Finite Group Databasebeta

← #10692961

Subgroups — #10692961

Order 768.

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

normal subgroups only

This group has 598 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#82373C211yesmaximal, characteristic
3#71619C211yesmaximal, characteristic
4#69924C212yesmaximal, complemented
5#67676C212yesmaximal, complemented
6#9470—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#935—21no—
9C22×C4⋊Dic3C2211yescharacteristic
10C3×C8×D4—24no—
11C3×#431—21no—
12C2×C3×C22⋊C8—21no—
13C2×C3×C22⋊C8C2212yes—
14C2×#834—21no—
15#5212—21no—
16#5212C2212yes—
17#5212C2211yescharacteristic
18#4981C412yescomplemented
19#4953C412yescomplemented
20#4943—21no—
21#4928—21no—
22#4551—24no—
23#4526—21no—
24#2972D311yescharacteristic
25#2486—31no—
26#1823—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—22no—
42C2×#217—21no—
43C2×#217—22no—
44#935—42no—
45#935—42no—
46C2×C4⋊Dic3—22no—
47C2×C4⋊Dic3—22no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3—22no—
51C2×C4⋊Dic3C2×C412yes—
52C2×Dic3⋊C4—22no—
53C3×C4⋊C8—22no—
54C3×C22⋊C8—22no—
55C3×C22⋊C8—22no—
56C3×C22⋊C8—22no—
57C3×C4×C8—22no—
58#834—42no—
59C3⋊#95—22no—
60C3⋊#95—22no—
61C3⋊#95—22no—
62C3⋊#95C2×C412yes—
63C2×#123—61no—
64C2×C4×D4C2×D311yescharacteristic
65C22×C4⋊C4—31no—
66C8×D4—24no—
67#431—21no—
68C2×#99—61no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8C2×D312yes—
71#398—32no—
72#397—32no—
73#389—61no—
74#381—61no—
75#380—31no—
76#380—32no—
77#380—61no—
78#342—61no—
79#340—64no—
80C24×C3C22⋊C411yescharacteristic
81C24×C3C2×D411yescharacteristic
82C2×C3×D4—22no—
83C2×C3×D4—22no—
84C2×C3×D4D812yes—
85C2×C3×D4SD1612yes—
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—42no—
89C22×C3×C4—42no—
90C22×C3×C4C22⋊C411yescharacteristic
91C22×C3×C4C22⋊C411yescharacteristic
92C22×C3×C4C22⋊C411yescharacteristic
93C22×C3×C4C22×C411yescharacteristic
94C22×C3×C4C2×D412yes—
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4#4112yes—
97C22×Dic3—22no—
98C22×Dic3—22no—
99C22×Dic3—42no—
100C22×Dic3—41no—
101C22×Dic3—41no—
102C22×Dic3—42no—
103C22×Dic3—42no—
104C2×C3×C8—24no—
105C2×C3×C8—44no—
106C2×C3×C8—82no—
107C3×C4⋊C4—22no—
108C3×C22⋊C4—22no—
109C3×C22⋊C4—22no—
110C3×C42—21no—
111C3×C42—21no—
112#217—44no—
113#217—41no—
114#217—41no—
115#217—42no—
116#217—42no—
117C4⋊Dic3—24no—
118C4⋊Dic3—42no—
119C4⋊Dic3—41no—
120C4⋊Dic3—41no—
121C4⋊Dic3—44no—
122Dic3⋊C4—44no—
123C22×D4D1211yescharacteristic
124C23×C4—61no—
125C23×C4#6711yescharacteristic
126C23×C4C22×D311yescharacteristic
127C22×C8—22no—
128C22×C8#6714yes—
129#123—122no—
130#123—122no—
131C4×D4—22no—
132C4×D4—22no—
133C2×C4⋊C4—32no—
134C2×C4⋊C4—62no—
135C2×C4⋊C4—62no—
136C2×C4⋊C4—62no—
137C2×C4⋊C4—61no—
138C2×C4⋊C4—61no—
139C2×C4⋊C4—62no—
140C2×C4⋊C4C4×D311yescharacteristic
141C2×C22⋊C4—62no—
142C2×C22⋊C4—61no—
143C2×C22⋊C4—62no—
144C2×C22⋊C4C4×D311yescharacteristic
145C2×C22⋊C4D1211yescharacteristic
146C2×C42#6711yescharacteristic
147C4⋊C8—22no—
148#99—122no—
149C22⋊C8—22no—
150C22⋊C8—22no—
151C22⋊C8—22no—
152C4×C8—22no—
153#95—32no—
154#95—62no—
155#95—62no—
156#95—62no—
157C23×C3—21no—
158C23×C3—21no—
159C23×C3—41no—
160C23×C3—41no—
161C23×C3—41no—
162C23×C3—41no—
163C23×C3#10212yes—
164C23×C3C2×C22⋊C411yescharacteristic
165C23×C3#12112yes—
166C23×C3C2×D811yescharacteristic
167C23×C3C2×SD1611yescharacteristic
168C23×C3#13512yes—
169C3×D4—24no—
170C3×D4—24no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—24no—
174C2×C3×C4—22no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—41no—
178C2×C3×C4—42no—
179C2×C3×C4—42no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4—81no—
183C2×C3×C4#10212yes—
184C2×C3×C4C4×D412yes—
185C2×C3×C4#12011yescharacteristic
186C2×C3×C4#12311yescharacteristic
187C2×Dic3—42no—
188C2×Dic3—42no—
189C2×Dic3—42no—
190C2×Dic3—42no—
191C2×Dic3—42no—
192C2×Dic3—42no—
193C2×Dic3—82no—
194C2×Dic3—82no—
195C2×Dic3—82no—
196C2×Dic3—82no—
197C2×Dic3—82no—
198C3×C8—44no—
199C24(C2×D3)⋊C411yescharacteristic
200C24C2×D1211yescharacteristic
201C2×D4—22no—
202C2×D4—22no—
203C2×D4#20412yes—
204C2×D4D2412yes—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—62no—
210C22×C4—62no—
211C22×C4—121no—
212C22×C4—121no—
213C22×C4—122no—
214C22×C4—122no—
215C22×C4—122no—
216C22×C4(C2×D3)⋊C411yescharacteristic
217C22×C4(C2×D3)⋊C411yescharacteristic
218C22×C4(C2×D3)⋊C411yescharacteristic
219C22×C4C2×C4×D311yescharacteristic
220C22×C4#23512yes—
221C22×C4C2×#6712yes—
222C22×C4C2×#6711yescharacteristic
223C2×C8—24no—
224C2×C8—44no—
225C2×C8—82no—
226C4⋊C4—22no—
227C4⋊C4—64no—
228C4⋊C4—122no—
229C4⋊C4—124no—
230C4⋊C4—124no—
231C4⋊C4—121no—
232C4⋊C4—121no—
233C22⋊C4—22no—
234C22⋊C4—22no—
235C22⋊C4—124no—
236C22⋊C4—121no—
237C22⋊C4—121no—
238C22⋊C4—122no—
239C22⋊C4—122no—
240C42—21no—
241C42—21no—
242C22×C3—21no—
243C22×C3—22no—
244C22×C3—21no—
245C22×C3—22no—
246C22×C3—22no—
247C22×C3—41no—
248C22×C3—41no—
249C22×C3—42no—
250C22×C3—42no—
251C22×C3—41no—
252C22×C3—41no—
253C22×C3—81no—
254C22×C3#38611yescharacteristic
255C22×C3C2×#10211yescharacteristic
256C22×C3#41611yescharacteristic
257C22×C3#46512yes—
258C22×C3#46612yes—
259C3×C4—21no—
260C3×C4—22no—
261C3×C4—22no—
262C3×C4—41no—
263C3×C4—41no—
264C3×C4#45812yes—
265Dic3—82no—
266Dic3—82no—
267Dic3—162no—
268C23—21no—
269C23—21no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23#82112yes—
275C23C2×#20411yescharacteristic
276C23C2×D2411yescharacteristic
277C23#90412yes—
278C23C2×(C2×D3)⋊C411yescharacteristic
279C23#93012yes—
280D4—24no—
281D4—24no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—24no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—81no—
294C2×C4—122no—
295C2×C4—122no—
296C2×C4—122no—
297C2×C4—122no—
298C2×C4—122no—
299C2×C4—122no—
300C2×C4—242no—
301C2×C4—242no—
302C2×C4—242no—
303C2×C4—242no—
304C2×C4—242no—
305C2×C4#82112yes—
306C2×C4C4×#6712yes—
307C2×C4#92911yescharacteristic
308C2×C4#95311yescharacteristic
309C8—44no—
310C2×C3—21no—
311C2×C3—22no—
312C2×C3—21no—
313C2×C3—22no—
314C2×C3—42no—
315C2×C3#189011yescharacteristic
316C2×C3#189311yescharacteristic
317C2×C3#190911yescharacteristic
318C22—21no—
319C22—22no—
320C22—21no—
321C22—22no—
322C22—22no—
323C22—41no—
324C22—41no—
325C22—42no—
326C22—42no—
327C22—41no—
328C22—41no—
329C22—81no—
330C22C2×#82111yescharacteristic
331C22#511311yescharacteristic
332C22#511412yes—
333C22#511512yes—
334C22#521311yescharacteristic
335C4—22no—
336C4—21no—
337C4—22no—
338C4—41no—
339C4—41no—
340C4—242no—
341C4—242no—
342C4—482no—
343C4#524312yes—
344C3#947011yescharacteristic, complemented, Sylow
345C2—21no—
346C2—22no—
347C2—21no—
348C2—22no—
349C2—42no—
350C2#7165511yescharacteristic
351C2#7197311yescharacteristic
352C2#7202411yescharacteristic
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#9470
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. #69924C2semidirect
5. #67676C2semidirect
18. #4981C4semidirect
19. #4953C4semidirect
344. C3#9470semidirect

Maximal subgroups

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

subgroupindex
C3×#29722
#823732
#716192
#699242
#676762
#94703