Finite Group Databasebeta

← #10692857

Subgroups — #10692857

Order 768.

490 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 107 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×#2566C211yesmaximal, characteristic
2#82302C211yesmaximal, characteristic
3#71736C212yesmaximal, complemented
4#71603C211yesmaximal, characteristic
5#67621C212yesmaximal, complemented
6#9449—31nomaximal, Sylow
7C2×C3×#125C2211yescharacteristic
8C2×C3×C4×D4C2211yescharacteristic
9C3×#396—21no—
10C3×#396—21no—
11C3×#390C411yescharacteristic
12C3×#390C411yescharacteristic
13C3×#385—21no—
14C3×#385—21no—
15C3×#384C411yescharacteristic
16C3×#382C411yescharacteristic
17C3×C4×C4⋊C4C2211yescharacteristic
18#5207—21no—
19C2×#830—21no—
20#5011—24no—
21#4926C2212yes—
22#4926C2212yes—
23#4543—24no—
24#4525—21no—
25#4525—21no—
26#2973—31no—
27#2566D311yescharacteristic
28#1534—32no—
29#1466—31no—
30#1285—32no—
31C22×C3×D4C2×C411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C23×C3×C4D411yescharacteristic
34C3×#125—22no—
35C3×#125—22no—
36C3×C4×D4—22no—
37C3×C4×D4—22no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4C2×C411yescharacteristic
44C2×C3×C4⋊C4C2×C411yescharacteristic
45C2×C3×C4⋊C4C2×C411yescharacteristic
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4D411yescharacteristic
51C2×C3×C22⋊C4D411yescharacteristic
52C2×C3×C42C2×C411yescharacteristic
53C2×C3×C42C2×C411yescharacteristic
54C2×C3×C42C2311yescharacteristic
55C2×#208—22no—
56C22×C3⋊C8—22no—
57C3×#95—21no—
58C3×#95—21no—
59C3×#95—21no—
60#830—22no—
61#830—22no—
62#830—22no—
63#830—22no—
64C4⋊(C3⋊C8)—22no—
65C4⋊(C3⋊C8)—22no—
66#803—22no—
67#803D414yes—
68C2×#125C2×D311yescharacteristic
69C2×C4×D4C2×D311yescharacteristic
70#435—64no—
71#432—32no—
72#432—32no—
73#407—61no—
74C2×C22⋊C8—61no—
75#396—21no—
76#396—21no—
77#390Dic311yescharacteristic
78#390Dic311yescharacteristic
79#385—21no—
80#385—21no—
81#384Dic311yescharacteristic
82#382Dic311yescharacteristic
83C4×C4⋊C4C2×D311yescharacteristic
84#330—64no—
85#323—61no—
86#323—61no—
87C24×C3C22⋊C411yescharacteristic
88C24×C3C22⋊C411yescharacteristic
89C2×C3×D4—22no—
90C2×C3×D4—22no—
91C2×C3×D4—41no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4—21no—
100C22×C3×C4—81no—
101C22×C3×C4—81no—
102C22×C3×C4C22⋊C411yescharacteristic
103C22×C3×C4C22⋊C411yescharacteristic
104C22×C3×C4C22×C411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C3×C4⋊C4—22no—
108C3×C4⋊C4—22no—
109C3×C4⋊C4—22no—
110C3×C4⋊C4—42no—
111C3×C4⋊C4—41no—
112C3×C4⋊C4—42no—
113C3×C4⋊C4—41no—
114C3×C4⋊C4—81no—
115C3×C4⋊C4—81no—
116C3×C22⋊C4—22no—
117C3×C22⋊C4—22no—
118C3×C22⋊C4—42no—
119C3×C22⋊C4—42no—
120C3×C22⋊C4—81no—
121C3×C22⋊C4—81no—
122C3×C42—41no—
123C3×C42—41no—
124C3×C42C2×D412yes—
125C3×C42#4112yes—
126#208—82no—
127C2×C3⋊C8—24no—
128C2×C3⋊C8—42no—
129C2×C3⋊C8—82no—
130C22×D4C2×Dic311yescharacteristic
131C23×C4#6711yescharacteristic
132C23×C4#6711yescharacteristic
133C2×M16—62no—
134C22×C8—62no—
135#125—22no—
136#125—22no—
137C4×D4—22no—
138C4×D4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4C2×Dic311yescharacteristic
145C2×C4⋊C4C2×Dic311yescharacteristic
146C2×C4⋊C4C2×Dic311yescharacteristic
147C2×C22⋊C4—21no—
148C2×C22⋊C4—21no—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4#6711yescharacteristic
152C2×C22⋊C4#6711yescharacteristic
153C2×C42C2×Dic311yescharacteristic
154C2×C42C2×Dic311yescharacteristic
155C2×C42C22×D311yescharacteristic
156C4⋊C8—62no—
157C4⋊C8—62no—
158C22⋊C8—62no—
159C22⋊C8—62no—
160C22⋊C8—62no—
161C22⋊C8—62no—
162#97—34no—
163#97—62no—
164#95—21no—
165#95—21no—
166#95—21no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3—21no—
172C23×C3—41no—
173C23×C3—41no—
174C23×C3—41no—
175C23×C3—41no—
176C23×C3#10112yes—
177C23×C3C2×C22⋊C411yescharacteristic
178C3×D4—44no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—22no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—42no—
187C2×C3×C4—42no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—81no—
202C2×C3×C4—81no—
203C2×C3×C4C4×D412yes—
204C2×C3×C4#12011yescharacteristic
205C2×C3×C4#12112yes—
206C2×C3×C4#12311yescharacteristic
207C3⋊C8—44no—
208C3⋊C8—82no—
209C24#21711yescharacteristic
210C24#21711yescharacteristic
211C2×D4—22no—
212C2×D4—22no—
213C2×D4—41no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4#21711yescharacteristic
225C22×C4#21711yescharacteristic
226C22×C4C22×Dic311yescharacteristic
227C22×C4C2×#6711yescharacteristic
228C22×C4C2×#6711yescharacteristic
229M16—242no—
230C2×C8—64no—
231C2×C8—122no—
232C2×C8—242no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—42no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—42no—
240C4⋊C4—81no—
241C4⋊C4—81no—
242C22⋊C4—22no—
243C22⋊C4—22no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—81no—
247C22⋊C4—81no—
248C42—41no—
249C42—41no—
250C42D3×D412yes—
251C42#23712yes—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—22no—
255C22×C3—42no—
256C22×C3—42no—
257C22×C3—41no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—81no—
263C22×C3—81no—
264C22×C3#38611yescharacteristic
265C22×C3C2×#10111yescharacteristic
266C22×C3#41311yescharacteristic
267C3×C4—22no—
268C3×C4—22no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—42no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4—81no—
275C3×C4—81no—
276C3×C4—81no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23#83612yes—
287C23C2×#21711yescharacteristic
288D4—44no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4D4×Dic312yes—
314C2×C4#93511yescharacteristic
315C2×C4#93711yescharacteristic
316C2×C4#93912yes—
317C8—124no—
318C8—242no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—42no—
322C2×C3—81no—
323C2×C3#188411yescharacteristic
324C2×C3#190111yescharacteristic
325C2×C3#191111yescharacteristic
326C22—21no—
327C22—21no—
328C22—22no—
329C22—42no—
330C22—42no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—81no—
337C22—81no—
338C22#522011yescharacteristic
339C22C2×#83611yescharacteristic
340C22#523711yescharacteristic
341C4—22no—
342C4—22no—
343C4—21no—
344C4—21no—
345C4—42no—
346C4—41no—
347C4—41no—
348C4—81no—
349C4—81no—
350C4—81no—
351C3#944911yescharacteristic, complemented, Sylow
352C2—21no—
353C2—21no—
354C2—42no—
355C2—81no—
356C2#7169811yescharacteristic
357C2#7181711yescharacteristic
358C2#7185311yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2566C2
socleC22×C3#386
prime radicalC22×C3×D4C2×C4
cosocleC2×C3×C42C23
hypercenter#2566D3
hypocenterC3#9449
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2566

Complements

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

NH ≅ G/Ntype
3. #71736C2semidirect
5. #67621C2semidirect
351. C3#9449semidirect

Maximal subgroups

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

subgroupindex
C3×#25662
#823022
#717362
#716032
#676212
#94493