Finite Group Databasebeta

← #10693034

Subgroups — #10693034

Order 768.

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

normal subgroups only

This group has 626 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#82416C211yesmaximal, characteristic
2#82301C211yesmaximal, characteristic, complemented
3C3×#1800C211yesmaximal, characteristic, complemented
4#70496C212yesmaximal, complemented
5#67713C212yesmaximal, complemented
6#9485—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#938—21no—
9C2×C4×D12C2211yescharacteristic
10C3×#423C2212yes—
11C2×C3×#100—21no—
12#5213—22no—
13C2×#830—22no—
14D4×C3⋊C8—24no—
15#4987—21no—
16#4960C412yes—
17#4953C412yes—
18#4936C2211yescharacteristic
19#4925C2212yes—
20C3×#343—21no—
21#4483—24no—
22#2972—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—42no—
40#938—42no—
41C2×(C2×D3)⋊C4—21no—
42C2×(C2×D3)⋊C4—22no—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3C2×C411yescharacteristic
45C22×C3⋊C8—24no—
46C4×D12—22no—
47C4×D12—22no—
48C3×C4⋊C8—22no—
49C3×C4⋊C8—22no—
50C3×#100—42no—
51C3⋊#95—22no—
52C3⋊#95—21no—
53C3⋊#95C2×C412yes—
54#830—24no—
55#830—22no—
56C4⋊(C3⋊C8)—22no—
57C4×C3⋊C8D414yescomplemented
58C2×#121—61no—
59C2×C4×D4—31no—
60C2×C4×D4C2×D311yescharacteristic
61C8×D4—64no—
62#431—32no—
63#423C2×D312yes—
64C2×#100—21no—
65C2×C22⋊C8—62no—
66#397—32no—
67#392—32no—
68#386—62no—
69#382—31no—
70#380—61no—
71#343—21no—
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—
97C3×M16—42no—
98C3×M16—42no—
99C2×C3×C8—42no—
100C3×C4⋊C4—22no—
101C3×C22⋊C4—24no—
102C3×C42C2×D412yes—
103C3×C42#4112yes—
104#217—44no—
105(C2×D3)⋊C4—41no—
106(C2×D3)⋊C4—41no—
107(C2×D3)⋊C4—82no—
108C4⋊Dic3—22no—
109C4⋊Dic3—41no—
110C4⋊Dic3—41no—
111C2×C3⋊C8—24no—
112C2×C3⋊C8—84no—
113C22×D4—31no—
114C22×D4—62no—
115C22×D4#6711yescharacteristic
116C23×C4—61no—
117C23×C4D1212yes—
118C2×M16—22no—
119C22×C8—64no—
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⋊C4#6711yescharacteristic
129C2×C22⋊C4—62no—
130C2×C22⋊C4—61no—
131C2×C22⋊C4—62no—
132C2×C22⋊C4C4×D312yes—
133C2×C42C22×D311yescharacteristic
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—62no—
137#100—42no—
138C22⋊C8—64no—
139C22⋊C8—62no—
140C4×C8—34no—
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—82no—
184C3⋊C8—44no—
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#21312yes—
195C2×D4#21512yes—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—62no—
199C22×C4—61no—
200C22×C4—82no—
201C22×C4—121no—
202C22×C4—122no—
203C22×C4—121no—
204C22×C4—122no—
205C22×C4(C2×D3)⋊C412yes—
206C22×C4C2×C4×D311yescharacteristic
207C22×C4C2×D1211yescharacteristic
208C22×C4C2×#6711yescharacteristic
209M16—42no—
210M16—42no—
211C2×C8—42no—
212C2×C8—64no—
213C2×C8—244no—
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—
223C42D3×D412yes—
224C42#23712yes—
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#80914yes—
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×C4#67⋊C412yes—
283C2×C4#88211yescharacteristic
284C2×C4#88611yescharacteristic
285C2×C4C2×#21311yescharacteristic
286C2×C4#93912yes—
287C2×C4C2×#21511yescharacteristic
288C2×C4#95012yes—
289C8—82no—
290C8—124no—
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—
312C22#495811yescharacteristic
313C22C2×#80911yescharacteristic
314C22#500211yescharacteristic
315C4—21no—
316C4—21no—
317C4—22no—
318C4—42no—
319C4—242no—
320C4—241no—
321C4#503812yes—
322C4#504812yes—
323C3#948511yescharacteristic, complemented, Sylow
324C2—21no—
325C2—21no—
326C2—44no—
327C2—242no—
328C2#6951511yescharacteristic
329C2#7007111yescharacteristic
330C2#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#9485
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. #82301C2semidirect
3. C3×#1800C2semidirect
4. #70496C2semidirect
5. #67713C2semidirect
24. #1800D3semidirect
57. C4×C3⋊C8D4semidirect
323. C3#9485semidirect

Maximal subgroups

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

subgroupindex
#824162
#823012
C3×#18002
#704962
#677132
#94853