Finite Group Databasebeta

← #10692632

Subgroups — #10692632

Order 768.

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

normal subgroups only

This group has 840 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
1C3×#2273C211yesmaximal, characteristic, complemented
2#82115C211yesmaximal, characteristic
3#71539C211yesmaximal, characteristic, complemented
4#69444C212yesmaximal, complemented
5#67618C212yesmaximal, complemented
6#9409—31nomaximal, Sylow
7C22×C3×C22⋊C4—21no—
8C22×C3×C22⋊C4C2211yescharacteristic
9C2×#953—21no—
10C2×#930C2211yescharacteristic
11C2×#929—21no—
12C3×C4×C22⋊C4—21no—
13C3×C4×C22⋊C4—24no—
14C3×C4×C22⋊C4C2212yes—
15C4×#217C2211yescharacteristic
16#5207C2212yes—
17#4960C412yescomplemented
18#4957—22no—
19#4949C412yescomplemented
20#4525—22no—
21#4451—24no—
22#2539—31no—
23#2273D311yescharacteristic, complemented
24#1507—32no—
25#1462—31no—
26#1297—32no—
27C25×C3D411yescharacteristic
28C23×C3×C4—22no—
29C23×C3×C4D411yescharacteristic
30C23×C3×C4C2311yescharacteristic
31C22×#67—21no—
32C22×#67—22no—
33C22×D12C2×C411yescharacteristic, complemented
34C2×C3×C22⋊C4—22no—
35C2×C3×C22⋊C4—28no—
36C2×C3×C22⋊C4—22no—
37C2×C3×C22⋊C4—22no—
38C2×C3×C22⋊C4—22no—
39C2×C3×C22⋊C4C2×C412yes—
40C2×C3×C22⋊C4D412yes—
41C2×C3×C42—22no—
42C2×C3×C42—22no—
43#953—44no—
44C2×#217—21no—
45C2×#217—22no—
46#930—22no—
47#930—22no—
48#929—44no—
49C2×(C2×D3)⋊C4—22no—
50C2×(C2×D3)⋊C4—21no—
51C2×C4⋊Dic3C2×C411yescharacteristic, complemented
52C2×Dic3⋊C4—22no—
53C2×C4×Dic3—21no—
54C2×#208—22no—
55C3×#95—22no—
56C3×#95D414yescomplemented
57C3⋊#95C2×C412yescomplemented
58#830—22no—
59#830—22no—
60C2×#123—61no—
61C2×#121—31no—
62C2×#120—61no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4C2×D311yescharacteristic
65#407—32no—
66#399—32no—
67#392—32no—
68#390—62no—
69C4×C22⋊C4—21no—
70C4×C22⋊C4—24no—
71C4×C22⋊C4—31no—
72C4×C22⋊C4C2×D312yes—
73#327—64no—
74#323—62no—
75C24×C3—21no—
76C24×C3—24no—
77C24×C3—21no—
78C24×C3—42no—
79C24×C3—81no—
80C24×C3C22⋊C412yes—
81C24×C3C2×D411yescharacteristic
82C23×D3—21no—
83C22×C3×C4—21no—
84C22×C3×C4—24no—
85C22×C3×C4—28no—
86C22×C3×C4—21no—
87C22×C3×C4—42no—
88C22×C3×C4—81no—
89C22×C3×C4—82no—
90C22×C3×C4—82no—
91C22×C3×C4C22⋊C412yes—
92C22×C3×C4C22×C411yescharacteristic
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4C2×D412yes—
95C22×C3×C4#4112yes—
96C2×#67—44no—
97C2×#67—42no—
98C2×#67—42no—
99C2×#67—44no—
100C2×#67—41no—
101C2×#67—41no—
102C2×#67—82no—
103C22×Dic3—22no—
104C22×Dic3—21no—
105C2×D12—22no—
106C2×D12—81no—
107C3×C22⋊C4—28no—
108C3×C22⋊C4—28no—
109C3×C22⋊C4—24no—
110C3×C22⋊C4—24no—
111C3×C42—82no—
112C3×C42—82no—
113#217—24no—
114#217—44no—
115(C2×D3)⋊C4—44no—
116(C2×D3)⋊C4—42no—
117C4⋊Dic3—22no—
118Dic3⋊C4—44no—
119C4×Dic3—81no—
120#208—82no—
121C2×C3⋊C8—42no—
122C25#6711yescharacteristic
123C22×D4—31no—
124C22×D4—61no—
125C22×D4—62no—
126C23×C4—22no—
127C23×C4#6711yescharacteristic
128C23×C4C22×D311yescharacteristic
129C2×M16—62no—
130#123—124no—
131#121—62no—
132#121—62no—
133#120—124no—
134C2×C4⋊C4—31no—
135C2×C4⋊C4—62no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—28no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—61no—
142C2×C22⋊C4—62no—
143C2×C22⋊C4—62no—
144C2×C22⋊C4—61no—
145C2×C22⋊C4C4×D312yes—
146C2×C22⋊C4D1212yes—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42—61no—
150C22⋊C8—62no—
151C22⋊C8—62no—
152#95—22no—
153#95—32no—
154#95#6714yescomplemented
155C23×C3—22no—
156C23×C3—22no—
157C23×C3—24no—
158C23×C3—22no—
159C23×C3—22no—
160C23×C3—22no—
161C23×C3—22no—
162C23×C3—41no—
163C23×C3—42no—
164C23×C3—42no—
165C23×C3—42no—
166C23×C3—42no—
167C23×C3—81no—
168C23×C3—81no—
169C23×C3—88no—
170C23×C3#10414yes—
171C23×C3C2×C22⋊C411yescharacteristic
172C23×C3#12011yescharacteristic
173C23×C3#12311yescharacteristic
174C22×D3—41no—
175C22×D3—42no—
176C22×D3—41no—
177C22×D3—41no—
178C22×D3—42no—
179C2×C3×C4—28no—
180C2×C3×C4—22no—
181C2×C3×C4—42no—
182C2×C3×C4—41no—
183C2×C3×C4—42no—
184C2×C3×C4—41no—
185C2×C3×C4—44no—
186C2×C3×C4—42no—
187C2×C3×C4—44no—
188C2×C3×C4—42no—
189C2×C3×C4—81no—
190C2×C3×C4—88no—
191C2×C3×C4C4×D412yes—
192C2×C3×C4#12112yes—
193#67—88no—
194#67—82no—
195#67—82no—
196C2×Dic3—24no—
197C2×Dic3—42no—
198C2×Dic3—42no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202D12—82no—
203C3⋊C8—82no—
204C24—21no—
205C24—24no—
206C24—21no—
207C24—42no—
208C24—61no—
209C24—81no—
210C24(C2×D3)⋊C412yes—
211C24C2×#6711yescharacteristic
212C2×D4—62no—
213C2×D4—124no—
214C2×D4—122no—
215C2×D4—122no—
216C2×D4—124no—
217C2×D4—121no—
218C2×D4—121no—
219C2×D4—241no—
220C2×D4—242no—
221C22×C4—21no—
222C22×C4—24no—
223C22×C4—28no—
224C22×C4—21no—
225C22×C4—42no—
226C22×C4—62no—
227C22×C4—61no—
228C22×C4—81no—
229C22×C4—82no—
230C22×C4—82no—
231C22×C4(C2×D3)⋊C412yes—
232C22×C4C2×C4×D311yescharacteristic
233C22×C4C2×D1211yescharacteristic
234C22×C4#23512yes—
235C22×C4C2×#6712yes—
236M16—242no—
237C2×C8—122no—
238C4⋊C4—62no—
239C4⋊C4—124no—
240C22⋊C4—28no—
241C22⋊C4—28no—
242C22⋊C4—24no—
243C22⋊C4—24no—
244C22⋊C4—64no—
245C22⋊C4—124no—
246C22⋊C4—124no—
247C22⋊C4—122no—
248C42—82no—
249C42—82no—
250C42—241no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—24no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3—44no—
260C22×C3—44no—
261C22×C3—44no—
262C22×C3—44no—
263C22×C3—42no—
264C22×C3—42no—
265C22×C3—82no—
266C22×C3—84no—
267C22×C3#38611yescharacteristic
268C22×C3C2×#10412yes—
269C22×C3#45312yes—
270C2×D3—82no—
271C2×D3—82no—
272C2×D3—81no—
273C2×D3—81no—
274C2×D3—81no—
275C3×C4—22no—
276C3×C4—22no—
277C3×C4—48no—
278C3×C4—42no—
279Dic3—44no—
280Dic3—81no—
281C23—22no—
282C23—22no—
283C23—24no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—41no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—81no—
294C23—81no—
295C23—88no—
296C23—121no—
297C23—121no—
298C23—122no—
299C23—122no—
300C23—121no—
301C23#80514yes—
302C23C2×(C2×D3)⋊C411yescharacteristic
303C23#92911yescharacteristic
304C23#95311yescharacteristic
305D4—242no—
306D4—248no—
307D4—242no—
308D4—242no—
309C2×C4—28no—
310C2×C4—22no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—64no—
320C2×C4—81no—
321C2×C4—88no—
322C2×C4—122no—
323C2×C4—122no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4C4×#6712yes—
328C2×C4#93012yes—
329C8—242no—
330C2×C3—22no—
331C2×C3—22no—
332C2×C3—41no—
333C2×C3—44no—
334C2×C3#188711yescharacteristic
335C2×C3#189611yescharacteristic
336C2×C3#191311yescharacteristic
337D3—161no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—24no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—44no—
347C22—44no—
348C22—44no—
349C22—44no—
350C22—42no—
351C22—42no—
352C22—82no—
353C22—84no—
354C22—242no—
355C22—242no—
356C22—241no—
357C22—241no—
358C22—241no—
359C22C2×#80512yes—
360C22#521311yescharacteristic
361C22#524112yes—
362C4—22no—
363C4—22no—
364C4—48no—
365C4—42no—
366C4—124no—
367C4—241no—
368C3#940911yescharacteristic, complemented, Sylow
369C2—22no—
370C2—22no—
371C2—41no—
372C2—44no—
373C2—481no—
374C2#7153211yescharacteristic
375C2#7165011yescharacteristic
376C2#7165611yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2273C2
socleC22×C3#386
prime radical#82115C2
cosocleC23×C3×C4C23
hypercenter#2273D3
hypocenterC3#9409
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2273

Complements

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

NH ≅ G/Ntype
1. C3×#2273C2semidirect
3. #71539C2semidirect
4. #69444C2semidirect
5. #67618C2semidirect
17. #4960C4semidirect
19. #4949C4semidirect
23. #2273D3semidirect
33. C22×D12C2×C4semidirect
51. C2×C4⋊Dic3C2×C4semidirect
56. C3×#95D4semidirect
57. C3⋊#95C2×C4semidirect
154. #95#67semidirect
368. C3#9409semidirect

Maximal subgroups

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

subgroupindex
C3×#22732
#821152
#715392
#694442
#676182
#94093