Finite Group Databasebeta

← #10693054

Subgroups — #10693054

Order 768.

602 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 111 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#82416C211yesmaximal, characteristic
2#82304C211yesmaximal, characteristic, complemented
3C3×#1800C211yesmaximal, characteristic, complemented
4#70496C212yesmaximal, complemented
5#67713C212yesmaximal, complemented
6#9489—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#938—21no—
9C2×C4×D12C2211yescharacteristic
10C3×#423C2212yes—
11C2×C3×#100—21no—
12#5213—22no—
13#5207—22no—
14#5012—24no—
15#4987—21no—
16#4960C412yes—
17#4953C412yes—
18#4936C2211yescharacteristic
19#4925C2212yes—
20C3×#343—21no—
21#4483—24no—
22#2975—31no—
23#2535—31no—
24#1800D311yescharacteristic, complemented
25#1486—32no—
26#1286—32no—
27C22×C3×D4D411yescharacteristic
28C23×C3×C4D412yes—
29C22×#67—22no—
30C22×D12C2×C411yescharacteristic
31C22×C4×D3—21no—
32C2×C3×M16—22no—
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—44no—
40C2×(C2×D3)⋊C4—21no—
41C2×(C2×D3)⋊C4—22no—
42C2×C4⋊Dic3—21no—
43C2×C4⋊Dic3C2×C411yescharacteristic
44C2×#208—24no—
45C4×D12—22no—
46C4×D12—22no—
47C3×C4⋊C8—22no—
48C3×C4⋊C8—22no—
49C3×#100—42no—
50C3⋊#95—22no—
51C3⋊#95—21no—
52C3⋊#95C2×C412yes—
53#830—24no—
54#830—22no—
55C4⋊(C3⋊C8)—22no—
56C4×C3⋊C8D414yescomplemented
57C2×#121—61no—
58C2×C4×D4—31no—
59C2×C4×D4C2×D311yescharacteristic
60#436—64no—
61#431—32no—
62#423C2×D312yes—
63C2×#100—21no—
64#407—62no—
65#397—32no—
66#392—32no—
67#386—62no—
68#382—31no—
69#380—61no—
70#343—21no—
71D4⋊C8—64no—
72C24×C3C22⋊C412yes—
73C23×D3—21no—
74C2×C3×D4—22no—
75C2×C3×D4—21no—
76C2×C3×D4—22no—
77C2×C3×D4—21no—
78C22×C3×C4—22no—
79C22×C3×C4—22no—
80C22×C3×C4—82no—
81C22×C3×C4C22⋊C412yes—
82C22×C3×C4C22×C411yescharacteristic
83C22×C3×C4C2×D411yescharacteristic
84C22×C3×C4C2×D411yescharacteristic
85C2×#67—44no—
86C2×#67—44no—
87C2×#67—82no—
88C22×Dic3—22no—
89C22×Dic3—21no—
90C2×D12—22no—
91C2×D12—42no—
92C2×C4×D3—42no—
93C2×C4×D3—42no—
94C2×C4×D3—41no—
95C2×C4×D3—41no—
96C3×M16—42no—
97C3×M16—42no—
98C2×C3×C8—42no—
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—42no—
109#208—84no—
110C2×C3⋊C8—24no—
111C22×D4—31no—
112C22×D4—62no—
113C22×D4#6711yescharacteristic
114C23×C4—61no—
115C23×C4D1212yes—
116C2×M16—22no—
117C2×M16—64no—
118#121—124no—
119C4×D4—22no—
120C4×D4—22no—
121C4×D4—62no—
122C4×D4—62no—
123C2×C4⋊C4—31no—
124C2×C4⋊C4—61no—
125C2×C4⋊C4#6711yescharacteristic
126C2×C22⋊C4—62no—
127C2×C22⋊C4—61no—
128C2×C22⋊C4—62no—
129C2×C22⋊C4C4×D312yes—
130C2×C42C22×D311yescharacteristic
131C4⋊C8—22no—
132C4⋊C8—22no—
133C4⋊C8—62no—
134#100—42no—
135C22⋊C8—64no—
136C22⋊C8—62no—
137C4×C8—34no—
138#95—32no—
139#95—62no—
140#95—61no—
141C23×C3—22no—
142C23×C3—22no—
143C23×C3—22no—
144C23×C3—42no—
145C23×C3—42no—
146C23×C3C2×C22⋊C411yescharacteristic
147C22×D3—22no—
148C22×D3—41no—
149C22×D3—41no—
150C22×D3—41no—
151C22×D3—41no—
152C22×D3—41no—
153C22×D3—41no—
154C3×D4—42no—
155C3×D4—42no—
156C2×C3×C4—22no—
157C2×C3×C4—41no—
158C2×C3×C4—41no—
159C2×C3×C4—44no—
160C2×C3×C4—42no—
161C2×C3×C4—82no—
162C2×C3×C4C4×D412yes—
163C2×C3×C4#12011yescharacteristic
164C2×C3×C4#12112yes—
165C2×C3×C4#12311yescharacteristic
166C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
167C2×C3×C4AGL(1,ℤ/8)12yes—
168C2×C3×C4#13711yescharacteristic
169#67—88no—
170C2×Dic3—42no—
171C2×Dic3—42no—
172C2×Dic3—41no—
173C2×Dic3—41no—
174C2×Dic3—42no—
175C2×Dic3—41no—
176D12—24no—
177D12—42no—
178C4×D3—82no—
179C4×D3—82no—
180C3×C8—82no—
181C3⋊C8—44no—
182C24—61no—
183C24(C2×D3)⋊C412yes—
184C2×D4—21no—
185C2×D4—22no—
186C2×D4—21no—
187C2×D4—22no—
188C2×D4—62no—
189C2×D4—122no—
190C2×D4—124no—
191C2×D4—124no—
192C2×D4—242no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—62no—
196C22×C4—61no—
197C22×C4—82no—
198C22×C4—121no—
199C22×C4—122no—
200C22×C4—121no—
201C22×C4—122no—
202C22×C4(C2×D3)⋊C412yes—
203C22×C4C2×C4×D311yescharacteristic
204C22×C4C2×D1211yescharacteristic
205C22×C4C2×#6711yescharacteristic
206M16—42no—
207M16—42no—
208M16—244no—
209C2×C8—42no—
210C2×C8—64no—
211C4⋊C4—22no—
212C4⋊C4—62no—
213C4⋊C4—122no—
214C22⋊C4—24no—
215C22⋊C4—124no—
216C22⋊C4—121no—
217C22⋊C4—121no—
218C22⋊C4—242no—
219C42D3×D412yes—
220C42#23712yes—
221C22×C3—21no—
222C22×C3—21no—
223C22×C3—44no—
224C22×C3—42no—
225C22×C3—42no—
226C22×C3—42no—
227C22×C3—82no—
228C22×C3#38611yescharacteristic
229C22×C3#41711yescharacteristic
230C22×C3#41811yescharacteristic
231C2×D3—42no—
232C2×D3—42no—
233C2×D3—42no—
234C2×D3—81no—
235C2×D3—81no—
236C2×D3—81no—
237C2×D3—81no—
238C3×C4—22no—
239C3×C4—21no—
240C3×C4—21no—
241C3×C4—42no—
242C3×C4#46812yes—
243C3×C4#46912yes—
244Dic3—82no—
245Dic3—81no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—42no—
250C23—42no—
251C23—62no—
252C23—121no—
253C23—121no—
254C23—121no—
255C23—121no—
256C23—121no—
257C23—121no—
258C23C2×(C2×D3)⋊C411yescharacteristic
259D4—42no—
260D4—42no—
261D4—64no—
262D4—122no—
263D4—248no—
264C2×C4—22no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—44no—
268C2×C4—42no—
269C2×C4—82no—
270C2×C4—122no—
271C2×C4—122no—
272C2×C4—121no—
273C2×C4—121no—
274C2×C4—121no—
275C2×C4—122no—
276C2×C4—242no—
277C2×C4—242no—
278C2×C4#67⋊C412yes—
279C2×C4#88211yescharacteristic
280C2×C4#88611yescharacteristic
281C2×C4#93211yescharacteristic
282C2×C4#93912yes—
283C2×C4#94211yescharacteristic
284C2×C4#94912yes—
285C8—82no—
286C8—124no—
287C2×C3—21no—
288C2×C3—21no—
289C2×C3—82no—
290C2×C3#189111yescharacteristic
291C2×C3#189511yescharacteristic
292C2×C3#191111yescharacteristic
293D3—82no—
294C22—21no—
295C22—21no—
296C22—44no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—82no—
301C22—122no—
302C22—122no—
303C22—122no—
304C22—241no—
305C22—241no—
306C22—241no—
307C22—241no—
308C22#495811yescharacteristic
309C22#496611yescharacteristic
310C22#500111yescharacteristic
311C4—21no—
312C4—21no—
313C4—22no—
314C4—42no—
315C4—242no—
316C4—241no—
317C4#503912yes—
318C4#504912yes—
319C3#948911yescharacteristic, complemented, Sylow
320C2—21no—
321C2—21no—
322C2—82no—
323C2—242no—
324C2#6951711yescharacteristic
325C2#7007111yescharacteristic
326C2#7013611yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1800C2
socleC22×C3#386
prime radical#82416C2
cosocleC2×C3×C42C23
hypercenter#1800D3
hypocenterC3#9489
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
2. #82304C2semidirect
3. C3×#1800C2semidirect
4. #70496C2semidirect
5. #67713C2semidirect
24. #1800D3semidirect
56. C4×C3⋊C8D4semidirect
319. C3#9489semidirect

Maximal subgroups

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

subgroupindex
#824162
#823042
C3×#18002
#704962
#677132
#94893