Finite Group Databasebeta

← #10824617

Subgroups — #10824617

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×C4×Dic12C211yesmaximal, characteristic, complemented
2C2×#4685C211yesmaximal, characteristic
3C4×#897C212yesmaximal
4C3×C4×#107C211yesmaximal, characteristic, complemented
5#70643C212yesmaximal, complemented
6#70635C212yesmaximal
7C4×#810C212yesmaximal, complemented
8#68888C211yesmaximal, characteristic
9#68864C211yesmaximal, characteristic, complemented
10#68839C211yesmaximal, characteristic, complemented
11#68831C211yesmaximal, characteristic
12#23698—31nomaximal, Sylow
13C22×Dic12C2211yescharacteristic
14C2×#897C2212yes—
15C2×C42×D3C2211yescharacteristic
16C2×C4×Dic6C2212yes—
17C2×C3×#107C2212yes—
18C2×C3×C4×C8C2211yescharacteristic
19C3×C4×C4⋊C4C2212yes—
20C2×(C2×D3)⋊C8C2211yescharacteristic
21C2×(C2×D3)⋊C8C2211yescharacteristic
22C2×#818C2211yescharacteristic
23C2×#816C2212yes—
24C2×Dic3⋊C8C2211yescharacteristic
25C2×Dic3⋊C8C2211yescharacteristic
26#4987C2212yes—
27#4984C2212yes—
28#4973C2212yes—
29C2×#810C2212yes—
30C2×#810C2212yes—
31C4×(C2×D3)⋊C4—22no—
32C4×C4⋊Dic3—22no—
33C2×C4×C3⋊C8C2211yescharacteristic
34C4×Dic12—24no—
35#4685—24no—
36#4668—22no—
37#4653—22no—
38C3×#340C2211yescharacteristic
39C3×#340C2211yescharacteristic
40#4529C2212yes—
41#4529C2212yes—
42C2×C4×Q16—31no—
43C2×#431—31no—
44C4×C22⋊Q8—32no—
45#2087—31no—
46#2076—31no—
47#2068—32no—
48#2056—31no—
49#2036—32no—
50#2001—31no—
51C4×#107D311yescharacteristic, complemented
52C4×#103—32no—
53C22×C4×D3D411yescharacteristic
54C22×C4×D3D411yescharacteristic
55C22×C4×D3C2311yescharacteristic
56C22×Dic6C2312yes—
57C22×C3×C8C2311yescharacteristic
58C22×C3×C8C2311yescharacteristic
59C2×C3×C4⋊C4C2312yes—
60C2×C3×C4⋊C4C2312yes—
61C2×C3×C42—22no—
62C2×C3×C42C2311yescharacteristic
63C2×(C2×D3)⋊C4—22no—
64C2×(C2×D3)⋊C4—22no—
65C2×C4⋊Dic3—22no—
66C2×C4⋊Dic3—22no—
67C2×C4⋊Dic3C2312yes—
68C2×Dic3⋊C4—22no—
69C2×C4×Dic3—22no—
70C2×C4×Dic3D411yescharacteristic
71C2×C4×Dic3D411yescharacteristic
72C2×C4×Dic3C2311yescharacteristic
73C22×C3⋊C8C2311yescharacteristic
74C22×C3⋊C8C2311yescharacteristic
75C2×Dic12—24no—
76C2×Dic12—22no—
77#897—24no—
78#897—24no—
79C42×D3—22no—
80C42×D3—22no—
81C4×Dic6—24no—
82C4×Dic6—24no—
83C3×#107—24no—
84C3×C4×C8D414yes—
85C3×#95—22no—
86C3⋊#95—22no—
87C3⋊#95—22no—
88C3⋊#95—22no—
89C3⋊#95—22no—
90(C2×D3)⋊C8—22no—
91(C2×D3)⋊C8—22no—
92#818—22no—
93#816—24no—
94Dic3⋊C8—22no—
95Dic3⋊C8—22no—
96#810—24no—
97#810—24no—
98C4×C3⋊C8—22no—
99C22×Q16—31no—
100C2×C22⋊Q8—32no—
101C2×C4×Q8—32no—
102C22×C42—31no—
103C4×Q16—64no—
104#431—64no—
105C2×#107—31no—
106C2×#107C2×D312yes—
107C2×C4⋊C8—31no—
108C2×C4⋊C8—31no—
109C2×#103—32no—
110C2×#103—32no—
111C2×#103—32no—
112C2×C22⋊C8—31no—
113C2×C22⋊C8—31no—
114C2×C4×C8—31no—
115C2×C4×C8C2×D311yescharacteristic
116#391—32no—
117#389—32no—
118#387—62no—
119#385—62no—
120#380—32no—
121C4×C4⋊C4—62no—
122C4×C4⋊C4C2×D312yes—
123C4×C22⋊C4—62no—
124#340—32no—
125#340—32no—
126#340C2×D311yescharacteristic
127#340C2×D311yescharacteristic
128C23×D3C2×D411yescharacteristic
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2411yescharacteristic
134C22×Dic3—22no—
135C22×Dic3—22no—
136C22×Dic3C2×D411yescharacteristic
137C22×Dic3C2×D411yescharacteristic
138C22×Dic3C2×D411yescharacteristic
139C2×C4×D3—22no—
140C2×C4×D3—22no—
141C2×C4×D3—22no—
142C2×C4×D3—21no—
143C2×C4×D3—21no—
144C2×C4×D3—22no—
145C2×C4×D3—22no—
146C2×C4×D3—42no—
147C2×C4×D3Q1614yescomplemented
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—22no—
156C2×C3×C8C2×D412yes—
157C2×C3×C8C2×D412yes—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4#4114yes—
162C3×C42—44no—
163C3×C42C2×D412yes—
164C3×C42#4112yes—
165(C2×D3)⋊C4—28no—
166(C2×D3)⋊C4—44no—
167C4⋊Dic3—28no—
168C4⋊Dic3—22no—
169C4⋊Dic3—22no—
170C4⋊Dic3—44no—
171Dic3⋊C4—44no—
172C4×Dic3—22no—
173C4×Dic3—22no—
174C4×Dic3—21no—
175C4×Dic3—21no—
176C4×Dic3—44no—
177C2×C3⋊C8—21no—
178C2×C3⋊C8—21no—
179C2×C3⋊C8—22no—
180C2×C3⋊C8—22no—
181Dic12—28no—
182C22×Q8—32no—
183C23×C4—31no—
184C23×C4—31no—
185C23×C4—31no—
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—62no—
203C2×C22⋊C4—62no—
204C2×C42—22no—
205C2×C42—31no—
206C2×C42—31no—
207C2×C42—31no—
208C2×C42—62no—
209C2×C42—62no—
210C2×C42—62no—
211C2×C42C22×D311yescharacteristic
212#107—24no—
213#107—62no—
214C4⋊C8—62no—
215C4⋊C8—62no—
216#103—64no—
217#103—64no—
218#103—64no—
219C22⋊C8—62no—
220C22⋊C8—62no—
221C4×C8—62no—
222C4×C8D1214yes—
223#95—22no—
224#95—62no—
225#95—62no—
226#95—62no—
227#95—62no—
228C23×C3C22×D411yescharacteristic
229C22×D3—21no—
230C22×D3—22no—
231C22×D3—21no—
232C22×D3—22no—
233C22×D3C2×Q1612yes—
234C2×C3×C4—22no—
235C2×C3×C4—24no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4#12112yes—
242C2×C3×C4#12112yes—
243C2×C3×C4C2×Q1612yes—
244C2×C3×C4#13512yes—
245C2×C3×C4C22×D411yescharacteristic
246C2×C3×C4C2×#4112yes—
247C2×C3×C4C2×#4111yescharacteristic
248C2×C3×C4C2×#4112yes—
249C2×Dic3—21no—
250C2×Dic3—21no—
251C2×Dic3—21no—
252C2×Dic3—21no—
253C2×Dic3—22no—
254C2×Dic3—24no—
255C2×Dic3—22no—
256C2×Dic3—42no—
257C2×Dic3—42no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3C2×Q1612yes—
262C4×D3—28no—
263C4×D3—44no—
264C4×D3—41no—
265C4×D3—42no—
266C4×D3—41no—
267Dic6—24no—
268Dic6—24no—
269Dic6—42no—
270Dic6—42no—
271C3×C8—22no—
272C3×C8—22no—
273C3⋊C8—41no—
274C3⋊C8—41no—
275C24—31no—
276C2×Q8—34no—
277C2×Q8—62no—
278C2×Q8—62no—
279C2×Q8—62no—
280C2×Q8—62no—
281C2×Q8—62no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—31no—
285C22×C4—34no—
286C22×C4—31no—
287C22×C4—31no—
288C22×C4—62no—
289C22×C4—62no—
290C22×C4—62no—
291C22×C4—62no—
292C22×C4—61no—
293C22×C4—62no—
294C22×C4—61no—
295C22×C4—62no—
296C22×C4—62no—
297C22×C4—122no—
298C22×C4D3×D411yescharacteristic
299C22×C4D3×D411yescharacteristic
300C22×C4C23×D311yescharacteristic
301Q16—68no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—61no—
305C2×C8—61no—
306C2×C8—62no—
307C2×C8—62no—
308C2×C8C2×D1212yes—
309C2×C8C2×D1212yes—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—68no—
314C4⋊C4—62no—
315C4⋊C4—62no—
316C4⋊C4—124no—
317C4⋊C4—124no—
318C4⋊C4#23514yes—
319C22⋊C4—68no—
320C22⋊C4—124no—
321C42—44no—
322C42—62no—
323C42—62no—
324C42—61no—
325C42—61no—
326C42—124no—
327C42C2×D1212yes—
328C42#23912yes—
329C22×C3C2×#12111yescharacteristic
330C22×C3#52512yes—
331C22×C3C22×Q1611yescharacteristic
332C22×C3C2×#13511yescharacteristic
333C22×C3#57512yes—
334C2×D3—24no—
335C2×D3—22no—
336C2×D3—42no—
337C2×D3—41no—
338C2×D3—41no—
339C3×C4—22no—
340C3×C4—44no—
341C3×C4—82no—
342C3×C4#46714yes—
343C3×C4#52511yescharacteristic
344C3×C4#52912yes—
345C3×C4#53211yescharacteristic
346Dic3—22no—
347Dic3—44no—
348Dic3—41no—
349Dic3—41no—
350Dic3—82no—
351C23—32no—
352C23—61no—
353C23—61no—
354C23—62no—
355C23—62no—
356C23C2×D3×D411yescharacteristic
357Q8—64no—
358Q8—64no—
359Q8—122no—
360Q8—122no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—32no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—68no—
372C2×C4—61no—
373C2×C4—61no—
374C2×C4—62no—
375C2×C4—64no—
376C2×C4—62no—
377C2×C4—124no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—122no—
383C2×C4—122no—
384C2×C4—122no—
385C2×C4—122no—
386C2×C4#89512yes—
387C2×C4#89512yes—
388C2×C4#91212yes—
389C2×C4D3×Q1612yes—
390C2×C4C22×D1211yescharacteristic
391C2×C4C2×#23512yes—
392C2×C4C2×#23911yescharacteristic
393C2×C4D3×#4112yes—
394C8—22no—
395C8—22no—
396C8—121no—
397C8—121no—
398C2×C3#239211yescharacteristic
399C2×C3C2×#46711yescharacteristic
400C2×C3#305411yescharacteristic
401C2×C3#310212yes—
402C2×C3#310412yes—
403D3—42no—
404C22—62no—
405C22—64no—
406C22—121no—
407C22—121no—
408C22—122no—
409C22C2×#89511yescharacteristic
410C22#554712yes—
411C22C2×#91211yescharacteristic
412C22C2×D3×Q1611yescharacteristic
413C22#576812yes—
414C4—22no—
415C4—44no—
416C4—62no—
417C4—82no—
418C4—121no—
419C4—124no—
420C4—121no—
421C4—242no—
422C4#488714yes—
423C4#547711yescharacteristic
424C4#552411yescharacteristic
425C4#558012yes—
426C3#2369811yescharacteristic, complemented, Sylow
427C2—122no—
428C2#7377611yescharacteristic
429C2C2×#488711yescharacteristic
430C2#7918111yescharacteristic
431C2#7959112yes—
432C2#7972212yes—
subgroupG/H
centerC22×C4D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#107C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenterC4×#107D3
hypocenterC3#23698
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#107

Complements

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

NH ≅ G/Ntype
1. C2×C4×Dic12C2semidirect
4. C3×C4×#107C2semidirect
5. #70643C2semidirect
7. C4×#810C2semidirect
9. #68864C2semidirect
10. #68839C2semidirect
51. C4×#107D3semidirect
147. C2×C4×D3Q16semidirect
426. C3#23698semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Dic122
C2×#46852
C4×#8972
C3×C4×#1072
#706432
#706352
C4×#8102
#688882
#688642
#688392
#688312
#236983