Finite Group Databasebeta

← #10691261

Subgroups — #10691261

Order 768.

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

normal subgroups only

This group has 690 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×#2617C211yesmaximal, characteristic
2#82093C211yesmaximal, characteristic, complemented
3#71675C212yesmaximal, complemented
4C2×#4525C211yesmaximal, characteristic, complemented
5#67625C212yesmaximal, complemented
6#9205—31nomaximal, Sylow
7C2×C3×#121C2211yescharacteristic
8C2×C3×#120—21no—
9C22×C3×C22⋊C4C2211yescharacteristic
10C3×#397C412yescomplemented
11C3×#394—22no—
12C3×#392C412yescomplemented
13C3×#386—21no—
14C3×#380C2211yescharacteristic
15#5247—21no—
16#5212C2212yes—
17C4×#217—21no—
18C2×#830C2212yes—
19#4951—24no—
20#4537—24no—
21#4525—22no—
22#2617D311yescharacteristic
23#2312—31no—
24#1519—32no—
25C2×#323—31no—
26#1289—32no—
27C25×C3D411yescharacteristic
28C22×C3×D4—21no—
29C22×C3×D4—22no—
30C22×C3×D4C2×C411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C23×C3×C4C2311yescharacteristic
33C3×#121—22no—
34C3×#121—22no—
35C3×#120—44no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4C2×C411yescharacteristic
38C2×C3×C22⋊C4—22no—
39C2×C3×C22⋊C4—22no—
40C2×C3×C22⋊C4—22no—
41C2×C3×C22⋊C4—21no—
42C2×C3×C22⋊C4—21no—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4C2×C412yes—
45C2×C3×C22⋊C4D412yes—
46C2×#217—22no—
47C2×#217—22no—
48C2×#217—22no—
49C2×C4⋊Dic3—22no—
50C2×C4×Dic3—22no—
51C22×C3⋊C8—22no—
52C3×#95—22no—
53C3×#95C2×C412yes—
54C3⋊#95—22no—
55C3⋊#95D414yescomplemented
56#830—22no—
57#830—22no—
58C2×#121C2×D311yescharacteristic
59C2×#120—21no—
60C22×C22⋊C4C2×D311yescharacteristic
61C2×C22⋊C8—32no—
62#397Dic312yescomplemented
63#394—22no—
64#392Dic312yescomplemented
65#388—64no—
66#386—21no—
67#380—32no—
68#380C2×D311yescharacteristic
69#379—61no—
70C4×C22⋊C4—61no—
71#327—64no—
72#323—62no—
73C24×C3—21no—
74C24×C3—21no—
75C24×C3—22no—
76C24×C3—22no—
77C24×C3—21no—
78C24×C3—81no—
79C24×C3—81no—
80C24×C3C22⋊C412yes—
81C24×C3C2×D411yescharacteristic
82C2×C3×D4—22no—
83C2×C3×D4—44no—
84C2×C3×D4—41no—
85C2×C3×D4—41no—
86C2×C3×D4—42no—
87C2×C3×D4—42no—
88C2×C3×D4—82no—
89C2×C3×D4—82no—
90C2×C3×D4—81no—
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4—22no—
95C22×C3×C4—81no—
96C22×C3×C4—81no—
97C22×C3×C4C22⋊C412yes—
98C22×C3×C4C22×C411yescharacteristic
99C22×C3×C4C2×D412yes—
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4#4112yes—
102C22×Dic3—24no—
103C3×C4⋊C4—22no—
104C3×C4⋊C4—81no—
105C3×C22⋊C4—24no—
106C3×C22⋊C4—24no—
107C3×C22⋊C4—44no—
108C3×C22⋊C4—42no—
109C3×C22⋊C4—42no—
110C3×C22⋊C4—82no—
111#217—42no—
112#217—42no—
113#217—82no—
114#217—82no—
115C4⋊Dic3—82no—
116C4×Dic3—82no—
117C2×C3⋊C8—42no—
118C2×C3⋊C8—82no—
119C25#6711yescharacteristic
120C22×D4—21no—
121C22×D4—22no—
122C22×D4C2×Dic311yescharacteristic
123C23×C4#6711yescharacteristic
124C23×C4C22×D311yescharacteristic
125C22×C8—62no—
126#121—22no—
127#121—22no—
128#120—44no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—62no—
131C2×C4⋊C4C2×Dic311yescharacteristic
132C2×C22⋊C4—22no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—62no—
139C2×C22⋊C4—62no—
140C2×C22⋊C4—62no—
141C2×C22⋊C4C2×Dic312yes—
142C2×C22⋊C4#6712yes—
143C2×C42—62no—
144C22⋊C8—62no—
145C22⋊C8—62no—
146#95—22no—
147#95—34no—
148#95—62no—
149#95C2×Dic312yes—
150C23×C3—22no—
151C23×C3—22no—
152C23×C3—22no—
153C23×C3—41no—
154C23×C3—41no—
155C23×C3—42no—
156C23×C3—42no—
157C23×C3—42no—
158C23×C3—42no—
159C23×C3—42no—
160C23×C3—42no—
161C23×C3—42no—
162C23×C3—42no—
163C23×C3—41no—
164C23×C3—41no—
165C23×C3—41no—
166C23×C3—81no—
167C23×C3—81no—
168C23×C3—84no—
169C23×C3—84no—
170C23×C3C2×C22⋊C411yescharacteristic
171C23×C3#12011yescharacteristic
172C23×C3#12311yescharacteristic
173C23×C3#13512yes—
174C23×C3AGL(1,ℤ/8)12yes—
175C3×D4—82no—
176C3×D4—88no—
177C3×D4—82no—
178C3×D4—82no—
179C2×C3×C4—22no—
180C2×C3×C4—42no—
181C2×C3×C4—42no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—42no—
185C2×C3×C4—42no—
186C2×C3×C4—42no—
187C2×C3×C4—42no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—42no—
191C2×C3×C4—41no—
192C2×C3×C4—81no—
193C2×C3×C4C4×D412yes—
194C2×C3×C4#12112yes—
195C2×Dic3—44no—
196C2×Dic3—44no—
197C2×Dic3—44no—
198C3⋊C8—82no—
199C24—21no—
200C24—22no—
201C24—21no—
202C24—22no—
203C24—21no—
204C24—81no—
205C24—81no—
206C24#21712yes—
207C24C2×#6711yescharacteristic
208C2×D4—22no—
209C2×D4—44no—
210C2×D4—42no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—42no—
214C2×D4—82no—
215C2×D4—82no—
216C2×D4—81no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—64no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4#21712yes—
225C22×C4D3×D412yes—
226C22×C4#23712yes—
227C22×C4C22×Dic311yescharacteristic
228C22×C4C2×#6711yescharacteristic
229C2×C8—122no—
230C2×C8—242no—
231C4⋊C4—22no—
232C4⋊C4—81no—
233C4⋊C4—242no—
234C22⋊C4—24no—
235C22⋊C4—24no—
236C22⋊C4—44no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—82no—
240C22⋊C4—122no—
241C22⋊C4—122no—
242C22⋊C4—242no—
243C22⋊C4—242no—
244C42—242no—
245C22×C3—21no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—21no—
251C22×C3—42no—
252C22×C3—42no—
253C22×C3—44no—
254C22×C3—42no—
255C22×C3—42no—
256C22×C3—42no—
257C22×C3—82no—
258C22×C3—84no—
259C22×C3—84no—
260C22×C3—82no—
261C22×C3—82no—
262C22×C3—81no—
263C22×C3—81no—
264C22×C3—81no—
265C22×C3#38611yescharacteristic
266C22×C3#41611yescharacteristic
267C22×C3#41811yescharacteristic
268C22×C3#44912yes—
269C22×C3#48212yes—
270C3×C4—22no—
271C3×C4—22no—
272C3×C4—42no—
273C3×C4—82no—
274C3×C4—81no—
275Dic3—84no—
276C23—22no—
277C23—22no—
278C23—22no—
279C23—41no—
280C23—41no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23—42no—
285C23—42no—
286C23—42no—
287C23—42no—
288C23—42no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—81no—
293C23—81no—
294C23—84no—
295C23—84no—
296C23#93212yes—
297C23#93511yescharacteristic
298C23#93711yescharacteristic
299C23#95012yes—
300C23C2×#21711yescharacteristic
301D4—82no—
302D4—88no—
303D4—82no—
304D4—82no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—81no—
319C2×C4—124no—
320C2×C4—124no—
321C2×C4—124no—
322C2×C4D4×Dic312yes—
323C2×C4#93912yes—
324C8—242no—
325C2×C3—22no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—22no—
329C2×C3—82no—
330C2×C3—161no—
331C2×C3#187511yescharacteristic
332C2×C3#191111yescharacteristic
333C2×C3#191311yescharacteristic
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—42no—
341C22—42no—
342C22—44no—
343C22—42no—
344C22—42no—
345C22—42no—
346C22—82no—
347C22—84no—
348C22—84no—
349C22—82no—
350C22—82no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22#503512yes—
355C22#503712yes—
356C22#521511yescharacteristic
357C22#522011yescharacteristic
358C22#523311yescharacteristic
359C4—22no—
360C4—22no—
361C4—42no—
362C4—82no—
363C4—81no—
364C4—244no—
365C3#920511yescharacteristic, complemented, Sylow
366C2—21no—
367C2—22no—
368C2—21no—
369C2—22no—
370C2—82no—
371C2—161no—
372C2#7167811yescharacteristic
373C2#7184811yescharacteristic
374C2#7185311yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#2617C2
socleC22×C3#386
cosocleC23×C3×C4C23
hypercenter#2617D3
hypocenterC3#9205
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2617

Complements

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

NH ≅ G/Ntype
2. #82093C2semidirect
3. #71675C2semidirect
4. C2×#4525C2semidirect
5. #67625C2semidirect
10. C3×#397C4semidirect
12. C3×#392C4semidirect
55. C3⋊#95D4semidirect
62. #397Dic3semidirect
64. #392Dic3semidirect
365. C3#9205semidirect

Maximal subgroups

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

subgroupindex
C3×#26172
#820932
#716752
C2×#45252
#676252
#92053