Finite Group Databasebeta

← #10690947

Subgroups — #10690947

Order 768.

630 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 147 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#382C211yesmaximal, characteristic
2#82075C211yesmaximal, characteristic
3#71531C211yesmaximal, characteristic
4C2×#4452C212yesmaximal
5#67612C212yesmaximal
6#9152—31nomaximal, Sylow
7C22×C3×C4⋊C4—21no—
8C22×C3×C42C2211yescharacteristic
9C2×#924C2211yescharacteristic
10C3×#382—24no—
11C2×C3×#95—21no—
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×#382D311yescharacteristic
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—42no—
34C2×C3×C4⋊C4—42no—
35C2×C3×C42—22no—
36C2×C3×C42—22no—
37C2×C3×C42C2×C412yes—
38C2×C3×C42D412yes—
39C2×#217—21no—
40C2×#217—21no—
41C2×C4⋊Dic3C2×C411yescharacteristic
42C2×C4⋊Dic3C2×C412yes—
43#924—22no—
44#924—22no—
45C2×Dic3⋊C4—22no—
46C2×Dic3⋊C4—21no—
47C22×C3⋊C8—22no—
48C3×#95—24no—
49C3×#95—22no—
50C3×#95D414yes—
51C3⋊#95—22no—
52C3⋊#95—22no—
53C3⋊#95—22no—
54C3⋊#95—21no—
55#830—22no—
56#830—22no—
57C2×C22⋊Q8—31no—
58C22×C4⋊C4—21no—
59C22×C42C2×D311yescharacteristic
60C2×C22⋊C8—32no—
61#391—32no—
62#389—32no—
63#387—62no—
64#385—61no—
65#382—24no—
66#381—61no—
67#380—31no—
68C2×#95—21no—
69C2×#95C2×D312yes—
70#336—62no—
71#328—64no—
72C24×C3C2×D411yescharacteristic
73C22×C3×C4—21no—
74C22×C3×C4—21no—
75C22×C3×C4—21no—
76C22×C3×C4—21no—
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C22×C3×C4—24no—
80C22×C3×C4—42no—
81C22×C3×C4—42no—
82C22×C3×C4—44no—
83C22×C3×C4—44no—
84C22×C3×C4—42no—
85C22×C3×C4—42no—
86C22×C3×C4C22⋊C412yes—
87C22×C3×C4C22⋊C412yes—
88C22×C3×C4SD1612yes—
89C22×C3×C4Q1612yes—
90C22×C3×C4C22×C411yescharacteristic
91C22×C3×C4C2×D411yescharacteristic
92C22×C3×C4C2×D412yes—
93C22×C3×C4#4112yes—
94C22×Dic3—22no—
95C22×Dic3—21no—
96C22×Dic3—21no—
97C2×Dic6—22no—
98C2×Dic6—42no—
99C3×C4⋊C4—48no—
100C3×C42—24no—
101C3×C42—24no—
102#217—41no—
103#217—41no—
104#217—41no—
105#217—41no—
106C4⋊Dic3—22no—
107C4⋊Dic3—42no—
108Dic3⋊C4—44no—
109Dic3⋊C4—42no—
110C2×C3⋊C8—44no—
111C2×C3⋊C8—42no—
112C22×Q8—31no—
113C23×C4—22no—
114C23×C4#6711yescharacteristic
115C23×C4#6711yescharacteristic
116C23×C4C22×D311yescharacteristic
117C22×C8—62no—
118C22⋊Q8—62no—
119C22⋊Q8—62no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—31no—
122C2×C4⋊C4—32no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—62no—
126C2×C4⋊C4—61no—
127C2×C22⋊C4—61no—
128C2×C22⋊C4—61no—
129C2×C42—22no—
130C2×C42—22no—
131C2×C42C4×D312yes—
132C2×C42D1212yes—
133C22⋊C8—62no—
134C22⋊C8—62no—
135#95—24no—
136#95—22no—
137#95—62no—
138#95—62no—
139#95—62no—
140#95—61no—
141#95#6714yes—
142C23×C3—22no—
143C23×C3—22no—
144C23×C3C2×C22⋊C411yescharacteristic
145C23×C3#12011yescharacteristic
146C23×C3#12311yescharacteristic
147C23×C3C2×SD1611yescharacteristic
148C23×C3C2×Q1611yescharacteristic
149C23×C3AGL(1,ℤ/8)11yescharacteristic
150C23×C3#13711yescharacteristic
151C2×C3×C4—22no—
152C2×C3×C4—22no—
153C2×C3×C4—22no—
154C2×C3×C4—22no—
155C2×C3×C4—22no—
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—42no—
163C2×C3×C4—42no—
164C2×C3×C4—42no—
165C2×C3×C4—42no—
166C2×C3×C4—44no—
167C2×C3×C4—44no—
168C2×C3×C4—44no—
169C2×C3×C4—84no—
170C2×C3×C4—84no—
171C2×C3×C4#10314yes—
172C2×C3×C4C4×D412yes—
173C2×C3×C4#12112yes—
174C2×Dic3—41no—
175C2×Dic3—41no—
176C2×Dic3—42no—
177C2×Dic3—42no—
178C2×Dic3—42no—
179C2×Dic3—42no—
180C2×Dic3—42no—
181Dic6—82no—
182C3⋊C8—82no—
183C24C2×#6711yescharacteristic
184C2×Q8—62no—
185C2×Q8—122no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—24no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—62no—
200C22×C4—61no—
201C22×C4—61no—
202C22×C4(C2×D3)⋊C412yes—
203C22×C4(C2×D3)⋊C412yes—
204C22×C4#21412yes—
205C22×C4#21612yes—
206C22×C4C2×C4×D311yescharacteristic
207C22×C4C2×D1211yescharacteristic
208C22×C4#23512yes—
209C22×C4C2×#6712yes—
210C2×C8—124no—
211C2×C8—122no—
212C4⋊C4—48no—
213C4⋊C4—62no—
214C4⋊C4—122no—
215C4⋊C4—124no—
216C4⋊C4—122no—
217C22⋊C4—121no—
218C22⋊C4—121no—
219C22⋊C4—121no—
220C22⋊C4—121no—
221C42—24no—
222C42—24no—
223C22×C3—21no—
224C22×C3—21no—
225C22×C3—21no—
226C22×C3—21no—
227C22×C3—22no—
228C22×C3—22no—
229C22×C3—22no—
230C22×C3—22no—
231C22×C3#38611yescharacteristic
232C22×C3#41012yes—
233C22×C3C2×#10311yescharacteristic
234C22×C3#41711yescharacteristic
235C22×C3#42112yes—
236C22×C3#45011yescharacteristic
237C22×C3#45111yescharacteristic
238C22×C3#48311yescharacteristic
239C22×C3#48411yescharacteristic
240C3×C4—24no—
241C3×C4—22no—
242C3×C4—22no—
243C3×C4—42no—
244C3×C4—84no—
245Dic3—81no—
246Dic3—81no—
247Dic3—82no—
248C23—22no—
249C23—22no—
250C23C2×(C2×D3)⋊C411yescharacteristic
251C23#92911yescharacteristic
252C23#93211yescharacteristic
253C23C2×#21411yescharacteristic
254C23#94211yescharacteristic
255C23C2×#21611yescharacteristic
256C23#95311yescharacteristic
257Q8—242no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—84no—
277C2×C4—84no—
278C2×C4—121no—
279C2×C4—121no—
280C2×C4—122no—
281C2×C4—122no—
282C2×C4—122no—
283C2×C4—122no—
284C2×C4—122no—
285C2×C4#81014yes—
286C2×C4C4×#6712yes—
287C2×C4#93012yes—
288C8—242no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3—21no—
292C2×C3—21no—
293C2×C3#151212yes—
294C2×C3#151812yes—
295C2×C3#187211yescharacteristic
296C2×C3#189711yescharacteristic
297C2×C3#189911yescharacteristic
298C22—21no—
299C22—21no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—21no—
305C22—21no—
306C22#496611yescharacteristic
307C22C2×#81011yescharacteristic
308C22#497011yescharacteristic
309C22#497111yescharacteristic
310C22#500512yes—
311C22#500612yes—
312C22#521311yescharacteristic
313C22#521811yescharacteristic
314C22#522811yescharacteristic
315C4—24no—
316C4—22no—
317C4—22no—
318C4—42no—
319C4—84no—
320C4—241no—
321C4—242no—
322C4—241no—
323C3#915211yescharacteristic, complemented, Sylow
324C2—21no—
325C2—21no—
326C2—21no—
327C2—21no—
328C2#6954412yes—
329C2#6954612yes—
330C2#7155911yescharacteristic
331C2#7159111yescharacteristic
332C2#7159511yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#382C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenterC2×#382D3
hypocenterC3#9152
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
18. #4929C4semidirect
19. #4928C4semidirect
323. C3#9152semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3822
#820752
#715312
C2×#44522
#676122
#91523