Finite Group Databasebeta

← #10693036

Subgroups — #10693036

Order 768.

626 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C3×#2972C211yesmaximal, characteristic, complemented
2#82416C211yesmaximal, characteristic
3#71617C211yesmaximal, characteristic, complemented
4#70762C212yesmaximal, complemented
5#67738C212yesmaximal, complemented
6#9485—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#938—21no—
9C2×C4×D12C2211yescharacteristic
10C3×C8×D4—24no—
11C3×#431C2212yes—
12C2×C3×C22⋊C8—22no—
13C2×#833—21no—
14#5213—22no—
15#4987—21no—
16#4960C412yes—
17#4953C412yes—
18#4936C2211yescharacteristic
19#4924C2212yes—
20#4530—21no—
21#4459—24no—
22#2972D311yescharacteristic, complemented
23#2535—31no—
24#1800—31no—
25#1486—32no—
26#1286—32no—
27C22×C3×D4D411yescharacteristic
28C23×C3×C4D412yes—
29C22×#67—22no—
30C22×D12C2×C411yescharacteristic
31C22×C4×D3—21no—
32C22×C3×C8—24no—
33C3×C4×D4—22no—
34C3×C4×D4—22no—
35C2×C3×C4⋊C4D411yescharacteristic
36C2×C3×C22⋊C4C2×C412yes—
37C2×C3×C42C2311yescharacteristic
38C2×#217—22no—
39#938—42no—
40#938—42no—
41C2×(C2×D3)⋊C4—21no—
42C2×(C2×D3)⋊C4—22no—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3C2×C411yescharacteristic
45C2×#208—22no—
46C4×D12—22no—
47C4×D12—22no—
48C3×C4⋊C8—22no—
49C3×C22⋊C8—24no—
50C3×C22⋊C8—22no—
51C3×C4×C8D414yescomplemented
52#833—42no—
53C3⋊#95—22no—
54C3⋊#95—21no—
55C3⋊#95C2×C412yes—
56C4⋊(C3⋊C8)—22no—
57C4⋊(C3⋊C8)—22no—
58C2×#121—61no—
59C2×C4×D4—31no—
60C2×C4×D4C2×D311yescharacteristic
61C8×D4—24no—
62#431C2×D312yes—
63#423—32no—
64C2×#100—61no—
65C2×C22⋊C8—22no—
66#397—32no—
67#392—32no—
68#386—62no—
69#382—31no—
70#380—61no—
71#343—61no—
72D4⋊C8—64no—
73C24×C3C22⋊C412yes—
74C23×D3—21no—
75C2×C3×D4—22no—
76C2×C3×D4—22no—
77C2×C3×D4D812yes—
78C2×C3×D4SD1612yes—
79C22×C3×C4—22no—
80C22×C3×C4—22no—
81C22×C3×C4—82no—
82C22×C3×C4C22⋊C412yes—
83C22×C3×C4C22×C411yescharacteristic
84C22×C3×C4C2×D411yescharacteristic
85C22×C3×C4C2×D411yescharacteristic
86C2×#67—44no—
87C2×#67—44no—
88C2×#67—82no—
89C22×Dic3—22no—
90C22×Dic3—21no—
91C2×D12—22no—
92C2×D12—42no—
93C2×C4×D3—42no—
94C2×C4×D3—41no—
95C2×C4×D3—41no—
96C2×C4×D3—42no—
97C2×C3×C8—24no—
98C2×C3×C8—84no—
99C3×C4⋊C4—22no—
100C3×C22⋊C4—24no—
101C3×C42C2×D412yes—
102C3×C42#4112yes—
103#217—44no—
104(C2×D3)⋊C4—41no—
105(C2×D3)⋊C4—41no—
106(C2×D3)⋊C4—82no—
107C4⋊Dic3—22no—
108C4⋊Dic3—41no—
109C4⋊Dic3—41no—
110#208—42no—
111#208—42no—
112C2×C3⋊C8—42no—
113C22×D4—31no—
114C22×D4—62no—
115C22×D4D1211yescharacteristic
116C23×C4—61no—
117C23×C4#6712yes—
118C2×M16—62no—
119C22×C8—24no—
120#121—122no—
121#121—122no—
122C4×D4—22no—
123C4×D4—22no—
124C4×D4—62no—
125C4×D4—62no—
126C2×C4⋊C4—31no—
127C2×C4⋊C4—61no—
128C2×C4⋊C4D1211yescharacteristic
129C2×C22⋊C4—62no—
130C2×C22⋊C4—61no—
131C2×C22⋊C4—62no—
132C2×C22⋊C4C4×D312yes—
133C2×C42C22×D311yescharacteristic
134C4⋊C8—22no—
135C4⋊C8—62no—
136C4⋊C8—62no—
137#100—122no—
138C22⋊C8—24no—
139C22⋊C8—22no—
140C4×C8#6714yescomplemented
141#95—32no—
142#95—62no—
143#95—61no—
144C23×C3—22no—
145C23×C3—22no—
146C23×C3—42no—
147C23×C3—42no—
148C23×C3#10214yes—
149C23×C3C2×C22⋊C411yescharacteristic
150C22×D3—22no—
151C22×D3—41no—
152C22×D3—41no—
153C22×D3—41no—
154C22×D3—41no—
155C22×D3—41no—
156C22×D3—41no—
157C3×D4—24no—
158C3×D4—24no—
159C2×C3×C4—22no—
160C2×C3×C4—41no—
161C2×C3×C4—41no—
162C2×C3×C4—44no—
163C2×C3×C4—42no—
164C2×C3×C4—82no—
165C2×C3×C4C4×D412yes—
166C2×C3×C4#12011yescharacteristic
167C2×C3×C4#12112yes—
168C2×C3×C4#12311yescharacteristic
169C2×C3×C4C2×D811yescharacteristic
170C2×C3×C4C2×SD1611yescharacteristic
171C2×C3×C4#13512yes—
172#67—88no—
173C2×Dic3—42no—
174C2×Dic3—42no—
175C2×Dic3—41no—
176C2×Dic3—42no—
177C2×Dic3—41no—
178C2×Dic3—41no—
179D12—24no—
180D12—42no—
181C4×D3—82no—
182C4×D3—82no—
183C3×C8—44no—
184C3⋊C8—82no—
185C24—61no—
186C24(C2×D3)⋊C412yes—
187C2×D4—22no—
188C2×D4—22no—
189C2×D4—62no—
190C2×D4—122no—
191C2×D4—124no—
192C2×D4—124no—
193C2×D4—242no—
194C2×D4#20412yes—
195C2×D4D2412yes—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—62no—
199C22×C4—61no—
200C22×C4—82no—
201C22×C4—121no—
202C22×C4—121no—
203C22×C4—122no—
204C22×C4—122no—
205C22×C4(C2×D3)⋊C412yes—
206C22×C4C2×C4×D311yescharacteristic
207C22×C4C2×D1211yescharacteristic
208C22×C4C2×#6711yescharacteristic
209M16—122no—
210M16—122no—
211C2×C8—24no—
212C2×C8—84no—
213C2×C8—122no—
214C4⋊C4—22no—
215C4⋊C4—62no—
216C4⋊C4—121no—
217C4⋊C4—121no—
218C22⋊C4—24no—
219C22⋊C4—124no—
220C22⋊C4—121no—
221C22⋊C4—121no—
222C22⋊C4—242no—
223C42#23512yes—
224C42C2×#6712yes—
225C22×C3—21no—
226C22×C3—21no—
227C22×C3—24no—
228C22×C3—44no—
229C22×C3—42no—
230C22×C3—42no—
231C22×C3—82no—
232C22×C3#38611yescharacteristic
233C22×C3C2×#10211yescharacteristic
234C22×C3#41611yescharacteristic
235C2×D3—42no—
236C2×D3—42no—
237C2×D3—42no—
238C2×D3—81no—
239C2×D3—81no—
240C2×D3—81no—
241C2×D3—81no—
242C3×C4—22no—
243C3×C4—21no—
244C3×C4—21no—
245C3×C4—42no—
246C3×C4#46512yes—
247C3×C4#46612yes—
248Dic3—82no—
249Dic3—81no—
250C23—22no—
251C23—22no—
252C23—42no—
253C23—42no—
254C23—62no—
255C23—121no—
256C23—121no—
257C23—121no—
258C23—121no—
259C23—121no—
260C23—121no—
261C23#82114yes—
262C23C2×(C2×D3)⋊C411yescharacteristic
263D4—24no—
264D4—24no—
265D4—64no—
266D4—122no—
267D4—248no—
268C2×C4—22no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—44no—
272C2×C4—42no—
273C2×C4—82no—
274C2×C4—122no—
275C2×C4—122no—
276C2×C4—121no—
277C2×C4—121no—
278C2×C4—122no—
279C2×C4—121no—
280C2×C4—242no—
281C2×C4—242no—
282C2×C4C2×#20411yescharacteristic
283C2×C4C2×D2411yescharacteristic
284C2×C4#90412yes—
285C2×C4C4×#6712yes—
286C2×C4#92911yescharacteristic
287C2×C4#93012yes—
288C2×C4#95311yescharacteristic
289C8—44no—
290C8—242no—
291C2×C3—21no—
292C2×C3—21no—
293C2×C3—44no—
294C2×C3#189111yescharacteristic
295C2×C3#189311yescharacteristic
296C2×C3#191111yescharacteristic
297D3—82no—
298C22—21no—
299C22—21no—
300C22—24no—
301C22—44no—
302C22—42no—
303C22—42no—
304C22—82no—
305C22—122no—
306C22—122no—
307C22—122no—
308C22—241no—
309C22—241no—
310C22—241no—
311C22—241no—
312C22C2×#82111yescharacteristic
313C22#511311yescharacteristic
314C22#521311yescharacteristic
315C4—22no—
316C4—21no—
317C4—21no—
318C4—42no—
319C4—242no—
320C4—241no—
321C4#511412yes—
322C4#511512yes—
323C3#948511yescharacteristic, complemented, Sylow
324C2—21no—
325C2—21no—
326C2—44no—
327C2—242no—
328C2#7165211yescharacteristic
329C2#7197311yescharacteristic
330C2#7202911yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2972C2
socleC22×C3#386
prime radical#82416C2
cosocleC2×C3×C42C23
hypercenter#2972D3
hypocenterC3#9485
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
1. C3×#2972C2semidirect
3. #71617C2semidirect
4. #70762C2semidirect
5. #67738C2semidirect
22. #2972D3semidirect
51. C3×C4×C8D4semidirect
140. C4×C8#67semidirect
323. C3#9485semidirect

Maximal subgroups

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

subgroupindex
C3×#29722
#824162
#716172
#707622
#677382
#94853