Finite Group Databasebeta

← #10690907

Subgroups — #10690907

Order 768.

738 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C4⋊C4C211yesmaximal, characteristic
2#82075C211yesmaximal, characteristic
3#71531C211yesmaximal, characteristic
4C2×#4452C212yesmaximal
5#67612C212yesmaximal
6#9144—31nomaximal, Sylow
7C22×C3×C4⋊C4—21no—
8C22×C3×C42—21no—
9C22×C3×C42C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C3×C4×C4⋊C4—24no—
12C2×C3×#95C2212yes—
13#5212C2211yescharacteristic
14#5211—21no—
15#5210—21no—
16C2×#830C2212yes—
17#4933—22no—
18#4929C412yescomplemented
19#4928C412yescomplemented
20#4524—22no—
21#4452—24no—
22#2855—31no—
23C2×C4×C4⋊C4D311yescharacteristic
24#1817—31no—
25C2×#328—32no—
26#1294—32no—
27C23×C3×C4—22no—
28C23×C3×C4D411yescharacteristic
29C23×C3×C4D411yescharacteristic
30C23×C3×C4C2311yescharacteristic
31C22×Dic6C2×C411yescharacteristic
32C2×C3×C4⋊C4—24no—
33C2×C3×C4⋊C4—28no—
34C2×C3×C42—22no—
35C2×C3×C42—22no—
36C2×C3×C42—24no—
37C2×C3×C42—42no—
38C2×C3×C42—42no—
39C2×C3×C42C2×C412yes—
40C2×C3×C42D412yes—
41C2×#217—21no—
42C2×#217—21no—
43C2×C4⋊Dic3C2×C411yescharacteristic
44C2×C4⋊Dic3C2×C412yes—
45#924—22no—
46#924—22no—
47C2×Dic3⋊C4—22no—
48C2×Dic3⋊C4—21no—
49C22×C3⋊C8—22no—
50C3×#95—22no—
51C3×#95D414yes—
52C3⋊#95—22no—
53C3⋊#95—22no—
54C3⋊#95—22no—
55C3⋊#95—21no—
56#830—22no—
57#830—22no—
58C2×C22⋊Q8—31no—
59C22×C4⋊C4—21no—
60C22×C42—21no—
61C22×C42C2×D311yescharacteristic
62C2×C22⋊C8—32no—
63#391—32no—
64#389—32no—
65#387—62no—
66#385—61no—
67#381—61no—
68#380—31no—
69C4×C4⋊C4—24no—
70C2×#95C2×D312yes—
71#336—62no—
72#328—64no—
73C24×C3C2×D411yescharacteristic
74C22×C3×C4—21no—
75C22×C3×C4—21no—
76C22×C3×C4—21no—
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C22×C3×C4—21no—
80C22×C3×C4—24no—
81C22×C3×C4—28no—
82C22×C3×C4—24no—
83C22×C3×C4—42no—
84C22×C3×C4—44no—
85C22×C3×C4—42no—
86C22×C3×C4—42no—
87C22×C3×C4C22⋊C412yes—
88C22×C3×C4C22⋊C412yes—
89C22×C3×C4SD1612yes—
90C22×C3×C4Q1612yes—
91C22×C3×C4C22×C411yescharacteristic
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4C2×D412yes—
94C22×C3×C4#4112yes—
95C22×Dic3—22no—
96C22×Dic3—21no—
97C22×Dic3—21no—
98C2×Dic6—22no—
99C2×Dic6—42no—
100C3×C4⋊C4—216no—
101C3×C42—24no—
102C3×C42—24no—
103C3×C42—48no—
104#217—41no—
105#217—41no—
106#217—41no—
107#217—41no—
108C4⋊Dic3—22no—
109C4⋊Dic3—42no—
110Dic3⋊C4—44no—
111Dic3⋊C4—42no—
112C2×C3⋊C8—44no—
113C2×C3⋊C8—42no—
114C22×Q8—31no—
115C23×C4—22no—
116C23×C4#6711yescharacteristic
117C23×C4#6711yescharacteristic
118C23×C4C22×D311yescharacteristic
119C22×C8—62no—
120C22⋊Q8—62no—
121C22⋊Q8—62no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—28no—
124C2×C4⋊C4—31no—
125C2×C4⋊C4—32no—
126C2×C4⋊C4—62no—
127C2×C4⋊C4—61no—
128C2×C22⋊C4—61no—
129C2×C22⋊C4—61no—
130C2×C42—22no—
131C2×C42—22no—
132C2×C42—24no—
133C2×C42—42no—
134C2×C42—42no—
135C2×C42C4×D312yes—
136C2×C42D1212yes—
137C22⋊C8—62no—
138C22⋊C8—62no—
139#95—22no—
140#95—62no—
141#95—62no—
142#95—62no—
143#95—61no—
144#95#6714yes—
145C23×C3—22no—
146C23×C3—22no—
147C23×C3C2×C22⋊C411yescharacteristic
148C23×C3#12011yescharacteristic
149C23×C3#12311yescharacteristic
150C23×C3C2×SD1611yescharacteristic
151C23×C3C2×Q1611yescharacteristic
152C23×C3AGL(1,ℤ/8)11yescharacteristic
153C23×C3#13711yescharacteristic
154C2×C3×C4—24no—
155C2×C3×C4—24no—
156C2×C3×C4—22no—
157C2×C3×C4—22no—
158C2×C3×C4—22no—
159C2×C3×C4—22no—
160C2×C3×C4—22no—
161C2×C3×C4—22no—
162C2×C3×C4—22no—
163C2×C3×C4—22no—
164C2×C3×C4—22no—
165C2×C3×C4—22no—
166C2×C3×C4—28no—
167C2×C3×C4—42no—
168C2×C3×C4—42no—
169C2×C3×C4—44no—
170C2×C3×C4—48no—
171C2×C3×C4—48no—
172C2×C3×C4—44no—
173C2×C3×C4#10314yes—
174C2×C3×C4#10414yes—
175C2×C3×C4C4×D412yes—
176C2×C3×C4#12112yes—
177C2×Dic3—41no—
178C2×Dic3—41no—
179C2×Dic3—42no—
180C2×Dic3—42no—
181C2×Dic3—42no—
182C2×Dic3—42no—
183C2×Dic3—42no—
184Dic6—82no—
185C3⋊C8—82no—
186C24C2×#6711yescharacteristic
187C2×Q8—62no—
188C2×Q8—122no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—24no—
196C22×C4—28no—
197C22×C4—24no—
198C22×C4—42no—
199C22×C4—44no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—62no—
203C22×C4—61no—
204C22×C4—61no—
205C22×C4(C2×D3)⋊C412yes—
206C22×C4(C2×D3)⋊C412yes—
207C22×C4#21412yes—
208C22×C4#21612yes—
209C22×C4C2×C4×D311yescharacteristic
210C22×C4C2×D1211yescharacteristic
211C22×C4#23512yes—
212C22×C4C2×#6712yes—
213C2×C8—124no—
214C2×C8—122no—
215C4⋊C4—216no—
216C4⋊C4—62no—
217C4⋊C4—122no—
218C4⋊C4—124no—
219C4⋊C4—122no—
220C22⋊C4—121no—
221C22⋊C4—121no—
222C22⋊C4—121no—
223C22⋊C4—121no—
224C42—24no—
225C42—24no—
226C42—48no—
227C22×C3—21no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3—22no—
232C22×C3—22no—
233C22×C3—22no—
234C22×C3—22no—
235C22×C3#38611yescharacteristic
236C22×C3#41012yes—
237C22×C3C2×#10311yescharacteristic
238C22×C3#41711yescharacteristic
239C22×C3C2×#10412yes—
240C22×C3#45011yescharacteristic
241C22×C3#45111yescharacteristic
242C22×C3#48311yescharacteristic
243C22×C3#48411yescharacteristic
244C3×C4—24no—
245C3×C4—24no—
246C3×C4—22no—
247C3×C4—22no—
248C3×C4—48no—
249Dic3—81no—
250Dic3—81no—
251Dic3—82no—
252C23—22no—
253C23—22no—
254C23C2×(C2×D3)⋊C411yescharacteristic
255C23#92911yescharacteristic
256C23#93211yescharacteristic
257C23C2×#21411yescharacteristic
258C23#94211yescharacteristic
259C23C2×#21611yescharacteristic
260C23#95311yescharacteristic
261Q8—242no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—28no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—48no—
278C2×C4—48no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—122no—
282C2×C4—122no—
283C2×C4—121no—
284C2×C4—121no—
285C2×C4—122no—
286C2×C4—122no—
287C2×C4—122no—
288C2×C4#80514yes—
289C2×C4#81014yes—
290C2×C4C4×#6712yes—
291C2×C4#93012yes—
292C8—242no—
293C2×C3—21no—
294C2×C3—21no—
295C2×C3—21no—
296C2×C3—21no—
297C2×C3#150312yes—
298C2×C3#150912yes—
299C2×C3#187211yescharacteristic
300C2×C3#189611yescharacteristic
301C2×C3#189911yescharacteristic
302C22—22no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22C2×#80512yes—
311C22#496611yescharacteristic
312C22C2×#81011yescharacteristic
313C22#497011yescharacteristic
314C22#497111yescharacteristic
315C22#500612yes—
316C22#521311yescharacteristic
317C22#521811yescharacteristic
318C22#522811yescharacteristic
319C4—24no—
320C4—24no—
321C4—22no—
322C4—22no—
323C4—48no—
324C4—242no—
325C4—241no—
326C4—241no—
327C3#914411yescharacteristic, complemented, Sylow
328C2—21no—
329C2—21no—
330C2—21no—
331C2—21no—
332C2#6947012yes—
333C2#6947112yes—
334C2#7153211yescharacteristic
335C2#7155911yescharacteristic
336C2#7159511yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×C4×C4⋊C4C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenterC2×C4×C4⋊C4D3
hypocenterC3#9144
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
18. #4929C4semidirect
19. #4928C4semidirect
327. C3#9144semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C4⋊C42
#820752
#715312
C2×#44522
#676122
#91443