Finite Group Databasebeta

← #10691256

Subgroups — #10691256

Order 768.

622 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 135 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×#2618C211yesmaximal, characteristic
2#82156C211yesmaximal, characteristic, complemented
3#71675C212yesmaximal
4C2×#4528C211yesmaximal, characteristic, complemented
5#67625C212yesmaximal, complemented
6#9204—31nomaximal, Sylow
7C2×C3×#123—21no—
8C2×C3×#121C2211yescharacteristic
9C22×C3×C4⋊C4C2211yescharacteristic
10C2×#926—21no—
11C3×#398C412yescomplemented
12C3×#396—22no—
13C3×#394C412yescomplemented
14C3×#386—21no—
15C3×#380C2211yescharacteristic
16#5212—21no—
17#5212C2212yes—
18C2×#830C2212yes—
19#4984—24no—
20#4537—24no—
21#4528—22no—
22#2618D311yescharacteristic
23#2316—31no—
24#1519—32no—
25C2×#324—31no—
26#1289—32no—
27C22×C3×D4—21no—
28C22×C3×D4C2×C411yescharacteristic
29C23×C3×C4D411yescharacteristic
30C23×C3×C4D411yescharacteristic
31C23×C3×C4C2311yescharacteristic
32C3×#123—42no—
33C3×#123—42no—
34C3×#121—22no—
35C3×#121—22no—
36C2×C3×C4⋊C4—22no—
37C2×C3×C4⋊C4—22no—
38C2×C3×C4⋊C4—22no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4C2×C411yescharacteristic
41C2×C3×C4⋊C4C2×C412yes—
42C2×C3×C4⋊C4D412yes—
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—22no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4—22no—
47C2×#217—22no—
48#926—42no—
49#926—42no—
50C2×C4⋊Dic3—22no—
51C2×C4⋊Dic3—22no—
52C2×C4⋊Dic3—22no—
53C2×C4×Dic3—22no—
54C22×C3⋊C8—22no—
55C3×#95—22no—
56C3×#95C2×C412yes—
57C3⋊#95—22no—
58C3⋊#95D414yes—
59#830—22no—
60#830—22no—
61C2×#123—21no—
62C2×#121C2×D311yescharacteristic
63C2×#117—61no—
64C22×C4⋊C4C2×D311yescharacteristic
65C2×C22⋊C8—32no—
66#398Dic312yescomplemented
67#396—22no—
68#394Dic312yescomplemented
69#389—64no—
70#386—21no—
71#380—32no—
72#380—61no—
73#380C2×D311yescharacteristic
74#327—64no—
75#324—62no—
76C24×C3—21no—
77C24×C3C2×D411yescharacteristic
78C2×C3×D4—22no—
79C2×C3×D4—41no—
80C2×C3×D4—41no—
81C2×C3×D4—41no—
82C2×C3×D4—41no—
83C2×C3×D4—41no—
84C2×C3×D4—41no—
85C2×C3×D4—81no—
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4—22no—
94C22×C3×C4—81no—
95C22×C3×C4—81no—
96C22×C3×C4C22⋊C412yes—
97C22×C3×C4C22⋊C412yes—
98C22×C3×C4D812yes—
99C22×C3×C4SD1612yes—
100C22×C3×C4C22×C411yescharacteristic
101C22×C3×C4C2×D412yes—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4#4112yes—
104C22×Dic3—24no—
105C3×C4⋊C4—24no—
106C3×C4⋊C4—24no—
107C3×C4⋊C4—22no—
108C3×C4⋊C4—44no—
109C3×C4⋊C4—81no—
110C3×C22⋊C4—41no—
111C3×C22⋊C4—41no—
112C3×C22⋊C4—44no—
113C3×C22⋊C4—42no—
114C3×C22⋊C4—82no—
115#217—42no—
116#217—42no—
117C4⋊Dic3—44no—
118C4⋊Dic3—82no—
119C4⋊Dic3—82no—
120C4×Dic3—44no—
121C2×C3⋊C8—42no—
122C2×C3⋊C8—82no—
123C22×D4—21no—
124C22×D4C2×Dic311yescharacteristic
125C23×C4#6711yescharacteristic
126C23×C4#6711yescharacteristic
127C23×C4C22×D311yescharacteristic
128C22×C8—62no—
129#123—42no—
130#123—42no—
131#121—22no—
132#121—22no—
133#117—122no—
134#117—122no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—62no—
140C2×C4⋊C4—62no—
141C2×C4⋊C4—62no—
142C2×C4⋊C4C2×Dic311yescharacteristic
143C2×C4⋊C4C2×Dic312yes—
144C2×C4⋊C4#6712yes—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—62no—
150C2×C42—62no—
151C22⋊C8—62no—
152C22⋊C8—62no—
153#95—22no—
154#95—34no—
155#95—62no—
156#95C2×Dic312yes—
157C23×C3—41no—
158C23×C3—41no—
159C23×C3—42no—
160C23×C3—42no—
161C23×C3—41no—
162C23×C3—41no—
163C23×C3—41no—
164C23×C3C2×C22⋊C411yescharacteristic
165C23×C3#12011yescharacteristic
166C23×C3#12311yescharacteristic
167C23×C3C2×D811yescharacteristic
168C23×C3C2×SD1611yescharacteristic
169C23×C3AGL(1,ℤ/8)11yescharacteristic
170C23×C3#13711yescharacteristic
171C3×D4—82no—
172C3×D4—81no—
173C3×D4—81no—
174C3×D4—81no—
175C3×D4—81no—
176C2×C3×C4—22no—
177C2×C3×C4—22no—
178C2×C3×C4—22no—
179C2×C3×C4—22no—
180C2×C3×C4—22no—
181C2×C3×C4—22no—
182C2×C3×C4—22no—
183C2×C3×C4—42no—
184C2×C3×C4—42no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—42no—
188C2×C3×C4—42no—
189C2×C3×C4—42no—
190C2×C3×C4—42no—
191C2×C3×C4—41no—
192C2×C3×C4—42no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—81no—
196C2×C3×C4#10214yes—
197C2×C3×C4C4×D412yes—
198C2×C3×C4#12112yes—
199C2×Dic3—44no—
200C2×Dic3—44no—
201C2×Dic3—44no—
202C3⋊C8—82no—
203C24—21no—
204C24C2×#6711yescharacteristic
205C2×D4—22no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—81no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—64no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4#21312yes—
225C22×C4#21412yes—
226C22×C4#21712yes—
227C22×C4#21712yes—
228C22×C4D3×D412yes—
229C22×C4#23712yes—
230C22×C4C22×Dic311yescharacteristic
231C22×C4C2×#6711yescharacteristic
232C2×C8—122no—
233C2×C8—242no—
234C4⋊C4—24no—
235C4⋊C4—24no—
236C4⋊C4—22no—
237C4⋊C4—44no—
238C4⋊C4—81no—
239C4⋊C4—124no—
240C4⋊C4—242no—
241C4⋊C4—242no—
242C22⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—44no—
245C22⋊C4—42no—
246C22⋊C4—82no—
247C22⋊C4—122no—
248C22⋊C4—122no—
249C42—124no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—41no—
257C22×C3—41no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—82no—
261C22×C3—82no—
262C22×C3—81no—
263C22×C3—81no—
264C22×C3—81no—
265C22×C3#38611yescharacteristic
266C22×C3C2×#10211yescharacteristic
267C22×C3#41711yescharacteristic
268C22×C3#44711yescharacteristic
269C22×C3#44811yescharacteristic
270C22×C3#48011yescharacteristic
271C22×C3#48111yescharacteristic
272C3×C4—24no—
273C3×C4—22no—
274C3×C4—22no—
275C3×C4—42no—
276C3×C4—82no—
277C3×C4—81no—
278Dic3—84no—
279C23—41no—
280C23—41no—
281C23—42no—
282C23—42no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23C2×#21311yescharacteristic
287C23C2×#21411yescharacteristic
288C23#93511yescharacteristic
289C23#93711yescharacteristic
290C23#94911yescharacteristic
291C23#95111yescharacteristic
292C23C2×#21711yescharacteristic
293D4—82no—
294D4—81no—
295D4—81no—
296D4—81no—
297D4—81no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—81no—
318C2×C4—124no—
319C2×C4—124no—
320C2×C4—124no—
321C2×C4#83214yes—
322C2×C4D4×Dic312yes—
323C2×C4#93912yes—
324C8—242no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—21no—
331C2×C3—161no—
332C2×C3#187411yescharacteristic
333C2×C3#191211yescharacteristic
334C2×C3#191311yescharacteristic
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—82no—
346C22—82no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22#503311yescharacteristic
351C22#503411yescharacteristic
352C22#503611yescharacteristic
353C22#504011yescharacteristic
354C22C2×#83211yescharacteristic
355C22#522011yescharacteristic
356C22#523211yescharacteristic
357C4—22no—
358C4—22no—
359C4—24no—
360C4—42no—
361C4—82no—
362C4—81no—
363C4—244no—
364C3#920411yescharacteristic, complemented, Sylow
365C2—21no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—21no—
370C2—21no—
371C2—161no—
372C2#7167711yescharacteristic
373C2#7184811yescharacteristic
374C2#7186411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2618C2
socleC22×C3#386
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenter#2618D3
hypocenterC3#9204
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2618

Complements

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

NH ≅ G/Ntype
2. #82156C2semidirect
4. C2×#4528C2semidirect
5. #67625C2semidirect
11. C3×#398C4semidirect
13. C3×#394C4semidirect
66. #398Dic3semidirect
68. #394Dic3semidirect
364. C3#9204semidirect

Maximal subgroups

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

subgroupindex
C3×#26182
#821562
#716752
C2×#45282
#676252
#92043