Finite Group Databasebeta

← #10692956

Subgroups — #10692956

Order 768.

578 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 135 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×#2972C211yesmaximal, characteristic
2#82374C211yesmaximal, characteristic
3#71619C211yesmaximal, characteristic
4#69925C212yesmaximal, complemented
5#67676C212yesmaximal, complemented
6#9469—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C22×C4⋊Dic3C2211yescharacteristic
9C3×C8×D4—24no—
10C3×#431—21no—
11C2×C3×C22⋊C8—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#2972D311yescharacteristic
25#2487—31no—
26#1824—32no—
27#1799—31no—
28#1289—32no—
29C22×C3×D4D411yescharacteristic
30C23×C3×C4D411yescharacteristic
31C23×C3×C4C2311yescharacteristic
32C23×Dic3—21no—
33C22×C3×C8—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×C4×C8—22no—
56#834—42no—
57C3⋊#95—22no—
58C3⋊#95—22no—
59C3⋊#95—22no—
60C3⋊#95C2×C412yes—
61C2×C4×D4C2×D311yescharacteristic
62C22×C4⋊C4—31no—
63C8×D4—24no—
64#431—21no—
65C2×#99—61no—
66C2×C22⋊C8—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—22no—
82C2×C3×D4SD1612yes—
83C2×C3×D4Q1612yes—
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—
100C2×C3×C8—24no—
101C2×C3×C8—44no—
102C2×C3×C8—82no—
103C3×C4⋊C4—22no—
104C3×C22⋊C4—22no—
105C3×C22⋊C4—22no—
106C3×C42—21no—
107C3×C42—21no—
108#217—42no—
109#217—42no—
110#217—42no—
111#217—82no—
112C4⋊Dic3—24no—
113C4⋊Dic3—42no—
114C4⋊Dic3—41no—
115C4⋊Dic3—41no—
116C4⋊Dic3—44no—
117Dic3⋊C4—82no—
118C22×D4D1211yescharacteristic
119C23×C4—61no—
120C23×C4#6711yescharacteristic
121C23×C4C22×D311yescharacteristic
122C22×C8—22no—
123C22×C8#6714yes—
124C4×D4—22no—
125C4×D4—22no—
126C2×C4⋊C4—32no—
127C2×C4⋊C4—62no—
128C2×C4⋊C4—62no—
129C2×C4⋊C4—61no—
130C2×C4⋊C4—61no—
131C2×C4⋊C4—62no—
132C2×C4⋊C4—62no—
133C2×C4⋊C4C4×D311yescharacteristic
134C2×C22⋊C4—61no—
135C2×C22⋊C4—62no—
136C2×C22⋊C4—62no—
137C2×C22⋊C4C4×D311yescharacteristic
138C2×C22⋊C4D1211yescharacteristic
139C2×C42#6711yescharacteristic
140C4⋊C8—22no—
141#99—122no—
142C22⋊C8—22no—
143C22⋊C8—22no—
144C22⋊C8—22no—
145C4×C8—22no—
146#95—32no—
147#95—62no—
148#95—62no—
149#95—62no—
150C23×C3—21no—
151C23×C3—21no—
152C23×C3—41no—
153C23×C3—41no—
154C23×C3—41no—
155C23×C3—41no—
156C23×C3#10312yes—
157C23×C3C2×C22⋊C411yescharacteristic
158C23×C3#12112yes—
159C23×C3C2×SD1611yescharacteristic
160C23×C3C2×Q1611yescharacteristic
161C23×C3#13512yes—
162C3×D4—24no—
163C3×D4—24no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—24no—
167C2×C3×C4—22no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—41no—
171C2×C3×C4—42no—
172C2×C3×C4—42no—
173C2×C3×C4—41no—
174C2×C3×C4—41no—
175C2×C3×C4—81no—
176C2×C3×C4#10312yes—
177C2×C3×C4C4×D412yes—
178C2×C3×C4#12011yescharacteristic
179C2×C3×C4#12311yescharacteristic
180C2×Dic3—42no—
181C2×Dic3—42no—
182C2×Dic3—42no—
183C2×Dic3—42no—
184C2×Dic3—42no—
185C2×Dic3—42no—
186C2×Dic3—84no—
187C2×Dic3—82no—
188C2×Dic3—82no—
189C2×Dic3—82no—
190C3×C8—44no—
191C24(C2×D3)⋊C411yescharacteristic
192C24C2×D1211yescharacteristic
193C2×D4—22no—
194C2×D4—22no—
195C2×D4#20412yes—
196C2×D4Dic1212yes—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—62no—
202C22×C4—62no—
203C22×C4—122no—
204C22×C4—122no—
205C22×C4—242no—
206C22×C4(C2×D3)⋊C411yescharacteristic
207C22×C4(C2×D3)⋊C411yescharacteristic
208C22×C4(C2×D3)⋊C411yescharacteristic
209C22×C4C2×C4×D311yescharacteristic
210C22×C4#23512yes—
211C22×C4C2×#6712yes—
212C22×C4C2×#6711yescharacteristic
213C2×C8—24no—
214C2×C8—44no—
215C2×C8—82no—
216C4⋊C4—22no—
217C4⋊C4—64no—
218C4⋊C4—122no—
219C4⋊C4—124no—
220C4⋊C4—121no—
221C4⋊C4—121no—
222C4⋊C4—242no—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C22⋊C4—122no—
226C22⋊C4—122no—
227C22⋊C4—122no—
228C22⋊C4—242no—
229C42—21no—
230C42—21no—
231C22×C3—21no—
232C22×C3—22no—
233C22×C3—21no—
234C22×C3—22no—
235C22×C3—22no—
236C22×C3—41no—
237C22×C3—41no—
238C22×C3—42no—
239C22×C3—42no—
240C22×C3—41no—
241C22×C3—41no—
242C22×C3—81no—
243C22×C3#38611yescharacteristic
244C22×C3C2×#10311yescharacteristic
245C22×C3#41611yescharacteristic
246C22×C3#46512yes—
247C22×C3#46712yes—
248C3×C4—21no—
249C3×C4—22no—
250C3×C4—22no—
251C3×C4—41no—
252C3×C4—41no—
253C3×C4#45812yes—
254Dic3—82no—
255Dic3—82no—
256Dic3—162no—
257C23—21no—
258C23—21no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23#81612yes—
264C23C2×#20411yescharacteristic
265C23#90412yes—
266C23C2×Dic1211yescharacteristic
267C23C2×(C2×D3)⋊C411yescharacteristic
268C23#93012yes—
269D4—24no—
270D4—24no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—81no—
283C2×C4—122no—
284C2×C4—122no—
285C2×C4—122no—
286C2×C4—122no—
287C2×C4—122no—
288C2×C4—122no—
289C2×C4—244no—
290C2×C4—242no—
291C2×C4—242no—
292C2×C4—242no—
293C2×C4#81612yes—
294C2×C4C4×#6712yes—
295C2×C4#92911yescharacteristic
296C2×C4#95311yescharacteristic
297C8—44no—
298C2×C3—21no—
299C2×C3—22no—
300C2×C3—21no—
301C2×C3—22no—
302C2×C3—42no—
303C2×C3#189011yescharacteristic
304C2×C3#189211yescharacteristic
305C2×C3#191011yescharacteristic
306C22—21no—
307C22—22no—
308C22—21no—
309C22—22no—
310C22—22no—
311C22—41no—
312C22—42no—
313C22—41no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—81no—
318C22C2×#81611yescharacteristic
319C22#511311yescharacteristic
320C22#511412yes—
321C22#511612yes—
322C22#521311yescharacteristic
323C4—22no—
324C4—21no—
325C4—22no—
326C4—41no—
327C4—41no—
328C4—242no—
329C4—242no—
330C4—482no—
331C4#524312yes—
332C3#946911yescharacteristic, complemented, Sylow
333C2—21no—
334C2—22no—
335C2—21no—
336C2—22no—
337C2—42no—
338C2#7165511yescharacteristic
339C2#7197211yescharacteristic
340C2#7202511yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2972C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC23×C3×C4C23
hypercenter#2972D3
hypocenterC3#9469
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2972

Complements

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

NH ≅ G/Ntype
4. #69925C2semidirect
5. #67676C2semidirect
18. #4986C4semidirect
19. #4949C4semidirect
332. C3#9469semidirect

Maximal subgroups

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

subgroupindex
C3×#29722
#823742
#716192
#699252
#676762
#94693