Finite Group Databasebeta

← #10824629

Subgroups — #10824629

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×#4833C211yesmaximal, characteristic
3#75965C212yesmaximal
4C3×#2086C211yesmaximal, characteristic, complemented
5#70636C211yesmaximal, characteristic
6#70487C211yesmaximal, characteristic, complemented
7#69634C212yesmaximal, complemented
8#68888C211yesmaximal, characteristic
9#68824C212yesmaximal
10#68784C211yesmaximal, characteristic, complemented
11#68781C212yesmaximal, complemented
12#23699—31nomaximal, Sylow
13C22×Dic12C2211yescharacteristic
14C2×#897C2212yes—
15C2×#887C2212yes—
16C2×C42×D3C2211yescharacteristic
17C2×C3×#107C2211yescharacteristic
18C2×C3×C4⋊C8C2211yescharacteristic
19C3×#389C2212yes—
20C2×(C2×D3)⋊C8C2211yescharacteristic
21C2×(C2×D3)⋊C8C2211yescharacteristic
22C2×#816C2212yes—
23C2×C8×Dic3C2211yescharacteristic
24C2×C8×Dic3C2211yescharacteristic
25#4989—22no—
26#4987C2212yes—
27Dic3×C4⋊C4—22no—
28Dic3×C4⋊C4C2212yes—
29C2×#810C2212yes—
30C2×#810C2212yes—
31C2×#808C2212yes—
32#4929C2212yes—
33C2×C4⋊(C3⋊C8)C2211yescharacteristic
34#4873—24no—
35#4833—24no—
36#4666—22no—
37#4654—22no—
38C3×#340C2211yescharacteristic
39C3×#340C2212yes—
40#4551C2211yescharacteristic
41#4529C2212yes—
42C2×C4×Q16—31no—
43C2×#431—31no—
44C4×C22⋊Q8—32no—
45#2086D311yescharacteristic, complemented
46#2075—31no—
47#2066—32no—
48#2056—31no—
49#2035—32no—
50#2001—31no—
51C4×#106—31no—
52C4×#103—32no—
53C22×C4×D3D411yescharacteristic
54C22×C4×D3D411yescharacteristic
55C22×C4×D3C2311yescharacteristic
56C22×Dic6C2312yes—
57C22×C3×C8C2311yescharacteristic
58C22×C3×C8C2311yescharacteristic
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4C2312yes—
61C2×C3×C4⋊C4C2312yes—
62C2×C3×C42C2311yescharacteristic
63C2×(C2×D3)⋊C4—22no—
64C2×(C2×D3)⋊C4—22no—
65C2×C4⋊Dic3—22no—
66C2×C4⋊Dic3C2312yes—
67C2×Dic3⋊C4—22no—
68C2×C4×Dic3—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—
79#887—24no—
80#887—24no—
81C42×D3—22no—
82C42×D3—22no—
83C3×#107—22no—
84C3×C4⋊C8—22no—
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#816—24no—
93C8×Dic3—22no—
94C8×Dic3D414yes—
95#810—24no—
96#810—24no—
97#808—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×#107C2×D311yescharacteristic
106C2×#106—32no—
107C2×C4⋊C8—31no—
108C2×C4⋊C8C2×D311yescharacteristic
109C2×#103—32no—
110C2×#103—32no—
111C2×#103—32no—
112C2×C22⋊C8—31no—
113C2×C22⋊C8—31no—
114C2×C4×C8—31no—
115C2×C4×C8—31no—
116#391—32no—
117#389C2×D312yes—
118#387—62no—
119#385—62no—
120#380—32no—
121C4×C4⋊C4—32no—
122C4×C4⋊C4—62no—
123C4×C22⋊C4—62no—
124#340—31no—
125#340—32no—
126#340C2×D311yescharacteristic
127#340C2×D312yes—
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—22no—
147C2×C4×D3—24no—
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—44no—
162C3×C4⋊C4#4114yes—
163C3×C42—21no—
164C3×C42—21no—
165(C2×D3)⋊C4—28no—
166(C2×D3)⋊C4—44no—
167C4⋊Dic3—28no—
168C4⋊Dic3—22no—
169C4⋊Dic3—22no—
170Dic3⋊C4—44no—
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×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—22no—
197C2×C4⋊C4—32no—
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—31no—
205C2×C42—31no—
206C2×C42—31no—
207C2×C42—62no—
208C2×C42—62no—
209C2×C42—62no—
210C2×C42—62no—
211C2×C42C22×D311yescharacteristic
212#107—22no—
213#106—64no—
214C4⋊C8—22no—
215C4⋊C8—62no—
216#103—64no—
217#103—64no—
218#103—64no—
219C22⋊C8—62no—
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×C3C22×D411yescharacteristic
229C22×D3—21no—
230C22×D3—22no—
231C22×D3—21no—
232C22×D3—22no—
233C22×D3#13512yes—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—24no—
238C2×C3×C4—22no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4#13512yes—
245C2×C3×C4C22×D411yescharacteristic
246C2×C3×C4C2×#4112yes—
247C2×C3×C4C2×#4112yes—
248C2×C3×C4C2×#4111yescharacteristic
249C2×Dic3—21no—
250C2×Dic3—21no—
251C2×Dic3—21no—
252C2×Dic3—22no—
253C2×Dic3—24no—
254C2×Dic3—42no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—42no—
258C2×Dic3—42no—
259C2×Dic3#12112yes—
260C2×Dic3#12112yes—
261C2×Dic3#13714yes—
262C4×D3—28no—
263C4×D3—41no—
264C4×D3—42no—
265C4×D3—41no—
266C4×D3—44no—
267Dic6—24no—
268Dic6—24no—
269Dic6—42no—
270Dic6—42no—
271C3×C8—22no—
272C3×C8—41no—
273C3⋊C8—22no—
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—31no—
286C22×C4—31no—
287C22×C4—62no—
288C22×C4—62no—
289C22×C4—64no—
290C22×C4—62no—
291C22×C4—62no—
292C22×C4—62no—
293C22×C4—61no—
294C22×C4—62no—
295C22×C4—61no—
296C22×C4—62no—
297C22×C4—62no—
298C22×C4D3×D411yescharacteristic
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—44no—
314C4⋊C4—68no—
315C4⋊C4—62no—
316C4⋊C4—62no—
317C4⋊C4—124no—
318C4⋊C4#23914yes—
319C22⋊C4—68no—
320C22⋊C4—124no—
321C42—21no—
322C42—21no—
323C42—32no—
324C42—32no—
325C42—62no—
326C42—62no—
327C42—124no—
328C42—124no—
329C22×C3C2×#12111yescharacteristic
330C22×C3#52512yes—
331C22×C3C2×#13511yescharacteristic
332C22×C3C2×#13511yescharacteristic
333C22×C3C2×#13712yes—
334C2×D3—24no—
335C2×D3—22no—
336C2×D3—42no—
337C2×D3—41no—
338C2×D3—41no—
339C3×C4—22no—
340C3×C4—44no—
341C3×C4—41no—
342C3×C4—82no—
343C3×C4#52511yescharacteristic
344C3×C4#52912yes—
345C3×C4#53211yescharacteristic
346Dic3—22no—
347Dic3—44no—
348Dic3—41no—
349Dic3—82no—
350Dic3#47014yes—
351C23—32no—
352C23—61no—
353C23—62no—
354C23—61no—
355C23—62no—
356C23C2×D3×D411yescharacteristic
357Q8—64no—
358Q8—64no—
359Q8—122no—
360Q8—122no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—24no—
366C2×C4—34no—
367C2×C4—32no—
368C2×C4—32no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—68no—
375C2×C4—61no—
376C2×C4—61no—
377C2×C4—61no—
378C2×C4—64no—
379C2×C4—62no—
380C2×C4—124no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—121no—
384C2×C4—122no—
385C2×C4—121no—
386C2×C4—122no—
387C2×C4—122no—
388C2×C4—122no—
389C2×C4#91212yes—
390C2×C4C2×D3×D411yescharacteristic
391C2×C4C2×#23912yes—
392C2×C4D3×#4112yes—
393C2×C4D3×#4111yescharacteristic
394C8—22no—
395C8—41no—
396C8—62no—
397C8—121no—
398C2×C3#239211yescharacteristic
399C2×C3#305011yescharacteristic
400C2×C3C2×#47011yescharacteristic
401C2×C3#310712yes—
402C2×C3#310812yes—
403D3—42no—
404C22—64no—
405C22—62no—
406C22—121no—
407C22—122no—
408C22—121no—
409C22D3×#12111yescharacteristic
410C22#562712yes—
411C22C2×#91211yescharacteristic
412C22D3×#13511yescharacteristic
413C22D3×#13712yes—
414C4—22no—
415C4—34no—
416C4—41no—
417C4—44no—
418C4—62no—
419C4—82no—
420C4—121no—
421C4—124no—
422C4—242no—
423C4#565611yescharacteristic
424C4#569112yes—
425C4#572511yescharacteristic
426C3#2369911yescharacteristic, complemented, Sylow
427C2—122no—
428C2#7773711yescharacteristic
429C2#8071411yescharacteristic
430C2D3×#47011yescharacteristic
431C2#8094912yes—
432C2#8120512yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2086C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2086D3
hypocenterC3#23699
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2086

Complements

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

NH ≅ G/Ntype
1. C2×#4873C2semidirect
4. C3×#2086C2semidirect
6. #70487C2semidirect
7. #69634C2semidirect
10. #68784C2semidirect
11. #68781C2semidirect
45. #2086D3semidirect
426. C3#23699semidirect

Maximal subgroups

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

subgroupindex
C2×#48732
C2×#48332
#759652
C3×#20862
#706362
#704872
#696342
#688882
#688242
#687842
#687812
#236993