Finite Group Databasebeta

← #10824614

Subgroups — #10824614

Order 768.

894 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4873C211yesmaximal, characteristic, complemented
2C2×#4718C211yesmaximal, characteristic
3#73866C212yesmaximal
4C3×#2076C211yesmaximal, characteristic
5#70622C211yesmaximal, characteristic, complemented
6#69964C211yesmaximal, characteristic
7#69527C212yesmaximal
8#68913C212yesmaximal, complemented
9#68864C211yesmaximal, characteristic, complemented
10#68785C211yesmaximal, characteristic, complemented
11#68781C212yesmaximal, complemented
12#23698—31nomaximal, Sylow
13C2×#924C2212yes—
14C22×C4×Dic3C2211yescharacteristic
15C22×Dic12C2211yescharacteristic
16C2×#887C2212yes—
17C2×C3×#107C2211yescharacteristic
18C2×C3×C22⋊C8C2211yescharacteristic
19C3×#380C2212yes—
20C2×#830C2211yescharacteristic
21C2×#816C2212yes—
22C2×Dic3⋊C8C2211yescharacteristic
23C2×Dic3⋊C8C2211yescharacteristic
24C2×C8×Dic3C2211yescharacteristic
25C2×C8×Dic3C2211yescharacteristic
26Dic3×C4⋊C4C2212yes—
27#4973C2212yes—
28#4972C2212yes—
29C2×#810C2212yes—
30C2×#810C2212yes—
31C2×#807C2212yes—
32#4946—22no—
33Dic3×C22⋊C4—22no—
34#4873—24no—
35#4718—24no—
36#4652—22no—
37#4649—22no—
38C3×#340C2211yescharacteristic
39C3×#340C2212yes—
40#4551C2211yescharacteristic
41#4529C2212yes—
42C2×C4×Q16—31no—
43C2×#431—31no—
44C4×C22⋊Q8—32no—
45#2087—31no—
46#2076D311yescharacteristic
47#2068—32no—
48#2056—31no—
49#2036—32no—
50#2001—31no—
51C4×#107—31no—
52C4×#103—32no—
53C23×C3×C4C2311yescharacteristic
54C23×Dic3D411yescharacteristic
55C23×Dic3D411yescharacteristic
56C22×Dic6C2312yes—
57C22×C3×C8C2311yescharacteristic
58C22×C3×C8C2311yescharacteristic
59C2×C3×C4⋊C4C2312yes—
60C2×C3×C4⋊C4C2312yes—
61C2×C3×C22⋊C4—22no—
62C2×#217—22no—
63C2×C4⋊Dic3C2312yes—
64#924—24no—
65#924—24no—
66C2×Dic3⋊C4—22no—
67C2×Dic3⋊C4—22no—
68C2×Dic3⋊C4—22no—
69C2×C4×Dic3—22no—
70C2×C4×Dic3—22no—
71C2×C4×Dic3—22no—
72C2×C4×Dic3—22no—
73C2×C4×Dic3D411yescharacteristic
74C2×C4×Dic3D411yescharacteristic
75C2×C4×Dic3C2311yescharacteristic
76C2×C4×Dic3C2311yescharacteristic
77C22×C3⋊C8C2311yescharacteristic
78C22×C3⋊C8C2311yescharacteristic
79C2×Dic12—24no—
80C2×Dic12—22no—
81#887—24no—
82#887—24no—
83C3×#107—22no—
84C3×C22⋊C8—22no—
85C3×#95—22no—
86C3⋊#95—22no—
87C3⋊#95—22no—
88C3⋊#95—22no—
89C3⋊#95—22no—
90#830—22no—
91#816—24no—
92Dic3⋊C8—22no—
93Dic3⋊C8—22no—
94C8×Dic3—22no—
95C8×Dic3D414yes—
96#810—24no—
97#810—24no—
98#807—24no—
99C22×Q16—31no—
100C2×C22⋊Q8—32no—
101C2×C4×Q8—32no—
102C22×C42—31no—
103C4×Q16—64no—
104#431—64no—
105C2×#107—32no—
106C2×#107C2×D311yescharacteristic
107C2×C4⋊C8—31no—
108C2×C4⋊C8—31no—
109C2×#103—32no—
110C2×#103—32no—
111C2×#103—32no—
112C2×C22⋊C8—31no—
113C2×C22⋊C8C2×D311yescharacteristic
114C2×C4×C8—31no—
115C2×C4×C8—31no—
116#391—32no—
117#389—32no—
118#387—62no—
119#385—62no—
120#380C2×D312yes—
121C4×C4⋊C4—32no—
122C4×C4⋊C4—62no—
123C4×C22⋊C4—62no—
124#340—31no—
125#340—32no—
126#340C2×D311yescharacteristic
127#340C2×D312yes—
128C24×C3C2×D411yescharacteristic
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2411yescharacteristic
137C22×Dic3—22no—
138C22×Dic3—22no—
139C22×Dic3—22no—
140C22×Dic3—22no—
141C22×Dic3—22no—
142C22×Dic3—42no—
143C22×Dic3Q1614yescomplemented
144C22×Dic3C2×D411yescharacteristic
145C22×Dic3C2×D411yescharacteristic
146C22×Dic3C2×D411yescharacteristic
147C22×Dic3C2×D411yescharacteristic
148C2×Dic6—22no—
149C2×Dic6—22no—
150C2×Dic6—22no—
151C2×Dic6—22no—
152C2×Dic6—22no—
153C2×Dic6#4114yes—
154C2×C3×C8—22no—
155C2×C3×C8—21no—
156C2×C3×C8—22no—
157C2×C3×C8C2×D412yes—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—22no—
162C3×C22⋊C4—44no—
163#217—28no—
164C4⋊Dic3—22no—
165C4⋊Dic3#4114yes—
166Dic3⋊C4—28no—
167Dic3⋊C4—44no—
168Dic3⋊C4—44no—
169C4×Dic3—21no—
170C4×Dic3—21no—
171C4×Dic3—22no—
172C4×Dic3—22no—
173C4×Dic3—44no—
174C4×Dic3—44no—
175C4×Dic3C2×D412yes—
176C4×Dic3#4112yes—
177C2×C3⋊C8—22no—
178C2×C3⋊C8—21no—
179C2×C3⋊C8—22no—
180C2×C3⋊C8C2×D412yes—
181Dic12—28no—
182C22×Q8—32no—
183C23×C4—31no—
184C23×C4—31no—
185C23×C4C22×D311yescharacteristic
186C2×Q16—64no—
187C2×Q16—62no—
188C22×C8—31no—
189C22×C8—31no—
190C22×C8C22×D311yescharacteristic
191C22×C8C22×D311yescharacteristic
192C22⋊Q8—64no—
193C22⋊Q8—64no—
194C4×Q8—64no—
195C4×Q8—64no—
196C2×C4⋊C4—32no—
197C2×C4⋊C4—62no—
198C2×C4⋊C4—62no—
199C2×C4⋊C4—62no—
200C2×C4⋊C4C22×D312yes—
201C2×C4⋊C4C22×D312yes—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—62no—
204C2×C42—31no—
205C2×C42—31no—
206C2×C42—31no—
207C2×C42—31no—
208C2×C42—62no—
209C2×C42—62no—
210C2×C42—62no—
211C2×C42—62no—
212#107—22no—
213#107—64no—
214C4⋊C8—62no—
215C4⋊C8—62no—
216#103—64no—
217#103—64no—
218#103—64no—
219C22⋊C8—22no—
220C22⋊C8—62no—
221C4×C8—34no—
222C4×C8—62no—
223#95—22no—
224#95—62no—
225#95—62no—
226#95—62no—
227#95—62no—
228C23×C3—21no—
229C23×C3—22no—
230C23×C3—21no—
231C23×C3—22no—
232C23×C3C2×Q1612yes—
233C23×C3C22×D411yescharacteristic
234C2×C3×C4—21no—
235C2×C3×C4—22no—
236C2×C3×C4—41no—
237C2×C3×C4—42no—
238C2×C3×C4—41no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4C22×D411yescharacteristic
246C2×C3×C4C2×#4112yes—
247C2×C3×C4C2×#4111yescharacteristic
248C2×C3×C4C2×#4112yes—
249C2×Dic3—21no—
250C2×Dic3—21no—
251C2×Dic3—22no—
252C2×Dic3—28no—
253C2×Dic3—21no—
254C2×Dic3—24no—
255C2×Dic3—24no—
256C2×Dic3—22no—
257C2×Dic3—44no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262C2×Dic3#12112yes—
263C2×Dic3#12112yes—
264C2×Dic3C2×Q1612yes—
265C2×Dic3C2×Q1612yes—
266C2×Dic3#13512yes—
267Dic6—24no—
268Dic6—24no—
269Dic6—42no—
270Dic6—42no—
271C3×C8—22no—
272C3×C8—41no—
273C3⋊C8—22no—
274C3⋊C8—41no—
275C24D3×D411yescharacteristic
276C2×Q8—34no—
277C2×Q8—62no—
278C2×Q8—62no—
279C2×Q8—62no—
280C2×Q8—62no—
281C2×Q8—62no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—31no—
289C22×C4—34no—
290C22×C4—31no—
291C22×C4—31no—
292C22×C4—31no—
293C22×C4—62no—
294C22×C4—62no—
295C22×C4—62no—
296C22×C4—62no—
297C22×C4—62no—
298C22×C4—122no—
299C22×C4D3×D411yescharacteristic
300C22×C4C23×D311yescharacteristic
301Q16—68no—
302C2×C8—22no—
303C2×C8—21no—
304C2×C8—22no—
305C2×C8—32no—
306C2×C8—62no—
307C2×C8—61no—
308C2×C8—62no—
309C2×C8D3×D412yes—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—34no—
315C4⋊C4—68no—
316C4⋊C4—62no—
317C4⋊C4—124no—
318C4⋊C4—124no—
319C22⋊C4—44no—
320C22⋊C4—68no—
321C42—32no—
322C42—32no—
323C42—62no—
324C42—62no—
325C42—61no—
326C42—61no—
327C42—124no—
328C42—124no—
329C22×C3—24no—
330C22×C3—22no—
331C22×C3—42no—
332C22×C3—41no—
333C22×C3—41no—
334C22×C3C2×#12111yescharacteristic
335C22×C3#52512yes—
336C22×C3C22×Q1611yescharacteristic
337C22×C3C2×#13511yescharacteristic
338C22×C3#57512yes—
339C3×C4—41no—
340C3×C4—82no—
341C3×C4—82no—
342C3×C4#52511yescharacteristic
343C3×C4#52912yes—
344C3×C4#53211yescharacteristic
345Dic3—22no—
346Dic3—22no—
347Dic3—44no—
348Dic3—44no—
349Dic3—41no—
350Dic3#46714yes—
351C23—22no—
352C23—21no—
353C23—22no—
354C23—21no—
355C23D3×Q1612yes—
356C23C2×D3×D411yescharacteristic
357Q8—64no—
358Q8—64no—
359Q8—122no—
360Q8—122no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—32no—
364C2×C4—32no—
365C2×C4—32no—
366C2×C4—32no—
367C2×C4—32no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—68no—
378C2×C4—61no—
379C2×C4—62no—
380C2×C4—61no—
381C2×C4—64no—
382C2×C4—64no—
383C2×C4—62no—
384C2×C4—61no—
385C2×C4—124no—
386C2×C4—122no—
387C2×C4—122no—
388C2×C4—122no—
389C2×C4—122no—
390C2×C4C2×D3×D411yescharacteristic
391C2×C4D3×#4112yes—
392C2×C4D3×#4112yes—
393C2×C4D3×#4111yescharacteristic
394C8—22no—
395C8—41no—
396C8—62no—
397C8—121no—
398C2×C3—42no—
399C2×C3#239211yescharacteristic
400C2×C3C2×#46711yescharacteristic
401C2×C3#305411yescharacteristic
402C2×C3#310212yes—
403C2×C3#310412yes—
404C22—24no—
405C22—22no—
406C22—41no—
407C22—42no—
408C22—41no—
409C22D3×#12111yescharacteristic
410C22#565612yes—
411C22C2×D3×Q1611yescharacteristic
412C22D3×#13511yescharacteristic
413C22#577312yes—
414C4—34no—
415C4—41no—
416C4—62no—
417C4—62no—
418C4—82no—
419C4—82no—
420C4—124no—
421C4—124no—
422C4—121no—
423C4#565611yescharacteristic
424C4#566811yescharacteristic
425C4#570612yes—
426C3#2369811yescharacteristic, complemented, Sylow
427C2—42no—
428C2#7745711yescharacteristic
429C2D3×#46711yescharacteristic
430C2#8086311yescharacteristic
431C2#8134812yes—
432C2#8145212yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2076C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2076D3
hypocenterC3#23698
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2076

Complements

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

NH ≅ G/Ntype
1. C2×#4873C2semidirect
5. #70622C2semidirect
8. #68913C2semidirect
9. #68864C2semidirect
10. #68785C2semidirect
11. #68781C2semidirect
143. C22×Dic3Q16semidirect
426. C3#23698semidirect

Maximal subgroups

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

subgroupindex
C2×#48732
C2×#47182
#738662
C3×#20762
#706222
#699642
#695272
#689132
#688642
#687852
#687812
#236983