Finite Group Databasebeta

← #10705559

Subgroups — #10705559

Order 768.

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

normal subgroups only

This group has 440 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#82346C211yesmaximal, characteristic
2C3×C4×#100C211yesmaximal, characteristic
3#71600C211yesmaximal, characteristic
4#67496C212yesmaximal, complemented
5#67495C212yesmaximal, complemented
6#11735—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#926C411yescharacteristic
9C2×C3×#100—21no—
10C3×C4×M16—21no—
11#5222—21no—
12C4×#217—21no—
13C4×#217C411yescharacteristic
14C2×#830—21no—
15#4986C2211yescharacteristic
16#4981C2211yescharacteristic
17#4953C411yescharacteristic
18#4952—21no—
19#4949C411yescharacteristic
20C4×Dic3⋊C4—21no—
21C3×#343C2212yes—
22#4528C2212yes—
23#4525—21no—
24#2395—31no—
25C4×#100D311yescharacteristic
26#1459—31no—
27#1353—32no—
28#1346—32no—
29C22×C3×D4D411yescharacteristic
30C23×C3×C4C2×C411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C2×C3×M16—22no—
33C3×C4×D4—41no—
34C3×C4×D4—41no—
35C2×C3×C4⋊C4C2311yescharacteristic
36C2×C3×C22⋊C4C2×C411yescharacteristic
37C2×C3×C22⋊C4D411yescharacteristic
38C2×C3×C42D411yescharacteristic
39C2×#217—21no—
40C2×#217—21no—
41C2×#217—21no—
42C2×#217—21no—
43#926—41no—
44#926—41no—
45C2×C4⋊Dic3C2×C411yescharacteristic
46C2×C4⋊Dic3C2×C411yescharacteristic
47C2×Dic3⋊C4—21no—
48C2×Dic3⋊C4—21no—
49C2×C4×Dic3—21no—
50C2×C4×Dic3—21no—
51C22×C3⋊C8—41no—
52C3×#100—24no—
53C3×#97—41no—
54C3×C4×C8—41no—
55C3⋊#95—21no—
56C3⋊#95—21no—
57C3⋊#95—21no—
58C3⋊#95—21no—
59C3⋊#95C2×C411yescharacteristic
60C3⋊#95C2×C411yescharacteristic
61#830—22no—
62#830—41no—
63C2×C4×D4C2×D311yescharacteristic
64C2×#117—31no—
65C2×#100—21no—
66C2×C22⋊C8—61no—
67C4×M16—21no—
68#400—31no—
69#399—31no—
70#398—31no—
71#397—31no—
72#397—61no—
73#394—61no—
74C4×C4⋊C4—61no—
75C4×C22⋊C4—31no—
76C4×C22⋊C4—61no—
77#343C2×D312yes—
78#324—32no—
79#323—61no—
80C24×C3C22⋊C411yescharacteristic
81C24×C3C22⋊C411yescharacteristic
82C2×C3×D4—22no—
83C2×C3×D4—22no—
84C2×C3×D4—41no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4—42no—
90C22×C3×C4—42no—
91C22×C3×C4C22⋊C411yescharacteristic
92C22×C3×C4C22⋊C411yescharacteristic
93C22×C3×C4C22×C411yescharacteristic
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×Dic3—21no—
97C22×Dic3—21no—
98C22×Dic3—21no—
99C22×Dic3—21no—
100C3×M16—24no—
101C3×M16—42no—
102C2×C3×C8—42no—
103C3×C4⋊C4—41no—
104C3×C22⋊C4—41no—
105C3×C22⋊C4—41no—
106C3×C42—21no—
107C3×C42—21no—
108#217—42no—
109#217—42no—
110#217—42no—
111#217—42no—
112C4⋊Dic3—41no—
113C4⋊Dic3—41no—
114Dic3⋊C4—42no—
115Dic3⋊C4—42no—
116C4×Dic3—42no—
117C4×Dic3—41no—
118C4×Dic3—41no—
119C2×C3⋊C8—42no—
120C2×C3⋊C8—82no—
121C22×D4#6711yescharacteristic
122C23×C4C4×D311yescharacteristic
123C23×C4D1211yescharacteristic
124C2×M16—22no—
125C22×C8—121no—
126C4×D4—41no—
127C4×D4—41no—
128#117—121no—
129#117—121no—
130C2×C4⋊C4—31no—
131C2×C4⋊C4—31no—
132C2×C4⋊C4—61no—
133C2×C4⋊C4—61no—
134C2×C4⋊C4C22×D311yescharacteristic
135C2×C22⋊C4—61no—
136C2×C22⋊C4—61no—
137C2×C22⋊C4—61no—
138C2×C22⋊C4—61no—
139C2×C22⋊C4C4×D311yescharacteristic
140C2×C22⋊C4D1211yescharacteristic
141C2×C42—61no—
142C2×C42—61no—
143C2×C42#6711yescharacteristic
144#100—24no—
145C22⋊C8—62no—
146C22⋊C8—121no—
147#97—41no—
148C4×C8—41no—
149#95—31no—
150#95—31no—
151#95—61no—
152#95—61no—
153#95—61no—
154#95—61no—
155C23×C3—21no—
156C23×C3—21no—
157C23×C3—21no—
158C23×C3—21no—
159C23×C3—21no—
160C23×C3—41no—
161C23×C3—41no—
162C23×C3—41no—
163C23×C3—41no—
164C23×C3#10412yes—
165C23×C3C2×C22⋊C411yescharacteristic
166C3×D4—44no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—22no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—42no—
174C2×C3×C4—42no—
175C2×C3×C4—42no—
176C2×C3×C4—41no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—81no—
180C2×C3×C4#9911yescharacteristic
181C2×C3×C4#9911yescharacteristic
182C2×C3×C4#13512yes—
183C2×Dic3—22no—
184C2×Dic3—41no—
185C2×Dic3—41no—
186C2×Dic3—41no—
187C2×Dic3—41no—
188C2×Dic3—41no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C2×Dic3—41no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C3×C8—44no—
196C3⋊C8—82no—
197C24(C2×D3)⋊C411yescharacteristic
198C24(C2×D3)⋊C411yescharacteristic
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—41no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—61no—
209C22×C4—61no—
210C22×C4—61no—
211C22×C4—61no—
212C22×C4(C2×D3)⋊C411yescharacteristic
213C22×C4(C2×D3)⋊C411yescharacteristic
214C22×C4C2×C4×D311yescharacteristic
215C22×C4C2×D1211yescharacteristic
216C22×C4C2×#6711yescharacteristic
217M16—24no—
218M16—42no—
219C2×C8—42no—
220C2×C8—122no—
221C2×C8—242no—
222C4⋊C4—41no—
223C4⋊C4—121no—
224C4⋊C4—121no—
225C4⋊C4—122no—
226C4⋊C4—122no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—122no—
230C22⋊C4—122no—
231C22⋊C4—122no—
232C22⋊C4—122no—
233C42—21no—
234C42—21no—
235C42—122no—
236C42—121no—
237C42—121no—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3—22no—
241C22×C3—42no—
242C22×C3—41no—
243C22×C3—41no—
244C22×C3—41no—
245C22×C3—41no—
246C22×C3—41no—
247C22×C3—81no—
248C22×C3—81no—
249C22×C3—81no—
250C22×C3C2×#9911yescharacteristic
251C22×C3#41611yescharacteristic
252C22×C3C2×#10411yescharacteristic
253C3×C4—21no—
254C3×C4—22no—
255C3×C4—41no—
256C3×C4—41no—
257C3×C4—41no—
258C3×C4#35212yes—
259Dic3—42no—
260Dic3—81no—
261Dic3—81no—
262Dic3—81no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23#80512yes—
273C23C2×(C2×D3)⋊C411yescharacteristic
274D4—44no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—62no—
288C2×C4—81no—
289C2×C4—121no—
290C2×C4—121no—
291C2×C4—121no—
292C2×C4—121no—
293C2×C4—121no—
294C2×C4—121no—
295C2×C4—121no—
296C2×C4—121no—
297C2×C4—121no—
298C2×C4—121no—
299C2×C4—121no—
300C2×C4#80611yescharacteristic
301C2×C4#80611yescharacteristic
302C2×C4#95012yes—
303C8—44no—
304C8—242no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—42no—
308C2×C3—81no—
309C2×C3#150111yescharacteristic
310C2×C3C2×#35211yescharacteristic
311C2×C3#213211yescharacteristic
312C22—21no—
313C22—21no—
314C22—22no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22C2×#80511yescharacteristic
325C22C2×#80611yescharacteristic
326C22#500211yescharacteristic
327C4—22no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—122no—
333C4—241no—
334C4—241no—
335C4—241no—
336C4#447612yes—
337C3#1173511yescharacteristic, complemented, Sylow
338C2—21no—
339C2—21no—
340C2—42no—
341C2—81no—
342C2#6944311yescharacteristic
343C2C2×#447611yescharacteristic
344C2#7014811yescharacteristic
subgroupG/H
centerC2×C4#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×C4×#100C2
socleC22×C3C2×#99
prime radicalC22×C3×D4D4
cosocleC2×C3×C4⋊C4C23
hypercenterC4×#100D3
hypocenterC3#11735
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
4. #67496C2semidirect
5. #67495C2semidirect
337. C3#11735semidirect

Maximal subgroups

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

subgroupindex
#823462
C3×C4×#1002
#716002
#674962
#674952
#117353