Finite Group Databasebeta

← #10714577

Subgroups — #10714577

Order 768.

436 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 93 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×#5183C211yesmaximal, characteristic
2C2×C3×#362C211yesmaximal, characteristic
3#72061C211yesmaximal, characteristic
4#71445C211yesmaximal, characteristic
5#70704C211yesmaximal, characteristic
6C2×#4496C211yesmaximal, characteristic
7#67810C211yesmaximal, characteristic
8#13415—31nomaximal, Sylow
9C22×C3×Q16C2211yescharacteristic
10C2×#944—21no—
11C22×#216—21no—
12C22×Dic12C2211yescharacteristic
13C2×C3×M32C2211yescharacteristic
14C2×C3×#108C2211yescharacteristic
15#5183—22no—
16#5183—22no—
17#5119—21no—
18#5106C2211yescharacteristic
19C2×C8×Dic3C2211yescharacteristic
20C2×#812C2211yescharacteristic
21C3×#362—22no—
22#4564—22no—
23#4555—21no—
24#4503—22no—
25#4496—22no—
26C2×#494—31no—
27C2×#362—31no—
28C2×#362D311yescharacteristic
29#2141—31no—
30#2141—31no—
31#1945—31no—
32#1376—31no—
33C22×C3×Q8—21no—
34C22×Dic6—21no—
35C2×C3×Q16—21no—
36C2×C3×Q16—21no—
37C2×C3×Q16—21no—
38C2×C3×Q16—21no—
39C2×C3×Q16—21no—
40C2×C3×Q16D412yes—
41C2×C3×M16—21no—
42C22×C3×C8C2311yescharacteristic
43#944—24no—
44#944—42no—
45C2×#216—24no—
46C2×#216—42no—
47C2×#216—42no—
48#926—21no—
49C2×Dic3⋊C4—42no—
50C2×C4×Dic3D411yescharacteristic
51C22×C3⋊C8D411yescharacteristic
52C2×Dic12—21no—
53C2×Dic12—21no—
54C2×Dic12—21no—
55C2×Dic12—21no—
56C2×Dic12—21no—
57C2×Dic12D412yes—
58C3×M32—21no—
59C3×M32—21no—
60C2×C3×C16—21no—
61C3×#108—21no—
62C3×#108—21no—
63#818D412yes—
64#817—21no—
65Dic3⋊C8—42no—
66C8×Dic3—21no—
67C8×Dic3—21no—
68#812—21no—
69#812—21no—
70C2×C3⋊C16—21no—
71C22×Q16—31no—
72C22×Q16—61no—
73C22×Q16C2×D311yescharacteristic
74C2×C4⋊Q8—61no—
75C2×M32—31no—
76C2×M32C2×D311yescharacteristic
77#494—62no—
78#494—62no—
79C2×#108C2×D311yescharacteristic
80#427—31no—
81#424—61no—
82C2×C4×C8—31no—
83#362—22no—
84#362—62no—
85#358—62no—
86#358—62no—
87#339—61no—
88C2×C3×Q8—22no—
89C2×C3×Q8—41no—
90C2×C3×Q8—41no—
91C2×C3×Q8—41no—
92C2×C3×Q8—41no—
93C2×C3×Q8—41no—
94C22×C3×C4—41no—
95C22×C3×C4C2×D411yescharacteristic
96C22×Dic3—21no—
97C22×Dic3—41no—
98C2×Dic6—22no—
99C2×Dic6—41no—
100C2×Dic6—41no—
101C2×Dic6—41no—
102C2×Dic6—41no—
103C2×Dic6—41no—
104C3×Q16—22no—
105C3×Q16—22no—
106C3×Q16—41no—
107C3×Q16—41no—
108C3×M16—41no—
109C3×M16—41no—
110C2×C3×C8—41no—
111C2×C3×C8C2×D411yescharacteristic
112C2×C3×C8C2×D411yescharacteristic
113C2×C3×C8C2×D411yescharacteristic
114C2×C3×C8#4111yescharacteristic
115C2×C3×C8#4111yescharacteristic
116C2×C3×C8#4111yescharacteristic
117#217—41no—
118#216—48no—
119C4⋊Dic3—22no—
120Dic3⋊C4—48no—
121C4×Dic3—21no—
122C4×Dic3—41no—
123C4×Dic3SD1612yes—
124C2×C3⋊C8—21no—
125C2×C3⋊C8—21no—
126C2×C3⋊C8D812yes—
127Dic12—22no—
128Dic12—22no—
129Dic12—41no—
130Dic12—41no—
131C3×C16—41no—
132C3⋊C16—41no—
133C22×Q8—21no—
134C22×Q8—61no—
135C2×Q16—21no—
136C2×Q16—21no—
137C2×Q16—21no—
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—32no—
141C2×Q16—64no—
142C2×Q16—61no—
143C2×Q16—61no—
144C2×Q16—61no—
145C2×Q16—61no—
146C2×Q16—61no—
147C2×Q16—122no—
148C2×Q16—122no—
149C2×Q16#6712yes—
150C2×M16—21no—
151C22×C8—31no—
152C22×C8C22×D311yescharacteristic
153C4⋊Q8—64no—
154C4⋊Q8—122no—
155#117—61no—
156C2×C4⋊C4—122no—
157C2×C42—31no—
158M32—21no—
159M32—21no—
160M32—61no—
161M32—61no—
162C2×C16—21no—
163C2×C16—61no—
164#108—21no—
165#108—21no—
166#107—32no—
167#106—61no—
168C4⋊C8—122no—
169C4×C8—61no—
170C4×C8—61no—
171C23×C3AGL(1,ℤ/8)11yescharacteristic
172C3×Q8—42no—
173C3×Q8—42no—
174C3×Q8—81no—
175C3×Q8—81no—
176C2×C3×C4—42no—
177C2×C3×C4—81no—
178C2×C3×C4—81no—
179C2×C3×C4#12011yescharacteristic
180C2×C3×C4#12111yescharacteristic
181C2×C3×C4#12411yescharacteristic
182C2×C3×C4C2×D811yescharacteristic
183C2×C3×C4C2×SD1611yescharacteristic
184C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
185C2×Dic3—24no—
186C2×Dic3—22no—
187C2×Dic3—42no—
188C2×Dic3—42no—
189C2×Dic3—81no—
190C2×Dic3—81no—
191Dic6—42no—
192Dic6—42no—
193Dic6—81no—
194Dic6—81no—
195C3×C8—81no—
196C3×C8#12111yescharacteristic
197C3×C8#12111yescharacteristic
198C3×C8#12311yescharacteristic
199C3×C8#12311yescharacteristic
200C3⋊C8—22no—
201C2×Q8—22no—
202C2×Q8—41no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8—41no—
206C2×Q8—41no—
207C2×Q8—62no—
208C2×Q8—121no—
209C2×Q8—121no—
210C2×Q8—121no—
211C2×Q8—121no—
212C2×Q8—121no—
213C22×C4—41no—
214C22×C4—61no—
215C22×C4—121no—
216C22×C4D3×D411yescharacteristic
217Q16—22no—
218Q16—22no—
219Q16—41no—
220Q16—41no—
221Q16—62no—
222Q16—62no—
223Q16—128no—
224Q16—121no—
225Q16—121no—
226M16—41no—
227M16—41no—
228C2×C8—32no—
229C2×C8—41no—
230C2×C8—61no—
231C2×C8—61no—
232C2×C8#23511yescharacteristic
233C2×C8#23511yescharacteristic
234C2×C8D3×D411yescharacteristic
235C2×C8D3×D411yescharacteristic
236C2×C8#23711yescharacteristic
237C2×C8C2×#6711yescharacteristic
238C4⋊C4—62no—
239C4⋊C4—128no—
240C22⋊C4—121no—
241C42—32no—
242C42—61no—
243C42—121no—
244C16—41no—
245C16—121no—
246C22×C3—22no—
247C22×C3#45011yescharacteristic
248C22×C3#45911yescharacteristic
249C22×C3#49011yescharacteristic
250C3×C4—82no—
251C3×C4#39411yescharacteristic
252C3×C4#44711yescharacteristic
253C3×C4#46011yescharacteristic
254C3×C4#48611yescharacteristic
255Dic3—24no—
256Dic3—82no—
257Dic3—82no—
258C23#91411yescharacteristic
259Q8—42no—
260Q8—42no—
261Q8—81no—
262Q8—81no—
263Q8—122no—
264Q8—122no—
265Q8—241no—
266Q8—241no—
267C2×C4—42no—
268C2×C4—64no—
269C2×C4—62no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—122no—
273C2×C4—122no—
274C2×C4—241no—
275C2×C4—241no—
276C2×C4#88411yescharacteristic
277C2×C4#88511yescharacteristic
278C2×C4D3×D811yescharacteristic
279C2×C4#91111yescharacteristic
280C2×C4D3×SD1611yescharacteristic
281C2×C4#93711yescharacteristic
282C8—62no—
283C8—81no—
284C8#88311yescharacteristic
285C8#88411yescharacteristic
286C8#92911yescharacteristic
287C8#93911yescharacteristic
288C2×C3—21no—
289C2×C3—21no—
290C2×C3#204111yescharacteristic
291C2×C3#221412yes—
292C22—22no—
293C22#476411yescharacteristic
294C22#478611yescharacteristic
295C22#516911yescharacteristic
296C4—64no—
297C4—82no—
298C4—242no—
299C4—242no—
300C4#476011yescharacteristic
301C4#476211yescharacteristic
302C4#495911yescharacteristic
303C4#515611yescharacteristic
304C3#1341511yescharacteristic, complemented, Sylow
305C2—21no—
306C2—21no—
307C2#6927312yes—
308C2#7021111yescharacteristic
subgroupG/H
centerC22#4786
commutator, cosocleC22×C3×C8C23
FrattiniC22×C8C22×D3
FittingC2×C3×#362C2
socleC22×C3#459
prime radicalC23×C3AGL(1,ℤ/8)
hypercenterC2×#362D3
hypocenterC3#13415
LCSG ▷ C22×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#362

Complements

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

NH ≅ G/Ntype
304. C3#13415semidirect

Maximal subgroups

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

subgroupindex
C2×#51832
C2×C3×#3622
#720612
#714452
#707042
C2×#44962
#678102
#134153