Finite Group Databasebeta

← #10704593

Subgroups — #10704593

Order 768.

434 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 87 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#82157C211yesmaximal, characteristic
2C3×#1463C211yesmaximal, characteristic
3#71552C211yesmaximal, characteristic
4#67453C212yesmaximal, complemented
5#67452C212yesmaximal, complemented
6#11514—31nomaximal, Sylow
7C22×C3×C4⋊C4C2211yescharacteristic
8C2×#926C411yescharacteristic
9C2×#926C411yescharacteristic
10C3×#407—21no—
11#5210—21no—
12#5210—21no—
13#5207—21no—
14#4977C411yescharacteristic
15#4977C2211yescharacteristic
16#4975C411yescharacteristic
17#4975C2211yescharacteristic
18#4972—21no—
19#4972—21no—
20C3×#324—21no—
21C3×#324C2212yes—
22#4528—21no—
23#4528C2212yes—
24#2317—31no—
25#1463—31no—
26#1463D311yescharacteristic
27#1325—32no—
28#1324—32no—
29C23×C3×C4C2×C411yescharacteristic
30C23×C3×C4D411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C2×C3×M16—41no—
33C2×C3×C4⋊C4—41no—
34C2×C3×C4⋊C4—41no—
35C2×C3×C4⋊C4C2×C411yescharacteristic
36C2×C3×C4⋊C4D411yescharacteristic
37C2×C3×C4⋊C4D411yescharacteristic
38C2×C3×C4⋊C4C2311yescharacteristic
39C2×#217—21no—
40C2×#217—21no—
41#926—22no—
42#926—22no—
43#926—41no—
44#926—41no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3C2×C411yescharacteristic
47C2×C4⋊Dic3C2×C411yescharacteristic
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—21no—
52C2×C4×Dic3—21no—
53C2×C4×Dic3C2×C411yescharacteristic
54C2×C4×Dic3C2×C411yescharacteristic
55C2×#208—41no—
56C3×C22⋊C8—22no—
57C3×C22⋊C8—41no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62#830—22no—
63#830—41no—
64C2×#117—31no—
65C2×#117—31no—
66C22×C4⋊C4C2×D311yescharacteristic
67#407—21no—
68#407—61no—
69#389—61no—
70#389—61no—
71#385—61no—
72#385—61no—
73#384—31no—
74#384—31no—
75#382—31no—
76#382—31no—
77#324—21no—
78#324—32no—
79#324—61no—
80#324C2×D312yes—
81C24×C3C22⋊C411yescharacteristic
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—22no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4—81no—
92C22×C3×C4—81no—
93C22×C3×C4C22⋊C411yescharacteristic
94C22×C3×C4C22⋊C411yescharacteristic
95C22×C3×C4C22⋊C411yescharacteristic
96C22×C3×C4C22×C411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—21no—
102C22×Dic3—21no—
103C3×M16—82no—
104C2×C3×C8—42no—
105C3×C4⋊C4—41no—
106C3×C4⋊C4—41no—
107C3×C4⋊C4—41no—
108C3×C4⋊C4—41no—
109#217—24no—
110#217—41no—
111#217—41no—
112C4⋊Dic3—24no—
113C4⋊Dic3—41no—
114C4⋊Dic3—41no—
115Dic3⋊C4—81no—
116Dic3⋊C4—81no—
117Dic3⋊C4—81no—
118Dic3⋊C4—81no—
119C4×Dic3—21no—
120C4×Dic3—21no—
121C4×Dic3—42no—
122C4×Dic3M1612yescomplemented
123C4×Dic3M1612yescomplemented
124#208—82no—
125C2×C3⋊C8—42no—
126C23×C4C4×D311yescharacteristic
127C23×C4D1211yescharacteristic
128C23×C4#6711yescharacteristic
129C2×M16—41no—
130C2×M16—121no—
131#117—62no—
132#117—62no—
133#117—121no—
134#117—121no—
135C2×C4⋊C4—31no—
136C2×C4⋊C4—31no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—61no—
140C2×C4⋊C4—61no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4C4×D311yescharacteristic
145C2×C4⋊C4D1211yescharacteristic
146C2×C4⋊C4#6711yescharacteristic
147C2×C4⋊C4C22×D311yescharacteristic
148C2×C22⋊C4—61no—
149C2×C22⋊C4—61no—
150C2×C42—31no—
151C2×C42—31no—
152C2×C42—61no—
153C22⋊C8—22no—
154C22⋊C8—41no—
155C22⋊C8—62no—
156C22⋊C8—121no—
157#95—61no—
158#95—61no—
159#95—61no—
160#95—61no—
161C23×C3—21no—
162C23×C3—21no—
163C23×C3—41no—
164C23×C3—41no—
165C23×C3#9911yescharacteristic
166C23×C3#9911yescharacteristic
167C23×C3C2×C22⋊C411yescharacteristic
168C2×C3×C4—22no—
169C2×C3×C4—22no—
170C2×C3×C4—22no—
171C2×C3×C4—22no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—42no—
181C2×C3×C4—42no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—81no—
187C2×C3×C4—81no—
188C2×C3×C4#10112yes—
189C2×C3×C4C2×M1611yescharacteristic
190C2×C3×C4C2×M1611yescharacteristic
191C2×Dic3—22no—
192C2×Dic3—22no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—42no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C3×C8—82no—
203C3⋊C8—82no—
204C24(C2×D3)⋊C411yescharacteristic
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—61no—
215C22×C4—61no—
216C22×C4—61no—
217C22×C4—61no—
218C22×C4—81no—
219C22×C4—81no—
220C22×C4(C2×D3)⋊C411yescharacteristic
221C22×C4(C2×D3)⋊C411yescharacteristic
222C22×C4(C2×D3)⋊C411yescharacteristic
223C22×C4C2×C4×D311yescharacteristic
224C22×C4C2×D1211yescharacteristic
225C22×C4C2×#6711yescharacteristic
226M16—82no—
227M16—242no—
228C2×C8—42no—
229C2×C8—122no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—64no—
235C4⋊C4—121no—
236C4⋊C4—121no—
237C4⋊C4—241no—
238C4⋊C4—241no—
239C4⋊C4—241no—
240C4⋊C4—241no—
241C22⋊C4—64no—
242C22⋊C4—121no—
243C22⋊C4—121no—
244C42—32no—
245C42—32no—
246C42—61no—
247C42—61no—
248C42—122no—
249C22×C3—21no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—21no—
257C22×C3—42no—
258C22×C3—42no—
259C22×C3#40711yescharacteristic
260C22×C3C2×#9911yescharacteristic
261C22×C3C2×#10111yescharacteristic
262C3×C4—22no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—41no—
266C3×C4—41no—
267C3×C4—41no—
268C3×C4—41no—
269Dic3—42no—
270Dic3—42no—
271Dic3—81no—
272Dic3—81no—
273C23—21no—
274C23—21no—
275C23—41no—
276C23—41no—
277C23#80611yescharacteristic
278C23#80611yescharacteristic
279C23C2×(C2×D3)⋊C411yescharacteristic
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—62no—
299C2×C4—62no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—121no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4—122no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4#82412yes—
312C2×C4D3×M1611yescharacteristic
313C2×C4D3×M1611yescharacteristic
314C8—82no—
315C8—242no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3—41no—
319C2×C3—41no—
320C2×C3#146311yescharacteristic
321C2×C3#212711yescharacteristic
322C2×C3#213311yescharacteristic
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—42no—
332C22—42no—
333C22C2×#80611yescharacteristic
334C22#512611yescharacteristic
335C22C2×#82411yescharacteristic
336C4—22no—
337C4—21no—
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—122no—
344C4—122no—
345C4—241no—
346C4—241no—
347C3#1151411yescharacteristic, complemented, Sylow
348C2—21no—
349C2—21no—
350C2—41no—
351C2—41no—
352C2#6997111yescharacteristic
353C2#7001911yescharacteristic
354C2#7012811yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1463C2
socleC22×C3C2×#99
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1463D3
hypocenterC3#11514
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
4. #67453C2semidirect
5. #67452C2semidirect
122. C4×Dic3M16semidirect
123. C4×Dic3M16semidirect
347. C3#11514semidirect

Maximal subgroups

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

subgroupindex
#821572
C3×#14632
#715522
#674532
#674522
#115143