Finite Group Databasebeta

← #10849222

Subgroups — #10849222

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×C3×#556C211yesmaximal, characteristic
2C2×#5079C211yesmaximal, characteristic, complemented
3#82475C211yesmaximal, characteristic, complemented
4C2×#5007C211yesmaximal, characteristic
5#71778C211yesmaximal, characteristic, complemented
6#71722C214yesmaximal
7#71634C211yesmaximal, characteristic, complemented
8#71628C214yesmaximal, complemented
9C4×#832C211yesmaximal, characteristic
10#25650—31nomaximal, Sylow
11C3×#556—24no—
12C3×#556—24no—
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#126—22no—
15C2×C3×#124C2212yes—
16C2×C3×C22⋊Q8—21no—
17C2×C3×C4×Q8C2212yes—
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×C3×#117—21no—
21C2×#944—21no—
22C2×#869C2211yescharacteristic
23C2×#835C2214yes—
24C2×#835C2214yes—
25C2×#832C2212yes—
26C2×#832C2212yes—
27#5079—24no—
28#5007—24no—
29#4981—22no—
30#4973—21no—
31C2×#807C2212yes—
32C4×C4⋊Dic3C2211yescharacteristic
33#4928C2214yes—
34C2×C4⋊(C3⋊C8)C2211yescharacteristic
35C2×C4×C3⋊C8C2211yescharacteristic
36#4529C2212yes—
37#4524C2214yes—
38C2×#556D311yescharacteristic
39C2×#487—31no—
40C2×#474—31no—
41#2546—31no—
42#2042—34no—
43#1984—31no—
44#1981—31no—
45#1863—34no—
46C4×#102—31no—
47C22×C3×Q8D411yescharacteristic
48C22×C3×Q8C2312yes—
49C22×C3×D4C2311yescharacteristic
50C23×C3×C4—21no—
51C22×Dic6—21no—
52C3×C4⋊Q8—22no—
53C3×C4⋊Q8D414yes—
54C3×#126—216no—
55C3×#124—24no—
56C3×#124—24no—
57C3×C22⋊Q8—28no—
58C3×C4×Q8—24no—
59C3×C4×Q8—24no—
60C3×C4×Q8—22no—
61C3×C4×Q8—22no—
62C3×C4×D4—22no—
63C3×C4×D4D414yes—
64C3×#117—28no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4D411yescharacteristic
69C2×C3×C4⋊C4D412yes—
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C22⋊C4—22no—
72C2×C3×C22⋊C4—21no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C42—22no—
75C2×C3×C42C2312yes—
76C2×C3×C42C2311yescharacteristic
77#944—44no—
78C2×C4⋊Dic3—21no—
79C2×C4⋊Dic3C2312yes—
80C2×C4⋊Dic3C2312yes—
81C2×Dic3⋊C4—24no—
82C2×C4×Dic3—21no—
83C22×C3⋊C8C2312yes—
84C22×C3⋊C8C2312yes—
85#869—22no—
86#869—22no—
87#835—28no—
88#835—28no—
89#832—24no—
90#832—24no—
91C3⋊#95—24no—
92C3⋊#95—21no—
93#807—24no—
94C4⋊(C3⋊C8)—22no—
95C4×C3⋊C8—22no—
96#556—24no—
97#556—24no—
98C2×C4⋊Q8—31no—
99C2×C4⋊Q8—61no—
100C2×C4⋊Q8C2×D311yescharacteristic
101C2×#126—22no—
102C2×#124C2×D312yes—
103C2×C22⋊Q8—21no—
104C2×C4×Q8C2×D312yes—
105C2×C4×Q8C2×D311yescharacteristic
106C2×C4×D4C2×D311yescharacteristic
107C2×#117—21no—
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—62no—
118#391—61no—
119#389—34no—
120C4×C4⋊C4—31no—
121#340—32no—
122#336—34no—
123C24×C3—21no—
124C2×C3×Q8—22no—
125C2×C3×Q8—22no—
126C2×C3×Q8—22no—
127C2×C3×Q8—22no—
128C2×C3×Q8—24no—
129C2×C3×Q8—24no—
130C2×C3×Q8C2×D414yes—
131C2×C3×D4—21no—
132C2×C3×D4—22no—
133C2×C3×D4—22no—
134C2×C3×D4C2×D412yes—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4—24no—
138C22×C3×C4—42no—
139C22×C3×C4—42no—
140C22×C3×C4C2×D412yes—
141C22×C3×C4C2×D412yes—
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2411yescharacteristic
145C22×Dic3—22no—
146C22×Dic3—22no—
147C2×Dic6—42no—
148C2×Dic6—42no—
149C2×Dic6—42no—
150C3×C4⋊C4—28no—
151C3×C4⋊C4—24no—
152C3×C4⋊C4—24no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—24no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4C2×D412yes—
157C3×C22⋊C4—28no—
158C3×C22⋊C4—28no—
159C3×C22⋊C4—24no—
160C3×C42—28no—
161C3×C42—22no—
162C3×C42—22no—
163C3×C42C2×D412yes—
164C3×C42C2×D412yes—
165C4⋊Dic3—22no—
166C4⋊Dic3—22no—
167C4⋊Dic3—22no—
168C4⋊Dic3—42no—
169C4⋊Dic3#4114yes—
170Dic3⋊C4—48no—
171C4×Dic3—42no—
172C2×C3⋊C8—22no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—22no—
175C2×C3⋊C8—24no—
176C2×C3⋊C8—22no—
177C22×Q8—61no—
178C22×Q8#6711yescharacteristic
179C22×Q8C22×D312yes—
180C22×D4C22×D311yescharacteristic
181C23×C4—21no—
182C22×C8—32no—
183C22×C8—32no—
184C4⋊Q8—22no—
185C4⋊Q8—62no—
186C4⋊Q8—62no—
187C4⋊Q8—124no—
188C4⋊Q8#6714yes—
189#126—216no—
190#124—24no—
191#124—24no—
192C22⋊Q8—28no—
193C4×Q8—24no—
194C4×Q8—24no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×D4—22no—
198C4×D4#6714yes—
199#117—28no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—32no—
204C2×C4⋊C4—32no—
205C2×C4⋊C4—61no—
206C2×C4⋊C4—64no—
207C2×C4⋊C4#6711yescharacteristic
208C2×C4⋊C4#6712yes—
209C2×C4⋊C4C22×D311yescharacteristic
210C2×C22⋊C4—22no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—22no—
213C2×C42—22no—
214C2×C42—61no—
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—61no—
225#95—64no—
226C23×C3—22no—
227C23×C3—42no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—42no—
231C23×C3C22×D411yescharacteristic
232C3×Q8—28no—
233C3×Q8—42no—
234C3×Q8—42no—
235C3×Q8—44no—
236C3×D4—22no—
237C3×D4—22no—
238C3×D4—42no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—24no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—24no—
247C2×C3×C4—44no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—48no—
251C2×C3×C4#12012yes—
252C2×C3×C4#12012yes—
253C2×C3×C4#13512yes—
254C2×C3×C4AGL(1,ℤ/8)12yes—
255C2×C3×C4C22×D411yescharacteristic
256C2×C3×C4C22×D411yescharacteristic
257C2×C3×C4C2×#4112yes—
258C2×C3×C42−1+412yes—
259C2×Dic3—24no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265Dic6—84no—
266C3⋊C8—42no—
267C3⋊C8—42no—
268C24—21no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—122no—
276C2×Q8—122no—
277C2×Q8—122no—
278C2×Q8C2×#6714yes—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—21no—
282C2×D4C2×#6712yes—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—62no—
289C22×C4—62no—
290C22×C4C2×#6712yes—
291C22×C4C2×#6712yes—
292C22×C4C2×#6711yescharacteristic
293C22×C4C2×#6711yescharacteristic
294C22×C4C23×D311yescharacteristic
295C2×C8—62no—
296C2×C8—64no—
297C2×C8—62no—
298C2×C8—62no—
299C2×C8—62no—
300C4⋊C4—28no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—22no—
304C4⋊C4—24no—
305C4⋊C4—21no—
306C4⋊C4—34no—
307C4⋊C4—62no—
308C4⋊C4—62no—
309C4⋊C4—62no—
310C4⋊C4—128no—
311C4⋊C4—122no—
312C4⋊C4C2×#6712yes—
313C22⋊C4—28no—
314C22⋊C4—28no—
315C22⋊C4—24no—
316C42—28no—
317C42—22no—
318C42—22no—
319C42—122no—
320C42C2×#6712yes—
321C42C2×#6712yes—
322C22×C3—42no—
323C22×C3—42no—
324C22×C3—42no—
325C22×C3—82no—
326C22×C3—82no—
327C22×C3C2×#12011yescharacteristic
328C22×C3#54812yes—
329C22×C3C2×#13511yescharacteristic
330C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
331C22×C3#57812yes—
332C3×C4—22no—
333C3×C4—44no—
334C3×C4—42no—
335C3×C4—42no—
336C3×C4—42no—
337C3×C4#44914yes—
338C3×C4#53211yescharacteristic
339C3×C4#53611yescharacteristic
340C3×C4#54712yes—
341Dic3—44no—
342Dic3—82no—
343C23—22no—
344C23—41no—
345C23—41no—
346C23—42no—
347C23—42no—
348C23C22×#6711yescharacteristic
349Q8—28no—
350Q8—44no—
351Q8—42no—
352Q8—42no—
353Q8—244no—
354D4—22no—
355D4—22no—
356D4—42no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—44no—
368C2×C4—48no—
369C2×C4—64no—
370C2×C4—122no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4—122no—
374C2×C4—122no—
375C2×C4#94912yes—
376C2×C4#95012yes—
377C2×C4#95312yes—
378C2×C4#95312yes—
379C2×C4D3×#4112yes—
380C2×C4#101012yes—
381C2×C4C22×#6711yescharacteristic
382C2×C4C22×#6711yescharacteristic
383C8—122no—
384C8—122no—
385C2×C3—82no—
386C2×C3#243211yescharacteristic
387C2×C3C2×#44911yescharacteristic
388C2×C3#301111yescharacteristic
389C2×C3#336412yes—
390C2×C3#336712yes—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—82no—
395C22—82no—
396C22C2×#94911yescharacteristic
397C22C2×#95011yescharacteristic
398C22#583112yes—
399C22#583812yes—
400C22C2×#95311yescharacteristic
401C4—22no—
402C4—42no—
403C4—42no—
404C4—44no—
405C4—42no—
406C4—124no—
407C4—242no—
408C4#523814yes—
409C4#582412yes—
410C4#582511yescharacteristic
411C4#583211yescharacteristic
412C3#2565011yescharacteristic, complemented, Sylow
413C2—82no—
414C2#8366812yes—
415C2#8367612yes—
416C2C2×#523811yescharacteristic
417C2#8409311yescharacteristic
418C2#8411711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#556C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×#556D3
hypocenterC3#25650
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
2. C2×#5079C2semidirect
3. #82475C2semidirect
5. #71778C2semidirect
7. #71634C2semidirect
8. #71628C2semidirect
412. C3#25650semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5562
C2×#50792
#824752
C2×#50072
#717782
#717222
#716342
#716282
C4×#8322
#256503