Finite Group Databasebeta

← #10691208

Subgroups — #10691208

Order 768.

524 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 127 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
1#82262C211yesmaximal, characteristic
2#82204C211yesmaximal, characteristic
3C3×#1458C211yesmaximal, characteristic
4#70778C212yesmaximal, complemented
5#67738C212yesmaximal, complemented
6#9195—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C2×#924—21no—
9C22×#208—21no—
10C2×#869C2211yescharacteristic
11C3×#431C2212yes—
12#4999—24no—
13#4981—21no—
14#4981C412yes—
15#4953—21no—
16#4953C412yes—
17#4936C2211yescharacteristic
18#4929—21no—
19#4924—21no—
20#4924C2212yes—
21C3×#324—21no—
22C3×#323—21no—
23#4486—24no—
24#2905—31no—
25#2621—31no—
26#1546—32no—
27#1458D311yescharacteristic
28#1286—32no—
29C23×C3×C4D411yescharacteristic
30C22×Dic6—21no—
31C3×#117—22no—
32C3×#117—22no—
33C2×C3×C4⋊C4C2×C411yescharacteristic
34C2×C3×C4⋊C4D411yescharacteristic
35C2×C3×C22⋊C4C2×C411yescharacteristic
36C2×C3×C22⋊C4D411yescharacteristic
37C2×C3×C42D411yescharacteristic
38C2×C3×C42C2311yescharacteristic
39C2×#217—22no—
40C2×C4⋊Dic3—21no—
41C2×C4⋊Dic3—21no—
42C2×C4⋊Dic3C2×C412yes—
43#924—42no—
44#924—42no—
45C2×Dic3⋊C4—22no—
46C2×#208—22no—
47C2×#208—22no—
48C2×#208—42no—
49C2×#208—42no—
50C22×C3⋊C8—22no—
51#869—22no—
52#869—22no—
53C3×C4⋊C8—22no—
54C3×C22⋊C8—22no—
55C3×C4×C8—22no—
56C3⋊#95—22no—
57C3⋊#95—22no—
58C3⋊#95—21no—
59C3⋊#95C2×C412yes—
60C4⋊(C3⋊C8)—24no—
61C4⋊(C3⋊C8)—22no—
62C4⋊(C3⋊C8)D414yes—
63C22×M16—61no—
64C2×C4⋊Q8—31no—
65C2×C22⋊Q8—61no—
66C2×#117C2×D311yescharacteristic
67#431C2×D312yes—
68#424—64no—
69#423—32no—
70#423—61no—
71#398—32no—
72#398—61no—
73#397—32no—
74#397—61no—
75#391—61no—
76#382—31no—
77#332—64no—
78#324—21no—
79#323—21no—
80C24×C3C22⋊C411yescharacteristic
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4C22⋊C411yescharacteristic
86C22×C3×C4C22⋊C411yescharacteristic
87C22×C3×C4C22⋊C411yescharacteristic
88C22×C3×C4D812yes—
89C22×C3×C4SD1612yes—
90C22×C3×C4C22×C411yescharacteristic
91C22×C3×C4C2×D411yescharacteristic
92C22×C3×C4C2×D411yescharacteristic
93C22×Dic3—22no—
94C22×Dic3—22no—
95C2×Dic6—41no—
96C2×Dic6—41no—
97C2×Dic6—41no—
98C2×Dic6—41no—
99C2×Dic6—41no—
100C2×Dic6—41no—
101C2×C3×C8—42no—
102C3×C4⋊C4—22no—
103C3×C4⋊C4—22no—
104C3×C22⋊C4—22no—
105C3×C22⋊C4—22no—
106C3×C42—21no—
107C3×C42—21no—
108C3×C42C2×D412yes—
109C3×C42#4112yes—
110#217—44no—
111C4⋊Dic3—24no—
112C4⋊Dic3—42no—
113C4⋊Dic3—41no—
114C4⋊Dic3—41no—
115Dic3⋊C4—44no—
116#208—44no—
117#208—42no—
118#208—42no—
119C2×C3⋊C8—24no—
120C2×C3⋊C8—44no—
121C22×Q8—61no—
122C23×C4D1211yescharacteristic
123C2×M16—62no—
124C2×M16—62no—
125C2×M16—122no—
126C2×M16—122no—
127C22×C8—62no—
128C4⋊Q8—62no—
129C4⋊Q8—62no—
130C22⋊Q8—122no—
131C22⋊Q8—122no—
132#117—22no—
133#117—22no—
134C2×C4⋊C4—32no—
135C2×C4⋊C4—62no—
136C2×C4⋊C4—61no—
137C2×C4⋊C4—61no—
138C2×C4⋊C4C4×D311yescharacteristic
139C2×C4⋊C4#6711yescharacteristic
140C2×C22⋊C4—62no—
141C2×C22⋊C4C4×D311yescharacteristic
142C2×C22⋊C4#6711yescharacteristic
143C2×C42D1211yescharacteristic
144C2×C42C22×D311yescharacteristic
145C4⋊C8—22no—
146C4⋊C8—34no—
147C4⋊C8—64no—
148C4⋊C8—62no—
149C22⋊C8—22no—
150C4×C8—22no—
151#95—32no—
152#95—62no—
153#95—62no—
154#95—61no—
155C23×C3—21no—
156C23×C3—21no—
157C23×C3—41no—
158C23×C3—41no—
159C23×C3#10212yes—
160C23×C3C2×C22⋊C411yescharacteristic
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—22no—
164C2×C3×C4—22no—
165C2×C3×C4—22no—
166C2×C3×C4—22no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—41no—
170C2×C3×C4—41no—
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×C3×C4#10212yes—
174C2×C3×C4C4×D412yes—
175C2×C3×C4#12011yescharacteristic
176C2×C3×C4#12112yes—
177C2×C3×C4#12311yescharacteristic
178C2×C3×C4C2×D811yescharacteristic
179C2×C3×C4C2×SD1611yescharacteristic
180C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
181C2×C3×C4#13711yescharacteristic
182C2×Dic3—42no—
183C2×Dic3—42no—
184C2×Dic3—42no—
185C2×Dic3—42no—
186C2×Dic3—42no—
187C2×Dic3—42no—
188Dic6—81no—
189Dic6—81no—
190Dic6—81no—
191Dic6—81no—
192C3×C8—82no—
193C3⋊C8—44no—
194C24(C2×D3)⋊C411yescharacteristic
195C2×Q8—121no—
196C2×Q8—121no—
197C2×Q8—121no—
198C2×Q8—121no—
199C2×Q8—121no—
200C2×Q8—121no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—62no—
206C22×C4—62no—
207C22×C4#20412yes—
208C22×C4D2412yes—
209C22×C4(C2×D3)⋊C411yescharacteristic
210C22×C4(C2×D3)⋊C411yescharacteristic
211C22×C4(C2×D3)⋊C411yescharacteristic
212C22×C4C2×C4×D311yescharacteristic
213C22×C4C2×D1211yescharacteristic
214C22×C4C2×#6711yescharacteristic
215M16—124no—
216M16—122no—
217M16—122no—
218C2×C8—42no—
219C2×C8—64no—
220C2×C8—124no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—64no—
224C4⋊C4—124no—
225C4⋊C4—121no—
226C4⋊C4—121no—
227C4⋊C4—122no—
228C22⋊C4—22no—
229C22⋊C4—22no—
230C22⋊C4—124no—
231C42—21no—
232C42—21no—
233C42D3×D412yes—
234C42#23712yes—
235C22×C3—21no—
236C22×C3—21no—
237C22×C3—22no—
238C22×C3—42no—
239C22×C3—41no—
240C22×C3—41no—
241C22×C3—81no—
242C22×C3#38611yescharacteristic
243C22×C3C2×#10211yescharacteristic
244C22×C3#41711yescharacteristic
245C3×C4—22no—
246C3×C4—22no—
247C3×C4—22no—
248C3×C4—41no—
249C3×C4—41no—
250C3×C4#44711yescharacteristic
251C3×C4#44811yescharacteristic
252C3×C4#48011yescharacteristic
253C3×C4#48111yescharacteristic
254Dic3—82no—
255Dic3—82no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23#82112yes—
261C23C2×(C2×D3)⋊C411yescharacteristic
262Q8—241no—
263Q8—241no—
264Q8—241no—
265Q8—241no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—122no—
279C2×C4—122no—
280C2×C4—122no—
281C2×C4—122no—
282C2×C4—122no—
283C2×C4—122no—
284C2×C4#82112yes—
285C2×C4#67⋊C412yes—
286C2×C4#88211yescharacteristic
287C2×C4#88611yescharacteristic
288C2×C4C2×#20411yescharacteristic
289C2×C4C2×D2411yescharacteristic
290C2×C4#90811yescharacteristic
291C2×C4#90911yescharacteristic
292C2×C4#93912yes—
293C8—82no—
294C8—124no—
295C2×C3—21no—
296C2×C3—21no—
297C2×C3—42no—
298C2×C3#187411yescharacteristic
299C2×C3#190911yescharacteristic
300C2×C3#191511yescharacteristic
301C22—21no—
302C22—21no—
303C22—22no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22#495811yescharacteristic
309C22C2×#82111yescharacteristic
310C22#513311yescharacteristic
311C4—22no—
312C4—22no—
313C4—22no—
314C4—41no—
315C4—41no—
316C4—242no—
317C4—242no—
318C4#473211yescharacteristic
319C4#473511yescharacteristic
320C4#473611yescharacteristic
321C4#473811yescharacteristic
322C3#919511yescharacteristic, complemented, Sylow
323C2—21no—
324C2—21no—
325C2—42no—
326C2#6993211yescharacteristic
327C2#7004611yescharacteristic
328C2#7013811yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1458C2
socleC22×C3#386
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1458D3
hypocenterC3#9195
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
4. #70778C2semidirect
5. #67738C2semidirect
322. C3#9195semidirect

Maximal subgroups

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

subgroupindex
#822622
#822042
C3×#14582
#707782
#677382
#91953