Finite Group Databasebeta

← #10900944

Subgroups — #10900944

Order 768.

1230 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#888C211yesmaximal, characteristic
2C2×#4971C211yesmaximal, characteristic
3C2×#4721C211yesmaximal, characteristic
4#74000C211yesmaximal, characteristic
5C3×#1823C211yesmaximal, characteristic
6#69910C211yesmaximal, characteristic
7#69501C211yesmaximal, characteristic
8#68910C214yesmaximal, complemented
9#68827C214yesmaximal, complemented
10#29641—31nomaximal, Sylow
11C23×Dic6—21no—
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C4⋊Dic3C2211yescharacteristic
14C2×#924C2211yescharacteristic
15C22×Dic3⋊C4—21no—
16C22×C4×Dic3C2211yescharacteristic
17C2×#888—24no—
18C2×#888—24no—
19C2×#888—22no—
20C2×#888C2214yes—
21C2×C3×C22⋊C8C2211yescharacteristic
22C3×#380C2211yescharacteristic
23C2×#830C2211yescharacteristic
24C2×#817C2212yes—
25C2×#816C2212yes—
26#4984C2212yes—
27#4973C2212yes—
28#4971—24no—
29C2×#810C2212yes—
30C2×#808C2212yes—
31#4950—21no—
32#4943—21no—
33#4721—24no—
34#4653—22no—
35C3×#340C2214yes—
36#4550C2214yes—
37#4529C2214yes—
38C22×C4⋊Q8—31no—
39C2×#483—31no—
40C2×#483—31no—
41#2855—31no—
42#2058—34no—
43#2024—34no—
44#1823—31no—
45#1823D311yescharacteristic
46#1817—31no—
47C23×C3×C4—21no—
48C23×C3×C4C2311yescharacteristic
49C23×Dic3—21no—
50C23×Dic3D411yescharacteristic
51C23×Dic3D411yescharacteristic
52C22×Dic6—28no—
53C22×Dic6—24no—
54C22×Dic6—44no—
55C22×Dic6—44no—
56C22×Dic6C2311yescharacteristic
57C22×C3×C8C2312yes—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4D414yes—
61C2×C3×C4⋊C4C2312yes—
62C2×C3×C4⋊C4C2312yes—
63C2×C3×C22⋊C4—21no—
64C2×#217—21no—
65C2×C4⋊Dic3—22no—
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3—22no—
68C2×C4⋊Dic3D414yes—
69C2×C4⋊Dic3C2311yescharacteristic
70C2×C4⋊Dic3C2312yes—
71#924—22no—
72#924—22no—
73C2×Dic3⋊C4—28no—
74C2×Dic3⋊C4—24no—
75C2×Dic3⋊C4—21no—
76C2×C4×Dic3—22no—
77C2×C4×Dic3—22no—
78C2×C4×Dic3D412yes—
79C2×C4×Dic3C2312yes—
80C22×C3⋊C8C2312yes—
81#888—216no—
82#888—28no—
83#888—28no—
84C3×C22⋊C8—22no—
85C3×#95—21no—
86C3⋊#95—22no—
87C3⋊#95—22no—
88C3⋊#95—22no—
89#830—22no—
90#817—24no—
91#816—24no—
92#810—24no—
93#808—24no—
94C23×Q8—61no—
95C2×C4⋊Q8—34no—
96C2×C4⋊Q8—62no—
97C2×C4⋊Q8—64no—
98C2×C4⋊Q8—64no—
99C2×C22⋊Q8—31no—
100C22×C4⋊C4—31no—
101C22×C4⋊C4—61no—
102C22×C4⋊C4C2×D311yescharacteristic
103C22×C42—31no—
104#483—64no—
105#483—64no—
106C2×#106—32no—
107C2×#106—32no—
108C2×#103—32no—
109C2×#103—32no—
110C2×C22⋊C8—31no—
111C2×C22⋊C8C2×D311yescharacteristic
112#391—32no—
113#389—32no—
114#387—62no—
115#385—61no—
116#381—61no—
117#380C2×D311yescharacteristic
118#340—34no—
119#340C2×D314yes—
120#336—34no—
121C24×C3C2×D411yescharacteristic
122C22×C3×C4—24no—
123C22×C3×C4—24no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—42no—
127C22×C3×C4—42no—
128C22×C3×C4C2×D412yes—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D412yes—
131C22×C3×C4C2411yescharacteristic
132C22×Dic3—22no—
133C22×Dic3—22no—
134C22×Dic3—22no—
135C22×Dic3—24no—
136C22×Dic3—24no—
137C22×Dic3—22no—
138C22×Dic3—21no—
139C22×Dic3—21no—
140C22×Dic3—42no—
141C22×Dic3—42no—
142C22×Dic3SD1614yes—
143C22×Dic3C2×D412yes—
144C22×Dic3C2×D412yes—
145C2×Dic6—216no—
146C2×Dic6—21no—
147C2×Dic6—21no—
148C2×Dic6—24no—
149C2×Dic6—48no—
150C2×Dic6—432no—
151C2×Dic6—48no—
152C2×C3×C8—22no—
153C2×C3×C8—24no—
154C3×C4⋊C4—24no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—24no—
157C3×C4⋊C4C2×D414yes—
158C3×C22⋊C4—42no—
159#217—42no—
160C4⋊Dic3—24no—
161C4⋊Dic3—22no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—44no—
165C4⋊Dic3C2×D414yes—
166Dic3⋊C4—216no—
167Dic3⋊C4—42no—
168C4×Dic3—24no—
169C4×Dic3—22no—
170C4×Dic3—22no—
171C2×C3⋊C8—22no—
172C2×C3⋊C8—24no—
173C22×Q8—31no—
174C22×Q8—64no—
175C22×Q8—68no—
176C22×Q8—124no—
177C22×Q8—124no—
178C23×C4—21no—
179C23×C4—31no—
180C23×C4—31no—
181C23×C4—61no—
182C23×C4C22×D311yescharacteristic
183C22×C8—32no—
184C22×C8C22×D312yes—
185C4⋊Q8—68no—
186C4⋊Q8—616no—
187C4⋊Q8—68no—
188C22⋊Q8—62no—
189C22⋊Q8—62no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—34no—
193C2×C4⋊C4—32no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4—64no—
196C2×C4⋊C4—68no—
197C2×C4⋊C4—62no—
198C2×C4⋊C4—61no—
199C2×C4⋊C4—62no—
200C2×C4⋊C4—61no—
201C2×C4⋊C4D1214yes—
202C2×C4⋊C4C22×D312yes—
203C2×C4⋊C4C22×D312yes—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—61no—
206C2×C42—32no—
207C2×C42—32no—
208C2×C42—62no—
209C2×C42—62no—
210#106—64no—
211#106—64no—
212#103—64no—
213#103—64no—
214C22⋊C8—22no—
215C22⋊C8—62no—
216#95—21no—
217#95—62no—
218#95—62no—
219#95—62no—
220C23×C3—22no—
221C23×C3—22no—
222C23×C3#12012yes—
223C23×C3C2×SD1612yes—
224C23×C3AGL(1,ℤ/8)12yes—
225C23×C3C22×D411yescharacteristic
226C2×C3×C4—22no—
227C2×C3×C4—24no—
228C2×C3×C4—24no—
229C2×C3×C4—24no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—48no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4#12012yes—
237C2×C3×C4C22×D411yescharacteristic
238C2×C3×C4C22×D411yescharacteristic
239C2×C3×C42−1+414yes—
240C2×Dic3—28no—
241C2×Dic3—22no—
242C2×Dic3—22no—
243C2×Dic3—22no—
244C2×Dic3—22no—
245C2×Dic3—28no—
246C2×Dic3—24no—
247C2×Dic3—42no—
248C2×Dic3—42no—
249C2×Dic3—48no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252C2×Dic3—42no—
253C2×Dic3—42no—
254C2×Dic3C2×SD1614yes—
255C2×Dic3#13714yes—
256Dic6—432no—
257Dic6—42no—
258Dic6—42no—
259C3×C8—42no—
260C3⋊C8—42no—
261C24D3×D411yescharacteristic
262C2×Q8—616no—
263C2×Q8—61no—
264C2×Q8—61no—
265C2×Q8—64no—
266C2×Q8—128no—
267C2×Q8—128no—
268C2×Q8—1232no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—34no—
274C22×C4—32no—
275C22×C4—32no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—64no—
279C22×C4—62no—
280C22×C4—62no—
281C22×C4—62no—
282C22×C4—64no—
283C22×C4—62no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4—122no—
287C22×C4—122no—
288C22×C4C2×D1212yes—
289C22×C4D3×D411yescharacteristic
290C22×C4D3×D412yes—
291C22×C4C23×D311yescharacteristic
292C2×C8—22no—
293C2×C8—24no—
294C2×C8—62no—
295C2×C8—64no—
296C4⋊C4—24no—
297C4⋊C4—22no—
298C4⋊C4—24no—
299C4⋊C4—34no—
300C4⋊C4—616no—
301C4⋊C4—64no—
302C4⋊C4—62no—
303C4⋊C4—61no—
304C4⋊C4—61no—
305C4⋊C4—124no—
306C4⋊C4—122no—
307C4⋊C4C2×D1214yes—
308C22⋊C4—42no—
309C22⋊C4—122no—
310C42—64no—
311C42—62no—
312C42—62no—
313C22×C3—22no—
314C22×C3—24no—
315C22×C3—24no—
316C22×C3—22no—
317C22×C3#45014yes—
318C22×C3C2×#12011yescharacteristic
319C22×C3#53612yes—
320C22×C3C22×SD1611yescharacteristic
321C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
322C22×C3C2×#13712yes—
323C3×C4—22no—
324C3×C4—44no—
325C3×C4—82no—
326C3×C4#53612yes—
327C3×C4#53612yes—
328Dic3—24no—
329Dic3—24no—
330Dic3—44no—
331Dic3—81no—
332Dic3—81no—
333C23—22no—
334C23—22no—
335C23#88212yes—
336C23#91112yes—
337C23D3×SD1612yes—
338C23C2×D3×D411yescharacteristic
339Q8—1232no—
340Q8—122no—
341Q8—122no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—34no—
347C2×C4—34no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—48no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—68no—
355C2×C4—62no—
356C2×C4—62no—
357C2×C4—62no—
358C2×C4—62no—
359C2×C4—68no—
360C2×C4—64no—
361C2×C4—122no—
362C2×C4—122no—
363C2×C4—128no—
364C2×C4—122no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4—122no—
368C2×C4#88212yes—
369C2×C4C22×D1211yescharacteristic
370C2×C4C2×D3×D411yescharacteristic
371C2×C4#101014yes—
372C8—42no—
373C8—122no—
374C2×C3—24no—
375C2×C3#261411yescharacteristic
376C2×C3C2×#45011yescharacteristic
377C2×C3#300411yescharacteristic
378C2×C3#322912yes—
379C2×C3#323812yes—
380C22—24no—
381C22—22no—
382C22—24no—
383C22—22no—
384C22#477614yes—
385C22C2×#88211yescharacteristic
386C22#564812yes—
387C22C2×#91111yescharacteristic
388C22C2×D3×SD1611yescharacteristic
389C22D3×#13712yes—
390C4—22no—
391C4—44no—
392C4—64no—
393C4—64no—
394C4—82no—
395C4—124no—
396C4—241no—
397C4—241no—
398C4#552612yes—
399C4#564812yes—
400C3#2964111yescharacteristic, complemented, Sylow
401C2—24no—
402C2#7413511yescharacteristic
403C2C2×#477611yescharacteristic
404C2#7895811yescharacteristic
405C2#8127712yes—
406C2#8141812yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1823C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1823D3
hypocenterC3#29641
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
8. #68910C2semidirect
9. #68827C2semidirect
400. C3#29641semidirect

Maximal subgroups

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

subgroupindex
C22×#8882
C2×#49712
C2×#47212
#740002
C3×#18232
#699102
#695012
#689102
#688272
#296413