Finite Group Databasebeta

← #10849225

Subgroups — #10849225

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
2#82475C211yesmaximal, characteristic, complemented
3C2×#4846C211yesmaximal, characteristic
4C2×#4681C211yesmaximal, characteristic, complemented
5C3×#1984C211yesmaximal, characteristic, complemented
6#70456C211yesmaximal, characteristic, complemented
7C4×#821C211yesmaximal, characteristic
8#68683C214yesmaximal, complemented
9#68630C214yesmaximal
10#25650—31nomaximal, Sylow
11C2×C3×C4×Q8C2211yescharacteristic
12C2×#946—21no—
13C2×#944—21no—
14#5584—24no—
15#5584—24no—
16C2×#898—22no—
17C2×#892—21no—
18C2×#875C2212yes—
19C2×C4×D12C2211yescharacteristic
20C2×#869C2211yescharacteristic
21C2×#869C2211yescharacteristic
22C2×C4×Dic6C2212yes—
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C4×C8C2211yescharacteristic
25C2×#821C2212yes—
26C2×#821C2212yes—
27C2×#818C2212yes—
28C2×#816C2214yes—
29C2×#816C2214yes—
30#4981—22no—
31#4973—21no—
32C4×C4⋊Dic3C2211yescharacteristic
33#4928C2214yes—
34#4846—24no—
35#4681—24no—
36C3×#336C2214yes—
37#4551C2212yes—
38C2×#556—31no—
39C2×#487—31no—
40C2×#474—31no—
41#2546—31no—
42#2042—34no—
43#1984D311yescharacteristic, complemented
44#1981—31no—
45#1863—34no—
46C4×#102—31no—
47C22×C3×Q8D411yescharacteristic
48C22×D12C2311yescharacteristic
49C22×C4×D3—21no—
50C22×Dic6—21no—
51C22×Dic6C2312yes—
52C22×C3×C8C2312yes—
53C22×C3×C8C2312yes—
54C3×C4×Q8—22no—
55C3×C4×Q8—22no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C4⋊C4D412yes—
58C2×C3×C42C2312yes—
59C2×C3×C42C2311yescharacteristic
60#946—28no—
61#944—44no—
62C2×(C2×D3)⋊C4—21no—
63C2×(C2×D3)⋊C4—22no—
64C2×(C2×D3)⋊C4—22no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3C2312yes—
69C2×C4⋊Dic3C2311yescharacteristic
70C2×C4⋊Dic3C2312yes—
71C2×Dic3⋊C4—22no—
72C2×Dic3⋊C4—24no—
73C2×C4×Dic3—22no—
74C2×C4×Dic3—21no—
75#898—216no—
76#892—28no—
77#875—24no—
78#875—24no—
79C4×D12—22no—
80C4×D12D414yes—
81#869—22no—
82#869—22no—
83#869—22no—
84#869D414yes—
85C4×Dic6—24no—
86C4×Dic6—24no—
87C3×C4⋊C8—22no—
88C3×C4×C8—22no—
89C3⋊#95—24no—
90C3⋊#95—21no—
91#821—24no—
92#821—24no—
93#818—24no—
94#816—28no—
95#816—28no—
96#556—64no—
97#556—64no—
98C2×C4⋊Q8—31no—
99C2×C4⋊Q8—31no—
100C2×C4⋊Q8—61no—
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⋊C8C2×D311yescharacteristic
112C2×#103—34no—
113C2×#103—34no—
114C2×#102—32no—
115C2×#102—32no—
116C2×C4×C8C2×D311yescharacteristic
117#398—62no—
118#391—61no—
119#389—34no—
120C4×C4⋊C4—31no—
121#340—32no—
122#336C2×D314yes—
123C23×D3—21no—
124C2×C3×Q8—22no—
125C2×C3×Q8—22no—
126C2×C3×Q8—22no—
127C22×C3×C4C2×D412yes—
128C22×C3×C4C2×D412yes—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2411yescharacteristic
132C22×Dic3—22no—
133C22×Dic3—21no—
134C22×Dic3—22no—
135C22×Dic3—22no—
136C2×D12—22no—
137C2×D12—22no—
138C2×D12—21no—
139C2×D12C2×D412yes—
140C2×C4×D3—24no—
141C2×C4×D3—42no—
142C2×C4×D3—42no—
143C2×Dic6—24no—
144C2×Dic6—24no—
145C2×Dic6—22no—
146C2×Dic6—42no—
147C2×Dic6—42no—
148C2×Dic6—42no—
149C2×Dic6C2×D414yes—
150C2×C3×C8—22no—
151C2×C3×C8—22no—
152C2×C3×C8—24no—
153C2×C3×C8—22no—
154C2×C3×C8—22no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—24no—
157C3×C42—22no—
158C3×C42—22no—
159C3×C42C2×D412yes—
160C3×C42C2×D412yes—
161(C2×D3)⋊C4—28no—
162(C2×D3)⋊C4—28no—
163(C2×D3)⋊C4—24no—
164C4⋊Dic3—24no—
165C4⋊Dic3—24no—
166C4⋊Dic3—21no—
167C4⋊Dic3—22no—
168C4⋊Dic3—22no—
169C4⋊Dic3—22no—
170C4⋊Dic3—42no—
171C4⋊Dic3C2×D412yes—
172C4⋊Dic3#4114yes—
173Dic3⋊C4—28no—
174Dic3⋊C4—48no—
175C4×Dic3—28no—
176C4×Dic3—42no—
177C22×Q8—32no—
178C22×Q8—61no—
179C22×Q8D1211yescharacteristic
180C22×D4—31no—
181C23×C4—61no—
182C22×C8C22×D312yes—
183C22×C8C22×D312yes—
184C4⋊Q8—34no—
185C4⋊Q8—62no—
186C4⋊Q8—62no—
187C4⋊Q8—62no—
188C4⋊Q8—124no—
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—31no—
201C2×C4⋊C4—32no—
202C2×C4⋊C4—32no—
203C2×C4⋊C4—61no—
204C2×C4⋊C4—62no—
205C2×C4⋊C4—61no—
206C2×C4⋊C4—61no—
207C2×C4⋊C4—64no—
208C2×C4⋊C4D1211yescharacteristic
209C2×C4⋊C4D1212yes—
210C2×C22⋊C4—62no—
211C2×C22⋊C4—61no—
212C2×C22⋊C4—62no—
213C2×C42—62no—
214C2×C42—61no—
215C2×C42C22×D312yes—
216C2×C42C22×D311yescharacteristic
217#107—64no—
218C4⋊C8—22no—
219#103—68no—
220#103—68no—
221#102—64no—
222#102—64no—
223C4×C8—22no—
224#95—61no—
225#95—64no—
226C23×C3C22×D411yescharacteristic
227C22×D3—22no—
228C22×D3—42no—
229C22×D3—41no—
230C22×D3—41no—
231C22×D3—42no—
232C3×Q8—44no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—24no—
239C2×C3×C4#12012yes—
240C2×C3×C4#12012yes—
241C2×C3×C4#13512yes—
242C2×C3×C4AGL(1,ℤ/8)12yes—
243C2×C3×C4C22×D411yescharacteristic
244C2×C3×C4C22×D411yescharacteristic
245C2×C3×C4C2×#4112yes—
246C2×C3×C42−1+412yes—
247C2×Dic3—22no—
248C2×Dic3—24no—
249C2×Dic3—24no—
250C2×Dic3—44no—
251C2×Dic3—41no—
252C2×Dic3—42no—
253C2×Dic3—41no—
254C2×Dic3—42no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—42no—
258D12—22no—
259D12—22no—
260D12—42no—
261C4×D3—48no—
262Dic6—28no—
263Dic6—42no—
264Dic6—42no—
265Dic6—84no—
266C3×C8—42no—
267C3×C8—42no—
268C24—61no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—34no—
273C2×Q8—62no—
274C2×Q8—64no—
275C2×Q8—64no—
276C2×Q8—122no—
277C2×Q8—122no—
278C2×Q8—122no—
279C2×D4—32no—
280C2×D4—61no—
281C2×D4—62no—
282C2×D4—62no—
283C22×C4—62no—
284C22×C4—61no—
285C22×C4—62no—
286C22×C4—64no—
287C22×C4—62no—
288C22×C4—122no—
289C22×C4—122no—
290C22×C4C2×D1212yes—
291C22×C4C2×D1212yes—
292C22×C4C2×D1211yescharacteristic
293C22×C4C2×D1211yescharacteristic
294C22×C4C23×D311yescharacteristic
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—24no—
300C4⋊C4—22no—
301C4⋊C4—24no—
302C4⋊C4—32no—
303C4⋊C4—34no—
304C4⋊C4—68no—
305C4⋊C4—64no—
306C4⋊C4—64no—
307C4⋊C4—61no—
308C4⋊C4—62no—
309C4⋊C4—62no—
310C4⋊C4—62no—
311C4⋊C4—128no—
312C4⋊C4—122no—
313C22⋊C4—68no—
314C22⋊C4—68no—
315C22⋊C4—64no—
316C42—22no—
317C42—22no—
318C42—68no—
319C42—122no—
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—42no—
334C3×C4—42no—
335C3×C4#44914yes—
336C3×C4#53211yescharacteristic
337C3×C4#53611yescharacteristic
338C3×C4#54712yes—
339Dic3—44no—
340Dic3—44no—
341Dic3—42no—
342Dic3—82no—
343C23—62no—
344C23—121no—
345C23—121no—
346C23—122no—
347C23—122no—
348C23C22×D1211yescharacteristic
349Q8—44no—
350Q8—68no—
351Q8—122no—
352Q8—122no—
353Q8—244no—
354D4—62no—
355D4—62no—
356D4—122no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—62no—
364C2×C4—64no—
365C2×C4—64no—
366C2×C4—121no—
367C2×C4—122no—
368C2×C4—121no—
369C2×C4—122no—
370C2×C4—124no—
371C2×C4—128no—
372C2×C4—122no—
373C2×C4—122no—
374C2×C4—122no—
375C2×C4#88212yes—
376C2×C4#88212yes—
377C2×C4#90412yes—
378C2×C4#90812yes—
379C2×C4C2×D3×D411yescharacteristic
380C2×C4C2×D3×D411yescharacteristic
381C2×C4#100712yes—
382C2×C4D3×#4112yes—
383C8—42no—
384C8—42no—
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×#88211yescharacteristic
397C22#557612yes—
398C22C2×#90411yescharacteristic
399C22C2×#90811yescharacteristic
400C22#575312yes—
401C4—22no—
402C4—42no—
403C4—42no—
404C4—122no—
405C4—124no—
406C4—124no—
407C4—242no—
408C4#473414yes—
409C4#563112yes—
410C4#566811yescharacteristic
411C4#572411yescharacteristic
412C3#2565011yescharacteristic, complemented, Sylow
413C2—242no—
414C2#7649811yescharacteristic
415C2C2×#473411yescharacteristic
416C2#7867911yescharacteristic
417C2#7993412yes—
418C2#7997012yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1984C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1984D3
hypocenterC3#25650
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #1984

Complements

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

NH ≅ G/Ntype
2. #82475C2semidirect
4. C2×#4681C2semidirect
5. C3×#1984C2semidirect
6. #70456C2semidirect
8. #68683C2semidirect
43. #1984D3semidirect
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
#824752
C2×#48462
C2×#46812
C3×#19842
#704562
C4×#8212
#686832
#686302
#256503