Finite Group Databasebeta

← #10849223

Subgroups — #10849223

Order 768.

1054 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5584C211yesmaximal, characteristic
2C3×#2546C211yesmaximal, characteristic, complemented
3C2×#5087C211yesmaximal, characteristic
4C2×#5093C211yesmaximal, characteristic, complemented
5#71634C211yesmaximal, characteristic, complemented
6#69631C214yesmaximal, complemented
7#69628C214yesmaximal
8#69486C211yesmaximal, characteristic, complemented
9C4×#809C211yesmaximal, characteristic
10#25650—31nomaximal, Sylow
11C2×C3×C4⋊Q8—21no—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#946—21no—
15#5584—24no—
16#5584—24no—
17C2×#898—22no—
18C2×#892—21no—
19C2×#875C2212yes—
20C2×C4×D12C2211yescharacteristic
21C2×#869C2211yescharacteristic
22C2×C4×Dic6C2212yes—
23C3×#398—22no—
24C3×#391—21no—
25C3×#389C2214yes—
26C3×C4×C4⋊C4C2211yescharacteristic
27#5093—24no—
28#5087—24no—
29C2×#810C2214yes—
30C2×#810C2214yes—
31C2×#809C2212yes—
32C2×#809C2212yes—
33C2×#807C2212yes—
34C2×C4⋊(C3⋊C8)C2211yescharacteristic
35C2×C4×C3⋊C8C2211yescharacteristic
36#4529C2212yes—
37#4524C2214yes—
38C2×#556—31no—
39C2×#487—31no—
40C2×#474—31no—
41#2546D311yescharacteristic, complemented
42#2042—34no—
43#1984—31no—
44#1981—31no—
45#1863—34no—
46C4×#102—31no—
47C22×C3×Q8—21no—
48C22×C3×Q8D411yescharacteristic
49C22×D12C2311yescharacteristic
50C22×C4×D3—21no—
51C22×Dic6C2312yes—
52C3×C4⋊Q8—22no—
53C3×C4⋊Q8—22no—
54C3×C4⋊Q8—44no—
55C3×C4×Q8—22no—
56C3×C4×Q8—22no—
57C2×C3×C4⋊C4—24no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C4⋊C4D412yes—
61C2×C3×C4⋊C4C2312yes—
62C2×C3×C4⋊C4C2312yes—
63C2×C3×C42—21no—
64C2×C3×C42C2312yes—
65C2×C3×C42C2311yescharacteristic
66#946—28no—
67C2×(C2×D3)⋊C4—21no—
68C2×(C2×D3)⋊C4—22no—
69C2×(C2×D3)⋊C4—22no—
70C2×C4⋊Dic3—21no—
71C2×C4⋊Dic3—21no—
72C2×C4⋊Dic3C2311yescharacteristic
73C2×Dic3⋊C4—22no—
74C2×C4×Dic3—22no—
75C22×C3⋊C8C2312yes—
76C22×C3⋊C8C2312yes—
77#898—216no—
78#892—28no—
79#875—24no—
80#875—24no—
81C4×D12—22no—
82C4×D12D414yes—
83#869—22no—
84#869D414yes—
85C4×Dic6—24no—
86C4×Dic6—24no—
87C3×#95—21no—
88C3×#95—24no—
89#810—28no—
90#810—28no—
91#809—24no—
92#809—24no—
93#807—24no—
94C4⋊(C3⋊C8)—22no—
95C4×C3⋊C8—22no—
96#556—64no—
97#556—64no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—31no—
100C2×C4⋊Q8C2×D311yescharacteristic
101C2×#126—62no—
102C2×#124—32no—
103C2×C22⋊Q8—61no—
104C2×C4×Q8—32no—
105C2×C4×Q8C2×D311yescharacteristic
106C2×C4×D4—31no—
107C2×#117—61no—
108#487—64no—
109#474—64no—
110C2×#107—32no—
111C2×C4⋊C8—31no—
112C2×#103—34no—
113C2×#103—34no—
114C2×#102—32no—
115C2×#102—32no—
116C2×C4×C8—31no—
117#398—22no—
118#391—21no—
119#389C2×D314yes—
120C4×C4⋊C4C2×D311yescharacteristic
121#340—32no—
122#336—34no—
123C23×D3—21no—
124C2×C3×Q8—22no—
125C2×C3×Q8—22no—
126C2×C3×Q8—22no—
127C2×C3×Q8—42no—
128C2×C3×Q8—42no—
129C2×C3×Q8—42no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4C2×D412yes—
133C22×C3×C4C2×D412yes—
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2411yescharacteristic
137C22×Dic3—21no—
138C22×Dic3—22no—
139C2×D12—22no—
140C2×D12—22no—
141C2×D12—21no—
142C2×D12C2×D412yes—
143C2×C4×D3—24no—
144C2×C4×D3—42no—
145C2×C4×D3—42no—
146C2×Dic6—24no—
147C2×Dic6—24no—
148C2×Dic6—22no—
149C2×Dic6C2×D414yes—
150C3×C4⋊C4—22no—
151C3×C4⋊C4—24no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—48no—
156C3×C4⋊C4—42no—
157C3×C4⋊C4#4114yes—
158C3×C42—22no—
159C3×C42—22no—
160C3×C42—42no—
161C3×C42C2×D412yes—
162C3×C42C2×D412yes—
163(C2×D3)⋊C4—28no—
164(C2×D3)⋊C4—28no—
165(C2×D3)⋊C4—24no—
166C4⋊Dic3—24no—
167C4⋊Dic3—24no—
168C4⋊Dic3—21no—
169C4⋊Dic3C2×D412yes—
170Dic3⋊C4—28no—
171C4×Dic3—28no—
172C2×C3⋊C8—22no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—22no—
175C2×C3⋊C8—24no—
176C2×C3⋊C8—22no—
177C22×Q8—21no—
178C22×Q8—32no—
179C22×Q8#6711yescharacteristic
180C22×D4—31no—
181C23×C4—61no—
182C22×C8—32no—
183C22×C8—32no—
184C4⋊Q8—22no—
185C4⋊Q8—22no—
186C4⋊Q8—34no—
187C4⋊Q8—44no—
188C4⋊Q8—62no—
189#126—616no—
190#124—64no—
191#124—64no—
192C22⋊Q8—68no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—64no—
196C4×Q8—64no—
197C4×D4—34no—
198C4×D4—62no—
199#117—68no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—31no—
203C2×C4⋊C4—61no—
204C2×C4⋊C4—62no—
205C2×C4⋊C4—61no—
206C2×C4⋊C4#6711yescharacteristic
207C2×C4⋊C4#6712yes—
208C2×C4⋊C4C22×D312yes—
209C2×C4⋊C4C22×D312yes—
210C2×C22⋊C4—62no—
211C2×C22⋊C4—61no—
212C2×C22⋊C4—62no—
213C2×C42—21no—
214C2×C42—62no—
215C2×C42C22×D312yes—
216C2×C42C22×D311yescharacteristic
217#107—64no—
218C4⋊C8—62no—
219#103—68no—
220#103—68no—
221#102—64no—
222#102—64no—
223C4×C8—62no—
224#95—21no—
225#95—24no—
226C23×C3C22×D411yescharacteristic
227C22×D3—22no—
228C22×D3—42no—
229C22×D3—41no—
230C22×D3—41no—
231C22×D3—42no—
232C3×Q8—44no—
233C3×Q8—84no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—24no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—24no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4#12012yes—
247C2×C3×C4#12012yes—
248C2×C3×C4#13512yes—
249C2×C3×C4AGL(1,ℤ/8)12yes—
250C2×C3×C4C22×D411yescharacteristic
251C2×C3×C4C22×D411yescharacteristic
252C2×C3×C4C2×#4112yes—
253C2×C3×C42−1+412yes—
254C2×Dic3—22no—
255C2×Dic3—24no—
256C2×Dic3—44no—
257C2×Dic3—41no—
258C2×Dic3—41no—
259D12—22no—
260D12—22no—
261D12—42no—
262C4×D3—48no—
263Dic6—28no—
264Dic6—42no—
265Dic6—42no—
266C3⋊C8—42no—
267C3⋊C8—42no—
268C24—61no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—34no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—62no—
277C2×Q8—64no—
278C2×Q8—64no—
279C2×D4—32no—
280C2×D4—61no—
281C2×D4—62no—
282C2×D4—62no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—61no—
286C22×C4—62no—
287C22×C4—64no—
288C22×C4—122no—
289C22×C4—122no—
290C22×C4C2×#6712yes—
291C22×C4C2×#6712yes—
292C22×C4C2×#6711yescharacteristic
293C22×C4C2×#6711yescharacteristic
294C22×C4C23×D311yescharacteristic
295C2×C8—62no—
296C2×C8—62no—
297C2×C8—62no—
298C2×C8—64no—
299C2×C8—62no—
300C4⋊C4—22no—
301C4⋊C4—24no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—32no—
306C4⋊C4—48no—
307C4⋊C4—42no—
308C4⋊C4—68no—
309C4⋊C4—64no—
310C4⋊C4—64no—
311C4⋊C4—61no—
312C4⋊C4#23914yes—
313C22⋊C4—68no—
314C22⋊C4—68no—
315C22⋊C4—64no—
316C42—22no—
317C42—22no—
318C42—42no—
319C42—68no—
320C42D3×D412yes—
321C42D3×D412yes—
322C22×C3C2×#12011yescharacteristic
323C22×C3#54812yes—
324C22×C3C2×#13511yescharacteristic
325C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
326C22×C3#57812yes—
327C2×D3—42no—
328C2×D3—42no—
329C2×D3—42no—
330C2×D3—82no—
331C2×D3—82no—
332C3×C4—22no—
333C3×C4—44no—
334C3×C4—42no—
335C3×C4—42no—
336C3×C4—82no—
337C3×C4#44914yes—
338C3×C4#53211yescharacteristic
339C3×C4#53611yescharacteristic
340C3×C4#54712yes—
341Dic3—44no—
342Dic3—42no—
343C23—62no—
344C23—121no—
345C23—122no—
346C23—121no—
347C23—122no—
348C23C22×#6711yescharacteristic
349Q8—44no—
350Q8—68no—
351Q8—84no—
352Q8—122no—
353Q8—122no—
354D4—62no—
355D4—62no—
356D4—122no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—62no—
370C2×C4—64no—
371C2×C4—121no—
372C2×C4—124no—
373C2×C4—121no—
374C2×C4—128no—
375C2×C4#93212yes—
376C2×C4#93712yes—
377C2×C4#93712yes—
378C2×C4#95012yes—
379C2×C4C2×D3×D411yescharacteristic
380C2×C4C2×D3×D411yescharacteristic
381C2×C4C2×#23912yes—
382C2×C4#100712yes—
383C8—122no—
384C8—122no—
385C2×C3#243211yescharacteristic
386C2×C3C2×#44911yescharacteristic
387C2×C3#301111yescharacteristic
388C2×C3#336412yes—
389C2×C3#336712yes—
390D3—82no—
391C22—122no—
392C22—122no—
393C22—122no—
394C22—242no—
395C22—242no—
396C22C2×#93211yescharacteristic
397C22C2×#93711yescharacteristic
398C22#581712yes—
399C22C2×#95011yescharacteristic
400C22#583812yes—
401C4—22no—
402C4—44no—
403C4—42no—
404C4—42no—
405C4—82no—
406C4—124no—
407C4—122no—
408C4#503714yes—
409C4#572411yescharacteristic
410C4#572511yescharacteristic
411C4#572612yes—
412C3#2565011yescharacteristic, complemented, Sylow
413C2—242no—
414C2C2×#503711yescharacteristic
415C2#8325511yescharacteristic
416C2#8341111yescharacteristic
417C2#8356312yes—
418C2#8368912yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2546C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2546D3
hypocenterC3#25650
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2546

Complements

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

NH ≅ G/Ntype
2. C3×#2546C2semidirect
4. C2×#5093C2semidirect
5. #71634C2semidirect
6. #69631C2semidirect
8. #69486C2semidirect
41. #2546D3semidirect
412. C3#25650semidirect

Maximal subgroups

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

subgroupindex
C2×#55842
C3×#25462
C2×#50872
C2×#50932
#716342
#696312
#696282
#694862
C4×#8092
#256503