Finite Group Databasebeta

← #10692976

Subgroups — #10692976

Order 768.

556 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 119 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×#2973C211yesmaximal, characteristic
2#82374C211yesmaximal, characteristic
3#71619C211yesmaximal, characteristic
4#69925C212yesmaximal, complemented
5#67676C212yesmaximal, complemented
6#9473—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C22×C4⋊Dic3C2211yescharacteristic
9C3×#435—24no—
10C3×#432—21no—
11C3×#407—21no—
12C2×C3×C22⋊C8C2212yes—
13#5220—21no—
14C2×#834—21no—
15#5212—21no—
16#5212C2212yes—
17#5212C2211yescharacteristic
18#4986C412yescomplemented
19#4949C412yescomplemented
20#4946—21no—
21#4928—21no—
22#4551—24no—
23#4526—21no—
24#2973D311yescharacteristic
25#2487—31no—
26#1824—32no—
27#1799—31no—
28#1289—32no—
29C22×C3×D4D411yescharacteristic
30C23×C3×C4D411yescharacteristic
31C23×C3×C4C2311yescharacteristic
32C23×Dic3—21no—
33C2×C3×M16—22no—
34C22×C3×C8D414yes—
35C3×C4×D4—22no—
36C3×C4×D4—22no—
37C2×C3×C4⋊C4C2×C411yescharacteristic
38C2×C3×C22⋊C4C2×C411yescharacteristic
39C2×C3×C22⋊C4D411yescharacteristic
40C2×C3×C42D411yescharacteristic
41C2×#217—21no—
42C2×#217—22no—
43C2×#217—22no—
44C2×C4⋊Dic3—22no—
45C2×C4⋊Dic3—22no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3—22no—
49C2×C4⋊Dic3C2×C412yes—
50C2×Dic3⋊C4—22no—
51C3×C4⋊C8—22no—
52C3×C22⋊C8—22no—
53C3×C22⋊C8—22no—
54C3×C22⋊C8—22no—
55C3×#97—22no—
56#834—42no—
57C3⋊#95—22no—
58C3⋊#95—22no—
59C3⋊#95—22no—
60C3⋊#95C2×C412yes—
61C2×C4×D4C2×D311yescharacteristic
62C22×C4⋊C4—31no—
63#435—24no—
64#432—21no—
65C2×#99—61no—
66#407—21no—
67C2×C22⋊C8C2×D312yes—
68#400—32no—
69#399—32no—
70#389—61no—
71#386—61no—
72#385—61no—
73#380—31no—
74#380—32no—
75#380—61no—
76#342—61no—
77#340—64no—
78C24×C3C22⋊C411yescharacteristic
79C24×C3C2×D411yescharacteristic
80C2×C3×D4—22no—
81C2×C3×D4—21no—
82C2×C3×D4—21no—
83C2×C3×D4—22no—
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—42no—
87C22×C3×C4—42no—
88C22×C3×C4C22⋊C411yescharacteristic
89C22×C3×C4C22⋊C411yescharacteristic
90C22×C3×C4C22⋊C411yescharacteristic
91C22×C3×C4C22×C411yescharacteristic
92C22×C3×C4C2×D412yes—
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4#4112yes—
95C22×Dic3—22no—
96C22×Dic3—22no—
97C22×Dic3—42no—
98C22×Dic3—42no—
99C22×Dic3—82no—
100C3×M16—82no—
101C2×C3×C8—24no—
102C2×C3×C8—44no—
103C3×C4⋊C4—22no—
104C3×C22⋊C4—22no—
105C3×C22⋊C4—22no—
106C3×C42—21no—
107C3×C42—21no—
108#217—41no—
109#217—41no—
110#217—42no—
111#217—42no—
112#217—82no—
113C4⋊Dic3—24no—
114C4⋊Dic3—42no—
115C4⋊Dic3—41no—
116C4⋊Dic3—41no—
117C4⋊Dic3—44no—
118Dic3⋊C4—82no—
119C22×D4D1211yescharacteristic
120C23×C4—61no—
121C23×C4#6711yescharacteristic
122C23×C4C22×D311yescharacteristic
123C2×M16—22no—
124C22×C8#6714yes—
125C4×D4—22no—
126C4×D4—22no—
127C2×C4⋊C4—32no—
128C2×C4⋊C4—62no—
129C2×C4⋊C4—62no—
130C2×C4⋊C4—61no—
131C2×C4⋊C4—61no—
132C2×C4⋊C4—62no—
133C2×C4⋊C4—62no—
134C2×C4⋊C4C4×D311yescharacteristic
135C2×C22⋊C4—61no—
136C2×C22⋊C4—62no—
137C2×C22⋊C4—62no—
138C2×C22⋊C4C4×D311yescharacteristic
139C2×C22⋊C4D1211yescharacteristic
140C2×C42#6711yescharacteristic
141C4⋊C8—22no—
142#99—122no—
143C22⋊C8—22no—
144C22⋊C8—22no—
145C22⋊C8—22no—
146#97—22no—
147#95—32no—
148#95—62no—
149#95—62no—
150#95—62no—
151C23×C3—21no—
152C23×C3—21no—
153C23×C3—21no—
154C23×C3—41no—
155C23×C3—41no—
156C23×C3—41no—
157C23×C3—41no—
158C23×C3C2×C22⋊C411yescharacteristic
159C23×C3#12112yes—
160C23×C3AGL(1,ℤ/8)11yescharacteristic
161C23×C3#13711yescharacteristic
162C23×C3#13712yes—
163C3×D4—42no—
164C3×D4—42no—
165C2×C3×C4—24no—
166C2×C3×C4—22no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—41no—
173C2×C3×C4—42no—
174C2×C3×C4—42no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4—81no—
178C2×C3×C4C4×D412yes—
179C2×C3×C4#12011yescharacteristic
180C2×C3×C4#12311yescharacteristic
181C2×Dic3—42no—
182C2×Dic3—42no—
183C2×Dic3—42no—
184C2×Dic3—42no—
185C2×Dic3—42no—
186C2×Dic3—42no—
187C2×Dic3—84no—
188C2×Dic3—82no—
189C2×Dic3—82no—
190C2×Dic3—82no—
191C3×C8—44no—
192C24(C2×D3)⋊C411yescharacteristic
193C24C2×D1211yescharacteristic
194C2×D4—21no—
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—62no—
203C22×C4—62no—
204C22×C4—122no—
205C22×C4—122no—
206C22×C4—242no—
207C22×C4(C2×D3)⋊C411yescharacteristic
208C22×C4(C2×D3)⋊C411yescharacteristic
209C22×C4(C2×D3)⋊C411yescharacteristic
210C22×C4C2×C4×D311yescharacteristic
211C22×C4#23512yes—
212C22×C4C2×#6712yes—
213C22×C4C2×#6711yescharacteristic
214M16—82no—
215C2×C8—24no—
216C2×C8—44no—
217C4⋊C4—22no—
218C4⋊C4—64no—
219C4⋊C4—122no—
220C4⋊C4—124no—
221C4⋊C4—121no—
222C4⋊C4—121no—
223C4⋊C4—242no—
224C22⋊C4—22no—
225C22⋊C4—22no—
226C22⋊C4—121no—
227C22⋊C4—121no—
228C22⋊C4—122no—
229C22⋊C4—122no—
230C22⋊C4—242no—
231C42—21no—
232C42—21no—
233C22×C3—21no—
234C22×C3—22no—
235C22×C3—21no—
236C22×C3—22no—
237C22×C3—41no—
238C22×C3—41no—
239C22×C3—42no—
240C22×C3—42no—
241C22×C3—41no—
242C22×C3—41no—
243C22×C3—41no—
244C22×C3—81no—
245C22×C3#38611yescharacteristic
246C22×C3#41711yescharacteristic
247C22×C3#41911yescharacteristic
248C22×C3#46812yes—
249C22×C3#47012yes—
250C3×C4—21no—
251C3×C4—22no—
252C3×C4—41no—
253C3×C4—41no—
254C3×C4—41no—
255C3×C4#45812yes—
256Dic3—82no—
257Dic3—82no—
258Dic3—162no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23#90811yescharacteristic
267C23#90912yes—
268C23#90911yescharacteristic
269C23C2×(C2×D3)⋊C411yescharacteristic
270C23#93012yes—
271D4—42no—
272D4—42no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—81no—
286C2×C4—122no—
287C2×C4—122no—
288C2×C4—122no—
289C2×C4—122no—
290C2×C4—122no—
291C2×C4—122no—
292C2×C4—244no—
293C2×C4—242no—
294C2×C4—242no—
295C2×C4—242no—
296C2×C4C4×#6712yes—
297C2×C4#92911yescharacteristic
298C2×C4#95311yescharacteristic
299C8—44no—
300C2×C3—21no—
301C2×C3—22no—
302C2×C3—21no—
303C2×C3—22no—
304C2×C3—81no—
305C2×C3#189011yescharacteristic
306C2×C3#189411yescharacteristic
307C2×C3#191011yescharacteristic
308C22—21no—
309C22—22no—
310C22—21no—
311C22—22no—
312C22—41no—
313C22—41no—
314C22—42no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—81no—
320C22#512011yescharacteristic
321C22#513311yescharacteristic
322C22#513412yes—
323C22#513712yes—
324C22#521311yescharacteristic
325C4—21no—
326C4—22no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—242no—
331C4—242no—
332C4—482no—
333C4#524312yes—
334C3#947311yescharacteristic, complemented, Sylow
335C2—21no—
336C2—22no—
337C2—21no—
338C2—22no—
339C2—81no—
340C2#7165511yescharacteristic
341C2#7198911yescharacteristic
342C2#7202511yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2973C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC23×C3×C4C23
hypercenter#2973D3
hypocenterC3#9473
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2973

Complements

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

NH ≅ G/Ntype
4. #69925C2semidirect
5. #67676C2semidirect
18. #4986C4semidirect
19. #4949C4semidirect
334. C3#9473semidirect

Maximal subgroups

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

subgroupindex
C3×#29732
#823742
#716192
#699252
#676762
#94733