Finite Group Databasebeta

← #10690282

Subgroups — #10690282

Order 768.

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

normal subgroups only

This group has 482 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×#2396C211yesmaximal, characteristic
2#82209C211yesmaximal, characteristic
3#71622C212yesmaximal, complemented
4#71571C211yesmaximal, characteristic
5#67620C212yesmaximal, complemented
6#9034—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×C3×#117C2211yescharacteristic
9C3×#400C411yescharacteristic
10C3×#399C411yescharacteristic
11C3×#398—21no—
12C3×#396—21no—
13C3×#395C411yescharacteristic
14C3×#393C411yescharacteristic
15C3×C4×C4⋊C4—21no—
16C3×C4×C4⋊C4C2211yescharacteristic
17C3×C4×C22⋊C4—21no—
18#5207—21no—
19#5000—24no—
20#4925C2212yes—
21#4924C2212yes—
22C4×#208—21no—
23#4535—24no—
24#4528—21no—
25#4525—21no—
26#2921—31no—
27#2396D311yescharacteristic
28#1487—32no—
29#1458—31no—
30#1303—32no—
31C22×C3×Q8C2×C411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C3×C4×Q8—22no—
34C3×C4×Q8—22no—
35C3×#117—22no—
36C3×#117—22no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4C2×C411yescharacteristic
42C2×C3×C4⋊C4C2×C411yescharacteristic
43C2×C3×C4⋊C4D411yescharacteristic
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4C2×C411yescharacteristic
47C2×C3×C22⋊C4D411yescharacteristic
48C2×C3×C42—21no—
49C2×C3×C42—21no—
50C2×C3×C42D411yescharacteristic
51C2×C3×C42C2311yescharacteristic
52C2×#208—22no—
53C3×#95—21no—
54C3×#95—21no—
55C3×#95—21no—
56C3×#95—21no—
57C3×#95C2×C411yescharacteristic
58C3×#95C2×C411yescharacteristic
59#830—24no—
60#830—22no—
61C4⋊(C3⋊C8)—22no—
62C4⋊(C3⋊C8)—22no—
63C4⋊(C3⋊C8)D414yes—
64#803—22no—
65C4×C3⋊C8—22no—
66C4×C3⋊C8—22no—
67C2×C4×Q8C2×D311yescharacteristic
68C2×#117C2×D311yescharacteristic
69#433—64no—
70#431—32no—
71#423—32no—
72#407—61no—
73C4×M16—61no—
74#400Dic311yescharacteristic
75#399Dic311yescharacteristic
76#398—21no—
77#396—21no—
78#395Dic311yescharacteristic
79#393Dic311yescharacteristic
80C4×C4⋊C4—21no—
81C4×C4⋊C4C2×D311yescharacteristic
82C4×C22⋊C4—21no—
83Q8⋊C8—64no—
84#324—61no—
85#323—61no—
86C24×C3C22⋊C411yescharacteristic
87C2×C3×Q8—22no—
88C2×C3×Q8—42no—
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4—22no—
94C22×C3×C4—22no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4C22⋊C411yescharacteristic
98C22×C3×C4C22⋊C411yescharacteristic
99C22×C3×C4C22⋊C411yescharacteristic
100C22×C3×C4C22×C411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C3×C4⋊C4—22no—
104C3×C4⋊C4—22no—
105C3×C4⋊C4—22no—
106C3×C4⋊C4—42no—
107C3×C4⋊C4—42no—
108C3×C4⋊C4—42no—
109C3×C4⋊C4—42no—
110C3×C22⋊C4—22no—
111C3×C22⋊C4—22no—
112C3×C22⋊C4—42no—
113C3×C22⋊C4—42no—
114C3×C42—21no—
115C3×C42—21no—
116C3×C42—41no—
117C3×C42—41no—
118C3×C42—42no—
119C3×C42C2×D412yes—
120C3×C42#4112yes—
121#208—42no—
122#208—42no—
123C2×C3⋊C8—24no—
124C2×C3⋊C8—42no—
125C22×Q8C2×Dic311yescharacteristic
126C23×C4#6711yescharacteristic
127C2×M16—62no—
128C4×Q8—22no—
129C4×Q8—22no—
130#117—22no—
131#117—22no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4C2×Dic311yescharacteristic
137C2×C4⋊C4C2×Dic311yescharacteristic
138C2×C4⋊C4#6711yescharacteristic
139C2×C22⋊C4—21no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4C2×Dic311yescharacteristic
142C2×C22⋊C4#6711yescharacteristic
143C2×C42—21no—
144C2×C42—21no—
145C2×C42#6711yescharacteristic
146C2×C42C22×D311yescharacteristic
147C4⋊C8—34no—
148C4⋊C8—62no—
149C4⋊C8—62no—
150C22⋊C8—64no—
151C22⋊C8—62no—
152#97—62no—
153C4×C8—62no—
154C4×C8—62no—
155#95—21no—
156#95—21no—
157#95—21no—
158#95—21no—
159#95C2×Dic311yescharacteristic
160#95C2×Dic311yescharacteristic
161C23×C3—21no—
162C23×C3—21no—
163C23×C3—21no—
164C23×C3—41no—
165C23×C3—41no—
166C23×C3C2×C22⋊C411yescharacteristic
167C3×Q8—24no—
168C3×Q8—42no—
169C2×C3×C4—22no—
170C2×C3×C4—22no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
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—42no—
183C2×C3×C4—42no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4C4×D412yes—
193C2×C3×C4#12011yescharacteristic
194C2×C3×C4#12112yes—
195C2×C3×C4#12311yescharacteristic
196C2×C3×C4#13512yes—
197C2×C3×C4#13712yes—
198C3⋊C8—44no—
199C3⋊C8—82no—
200C24#21711yescharacteristic
201C2×Q8—22no—
202C2×Q8—42no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4#21711yescharacteristic
212C22×C4#21711yescharacteristic
213C22×C4#21711yescharacteristic
214C22×C4C22×Dic311yescharacteristic
215C22×C4C2×#6711yescharacteristic
216C22×C4C2×#6711yescharacteristic
217M16—122no—
218M16—122no—
219C2×C8—64no—
220C2×C8—122no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C22⋊C4—22no—
229C22⋊C4—22no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C42—21no—
233C42—21no—
234C42—41no—
235C42—41no—
236C42—42no—
237C42D3×D412yes—
238C42#23712yes—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3—42no—
242C22×C3—41no—
243C22×C3—41no—
244C22×C3—41no—
245C22×C3—81no—
246C22×C3#38611yescharacteristic
247C22×C3#41611yescharacteristic
248C22×C3#41911yescharacteristic
249C3×C4—22no—
250C3×C4—22no—
251C3×C4—42no—
252C3×C4—41no—
253C3×C4—41no—
254C3×C4—41no—
255C3×C4—42no—
256C3×C4—81no—
257C3×C4—81no—
258C3×C4#45212yes—
259C3×C4#48512yes—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—41no—
264C23—41no—
265C23C2×#21711yescharacteristic
266Q8—24no—
267Q8—42no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4D4×Dic312yes—
292C2×C4#93511yescharacteristic
293C2×C4#93711yescharacteristic
294C2×C4#93912yes—
295C2×C4#94212yes—
296C2×C4#95012yes—
297C8—124no—
298C8—242no—
299C2×C3—21no—
300C2×C3—21no—
301C2×C3—81no—
302C2×C3#187311yescharacteristic
303C2×C3#188411yescharacteristic
304C2×C3#189111yescharacteristic
305C22—21no—
306C22—21no—
307C22—42no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—81no—
312C22#522011yescharacteristic
313C22#522511yescharacteristic
314C22#523311yescharacteristic
315C4—22no—
316C4—22no—
317C4—42no—
318C4—41no—
319C4—42no—
320C4—41no—
321C4—41no—
322C4—81no—
323C4—81no—
324C4#504512yes—
325C4#504712yes—
326C3#903411yescharacteristic, complemented, Sylow
327C2—21no—
328C2—21no—
329C2—81no—
330C2#7169311yescharacteristic
331C2#7171411yescharacteristic
332C2#7181711yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2396C2
socleC22×C3#386
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2396D3
hypocenterC3#9034
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2396

Complements

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

NH ≅ G/Ntype
3. #71622C2semidirect
5. #67620C2semidirect
326. C3#9034semidirect

Maximal subgroups

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

subgroupindex
C3×#23962
#822092
#716222
#715712
#676202
#90343