Finite Group Databasebeta

← #10690827

Subgroups — #10690827

Order 768.

588 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#431C211yesmaximal, characteristic
2#82076C211yesmaximal, characteristic
3#71530C211yesmaximal, characteristic
4C2×#4481C212yesmaximal, complemented
5#67702C212yesmaximal, complemented
6#9128—31nomaximal, Sylow
7C22×C3×C42C2211yescharacteristic
8C2×#924C412yes—
9C2×#869C2211yescharacteristic
10C3×#431—24no—
11C2×C3×C4⋊C8C2212yes—
12C2×C3×C22⋊C8—21no—
13C2×C3×C4×C8—21no—
14#5212—21no—
15#5212—21no—
16C2×#830—21no—
17#4935C2211yescharacteristic
18#4929C412yes—
19#4928—21no—
20#4928—21no—
21C2×C4⋊(C3⋊C8)C2212yes—
22#4524—21no—
23#4481—24no—
24C2×#431D311yescharacteristic
25#2857—31no—
26#1797—31no—
27C2×#333—32no—
28#1296—32no—
29C23×C3×C4C2×C411yescharacteristic
30C23×C3×C4D411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C22×Dic6C2×C412yes—
33C22×C3×C8—22no—
34C2×C3×C42—22no—
35C2×C3×C42—22no—
36C2×C3×C42C2×C411yescharacteristic
37C2×C3×C42D411yescharacteristic
38C2×C3×C42D411yescharacteristic
39C2×C3×C42C2311yescharacteristic
40C2×#217—22no—
41C2×C4⋊Dic3—21no—
42C2×C4⋊Dic3—21no—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3C2×C412yes—
46#924—44no—
47C2×Dic3⋊C4—22no—
48C22×C3⋊C8—22no—
49#869—22no—
50#869—22no—
51C3×C4⋊C8—22no—
52C3×C4⋊C8D414yes—
53C3×C22⋊C8—24no—
54C3×C4×C8—24no—
55C3⋊#95—22no—
56C3⋊#95—22no—
57#830—42no—
58C4⋊(C3⋊C8)—22no—
59C4⋊(C3⋊C8)—22no—
60C2×C4⋊Q8—31no—
61C2×C22⋊Q8—32no—
62C22×C42C2×D311yescharacteristic
63#431—24no—
64C2×C4⋊C8—32no—
65C2×C4⋊C8C2×D312yes—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—61no—
68C2×C4×C8—21no—
69#391—32no—
70#389—61no—
71#389—61no—
72#384—31no—
73#380—61no—
74#380—61no—
75#336—61no—
76#333—64no—
77C24×C3C22⋊C411yescharacteristic
78C22×C3×C4—22no—
79C22×C3×C4—22no—
80C22×C3×C4—22no—
81C22×C3×C4—22no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—22no—
85C22×C3×C4—22no—
86C22×C3×C4—41no—
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×D411yescharacteristic
95C22×Dic3—22no—
96C22×Dic3—22no—
97C2×Dic6—24no—
98C2×Dic6—42no—
99C2×Dic6—42no—
100C2×C3×C8—24no—
101C2×C3×C8—42no—
102C2×C3×C8—42no—
103C3×C42—22no—
104C3×C42—21no—
105C3×C42—22no—
106C3×C42—21no—
107C3×C42C2×D412yes—
108C3×C42#4112yes—
109#217—44no—
110C4⋊Dic3—41no—
111C4⋊Dic3—42no—
112C4⋊Dic3—41no—
113C4⋊Dic3—42no—
114C4⋊Dic3—42no—
115C4⋊Dic3—42no—
116Dic3⋊C4—44no—
117C2×C3⋊C8—42no—
118C2×C3⋊C8—42no—
119C2×C3⋊C8—42no—
120C22×Q8—32no—
121C23×C4C4×D311yescharacteristic
122C23×C4D1211yescharacteristic
123C23×C4#6711yescharacteristic
124C22×C8—22no—
125C22×C8—62no—
126C4⋊Q8—62no—
127C4⋊Q8—62no—
128C22⋊Q8—124no—
129C2×C4⋊C4—32no—
130C2×C4⋊C4—62no—
131C2×C4⋊C4—61no—
132C2×C4⋊C4—61no—
133C2×C4⋊C4—61no—
134C2×C4⋊C4—61no—
135C2×C22⋊C4—62no—
136C2×C42—22no—
137C2×C42—22no—
138C2×C42C4×D311yescharacteristic
139C2×C42D1211yescharacteristic
140C2×C42#6711yescharacteristic
141C2×C42C22×D311yescharacteristic
142C4⋊C8—22no—
143C4⋊C8—62no—
144C4⋊C8—62no—
145C4⋊C8#6714yes—
146C22⋊C8—24no—
147C22⋊C8—122no—
148C4×C8—24no—
149#95—62no—
150#95—62no—
151C23×C3—21no—
152C23×C3—21no—
153C23×C3—21no—
154C23×C3—21no—
155C23×C3—21no—
156C23×C3—21no—
157C23×C3#10312yes—
158C23×C3C2×C22⋊C411yescharacteristic
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—24no—
166C2×C3×C4—24no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—44no—
171C2×C3×C4—42no—
172C2×C3×C4—42no—
173C2×C3×C4#10012yes—
174C2×C3×C4#10312yes—
175C2×C3×C4C4×D412yes—
176C2×C3×C4#12011yescharacteristic
177C2×C3×C4#12112yes—
178C2×C3×C4#12311yescharacteristic
179C2×C3×C4C2×SD1611yescharacteristic
180C2×C3×C4C2×Q1611yescharacteristic
181C2×C3×C4#13512yes—
182C2×C3×C4#13512yes—
183C2×C3×C4#13712yes—
184C2×Dic3—42no—
185C2×Dic3—42no—
186C2×Dic3—42no—
187C2×Dic3—42no—
188C2×Dic3—42no—
189C2×Dic3—42no—
190Dic6—84no—
191C3×C8—44no—
192C3⋊C8—82no—
193C24(C2×D3)⋊C411yescharacteristic
194C2×Q8—64no—
195C2×Q8—122no—
196C2×Q8—122no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—41no—
206C22×C4—62no—
207C22×C4—62no—
208C22×C4#20412yes—
209C22×C4Dic1212yes—
210C22×C4(C2×D3)⋊C411yescharacteristic
211C22×C4(C2×D3)⋊C411yescharacteristic
212C22×C4(C2×D3)⋊C411yescharacteristic
213C22×C4C2×C4×D311yescharacteristic
214C22×C4C2×D1211yescharacteristic
215C22×C4C2×#6711yescharacteristic
216C2×C8—24no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—122no—
220C2×C8—122no—
221C2×C8—122no—
222C4⋊C4—122no—
223C4⋊C4—124no—
224C4⋊C4—122no—
225C4⋊C4—121no—
226C4⋊C4—122no—
227C4⋊C4—122no—
228C4⋊C4—121no—
229C22⋊C4—124no—
230C42—22no—
231C42—21no—
232C42—22no—
233C42—21no—
234C42#23512yes—
235C42C2×#6712yes—
236C22×C3—22no—
237C22×C3—22no—
238C22×C3—22no—
239C22×C3—41no—
240C22×C3—41no—
241C22×C3—41no—
242C22×C3—41no—
243C22×C3#38611yescharacteristic
244C22×C3C2×#10011yescharacteristic
245C22×C3#41311yescharacteristic
246C22×C3C2×#10311yescharacteristic
247C22×C3#41611yescharacteristic
248C22×C3#41611yescharacteristic
249C22×C3#41911yescharacteristic
250C3×C4—22no—
251C3×C4—22no—
252C3×C4—22no—
253C3×C4—41no—
254C3×C4#45212yes—
255C3×C4#46512yes—
256C3×C4#46712yes—
257C3×C4#48512yes—
258Dic3—82no—
259Dic3—82no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23#81612yes—
267C23C2×(C2×D3)⋊C411yescharacteristic
268Q8—244no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—44no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—122no—
284C2×C4—122no—
285C2×C4—122no—
286C2×C4—122no—
287C2×C4—122no—
288C2×C4—122no—
289C2×C4#81612yes—
290C2×C4#82312yes—
291C2×C4C2×#20411yescharacteristic
292C2×C4#90412yes—
293C2×C4C2×Dic1211yescharacteristic
294C2×C4C4×#6712yes—
295C2×C4#92911yescharacteristic
296C2×C4#93012yes—
297C2×C4#95012yes—
298C2×C4#95112yes—
299C2×C4#95311yescharacteristic
300C8—44no—
301C8—242no—
302C2×C3—42no—
303C2×C3#154211yescharacteristic
304C2×C3#154311yescharacteristic
305C2×C3#154811yescharacteristic
306C2×C3#155411yescharacteristic
307C2×C3#187311yescharacteristic
308C2×C3#189211yescharacteristic
309C2×C3#190011yescharacteristic
310C22—22no—
311C22—22no—
312C22—22no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22#500211yescharacteristic
318C22#500411yescharacteristic
319C22C2×#81611yescharacteristic
320C22#511311yescharacteristic
321C22C2×#82311yescharacteristic
322C22#513211yescharacteristic
323C22#521311yescharacteristic
324C4—22no—
325C4—22no—
326C4—22no—
327C4—41no—
328C4—242no—
329C4—242no—
330C4#499612yes—
331C4#511412yes—
332C4#511612yes—
333C4#523912yes—
334C3#912811yescharacteristic, complemented, Sylow
335C2—42no—
336C2#7077311yescharacteristic
337C2#7077611yescharacteristic
338C2#7077911yescharacteristic
339C2#7078111yescharacteristic
340C2#7158511yescharacteristic
341C2#7197211yescharacteristic
342C2#7200211yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×#431C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenterC2×#431D3
hypocenterC3#9128
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#431

Complements

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

NH ≅ G/Ntype
4. C2×#4481C2semidirect
5. #67702C2semidirect
334. C3#9128semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4312
#820762
#715302
C2×#44812
#677022
#91283