Finite Group Databasebeta

← #10694322

Subgroups — #10694322

Order 768.

624 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82076C211yesmaximal, characteristic
2C3×#1434C211yesmaximal, characteristic
3#71530C211yesmaximal, characteristic
4C2×#4452C214yesmaximal
5#9714—31nomaximal, Sylow
6C22×C3×C42C2211yescharacteristic
7C2×#924C2212yes—
8C2×#869C411yescharacteristic, complemented
9C2×C3×#95—21no—
10C2×C3×#95—21no—
11C2×C3×#95C2212yes—
12#5212—21no—
13#5212—21no—
14C2×#830C2212yes—
15#4935C411yescharacteristic, complemented
16#4929C412yescomplemented
17#4928—21no—
18#4928—21no—
19C2×C4⋊(C3⋊C8)—21no—
20#4524—21no—
21#4452—28no—
22#2857—31no—
23#1797—31no—
24C2×#328—34no—
25#1434D311yescharacteristic
26C23×C3×C4—22no—
27C23×C3×C4D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C22×Dic6C2×C412yes—
31C2×C3×C42—42no—
32C2×C3×C42C2×C411yescharacteristic
33C2×C3×C42C2×C411yescharacteristic
34C2×C3×C42D411yescharacteristic
35C2×C3×C42D411yescharacteristic
36C2×#217—22no—
37C2×C4⋊Dic3—21no—
38C2×C4⋊Dic3—21no—
39C2×C4⋊Dic3—21no—
40C2×C4⋊Dic3—21no—
41C2×C4⋊Dic3C2×C412yes—
42#924—24no—
43#924—24no—
44C2×Dic3⋊C4—22no—
45C22×C3⋊C8—22no—
46#869—42no—
47C3×#95—22no—
48C3×#95—42no—
49C3×#95—42no—
50C3×#95D414yes—
51C3⋊#95—22no—
52C3⋊#95—22no—
53#830—24no—
54C4⋊(C3⋊C8)—42no—
55C2×C4⋊Q8—31no—
56C2×C22⋊Q8—32no—
57C22×C42C2×D311yescharacteristic
58C2×C4⋊C8—61no—
59C2×C22⋊C8—32no—
60#391—32no—
61#389—61no—
62#389—61no—
63#384—31no—
64#380—61no—
65#380—61no—
66C2×#95—21no—
67C2×#95—21no—
68C2×#95C2×D312yes—
69#336—61no—
70#328—68no—
71C24×C3C2×D411yescharacteristic
72C22×C3×C4—22no—
73C22×C3×C4—22no—
74C22×C3×C4—24no—
75C22×C3×C4—41no—
76C22×C3×C4—41no—
77C22×C3×C4—41no—
78C22×C3×C4—41no—
79C22×C3×C4—41no—
80C22×C3×C4—42no—
81C22×C3×C4—42no—
82C22×C3×C4—42no—
83C22×C3×C4—42no—
84C22×C3×C4—42no—
85C22×C3×C4—42no—
86C22×C3×C4C22⋊C411yescharacteristic
87C22×C3×C4C22⋊C411yescharacteristic
88C22×C3×C4C22⋊C411yescharacteristic
89C22×C3×C4C22⋊C411yescharacteristic
90C22×C3×C4SD1612yes—
91C22×C3×C4Q1612yes—
92C22×C3×C4C22×C411yescharacteristic
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4C2×D412yes—
95C22×C3×C4#4112yes—
96C22×Dic3—22no—
97C22×Dic3—22no—
98C2×Dic6—24no—
99C2×Dic6—42no—
100C2×Dic6—42no—
101C3×C42—22no—
102C3×C42—22no—
103C3×C42—41no—
104C3×C42—41no—
105#217—44no—
106C4⋊Dic3—24no—
107C4⋊Dic3—42no—
108C4⋊Dic3—42no—
109C4⋊Dic3—42no—
110C4⋊Dic3—42no—
111Dic3⋊C4—42no—
112Dic3⋊C4—42no—
113C2×C3⋊C8—42no—
114C2×C3⋊C8—42no—
115C2×C3⋊C8—42no—
116C22×Q8—32no—
117C23×C4—22no—
118C23×C4#6711yescharacteristic
119C23×C4#6711yescharacteristic
120C23×C4C22×D311yescharacteristic
121C22×C8—62no—
122C4⋊Q8—122no—
123C22⋊Q8—64no—
124C22⋊Q8—64no—
125C2×C4⋊C4—32no—
126C2×C4⋊C4—61no—
127C2×C4⋊C4—61no—
128C2×C4⋊C4—61no—
129C2×C4⋊C4—61no—
130C2×C4⋊C4—62no—
131C2×C22⋊C4—62no—
132C2×C42—42no—
133C2×C42C4×D311yescharacteristic
134C2×C42C4×D311yescharacteristic
135C2×C42D1211yescharacteristic
136C2×C42D1211yescharacteristic
137C4⋊C8—122no—
138C22⋊C8—64no—
139#95—22no—
140#95—42no—
141#95—42no—
142#95—62no—
143#95—62no—
144#95D1214yes—
145C23×C3—21no—
146C23×C3—21no—
147C23×C3—21no—
148C23×C3—21no—
149C23×C3C2×C22⋊C411yescharacteristic
150C23×C3#12112yes—
151C23×C3C2×SD1611yescharacteristic
152C23×C3C2×Q1611yescharacteristic
153C23×C3#13712yes—
154C2×C3×C4—21no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—21no—
158C2×C3×C4—21no—
159C2×C3×C4—21no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—42no—
168C2×C3×C4—42no—
169C2×C3×C4—42no—
170C2×C3×C4—42no—
171C2×C3×C4—44no—
172C2×C3×C4—44no—
173C2×C3×C4—44no—
174C2×C3×C4—44no—
175C2×C3×C4—82no—
176C2×C3×C4—82no—
177C2×C3×C4—82no—
178C2×C3×C4—82no—
179C2×C3×C4#9912yes—
180C2×C3×C4#10312yes—
181C2×C3×C4#10312yes—
182C2×C3×C4C4×D412yes—
183C2×C3×C4#12411yescharacteristic
184C2×C3×C4#12711yescharacteristic
185C2×Dic3—42no—
186C2×Dic3—42no—
187C2×Dic3—42no—
188C2×Dic3—42no—
189C2×Dic3—42no—
190C2×Dic3—42no—
191Dic6—84no—
192C3⋊C8—82no—
193C24C2×#6711yescharacteristic
194C2×Q8—64no—
195C2×Q8—122no—
196C2×Q8—122no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—24no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—62no—
212C22×C4—62no—
213C22×C4(C2×D3)⋊C411yescharacteristic
214C22×C4(C2×D3)⋊C411yescharacteristic
215C22×C4(C2×D3)⋊C411yescharacteristic
216C22×C4(C2×D3)⋊C411yescharacteristic
217C22×C4#21412yes—
218C22×C4#21612yes—
219C22×C4C2×C4×D311yescharacteristic
220C22×C4C2×D1211yescharacteristic
221C22×C4C2×D1212yes—
222C22×C4#23512yes—
223C2×C8—122no—
224C2×C8—122no—
225C2×C8—122no—
226C4⋊C4—64no—
227C4⋊C4—122no—
228C4⋊C4—122no—
229C4⋊C4—122no—
230C4⋊C4—122no—
231C4⋊C4—122no—
232C4⋊C4—122no—
233C22⋊C4—124no—
234C42—22no—
235C42—22no—
236C42—41no—
237C42—41no—
238C22×C3—22no—
239C22×C3—22no—
240C22×C3—22no—
241C22×C3—22no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3#39011yescharacteristic
247C22×C3C2×#9911yescharacteristic
248C22×C3#41011yescharacteristic
249C22×C3C2×#10311yescharacteristic
250C22×C3#41911yescharacteristic
251C22×C3#42111yescharacteristic
252C22×C3#42111yescharacteristic
253C22×C3#46112yes—
254C22×C3#46212yes—
255C3×C4—21no—
256C3×C4—22no—
257C3×C4—21no—
258C3×C4—22no—
259C3×C4—22no—
260C3×C4—22no—
261C3×C4—41no—
262C3×C4—84no—
263Dic3—82no—
264Dic3—82no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23C2×(C2×D3)⋊C411yescharacteristic
270C23#93012yes—
271C23C2×#21411yescharacteristic
272C23C2×#21611yescharacteristic
273C23#95112yes—
274Q8—244no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—122no—
301C2×C4—122no—
302C2×C4—122no—
303C2×C4—122no—
304C2×C4—122no—
305C2×C4—122no—
306C2×C4#80612yes—
307C2×C4#81012yes—
308C2×C4#81012yes—
309C2×C4C4×D1212yes—
310C2×C4#87411yescharacteristic
311C2×C4#87511yescharacteristic
312C8—242no—
313C2×C3—22no—
314C2×C3—22no—
315C2×C3#151111yescharacteristic
316C2×C3#151511yescharacteristic
317C2×C3#151811yescharacteristic
318C2×C3#152011yescharacteristic
319C2×C3#196311yescharacteristic
320C2×C3#196411yescharacteristic
321C2×C3#198111yescharacteristic
322C22—22no—
323C22—22no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—21no—
328C22—21no—
329C22—22no—
330C22#493911yescharacteristic
331C22C2×#80611yescharacteristic
332C22C2×#81011yescharacteristic
333C22#500411yescharacteristic
334C22#500511yescharacteristic
335C22#500511yescharacteristic
336C22#500611yescharacteristic
337C22#501912yes—
338C22#503112yes—
339C4—21no—
340C4—22no—
341C4—21no—
342C4—22no—
343C4—22no—
344C4—22no—
345C4—41no—
346C4—84no—
347C4—242no—
348C4—242no—
349C3#971411yescharacteristic, complemented, Sylow
350C2—22no—
351C2—22no—
352C2#6944011yescharacteristic
353C2#6945611yescharacteristic
354C2#6948611yescharacteristic
355C2#6951011yescharacteristic
356C2#6953311yescharacteristic
357C2#6954611yescharacteristic
358C2#6956611yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1434C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1434D3
hypocenterC3#9714
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1434

Complements

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

NH ≅ G/Ntype
8. C2×#869C4semidirect
15. #4935C4semidirect
16. #4929C4semidirect
349. C3#9714semidirect

Maximal subgroups

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

subgroupindex
#820762
C3×#14342
#715302
C2×#44522
#97143