Finite Group Databasebeta

← #10690211

Subgroups — #10690211

Order 768.

1224 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 1224 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×#2312C211yesmaximal, characteristic, complemented
2C2×#4960C211yesmaximal, characteristic
3C2×#4525C211yesmaximal, characteristic, complemented
4#69521C212yesmaximal, complemented
5#67624C212yesmaximal, complemented
6#9020—31nomaximal, Sylow
7C23×#67—21no—
8C23×D12C2211yescharacteristic
9C22×C3×C22⋊C4C2211yescharacteristic
10C22×(C2×D3)⋊C4—21no—
11C3×#388—24no—
12C3×#380C2212yes—
13C3×#379—21no—
14C3×C4×C22⋊C4—21no—
15C2×C3⋊#95C2211yescharacteristic
16C2×#830C2212yes—
17C2×#809—24no—
18#4960—22no—
19#4960C414yescomplemented
20#4525—22no—
21C2×#392—31no—
22#2312D311yescharacteristic, complemented
23#1786—32no—
24C2×#323—31no—
25#1294—32no—
26C25×C3D411yescharacteristic
27C24×D3—21no—
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C22×#67—24no—
31C22×#67—44no—
32C22×#67—44no—
33C22×#67—44no—
34C23×Dic3—21no—
35C22×D12—22no—
36C22×D12—22no—
37C22×D12—21no—
38C22×D12—44no—
39C22×D12C2×C412yes—
40C2×C3×C4⋊C4D414yescomplemented
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4—22no—
45C2×C3×C22⋊C4—22no—
46C2×C3×C22⋊C4C2×C412yes—
47C2×C3×C22⋊C4D412yes—
48C2×C3×C42—22no—
49C2×(C2×D3)⋊C4—24no—
50C2×(C2×D3)⋊C4—44no—
51C22×C3⋊C8—22no—
52C3×#95—22no—
53C3×#95—22no—
54C3⋊#95—21no—
55C3⋊#95C2×C412yes—
56#830—22no—
57#830—22no—
58#809—48no—
59C23×D4—31no—
60C23×D4—61no—
61C22×C22⋊C4—61no—
62C22×C22⋊C4C2×D311yescharacteristic
63C2×#102—64no—
64C2×C22⋊C8—32no—
65#392—34no—
66#392—62no—
67#388—24no—
68#380C2×D312yes—
69#379—21no—
70C4×C22⋊C4—21no—
71C2×#95—31no—
72#323—62no—
73C24×C3—21no—
74C24×C3—22no—
75C24×C3—21no—
76C24×C3—22no—
77C24×C3—42no—
78C24×C3—42no—
79C24×C3C22⋊C412yes—
80C24×C3C2×D411yescharacteristic
81C23×D3—24no—
82C23×D3—22no—
83C23×D3—42no—
84C23×D3—42no—
85C23×D3—42no—
86C23×D3—41no—
87C23×D3—41no—
88C23×D3—42no—
89C23×D3—42no—
90C22×C3×C4—21no—
91C22×C3×C4—21no—
92C22×C3×C4—24no—
93C22×C3×C4—42no—
94C22×C3×C4—42no—
95C22×C3×C4C22⋊C412yes—
96C22×C3×C4C22×C411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D412yes—
99C22×C3×C4#4112yes—
100C2×#67—48no—
101C2×#67—48no—
102C2×#67—48no—
103C2×#67—432no—
104C22×Dic3—24no—
105C22×Dic3—22no—
106C22×Dic3—42no—
107C22×Dic3—42no—
108C2×D12—28no—
109C2×D12—24no—
110C2×D12—44no—
111C2×D12—48no—
112C2×D12—44no—
113C2×D12—44no—
114C2×D12—42no—
115C3×C4⋊C4—44no—
116C3×C22⋊C4—24no—
117C3×C22⋊C4—24no—
118C3×C22⋊C4—42no—
119C3×C22⋊C4—42no—
120C3×C22⋊C4—82no—
121C3×C22⋊C4—82no—
122C3×C42—82no—
123(C2×D3)⋊C4—48no—
124C2×C3⋊C8—44no—
125C2×C3⋊C8—42no—
126C25—61no—
127C25#6711yescharacteristic
128C22×D4—32no—
129C22×D4—64no—
130C22×D4—61no—
131C22×D4—62no—
132C22×D4—62no—
133C22×D4—124no—
134C22×D4—124no—
135C22×D4—124no—
136C22×D4—124no—
137C23×C4—61no—
138C23×C4#6711yescharacteristic
139C23×C4C22×D311yescharacteristic
140C22×C8—62no—
141C2×C4⋊C4#6714yescomplemented
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—64no—
148C2×C22⋊C4—124no—
149C2×C22⋊C4C4×D312yes—
150C2×C22⋊C4D1212yes—
151C2×C42—22no—
152#102—128no—
153C22⋊C8—62no—
154C22⋊C8—62no—
155#95—22no—
156#95—22no—
157#95—32no—
158#95—61no—
159C23×C3—22no—
160C23×C3—22no—
161C23×C3—22no—
162C23×C3—22no—
163C23×C3—22no—
164C23×C3—42no—
165C23×C3—42no—
166C23×C3—48no—
167C23×C3—42no—
168C23×C3—42no—
169C23×C3—42no—
170C23×C3—42no—
171C23×C3—81no—
172C23×C3—81no—
173C23×C3—84no—
174C23×C3C2×C22⋊C411yescharacteristic
175C23×C3#12011yescharacteristic
176C23×C3#12311yescharacteristic
177C23×C3#13512yes—
178C23×C3AGL(1,ℤ/8)12yes—
179C22×D3—24no—
180C22×D3—42no—
181C22×D3—42no—
182C22×D3—44no—
183C22×D3—48no—
184C22×D3—42no—
185C22×D3—48no—
186C22×D3—42no—
187C22×D3—42no—
188C22×D3—42no—
189C22×D3—44no—
190C22×D3—84no—
191C22×D3—84no—
192C22×D3—82no—
193C22×D3—82no—
194C22×D3—82no—
195C22×D3—82no—
196C2×C3×C4—24no—
197C2×C3×C4—22no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4—42no—
203C2×C3×C4—44no—
204C2×C3×C4—44no—
205C2×C3×C4—44no—
206C2×C3×C4—81no—
207C2×C3×C4C4×D412yes—
208C2×C3×C4#12112yes—
209#67—432no—
210C2×Dic3—24no—
211C2×Dic3—48no—
212C2×Dic3—42no—
213C2×Dic3—42no—
214D12—28no—
215D12—48no—
216D12—44no—
217C3⋊C8—82no—
218C24—22no—
219C24—21no—
220C24—21no—
221C24—22no—
222C24—42no—
223C24—42no—
224C24—64no—
225C24—62no—
226C24—122no—
227C24—122no—
228C24—122no—
229C24—121no—
230C24—122no—
231C24—122no—
232C24—121no—
233C24(C2×D3)⋊C412yes—
234C24C2×#6711yescharacteristic
235C2×D4—68no—
236C2×D4—64no—
237C2×D4—124no—
238C2×D4—124no—
239C2×D4—128no—
240C2×D4—124no—
241C2×D4—128no—
242C2×D4—128no—
243C2×D4—128no—
244C2×D4—1232no—
245C2×D4—122no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—24no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—64no—
252C22×C4—62no—
253C22×C4—122no—
254C22×C4—122no—
255C22×C4(C2×D3)⋊C412yes—
256C22×C4C2×C4×D311yescharacteristic
257C22×C4C2×D1211yescharacteristic
258C22×C4#23512yes—
259C22×C4C2×#6712yes—
260C2×C8—124no—
261C2×C8—122no—
262C4⋊C4—44no—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—82no—
268C22⋊C4—82no—
269C22⋊C4—128no—
270C42—82no—
271C22×C3—24no—
272C22×C3—24no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—44no—
278C22×C3—42no—
279C22×C3—42no—
280C22×C3—42no—
281C22×C3—82no—
282C22×C3—84no—
283C22×C3—84no—
284C22×C3#38611yescharacteristic
285C22×C3#41611yescharacteristic
286C22×C3#41811yescharacteristic
287C22×C3#44912yes—
288C22×C3#45312yes—
289C22×C3B(4,𝔽2)12yes—
290C22×C3#48212yes—
291C2×D3—44no—
292C2×D3—44no—
293C2×D3—44no—
294C2×D3—82no—
295C2×D3—82no—
296C2×D3—88no—
297C2×D3—88no—
298C2×D3—82no—
299C2×D3—82no—
300C3×C4—22no—
301C3×C4—22no—
302C3×C4—42no—
303C3×C4—84no—
304Dic3—44no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—42no—
311C23—42no—
312C23—48no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—64no—
318C23—81no—
319C23—81no—
320C23—84no—
321C23—122no—
322C23—128no—
323C23—122no—
324C23—122no—
325C23—124no—
326C23—128no—
327C23—122no—
328C23—124no—
329C23—122no—
330C23—122no—
331C23—244no—
332C23—244no—
333C23—242no—
334C23—242no—
335C23—242no—
336C23—242no—
337C23C2×(C2×D3)⋊C411yescharacteristic
338C23#92911yescharacteristic
339C23#94912yes—
340C23#95012yes—
341C23#95311yescharacteristic
342D4—68no—
343D4—128no—
344D4—124no—
345D4—1232no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—64no—
357C2×C4—81no—
358C2×C4—128no—
359C2×C4—122no—
360C2×C4—122no—
361C2×C4C4×#6712yes—
362C2×C4#93012yes—
363C8—242no—
364C2×C3—21no—
365C2×C3—21no—
366C2×C3—22no—
367C2×C3—82no—
368C2×C3#160014yes—
369C2×C3#187511yescharacteristic
370C2×C3#188711yescharacteristic
371C2×C3#188911yescharacteristic
372D3—84no—
373C22—24no—
374C22—24no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—44no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—82no—
384C22—84no—
385C22—84no—
386C22—124no—
387C22—124no—
388C22—124no—
389C22—242no—
390C22—242no—
391C22—248no—
392C22—242no—
393C22—248no—
394C22—242no—
395C22#500111yescharacteristic
396C22#500211yescharacteristic
397C22#500312yes—
398C22#521311yescharacteristic
399C22#523812yes—
400C22#524112yes—
401C22#524412yes—
402C4—22no—
403C4—22no—
404C4—42no—
405C4—84no—
406C4—124no—
407C3#902011yescharacteristic, complemented, Sylow
408C2—21no—
409C2—21no—
410C2—22no—
411C2—82no—
412C2—244no—
413C2#7158611yescharacteristic
414C2#7165011yescharacteristic
415C2#7165411yescharacteristic
416C2#7188414yes—
subgroupG/H
centerC23#953
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2312C2
socleC23×C3#120
prime radicalC2×#4960C2
cosocleC23×C3×C4C23
hypercenter#2312D3
hypocenterC3#9020
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
1. C3×#2312C2semidirect
3. C2×#4525C2semidirect
4. #69521C2semidirect
5. #67624C2semidirect
19. #4960C4semidirect
22. #2312D3semidirect
40. C2×C3×C4⋊C4D4semidirect
141. C2×C4⋊C4#67semidirect
407. C3#9020semidirect

Maximal subgroups

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

subgroupindex
C3×#23122
C2×#49602
C2×#45252
#695212
#676242
#90203