Finite Group Databasebeta

← #10849226

Subgroups — #10849226

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×#5542C211yesmaximal, characteristic
2C2×#5007C211yesmaximal, characteristic
3C2×#4681C211yesmaximal, characteristic, complemented
4#73711C211yesmaximal, characteristic, complemented
5C3×C4×#102C211yesmaximal, characteristic
6#70178C214yesmaximal
7#69788C211yesmaximal, characteristic, complemented
8#69486C211yesmaximal, characteristic, complemented
9#68906C214yesmaximal, complemented
10#25650—31nomaximal, Sylow
11C2×C3×C4×D4C2211yescharacteristic
12C2×#936C2212yes—
13C2×#926—21no—
14C2×#924—21no—
15#5542—24no—
16#5542—24no—
17C2×#887C2212yes—
18C2×#879—22no—
19C2×#869—21no—
20C2×#869C2211yescharacteristic
21C2×#869C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×#102C2212yes—
24C2×C3×C4×C8C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26C2×#832C2212yes—
27C2×#816C2214yes—
28#5007—24no—
29#4984C2214yes—
30C2×#810C2214yes—
31C2×#807C2212yes—
32#4929—21no—
33C2×C4⋊(C3⋊C8)C2211yescharacteristic
34#4681—24no—
35#4661—22no—
36C3×#340C2212yes—
37#4550C2214yes—
38C2×#556—31no—
39C2×#487—31no—
40C2×#474—31no—
41#2546—31no—
42#2042—34no—
43#1984—31no—
44#1981—31no—
45#1863—34no—
46C4×#102D311yescharacteristic
47C22×C3×D4C2311yescharacteristic
48C23×C3×C4—21no—
49C22×Dic6—21no—
50C22×Dic6D411yescharacteristic
51C22×Dic6C2312yes—
52C22×C3×C8C2312yes—
53C3×C4×D4—22no—
54C3×C4×D4D414yes—
55C2×C3×C4⋊C4C2311yescharacteristic
56C2×C3×C4⋊C4C2312yes—
57C2×C3×C22⋊C4—21no—
58C2×C3×C42—21no—
59C2×C3×C42C2311yescharacteristic
60C2×#217—22no—
61C2×#217—22no—
62#936—24no—
63#936—24no—
64#926—28no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3—24no—
69C2×C4⋊Dic3D411yescharacteristic
70C2×C4⋊Dic3C2312yes—
71#924—28no—
72C2×Dic3⋊C4—22no—
73C2×Dic3⋊C4D412yes—
74C2×C4×Dic3—22no—
75C2×C4×Dic3C2312yes—
76C22×C3⋊C8C2312yes—
77#887—24no—
78#887—24no—
79#879—216no—
80#869—22no—
81#869—22no—
82#869—22no—
83#869—44no—
84#869D414yes—
85C4×Dic6—22no—
86C4×Dic6—22no—
87C3×#102—24no—
88C3×C4×C8—22no—
89C3×#95—21no—
90#832—24no—
91C3⋊#95—24no—
92#816—28no—
93#810—28no—
94#807—24no—
95C4⋊(C3⋊C8)—22no—
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—31no—
105C2×C4×Q8—32no—
106C2×C4×D4C2×D311yescharacteristic
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×#102C2×D312yes—
116C2×C4×C8C2×D311yescharacteristic
117#398—62no—
118#391—61no—
119#389—34no—
120C4×C4⋊C4C2×D311yescharacteristic
121#340C2×D312yes—
122#336—34no—
123C24×C3—21no—
124C2×C3×D4—21no—
125C2×C3×D4—22no—
126C2×C3×D4—22no—
127C2×C3×D4C2×D412yes—
128C22×C3×C4—22no—
129C22×C3×C4—21no—
130C22×C3×C4—24no—
131C22×C3×C4—42no—
132C22×C3×C4—42no—
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2411yescharacteristic
136C22×Dic3—22no—
137C22×Dic3—22no—
138C22×Dic3C2×D412yes—
139C22×Dic3C2×D412yes—
140C2×Dic6—24no—
141C2×Dic6—24no—
142C2×Dic6—22no—
143C2×Dic6—22no—
144C2×Dic6—22no—
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—22no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4C2×D412yes—
156C3×C4⋊C4#4114yes—
157C3×C22⋊C4—24no—
158C3×C42—42no—
159C3×C42C2×D412yes—
160C3×C42C2×D412yes—
161#217—28no—
162#217—28no—
163C4⋊Dic3—22no—
164C4⋊Dic3—24no—
165C4⋊Dic3—24no—
166C4⋊Dic3—22no—
167C4⋊Dic3—22no—
168C4⋊Dic3—48no—
169C4⋊Dic3—42no—
170Dic3⋊C4—28no—
171Dic3⋊C4—24no—
172C4×Dic3—28no—
173C4×Dic3—22no—
174C4×Dic3—22no—
175C2×C3⋊C8—22no—
176C2×C3⋊C8—24no—
177C22×Q8—31no—
178C22×Q8—32no—
179C22×Q8—61no—
180C22×D4C22×D311yescharacteristic
181C23×C4—21no—
182C22×C8—32no—
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—64no—
194C4×Q8—64no—
195C4×Q8—62no—
196C4×Q8—62no—
197C4×D4—22no—
198C4×D4D1214yes—
199#117—68no—
200C2×C4⋊C4—31no—
201C2×C4⋊C4—32no—
202C2×C4⋊C4—32no—
203C2×C4⋊C4—62no—
204C2×C4⋊C4—61no—
205C2×C4⋊C4—61no—
206C2×C4⋊C4—64no—
207C2×C4⋊C4—61no—
208C2×C4⋊C4C22×D311yescharacteristic
209C2×C4⋊C4C22×D312yes—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—62no—
212C2×C22⋊C4—62no—
213C2×C42—21no—
214C2×C42—32no—
215C2×C42—62no—
216C2×C42C22×D311yescharacteristic
217#107—64no—
218C4⋊C8—62no—
219#103—68no—
220#103—68no—
221#102—24no—
222#102—64no—
223C4×C8—22no—
224#95—21no—
225#95—64no—
226C23×C3—22no—
227C23×C3—42no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—42no—
231C23×C3C22×D411yescharacteristic
232C3×D4—22no—
233C3×D4—22no—
234C3×D4—42no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—24no—
238C2×C3×C4—41no—
239C2×C3×C4—42no—
240C2×C3×C4—41no—
241C2×C3×C4—42no—
242C2×C3×C4—48no—
243C2×C3×C4#12012yes—
244C2×C3×C4#12012yes—
245C2×C3×C4#13512yes—
246C2×C3×C4AGL(1,ℤ/8)12yes—
247C2×C3×C4C22×D411yescharacteristic
248C2×C3×C4C22×D411yescharacteristic
249C2×C3×C4C2×#4112yes—
250C2×C3×C42−1+412yes—
251C2×Dic3—22no—
252C2×Dic3—22no—
253C2×Dic3—24no—
254C2×Dic3—22no—
255C2×Dic3—22no—
256C2×Dic3—24no—
257C2×Dic3—44no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261Dic6—28no—
262Dic6—42no—
263Dic6—42no—
264Dic6—44no—
265Dic6—84no—
266C3×C8—42no—
267C3⋊C8—42no—
268C24—21no—
269C2×Q8—34no—
270C2×Q8—62no—
271C2×Q8—62no—
272C2×Q8—62no—
273C2×Q8—62no—
274C2×Q8—64no—
275C2×Q8—64no—
276C2×Q8—122no—
277C2×Q8—122no—
278C2×Q8—122no—
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—22no—
282C2×D4C2×D1212yes—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—24no—
286C22×C4—32no—
287C22×C4—32no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—62no—
291C22×C4—62no—
292C22×C4D3×D411yescharacteristic
293C22×C4D3×D411yescharacteristic
294C22×C4C23×D311yescharacteristic
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—62no—
299C2×C8—64no—
300C4⋊C4—21no—
301C4⋊C4—22no—
302C4⋊C4—68no—
303C4⋊C4—62no—
304C4⋊C4—64no—
305C4⋊C4—64no—
306C4⋊C4—64no—
307C4⋊C4—62no—
308C4⋊C4—62no—
309C4⋊C4—128no—
310C4⋊C4—122no—
311C4⋊C4C2×D1212yes—
312C4⋊C4#23514yes—
313C22⋊C4—24no—
314C22⋊C4—68no—
315C22⋊C4—68no—
316C42—42no—
317C42—68no—
318C42—62no—
319C42—62no—
320C42C2×D1212yes—
321C42D3×D412yes—
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#44914yes—
336C3×C4#53211yescharacteristic
337C3×C4#53611yescharacteristic
338C3×C4#54712yes—
339Dic3—44no—
340Dic3—42no—
341Dic3—42no—
342Dic3—82no—
343C23—22no—
344C23—41no—
345C23—41no—
346C23—42no—
347C23—42no—
348C23C2×D3×D411yescharacteristic
349Q8—68no—
350Q8—122no—
351Q8—122no—
352Q8—124no—
353Q8—244no—
354D4—22no—
355D4—22no—
356D4—42no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—48no—
365C2×C4—62no—
366C2×C4—62no—
367C2×C4—64no—
368C2×C4—62no—
369C2×C4—62no—
370C2×C4—64no—
371C2×C4—124no—
372C2×C4—122no—
373C2×C4—122no—
374C2×C4—122no—
375C2×C4#88212yes—
376C2×C4#88212yes—
377C2×C4#91112yes—
378C2×C4#91612yes—
379C2×C4C22×D1211yescharacteristic
380C2×C4C2×#23512yes—
381C2×C4C2×D3×D411yescharacteristic
382C2×C4#101012yes—
383C8—42no—
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×#88211yescharacteristic
397C22#555212yes—
398C22C2×#91111yescharacteristic
399C22C2×#91611yescharacteristic
400C22#577112yes—
401C4—22no—
402C4—42no—
403C4—44no—
404C4—124no—
405C4—122no—
406C4—122no—
407C4—242no—
408C4#478114yes—
409C4#547711yescharacteristic
410C4#552611yescharacteristic
411C4#555112yes—
412C3#2565011yescharacteristic, complemented, Sylow
413C2—82no—
414C2#7348911yescharacteristic
415C2C2×#478111yescharacteristic
416C2#7883711yescharacteristic
417C2#7957112yes—
418C2#7962312yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#102C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC4×#102D3
hypocenterC3#25650
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
3. C2×#4681C2semidirect
4. #73711C2semidirect
7. #69788C2semidirect
8. #69486C2semidirect
9. #68906C2semidirect
412. C3#25650semidirect

Maximal subgroups

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

subgroupindex
C2×#55422
C2×#50072
C2×#46812
#737112
C3×C4×#1022
#701782
#697882
#694862
#689062
#256503