Finite Group Databasebeta

← #10693006

Subgroups — #10693006

Order 768.

838 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 838 conjugacy classes of subgroups — too many to draw. Only its 103 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
2#82125C211yesmaximal, characteristic
3#71539C211yesmaximal, characteristic, complemented
4#69454C212yesmaximal, complemented
5#67616C212yesmaximal, complemented
6#9479—31nomaximal, Sylow
7C22×#235C2211yescharacteristic
8C22×C3×C22⋊C4C2211yescharacteristic
9C2×#953—21no—
10C3×#388—24no—
11C3×#380—21no—
12C3×#379—21no—
13C3×C4×C22⋊C4C2212yes—
14#5213—21no—
15#5211C2211yescharacteristic
16#5207C2212yes—
17#4960C412yescomplemented
18#4956—22no—
19#4947C412yescomplemented
20C2×#805—24no—
21#4525—22no—
22#2404—31no—
23#2312D311yescharacteristic, complemented
24#1789—32no—
25#1462—31no—
26#1284—32no—
27C25×C3D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C22×#67—21no—
31C22×#67—22no—
32C2×#235—22no—
33C2×#235—22no—
34C2×#235—44no—
35C22×D12C2×C411yescharacteristic
36C22×C4×D3—21no—
37C22×Dic6C2×C411yescharacteristic
38C2×C3×C4⋊C4—22no—
39C2×C3×C22⋊C4—22no—
40C2×C3×C22⋊C4—22no—
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4C2×C412yes—
45C2×C3×C22⋊C4D412yes—
46C2×C3×C42D414yescomplemented
47#953—44no—
48C2×#217—22no—
49C2×#217—21no—
50C2×(C2×D3)⋊C4—22no—
51C2×#208—22no—
52C3×#95—22no—
53C3×#95—22no—
54C3⋊#95—22no—
55C3⋊#95—21no—
56C3⋊#95C2×C412yes—
57#830—22no—
58#830—22no—
59#805—48no—
60C22×#41—31no—
61C2×#120—61no—
62C22×C22⋊C4C2×D311yescharacteristic
63C2×#104—64no—
64#407—32no—
65#393—32no—
66#392—32no—
67#388—24no—
68#386—61no—
69#386—62no—
70#381—31no—
71#380—21no—
72#379—21no—
73C4×C22⋊C4C2×D312yes—
74#323—62no—
75C24×C3—21no—
76C24×C3—21no—
77C24×C3—42no—
78C24×C3—42no—
79C24×C3—81no—
80C24×C3C22⋊C412yes—
81C24×C3C2×D411yescharacteristic
82C23×D3—21no—
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4—24no—
86C22×C3×C4—42no—
87C22×C3×C4—81no—
88C22×C3×C4C22⋊C412yes—
89C22×C3×C4C22×C411yescharacteristic
90C22×C3×C4C2×D411yescharacteristic
91C22×C3×C4C2×D412yes—
92C22×C3×C4#4112yes—
93C2×#67—24no—
94C2×#67—42no—
95C2×#67—41no—
96C2×#67—41no—
97C2×#67—44no—
98C2×#67—44no—
99C2×#67—82no—
100C22×Dic3—21no—
101C22×Dic3—22no—
102#235—28no—
103#235—44no—
104#235—44no—
105#235—44no—
106C2×D12—22no—
107C2×D12—42no—
108C2×C4×D3—42no—
109C2×C4×D3—42no—
110C2×C4×D3—42no—
111C2×Dic6—22no—
112C2×Dic6—42no—
113C3×C4⋊C4—82no—
114C3×C22⋊C4—24no—
115C3×C22⋊C4—24no—
116C3×C22⋊C4—44no—
117C3×C22⋊C4—82no—
118C3×C22⋊C4—82no—
119C3×C42—44no—
120#217—44no—
121#217—42no—
122(C2×D3)⋊C4—82no—
123#208—44no—
124C2×C3⋊C8—42no—
125C25#6711yescharacteristic
126C2×#41—62no—
127C2×#41—62no—
128C2×#41—124no—
129C22×Q8—31no—
130C22×D4—31no—
131C22×D4—61no—
132C22×D4—62no—
133C23×C4—61no—
134C23×C4#6711yescharacteristic
135C23×C4C22×D311yescharacteristic
136C2×M16—62no—
137#120—124no—
138C2×C4⋊C4—22no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—62no—
145C2×C22⋊C4—61no—
146C2×C22⋊C4—62no—
147C2×C22⋊C4C4×D312yes—
148C2×C22⋊C4D1212yes—
149C2×C42#6714yescomplemented
150#104—128no—
151C22⋊C8—62no—
152C22⋊C8—62no—
153#95—22no—
154#95—22no—
155#95—32no—
156#95—62no—
157#95—61no—
158C23×C3—22no—
159C23×C3—22no—
160C23×C3—22no—
161C23×C3—22no—
162C23×C3—22no—
163C23×C3—41no—
164C23×C3—42no—
165C23×C3—42no—
166C23×C3—42no—
167C23×C3—42no—
168C23×C3—42no—
169C23×C3—42no—
170C23×C3—81no—
171C23×C3—81no—
172C23×C3—88no—
173C23×C3C2×C22⋊C411yescharacteristic
174C23×C3#12011yescharacteristic
175C23×C3#12311yescharacteristic
176C22×D3—22no—
177C22×D3—42no—
178C22×D3—41no—
179C22×D3—41no—
180C22×D3—42no—
181C2×C3×C4—24no—
182C2×C3×C4—22no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—42no—
186C2×C3×C4—42no—
187C2×C3×C4—42no—
188C2×C3×C4—44no—
189C2×C3×C4—44no—
190C2×C3×C4—44no—
191C2×C3×C4—81no—
192C2×C3×C4C4×D412yes—
193C2×C3×C4#12112yes—
194#67—44no—
195#67—44no—
196#67—88no—
197C2×Dic3—22no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—42no—
201C2×Dic3—42no—
202C2×Dic3—42no—
203D12—24no—
204D12—42no—
205C4×D3—48no—
206Dic6—24no—
207Dic6—42no—
208C3⋊C8—82no—
209C24—21no—
210C24—21no—
211C24—42no—
212C24—42no—
213C24—61no—
214C24—81no—
215C24(C2×D3)⋊C412yes—
216C24C2×#6711yescharacteristic
217#41—68no—
218#41—124no—
219#41—124no—
220#41—124no—
221C2×Q8—62no—
222C2×Q8—122no—
223C2×D4—64no—
224C2×D4—62no—
225C2×D4—122no—
226C2×D4—122no—
227C2×D4—121no—
228C2×D4—121no—
229C2×D4—124no—
230C2×D4—124no—
231C2×D4—242no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—24no—
235C22×C4—42no—
236C22×C4—61no—
237C22×C4—62no—
238C22×C4—81no—
239C22×C4—122no—
240C22×C4—122no—
241C22×C4—122no—
242C22×C4(C2×D3)⋊C412yes—
243C22×C4C2×C4×D311yescharacteristic
244C22×C4C2×D1211yescharacteristic
245C22×C4#23512yes—
246C22×C4C2×#6712yes—
247M16—124no—
248C2×C8—122no—
249C4⋊C4—82no—
250C22⋊C4—24no—
251C22⋊C4—24no—
252C22⋊C4—44no—
253C22⋊C4—82no—
254C22⋊C4—82no—
255C22⋊C4—124no—
256C22⋊C4—122no—
257C22⋊C4—242no—
258C42—44no—
259C22×C3—21no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—44no—
267C22×C3—44no—
268C22×C3—42no—
269C22×C3—42no—
270C22×C3—42no—
271C22×C3—82no—
272C22×C3—84no—
273C22×C3—84no—
274C22×C3#38611yescharacteristic
275C22×C3#42112yes—
276C22×C3B(4,𝔽2)12yes—
277C2×D3—44no—
278C2×D3—42no—
279C2×D3—82no—
280C2×D3—81no—
281C2×D3—81no—
282C3×C4—22no—
283C3×C4—22no—
284C3×C4—42no—
285C3×C4—84no—
286Dic3—42no—
287Dic3—82no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—22no—
293C23—41no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23—42no—
300C23—62no—
301C23—81no—
302C23—81no—
303C23—88no—
304C23—122no—
305C23—121no—
306C23—121no—
307C23—122no—
308C23C2×(C2×D3)⋊C411yescharacteristic
309C23#92911yescharacteristic
310C23#95311yescharacteristic
311Q8—64no—
312Q8—122no—
313D4—64no—
314D4—122no—
315D4—124no—
316D4—124no—
317D4—248no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—62no—
329C2×C4—81no—
330C2×C4—128no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—122no—
334C2×C4—122no—
335C2×C4—122no—
336C2×C4C4×#6712yes—
337C2×C4#93012yes—
338C8—242no—
339C2×C3—22no—
340C2×C3—22no—
341C2×C3—41no—
342C2×C3—82no—
343C2×C3#188911yescharacteristic
344C2×C3#189711yescharacteristic
345C2×C3#191111yescharacteristic
346D3—82no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—44no—
355C22—44no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—82no—
360C22—84no—
361C22—84no—
362C22—124no—
363C22—122no—
364C22—242no—
365C22—241no—
366C22—241no—
367C22#500512yes—
368C22#521311yescharacteristic
369C22#524412yes—
370C4—22no—
371C4—22no—
372C4—42no—
373C4—84no—
374C4—122no—
375C4—242no—
376C3#947911yescharacteristic, complemented, Sylow
377C2—22no—
378C2—22no—
379C2—41no—
380C2—82no—
381C2—242no—
382C2#7159111yescharacteristic
383C2#7165211yescharacteristic
384C2#7165411yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2312C2
socleC22×C3#386
prime radical#82125C2
cosocleC23×C3×C4C23
hypercenter#2312D3
hypocenterC3#9479
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
1. C3×#2312C2semidirect
3. #71539C2semidirect
4. #69454C2semidirect
5. #67616C2semidirect
17. #4960C4semidirect
19. #4947C4semidirect
23. #2312D3semidirect
46. C2×C3×C42D4semidirect
149. C2×C42#67semidirect
376. C3#9479semidirect

Maximal subgroups

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

subgroupindex
C3×#23122
#821252
#715392
#694542
#676162
#94793