Finite Group Databasebeta

← #10900950

Subgroups — #10900950

Order 768.

1230 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#888C211yesmaximal, characteristic
2C2×C3×#484C211yesmaximal, characteristic
3#82810C211yesmaximal, characteristic
4C2×#5044C211yesmaximal, characteristic
5#71554C211yesmaximal, characteristic
6#70617C214yesmaximal, complemented
7#70231C214yesmaximal, complemented
8#69925C211yesmaximal, characteristic
9#69910C211yesmaximal, characteristic
10#29642—31nomaximal, Sylow
11C23×Dic6—21no—
12C2×C3×C22⋊Q8C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C4⋊Dic3C2211yescharacteristic
15C22×Dic3⋊C4—21no—
16C22×C4×Dic3C2211yescharacteristic
17C2×#888—24no—
18C2×#888—24no—
19C2×#888—22no—
20C2×#888C2214yes—
21C3×#484—24no—
22C2×C3×#107C2212yes—
23C2×C3×#103C2212yes—
24C2×C3×C22⋊C8C2211yescharacteristic
25#5229C2212yes—
26C2×#835C2212yes—
27#5212C2211yescharacteristic
28C2×#830C2211yescharacteristic
29#5044—24no—
30#4978—22no—
31#4972C2212yes—
32C2×#807C2212yes—
33#4946—21no—
34#4943—21no—
35#4551C2214yes—
36#4550C2214yes—
37#4529C2214yes—
38C22×C4⋊Q8—31no—
39C2×#484—31no—
40C2×#484D311yescharacteristic
41#2855—31no—
42#2057—34no—
43#2024—34no—
44#1824—31no—
45#1824—31no—
46#1817—31no—
47C22×C3×Q8C2311yescharacteristic
48C23×C3×C4—21no—
49C23×C3×C4C2311yescharacteristic
50C23×Dic3—21no—
51C23×Dic3D411yescharacteristic
52C23×Dic3D411yescharacteristic
53C22×Dic6—28no—
54C22×Dic6—24no—
55C22×Dic6—48no—
56C22×C3×C8C2312yes—
57C3×C22⋊Q8—22no—
58C3×C22⋊Q8—22no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4D414yes—
63C2×C3×C4⋊C4C2311yescharacteristic
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C22⋊C4—21no—
66C2×#217—21no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3—22no—
69C2×C4⋊Dic3D414yes—
70C2×C4⋊Dic3C2312yes—
71C2×C4⋊Dic3C2312yes—
72C2×Dic3⋊C4—28no—
73C2×Dic3⋊C4—24no—
74C2×Dic3⋊C4—22no—
75C2×C4×Dic3—22no—
76C2×C4×Dic3—22no—
77C2×C4×Dic3D412yes—
78C2×C4×Dic3C2312yes—
79C22×C3⋊C8C2312yes—
80#888—216no—
81#888—28no—
82#888—28no—
83C3×#107—24no—
84C3×#103—24no—
85C3×C22⋊C8—22no—
86#835—24no—
87C3⋊#95—21no—
88C3⋊#95—22no—
89C3⋊#95—22no—
90C3⋊#95—22no—
91#830—22no—
92#807—24no—
93C23×Q8—61no—
94C2×C4⋊Q8—34no—
95C2×C4⋊Q8—62no—
96C2×C4⋊Q8—64no—
97C2×C4⋊Q8—64no—
98C2×C22⋊Q8C2×D311yescharacteristic
99C22×C4⋊C4—31no—
100C22×C4⋊C4—61no—
101C22×C4⋊C4C2×D311yescharacteristic
102C22×C42—31no—
103#484—24no—
104#484—64no—
105C2×#107—32no—
106C2×#107C2×D312yes—
107C2×#103—32no—
108C2×#103C2×D312yes—
109C2×C22⋊C8—31no—
110C2×C22⋊C8C2×D311yescharacteristic
111#391—32no—
112#389—32no—
113#387—62no—
114#385—61no—
115#381—61no—
116#380—31no—
117#340—34no—
118#340—34no—
119#336—34no—
120C24×C3C2×D411yescharacteristic
121C2×C3×Q8—21no—
122C2×C3×Q8—21no—
123C2×C3×Q8—24no—
124C22×C3×C4—24no—
125C22×C3×C4—24no—
126C22×C3×C4—22no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—44no—
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D412yes—
132C22×C3×C4C2×D412yes—
133C22×C3×C4C2411yescharacteristic
134C22×Dic3—22no—
135C22×Dic3—22no—
136C22×Dic3—22no—
137C22×Dic3—24no—
138C22×Dic3—24no—
139C22×Dic3—22no—
140C22×Dic3—22no—
141C22×Dic3—44no—
142C22×Dic3Q1614yes—
143C22×Dic3C2×D412yes—
144C22×Dic3C2×D412yes—
145C2×Dic6—216no—
146C2×Dic6—432no—
147C2×Dic6—48no—
148C2×Dic6—48no—
149C2×C3×C8—22no—
150C2×C3×C8—24no—
151C3×C4⋊C4—24no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—42no—
156C3×C4⋊C4C2×D414yes—
157C3×C22⋊C4—42no—
158#217—42no—
159C4⋊Dic3—24no—
160C4⋊Dic3—22no—
161C4⋊Dic3—24no—
162C4⋊Dic3C2×D414yes—
163Dic3⋊C4—216no—
164Dic3⋊C4—44no—
165C4×Dic3—24no—
166C4×Dic3—22no—
167C4×Dic3—22no—
168C2×C3⋊C8—22no—
169C2×C3⋊C8—24no—
170C22×Q8—64no—
171C22×Q8—68no—
172C22×Q8—128no—
173C22×Q8C22×D311yescharacteristic
174C23×C4—21no—
175C23×C4—31no—
176C23×C4—31no—
177C23×C4—61no—
178C23×C4C22×D311yescharacteristic
179C22×C8—32no—
180C22×C8C22×D312yes—
181C4⋊Q8—68no—
182C4⋊Q8—616no—
183C4⋊Q8—68no—
184C22⋊Q8—22no—
185C22⋊Q8—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—34no—
190C2×C4⋊C4—32no—
191C2×C4⋊C4—32no—
192C2×C4⋊C4—64no—
193C2×C4⋊C4—68no—
194C2×C4⋊C4—61no—
195C2×C4⋊C4—62no—
196C2×C4⋊C4—62no—
197C2×C4⋊C4#6714yes—
198C2×C4⋊C4C22×D311yescharacteristic
199C2×C4⋊C4C22×D312yes—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—61no—
202C2×C42—32no—
203C2×C42—32no—
204C2×C42—62no—
205C2×C42—62no—
206#107—24no—
207#107—64no—
208#103—24no—
209#103—64no—
210C22⋊C8—22no—
211C22⋊C8—62no—
212#95—62no—
213#95—62no—
214#95—62no—
215#95—61no—
216C23×C3—22no—
217C23×C3—22no—
218C23×C3#12012yes—
219C23×C3C2×Q1612yes—
220C23×C3#13712yes—
221C23×C3C22×D411yescharacteristic
222C3×Q8—42no—
223C3×Q8—42no—
224C2×C3×C4—22no—
225C2×C3×C4—24no—
226C2×C3×C4—24no—
227C2×C3×C4—24no—
228C2×C3×C4—48no—
229C2×C3×C4—44no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4#12012yes—
235C2×C3×C4C22×D411yescharacteristic
236C2×C3×C4C22×D411yescharacteristic
237C2×C3×C42−1+414yes—
238C2×Dic3—28no—
239C2×Dic3—22no—
240C2×Dic3—22no—
241C2×Dic3—22no—
242C2×Dic3—22no—
243C2×Dic3—28no—
244C2×Dic3—24no—
245C2×Dic3—48no—
246C2×Dic3—44no—
247C2×Dic3—42no—
248C2×Dic3—42no—
249C2×Dic3—42no—
250C2×Dic3C2×Q1614yes—
251C2×Dic3AGL(1,ℤ/8)14yes—
252Dic6—432no—
253C3×C8—42no—
254C3⋊C8—42no—
255C24D3×D411yescharacteristic
256C2×Q8—21no—
257C2×Q8—24no—
258C2×Q8—21no—
259C2×Q8—616no—
260C2×Q8—128no—
261C2×Q8—1232no—
262C2×Q8—128no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—34no—
269C22×C4—32no—
270C22×C4—32no—
271C22×C4—44no—
272C22×C4—64no—
273C22×C4—62no—
274C22×C4—62no—
275C22×C4—62no—
276C22×C4—64no—
277C22×C4—62no—
278C22×C4—62no—
279C22×C4—124no—
280C22×C4D3×D411yescharacteristic
281C22×C4D3×D412yes—
282C22×C4C2×#6712yes—
283C22×C4C23×D311yescharacteristic
284C2×C8—24no—
285C2×C8—22no—
286C2×C8—62no—
287C2×C8—64no—
288C4⋊C4—24no—
289C4⋊C4—22no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—34no—
293C4⋊C4—42no—
294C4⋊C4—616no—
295C4⋊C4—64no—
296C4⋊C4—62no—
297C4⋊C4—64no—
298C4⋊C4—124no—
299C4⋊C4C2×#6714yes—
300C22⋊C4—42no—
301C22⋊C4—122no—
302C42—64no—
303C42—62no—
304C42—62no—
305C22×C3—22no—
306C22×C3—24no—
307C22×C3—24no—
308C22×C3—22no—
309C22×C3#45114yes—
310C22×C3C2×#12011yescharacteristic
311C22×C3#53612yes—
312C22×C3C22×Q1611yescharacteristic
313C22×C3C2×AGL(1,ℤ/8)12yes—
314C22×C3C2×#13711yescharacteristic
315C3×C4—22no—
316C3×C4—44no—
317C3×C4—81no—
318C3×C4—81no—
319C3×C4#53612yes—
320C3×C4#53612yes—
321Dic3—24no—
322Dic3—24no—
323Dic3—44no—
324Dic3—82no—
325C23—22no—
326C23—22no—
327C23D3×Q1612yes—
328C23#91812yes—
329C23#93712yes—
330C23C2×D3×D411yescharacteristic
331Q8—42no—
332Q8—42no—
333Q8—1232no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—34no—
339C2×C4—34no—
340C2×C4—48no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—68no—
347C2×C4—68no—
348C2×C4—62no—
349C2×C4—62no—
350C2×C4—62no—
351C2×C4—62no—
352C2×C4—64no—
353C2×C4—128no—
354C2×C4—124no—
355C2×C4—122no—
356C2×C4—122no—
357C2×C4—122no—
358C2×C4#93712yes—
359C2×C4C2×D3×D411yescharacteristic
360C2×C4#101014yes—
361C2×C4C22×#6711yescharacteristic
362C8—42no—
363C8—122no—
364C2×C3—24no—
365C2×C3#261411yescharacteristic
366C2×C3C2×#45111yescharacteristic
367C2×C3#300311yescharacteristic
368C2×C3#323012yes—
369C2×C3#323912yes—
370C22—24no—
371C22—24no—
372C22—22no—
373C22—22no—
374C22#518614yes—
375C22#564812yes—
376C22C2×D3×Q1611yescharacteristic
377C22C2×#91811yescharacteristic
378C22D3×AGL(1,ℤ/8)12yes—
379C22C2×#93711yescharacteristic
380C4—22no—
381C4—44no—
382C4—64no—
383C4—64no—
384C4—81no—
385C4—81no—
386C4—124no—
387C4—242no—
388C4#564812yes—
389C4#582512yes—
390C3#2964211yescharacteristic, complemented, Sylow
391C2—24no—
392C2#8145812yes—
393C2#8145912yes—
394C2#8286411yescharacteristic
395C2C2×#518611yescharacteristic
396C2#8394411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#484C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#484D3
hypocenterC3#29642
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#484

Complements

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

NH ≅ G/Ntype
6. #70617C2semidirect
7. #70231C2semidirect
390. C3#29642semidirect

Maximal subgroups

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

subgroupindex
C22×#8882
C2×C3×#4842
#828102
C2×#50442
#715542
#706172
#702312
#699252
#699102
#296423