Finite Group Databasebeta

← #10690837

Subgroups — #10690837

Order 768.

588 conjugacy classes of subgroups.

The subgroup lattice has 352 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×#4486C212yesmaximal, complemented
5#67702C212yesmaximal, complemented
6#9130—31nomaximal, Sylow
7C22×C3×C42C2211yescharacteristic
8C2×#924—21no—
9C2×#869C2211yescharacteristic
10C3×#431—24no—
11C2×C3×C4⋊C8C2212yes—
12C2×C3×C22⋊C8—21no—
13C2×C3×C4×C8—21no—
14#5212—21no—
15#5212C412yes—
16C2×#830—21no—
17#4935C2211yescharacteristic
18#4929—21no—
19#4928—21no—
20#4928C412yes—
21C2×C4⋊(C3⋊C8)C2212yes—
22#4524—21no—
23#4486—24no—
24C2×#431D311yescharacteristic
25#2857—31no—
26#1797—31no—
27C2×#332—32no—
28#1296—32no—
29C23×C3×C4C2×C411yescharacteristic
30C23×C3×C4D411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C22×Dic6—21no—
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⋊Dic3C2×C412yes—
45C2×C4⋊Dic3C2×C412yes—
46#924—42no—
47#924—42no—
48C2×Dic3⋊C4—22no—
49C22×C3⋊C8—22no—
50#869—22no—
51#869—22no—
52C3×C4⋊C8—22no—
53C3×C4⋊C8D414yes—
54C3×C22⋊C8—24no—
55C3×C4×C8—24no—
56C3⋊#95—22no—
57C3⋊#95—22no—
58#830—42no—
59C4⋊(C3⋊C8)—22no—
60C4⋊(C3⋊C8)—22no—
61C2×C4⋊Q8—31no—
62C2×C22⋊Q8—61no—
63C22×C42C2×D311yescharacteristic
64#431—24no—
65C2×C4⋊C8—32no—
66C2×C4⋊C8C2×D312yes—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—61no—
69C2×C4×C8—21no—
70#391—61no—
71#389—32no—
72#389—61no—
73#384—31no—
74#380—32no—
75#380—61no—
76#336—61no—
77#332—64no—
78C24×C3C22⋊C411yescharacteristic
79C22×C3×C4—22no—
80C22×C3×C4—22no—
81C22×C3×C4—22no—
82C22×C3×C4—22no—
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4—22no—
86C22×C3×C4—22no—
87C22×C3×C4—41no—
88C22×C3×C4C22⋊C411yescharacteristic
89C22×C3×C4C22⋊C411yescharacteristic
90C22×C3×C4C22⋊C411yescharacteristic
91C22×C3×C4D812yes—
92C22×C3×C4SD1612yes—
93C22×C3×C4C22×C411yescharacteristic
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×Dic3—22no—
97C22×Dic3—22no—
98C2×Dic6—42no—
99C2×Dic6—41no—
100C2×Dic6—41no—
101C2×Dic6—42no—
102C2×C3×C8—24no—
103C2×C3×C8—42no—
104C2×C3×C8—42no—
105C3×C42—22no—
106C3×C42—21no—
107C3×C42—22no—
108C3×C42—21no—
109C3×C42C2×D412yes—
110C3×C42#4112yes—
111#217—44no—
112C4⋊Dic3—24no—
113C4⋊Dic3—41no—
114C4⋊Dic3—41no—
115C4⋊Dic3—41no—
116C4⋊Dic3—41no—
117C4⋊Dic3—42no—
118C4⋊Dic3—42no—
119Dic3⋊C4—44no—
120C2×C3⋊C8—42no—
121C2×C3⋊C8—42no—
122C2×C3⋊C8—42no—
123C22×Q8—61no—
124C23×C4C4×D311yescharacteristic
125C23×C4D1211yescharacteristic
126C23×C4#6711yescharacteristic
127C22×C8—22no—
128C22×C8—62no—
129C4⋊Q8—62no—
130C4⋊Q8—62no—
131C22⋊Q8—122no—
132C22⋊Q8—122no—
133C2×C4⋊C4—32no—
134C2×C4⋊C4—32no—
135C2×C4⋊C4—62no—
136C2×C4⋊C4—61no—
137C2×C4⋊C4—61no—
138C2×C4⋊C4—61no—
139C2×C22⋊C4—62no—
140C2×C42—22no—
141C2×C42—22no—
142C2×C42C4×D311yescharacteristic
143C2×C42D1211yescharacteristic
144C2×C42#6711yescharacteristic
145C2×C42C22×D311yescharacteristic
146C4⋊C8—22no—
147C4⋊C8—62no—
148C4⋊C8—62no—
149C4⋊C8#6714yes—
150C22⋊C8—24no—
151C22⋊C8—122no—
152C4×C8—24no—
153#95—62no—
154#95—62no—
155C23×C3—21no—
156C23×C3—21no—
157C23×C3—21no—
158C23×C3—21no—
159C23×C3—21no—
160C23×C3—21no—
161C23×C3#10212yes—
162C23×C3C2×C22⋊C411yescharacteristic
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—24no—
170C2×C3×C4—24no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—44no—
175C2×C3×C4—42no—
176C2×C3×C4—42no—
177C2×C3×C4#10112yes—
178C2×C3×C4#10212yes—
179C2×C3×C4C4×D412yes—
180C2×C3×C4#12011yescharacteristic
181C2×C3×C4#12112yes—
182C2×C3×C4#12311yescharacteristic
183C2×C3×C4C2×D811yescharacteristic
184C2×C3×C4C2×SD1611yescharacteristic
185C2×C3×C4#13512yes—
186C2×C3×C4#13512yes—
187C2×C3×C4#13712yes—
188C2×Dic3—42no—
189C2×Dic3—42no—
190C2×Dic3—42no—
191C2×Dic3—42no—
192C2×Dic3—42no—
193C2×Dic3—42no—
194Dic6—82no—
195Dic6—82no—
196C3×C8—44no—
197C3⋊C8—82no—
198C24(C2×D3)⋊C411yescharacteristic
199C2×Q8—121no—
200C2×Q8—121no—
201C2×Q8—122no—
202C2×Q8—122no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—41no—
212C22×C4—62no—
213C22×C4—62no—
214C22×C4#20412yes—
215C22×C4D2412yes—
216C22×C4(C2×D3)⋊C411yescharacteristic
217C22×C4(C2×D3)⋊C411yescharacteristic
218C22×C4(C2×D3)⋊C411yescharacteristic
219C22×C4C2×C4×D311yescharacteristic
220C22×C4C2×D1211yescharacteristic
221C22×C4C2×#6711yescharacteristic
222C2×C8—24no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—122no—
226C2×C8—122no—
227C2×C8—122no—
228C4⋊C4—64no—
229C4⋊C4—122no—
230C4⋊C4—124no—
231C4⋊C4—121no—
232C4⋊C4—121no—
233C4⋊C4—121no—
234C4⋊C4—122no—
235C4⋊C4—121no—
236C22⋊C4—124no—
237C42—22no—
238C42—21no—
239C42—22no—
240C42—21no—
241C42#23512yes—
242C42C2×#6712yes—
243C22×C3—22no—
244C22×C3—22no—
245C22×C3—22no—
246C22×C3—41no—
247C22×C3—41no—
248C22×C3—41no—
249C22×C3—41no—
250C22×C3#38611yescharacteristic
251C22×C3C2×#10111yescharacteristic
252C22×C3#41311yescharacteristic
253C22×C3C2×#10211yescharacteristic
254C22×C3#41611yescharacteristic
255C22×C3#41611yescharacteristic
256C22×C3#41911yescharacteristic
257C3×C4—22no—
258C3×C4—22no—
259C3×C4—22no—
260C3×C4—41no—
261C3×C4#45212yes—
262C3×C4#46512yes—
263C3×C4#46612yes—
264C3×C4#48512yes—
265Dic3—82no—
266Dic3—82no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23#82112yes—
274C23C2×(C2×D3)⋊C411yescharacteristic
275Q8—242no—
276Q8—242no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—122no—
292C2×C4—122no—
293C2×C4—122no—
294C2×C4—122no—
295C2×C4—122no—
296C2×C4—122no—
297C2×C4#82112yes—
298C2×C4#82412yes—
299C2×C4C2×#20411yescharacteristic
300C2×C4C2×D2411yescharacteristic
301C2×C4#90412yes—
302C2×C4C4×#6712yes—
303C2×C4#92911yescharacteristic
304C2×C4#93012yes—
305C2×C4#95012yes—
306C2×C4#95112yes—
307C2×C4#95311yescharacteristic
308C8—44no—
309C8—242no—
310C2×C3—42no—
311C2×C3#154011yescharacteristic
312C2×C3#154411yescharacteristic
313C2×C3#154611yescharacteristic
314C2×C3#155411yescharacteristic
315C2×C3#187311yescharacteristic
316C2×C3#189311yescharacteristic
317C2×C3#190111yescharacteristic
318C22—22no—
319C22—22no—
320C22—22no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22#500211yescharacteristic
326C22#500411yescharacteristic
327C22C2×#82111yescharacteristic
328C22#511311yescharacteristic
329C22#513211yescharacteristic
330C22C2×#82411yescharacteristic
331C22#521311yescharacteristic
332C4—22no—
333C4—22no—
334C4—22no—
335C4—41no—
336C4—242no—
337C4—242no—
338C4#499612yes—
339C4#511412yes—
340C4#511512yes—
341C4#523912yes—
342C3#913011yescharacteristic, complemented, Sylow
343C2—42no—
344C2#7077211yescharacteristic
345C2#7077711yescharacteristic
346C2#7077811yescharacteristic
347C2#7078111yescharacteristic
348C2#7158511yescharacteristic
349C2#7197311yescharacteristic
350C2#7200311yescharacteristic
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#9130
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×#4486C2semidirect
5. #67702C2semidirect
342. C3#9130semidirect

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×#44862
#677022
#91303