#520511yescharacteristic
299C4—21no—
300C4—21no—
301C4—41no—
302C4—41no—
303C4—81no—
304C4—81no—
305C4—81no—
306C4—81no—
307C4—122no—
308C4—241no—
309C4—241no—
310C4—241no—
311C3#1225111yescharacteristic, complemented, Sylow
312C2—21no—
313C2—21no—
314C2#7097511yescharacteristic
315C2#7099311yescharacteristic
316C2#7100311yescharacteristic
subgroupG/H
centerC22#5032
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1611C2
socleC22×C3B(4,𝔽2)
prime radicalC23×C3#120
hypercenter#1611D3
hypocenterC3#12251
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1611

Complements

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

NH ≅ G/Ntype
311. C3#12251semidirect

Maximal subgroups

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

subgroupindex
#779272
C3×#16112
#719012
#718982
#676502
#674582
#674522
#122513