Finite Group Databasebeta

← #10698800

Subgroups — #10698800

Order 768.

408 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 95 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×#5090C211yesmaximal, characteristic
2C3×#1546C211yesmaximal, characteristic
3#71773C211yesmaximal, characteristic
4#70767C211yesmaximal, characteristic
5#70528C211yesmaximal, characteristic
6#70493C211yesmaximal, characteristic
7#67745C211yesmaximal, characteristic
8#10447—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C22×#216—21no—
11C2×C4×Dic6C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C2×#835—21no—
15C2×#824—21no—
16#5090—22no—
17#5090—22no—
18#4925C2211yescharacteristic
19#4924C2211yescharacteristic
20C2×C4⋊(C3⋊C8)C2211yescharacteristic
21C3×#332—22no—
22#4554—21no—
23#4547—22no—
24#4480—22no—
25#4456—22no—
26C2×#462—31no—
27#1801—31no—
28#1570—31no—
29#1546—31no—
30#1546D311yescharacteristic
31#1479—31no—
32#1301—31no—
33C22×C3×Q8—21no—
34C22×Dic6D411yescharacteristic
35C2×C3×M16—21no—
36C3×C4⋊Q8—21no—
37C3×C4⋊Q8—21no—
38C3×C4⋊Q8—21no—
39C3×C4⋊Q8D412yes—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C42C2311yescharacteristic
43C2×#216—24no—
44C2×#216—42no—
45C2×#216—42no—
46C2×C4⋊Dic3D411yescharacteristic
47C2×Dic3⋊C4—21no—
48C2×C4×Dic3—21no—
49C2×#208—21no—
50C22×C3⋊C8—21no—
51C4×Dic6—21no—
52C4×Dic6—21no—
53C4×Dic6—21no—
54C4×Dic6—21no—
55C3×C4⋊C8—21no—
56C3×C4⋊C8—21no—
57C3×C4⋊C8—21no—
58C3×C22⋊C8—21no—
59C3×C4×C8—21no—
60#835—24no—
61#830—21no—
62#824—42no—
63C4⋊(C3⋊C8)—21no—
64C4⋊(C3⋊C8)—21no—
65C4⋊(C3⋊C8)—21no—
66C4⋊(C3⋊C8)D412yes—
67C4⋊(C3⋊C8)D412yes—
68C4×C3⋊C8—21no—
69C22×Q16—61no—
70C2×C4⋊Q8C2×D311yescharacteristic
71C2×C4×Q8—31no—
72#462—62no—
73#462—62no—
74#431—31no—
75#431C2×D311yescharacteristic
76#423—31no—
77#423C2×D311yescharacteristic
78C2×C4⋊C8—31no—
79C2×#103—61no—
80C2×#101—61no—
81#343—61no—
82#334—62no—
83#332—22no—
84#332—62no—
85Q8⋊C8—62no—
86C2×C3×Q8—22no—
87C2×C3×Q8—41no—
88C2×C3×Q8—41no—
89C2×C3×Q8—41no—
90C2×C3×Q8—41no—
91C2×C3×Q8—41no—
92C22×C3×C4—41no—
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4#4111yescharacteristic
95C22×C3×C4#4111yescharacteristic
96C22×Dic3—21no—
97C22×Dic3—21no—
98C2×Dic6—21no—
99C2×Dic6—21no—
100C2×Dic6—21no—
101C2×Dic6—21no—
102C2×Dic6—21no—
103C2×Dic6SD1612yes—
104C3×M16—41no—
105C3×M16—41no—
106C2×C3×C8—41no—
107C2×C3×C8—41no—
108C3×C4⋊C4—22no—
109C3×C4⋊C4—22no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—41no—
112C3×C42C2×D411yescharacteristic
113C3×C42C2×D411yescharacteristic
114C3×C42C2×D411yescharacteristic
115C3×C42#4111yescharacteristic
116#216—48no—
117C4⋊Dic3—21no—
118C4⋊Dic3—21no—
119Dic3⋊C4—41no—
120Dic3⋊C4—41no—
121C4×Dic3—41no—
122C4×Dic3—41no—
123#208—41no—
124#208—41no—
125C2×C3⋊C8—22no—
126C2×C3⋊C8—22no—
127C2×C3⋊C8—41no—
128C2×C3⋊C8—41no—
129C2×C3⋊C8—41no—
130C22×Q8—21no—
131C22×Q8—31no—
132C2×Q16—64no—
133C2×Q16—122no—
134C2×Q16—122no—
135C2×M16—21no—
136C2×M16—61no—
137C22×C8—61no—
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8#6712yes—
142C4×Q8—61no—
143C4×Q8—61no—
144C4×Q8—61no—
145C4×Q8—61no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—31no—
149C2×C4⋊C4—61no—
150C2×C42—61no—
151C2×C42C22×D311yescharacteristic
152C4⋊C8—21no—
153C4⋊C8—21no—
154C4⋊C8—21no—
155C4⋊C8—32no—
156C4⋊C8—32no—
157C4⋊C8—61no—
158C4⋊C8—61no—
159C4⋊C8—61no—
160#103—64no—
161#101—122no—
162C22⋊C8—21no—
163C22⋊C8—61no—
164C4×C8—21no—
165C4×C8—61no—
166C23×C3#12311yescharacteristic
167C3×Q8—42no—
168C3×Q8—42no—
169C3×Q8—81no—
170C3×Q8—81no—
171C2×C3×C4—22no—
172C2×C3×C4—41no—
173C2×C3×C4—42no—
174C2×C3×C4—81no—
175C2×C3×C4—81no—
176C2×C3×C4#12011yescharacteristic
177C2×C3×C4#12111yescharacteristic
178C2×C3×C4#12111yescharacteristic
179C2×C3×C4#12111yescharacteristic
180C2×C3×C4#12311yescharacteristic
181C2×C3×C4#12411yescharacteristic
182C2×C3×C4C2×SD1611yescharacteristic
183C2×C3×C4#13511yescharacteristic
184C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
185C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
186C2×Dic3—22no—
187C2×Dic3—41no—
188C2×Dic3—41no—
189C2×Dic3—41no—
190C2×Dic3—81no—
191Dic6—22no—
192Dic6—22no—
193Dic6—41no—
194Dic6—41no—
195C3×C8—81no—
196C3×C8—81no—
197C3⋊C8—42no—
198C3⋊C8—42no—
199C3⋊C8—81no—
200C2×Q8—22no—
201C2×Q8—32no—
202C2×Q8—41no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8—41no—
206C2×Q8—41no—
207C2×Q8—61no—
208C2×Q8—61no—
209C2×Q8—61no—
210C2×Q8—61no—
211C2×Q8—61no—
212C22×C4—41no—
213C22×C4—61no—
214C22×C4—61no—
215C22×C4#23511yescharacteristic
216C22×C4D3×D411yescharacteristic
217C22×C4#23711yescharacteristic
218Q16—128no—
219M16—41no—
220M16—41no—
221M16—121no—
222M16—121no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—62no—
226C2×C8—62no—
227C2×C8—121no—
228C2×C8—121no—
229C2×C8—121no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4—121no—
237C4⋊C4—121no—
238C42—121no—
239C42—121no—
240C42#23511yescharacteristic
241C42D3×D411yescharacteristic
242C42D3×D411yescharacteristic
243C42C2×#6711yescharacteristic
244C22×C3—21no—
245C22×C3—21no—
246C22×C3#39411yescharacteristic
247C22×C3#48111yescharacteristic
248C22×C3#48211yescharacteristic
249C3×C4—21no—
250C3×C4—21no—
251C3×C4—21no—
252C3×C4—82no—
253C3×C4#44911yescharacteristic
254C3×C4#45011yescharacteristic
255C3×C4#46511yescharacteristic
256C3×C4#46911yescharacteristic
257C3×C4#47312yes—
258Dic3—42no—
259Dic3—81no—
260C23#88311yescharacteristic
261Q8—42no—
262Q8—42no—
263Q8—62no—
264Q8—62no—
265Q8—81no—
266Q8—81no—
267Q8—121no—
268Q8—121no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—62no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—121no—
276C2×C4—121no—
277C2×C4—121no—
278C2×C4—241no—
279C2×C4#88411yescharacteristic
280C2×C4#88411yescharacteristic
281C2×C4#88511yescharacteristic
282C2×C4#91111yescharacteristic
283C2×C4#91211yescharacteristic
284C2×C4D3×SD1611yescharacteristic
285C2×C4#91411yescharacteristic
286C2×C4#92911yescharacteristic
287C2×C4#93711yescharacteristic
288C2×C4#93911yescharacteristic
289C8—81no—
290C8—81no—
291C8—122no—
292C8—122no—
293C8—241no—
294C2×C3—21no—
295C2×C3—21no—
296C2×C3#203411yescharacteristic
297C2×C3#204711yescharacteristic
298C2×C3#204911yescharacteristic
299C22—21no—
300C22—21no—
301C22#477711yescharacteristic
302C22#478411yescharacteristic
303C22#495911yescharacteristic
304C4—21no—
305C4—21no—
306C4—21no—
307C4—82no—
308C4—122no—
309C4—241no—
310C4#475512yes—
311C4#477811yescharacteristic
312C4#478211yescharacteristic
313C4#515811yescharacteristic
314C4#516911yescharacteristic
315C3#1044711yescharacteristic, complemented, Sylow
316C2—21no—
317C2—21no—
318C2#7012011yescharacteristic
319C2#7020711yescharacteristic
320C2#7035211yescharacteristic
subgroupG/H
centerC22#4959
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1546C2
socleC22×C3#394
prime radicalC23×C3#123
hypercenter#1546D3
hypocenterC3#10447
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
315. C3#10447semidirect

Maximal subgroups

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

subgroupindex
C2×#50902
C3×#15462
#717732
#707672
#705282
#704932
#677452
#104473