Finite Group Databasebeta

← #10694321

Subgroups — #10694321

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×#1797C211yesmaximal, characteristic
3#71534C211yesmaximal, characteristic
4C2×#4467C214yesmaximal
5#9714—31nomaximal, Sylow
6C22×C3×C42C2211yescharacteristic
7C2×#924C2212yes—
8C2×#869C411yescharacteristic, complemented
9C2×C3×C4⋊C8—21no—
10C2×C3×C22⋊C8C2212yes—
11#5212—21no—
12#5212—21no—
13C2×C3⋊#95—21no—
14C2×C3⋊#95—21no—
15C2×C3⋊#95C2212yes—
16#4935C411yescharacteristic, complemented
17#4929C412yescomplemented
18#4928—21no—
19#4928—21no—
20C3×#336—21no—
21#4467—28no—
22#2857—31no—
23#1797D311yescharacteristic
24C2×#328—34no—
25#1434—31no—
26C23×C3×C4D411yescharacteristic
27C23×C3×C4D411yescharacteristic
28C23×C3×C4C2311yescharacteristic
29C23×Dic3—22no—
30C22×Dic6C2×C412yes—
31C22×C3×C8—22no—
32C2×C3×C42—42no—
33C2×C3×C42C2×C411yescharacteristic
34C2×C3×C42C2×C411yescharacteristic
35C2×C3×C42D411yescharacteristic
36C2×C3×C42D411yescharacteristic
37C2×#217—22no—
38C2×C4⋊Dic3—21no—
39C2×C4⋊Dic3—21no—
40C2×C4⋊Dic3—21no—
41C2×C4⋊Dic3—21no—
42C2×C4⋊Dic3C2×C412yes—
43#924—24no—
44#924—24no—
45C2×Dic3⋊C4—22no—
46#869—42no—
47C3×C4⋊C8—42no—
48C3×C22⋊C8—24no—
49C3⋊#95—22no—
50C3⋊#95—22no—
51C3⋊#95—22no—
52C3⋊#95—42no—
53C3⋊#95—42no—
54C3⋊#95D414yes—
55C2×C4⋊Q8—31no—
56C2×C22⋊Q8—32no—
57C22×C42C2×D311yescharacteristic
58C2×C4⋊C8—21no—
59C2×C22⋊C8C2×D312yes—
60#391—32no—
61#389—61no—
62#389—61no—
63#384—31no—
64#380—61no—
65#380—61no—
66C2×#95—32no—
67C2×#95—61no—
68C2×#95—61no—
69#336—21no—
70#328—68no—
71C24×C3C2×D411yescharacteristic
72C22×C3×C4—22no—
73C22×C3×C4—22no—
74C22×C3×C4—41no—
75C22×C3×C4—41no—
76C22×C3×C4—41no—
77C22×C3×C4—41no—
78C22×C3×C4—41no—
79C22×C3×C4C22⋊C411yescharacteristic
80C22×C3×C4C22⋊C411yescharacteristic
81C22×C3×C4C22⋊C411yescharacteristic
82C22×C3×C4C22⋊C411yescharacteristic
83C22×C3×C4SD1612yes—
84C22×C3×C4Q1612yes—
85C22×C3×C4C22×C411yescharacteristic
86C22×C3×C4C2×D412yes—
87C22×C3×C4C2×D411yescharacteristic
88C22×C3×C4#4112yes—
89C22×Dic3—22no—
90C22×Dic3—22no—
91C22×Dic3—24no—
92C22×Dic3—42no—
93C22×Dic3—42no—
94C22×Dic3—42no—
95C22×Dic3—42no—
96C22×Dic3—42no—
97C22×Dic3—42no—
98C2×Dic6—24no—
99C2×Dic6—42no—
100C2×Dic6—42no—
101C2×C3×C8—42no—
102C2×C3×C8—42no—
103C2×C3×C8—42no—
104C3×C42—22no—
105C3×C42—22no—
106C3×C42—41no—
107C3×C42—41no—
108#217—44no—
109C4⋊Dic3—24no—
110C4⋊Dic3—42no—
111C4⋊Dic3—42no—
112C4⋊Dic3—42no—
113C4⋊Dic3—42no—
114Dic3⋊C4—42no—
115Dic3⋊C4—42no—
116C22×Q8—32no—
117C23×C4—62no—
118C23×C4D1211yescharacteristic
119C23×C4D1211yescharacteristic
120C23×C4C22×D311yescharacteristic
121C22×C8—22no—
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×C42#6711yescharacteristic
136C2×C42#6711yescharacteristic
137C4⋊C8—42no—
138C22⋊C8—24no—
139#95—34no—
140#95—62no—
141#95—62no—
142#95—62no—
143#95—122no—
144#95—122no—
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#9912yes—
173C2×C3×C4#10312yes—
174C2×C3×C4#10312yes—
175C2×C3×C4C4×D412yes—
176C2×C3×C4#12411yescharacteristic
177C2×C3×C4#12711yescharacteristic
178C2×Dic3—44no—
179C2×Dic3—44no—
180C2×Dic3—42no—
181C2×Dic3—42no—
182C2×Dic3—42no—
183C2×Dic3—42no—
184C2×Dic3—42no—
185C2×Dic3—42no—
186C2×Dic3—44no—
187C2×Dic3—82no—
188C2×Dic3—82no—
189C2×Dic3—82no—
190C2×Dic3—82no—
191Dic6—84no—
192C3×C8—82no—
193C24C2×D1211yescharacteristic
194C2×Q8—64no—
195C2×Q8—122no—
196C2×Q8—122no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—64no—
205C22×C4—62no—
206C22×C4—62no—
207C22×C4—122no—
208C22×C4—122no—
209C22×C4—122no—
210C22×C4—122no—
211C22×C4—122no—
212C22×C4—122no—
213C22×C4#20412yes—
214C22×C4Dic1212yes—
215C22×C4(C2×D3)⋊C411yescharacteristic
216C22×C4(C2×D3)⋊C411yescharacteristic
217C22×C4(C2×D3)⋊C411yescharacteristic
218C22×C4(C2×D3)⋊C411yescharacteristic
219C22×C4C2×C4×D311yescharacteristic
220C22×C4D3×D412yes—
221C22×C4#23912yes—
222C22×C4C2×#6711yescharacteristic
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
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—
262Dic3—84no—
263Dic3—82no—
264Dic3—82no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23#89512yes—
270C23C2×#20411yescharacteristic
271C23C2×Dic1211yescharacteristic
272C23#90912yes—
273C23C2×(C2×D3)⋊C411yescharacteristic
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—124no—
294C2×C4—124no—
295C2×C4—122no—
296C2×C4—122no—
297C2×C4—122no—
298C2×C4—122no—
299C2×C4—122no—
300C2×C4—122no—
301C2×C4—124no—
302C2×C4—242no—
303C2×C4—242no—
304C2×C4—242no—
305C2×C4—242no—
306C2×C4#80612yes—
307C2×C4#81612yes—
308C2×C4#81612yes—
309C2×C4#89312yes—
310C2×C4#94011yescharacteristic
311C2×C4#94711yescharacteristic
312C8—82no—
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—21no—
323C22—21no—
324C22—22no—
325C22—22no—
326C22—21no—
327C22—21no—
328C22—22no—
329C22—22no—
330C22#484112yes—
331C22#484912yes—
332C22C2×#80611yescharacteristic
333C22#498811yescharacteristic
334C22#500611yescharacteristic
335C22C2×#81611yescharacteristic
336C22#512011yescharacteristic
337C22#513911yescharacteristic
338C22#513911yescharacteristic
339C4—22no—
340C4—22no—
341C4—22no—
342C4—21no—
343C4—22no—
344C4—21no—
345C4—41no—
346C4—244no—
347C4—242no—
348C4—242no—
349C3#971411yescharacteristic, complemented, Sylow
350C2—22no—
351C2—22no—
352C2#6952811yescharacteristic
353C2#6994611yescharacteristic
354C2#6998111yescharacteristic
355C2#7045611yescharacteristic
356C2#7051311yescharacteristic
357C2#7079311yescharacteristic
358C2#7083711yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1797C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1797D3
hypocenterC3#9714
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
8. C2×#869C4semidirect
16. #4935C4semidirect
17. #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×#17972
#715342
C2×#44672
#97143