Finite Group Databasebeta

← #10823960

Subgroups — #10823960

Order 768.

1010 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4802C212yesmaximal
2D3×#391C211yesmaximal, characteristic
3#74148C211yesmaximal, characteristic
4C3×#2074C211yesmaximal, characteristic, complemented
5#70233C212yesmaximal, complemented
6#69671C211yesmaximal, characteristic, complemented
7#69560C211yesmaximal, characteristic, complemented
8#68914C212yesmaximal, complemented
9#68867C212yesmaximal
10#68660C211yesmaximal, characteristic
11#68638C211yesmaximal, characteristic, complemented
12#23629—31nomaximal, Sylow
13C22×D3×Q8C2211yescharacteristic
14C22×ΓL(1,𝔽25)C2212yes—
15C2×#892C2212yes—
16C2×D3×C4⋊C4C2211yescharacteristic
17C2×C42×D3C2211yescharacteristic
18C2×C3×#103C2212yes—
19C3×#391C2211yescharacteristic
20C3×#382C2211yescharacteristic
21#5229C2211yescharacteristic
22C2×#835C2212yes—
23C2×#816C2212yes—
24#4975C2212yes—
25#4973C2211yescharacteristic
26C2×#810C2212yes—
27#4936C2211yescharacteristic
28#4929C2211yescharacteristic
29#4802—28no—
30#4667—22no—
31D3×#95—22no—
32#4649—22no—
33#4646—22no—
34C3×#340C2212yes—
35C3×#336C2211yescharacteristic
36#4551C2212yes—
37#4550C2212yes—
38#4529C2212yes—
39#4529C2212yes—
40#4524C2211yescharacteristic
41C2×#419—32no—
42C2×#391—31no—
43#2285—31no—
44#2074—31no—
45#2074—31no—
46#2074—31no—
47#2074D311yescharacteristic, complemented
48#2072—32no—
49#2072—32no—
50#1736—32no—
51#1728—31no—
52C22×C3×Q8C2311yescharacteristic
53C2×D3×Q8—22no—
54C2×D3×Q8—24no—
55C2×D3×Q8—24no—
56C2×D3×Q8D414yes—
57C22×C4×D3—21no—
58C22×C4×D3—21no—
59C22×C4×D3D412yes—
60C22×C4×D3C2311yescharacteristic
61C22×Dic6—21no—
62C22×Dic6C2311yescharacteristic
63C22×C3×C8C2312yes—
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C4⋊C4C2311yescharacteristic
66C2×C3×C42C2311yescharacteristic
67C2×(C2×D3)⋊C4—22no—
68C2×C4⋊Dic3C2311yescharacteristic
69C2×C4⋊Dic3C2312yes—
70C2×Dic3⋊C4—21no—
71C2×Dic3⋊C4—22no—
72C2×Dic3⋊C4—22no—
73C2×C4×Dic3—22no—
74C2×C4×Dic3D412yes—
75C2×C4×Dic3C2311yescharacteristic
76C22×C3⋊C8C2312yes—
77C2×ΓL(1,𝔽25)—24no—
78C2×ΓL(1,𝔽25)—24no—
79C2×ΓL(1,𝔽25)—24no—
80#892—24no—
81#892—24no—
82D3×C4⋊C4—24no—
83C42×D3—24no—
84C3×#103—24no—
85C3×#95—22no—
86C3×#95—22no—
87#835—24no—
88C3⋊#95—22no—
89C3⋊#95—22no—
90C3⋊#95—22no—
91C3⋊#95—22no—
92#816—24no—
93#810—24no—
94C23×Q8—31no—
95C22×M16—32no—
96C2×#117—32no—
97C22×C4⋊C4—31no—
98C22×C42—31no—
99#419—68no—
100C2×#103—32no—
101C2×#103—32no—
102C2×#103—32no—
103C2×#103C2×D312yes—
104#391—31no—
105#391—31no—
106#391—31no—
107#391—62no—
108#391C2×D311yescharacteristic
109#384—32no—
110#382—31no—
111#382C2×D311yescharacteristic
112#380—62no—
113C4×C4⋊C4—62no—
114C2×#95—62no—
115#340—32no—
116#340—32no—
117#340—32no—
118#340C2×D312yes—
119#336—32no—
120#336—31no—
121#336C2×D311yescharacteristic
122C23×D3C2×D411yescharacteristic
123C2×C3×Q8—21no—
124C2×C3×Q8—22no—
125C2×C3×Q8—22no—
126C2×C3×Q8C2×D412yes—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—22no—
130C22×C3×C4C2×D412yes—
131C22×C3×C4C2411yescharacteristic
132C22×Dic3—21no—
133C22×Dic3—21no—
134C22×Dic3—22no—
135C22×Dic3C2×D412yes—
136C22×Dic3C2×D411yescharacteristic
137D3×Q8—216no—
138D3×Q8—48no—
139C2×C4×D3—24no—
140C2×C4×D3—24no—
141C2×C4×D3—24no—
142C2×C4×D3—42no—
143C2×C4×D3—42no—
144C2×C4×D3—42no—
145C2×C4×D3—44no—
146C2×C4×D3—42no—
147C2×C4×D3C2×D412yes—
148C2×C4×D3#4114yes—
149C2×C4×D3#4114yes—
150C2×C4×D3#4112yes—
151C2×Dic6—22no—
152C2×Dic6—22no—
153C2×Dic6—21no—
154C2×Dic6—24no—
155C2×Dic6—42no—
156C2×Dic6—42no—
157C2×Dic6C2×D412yes—
158C2×C3×C8—22no—
159C2×C3×C8—22no—
160C2×C3×C8—22no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—22no—
164C3×C42—22no—
165(C2×D3)⋊C4—44no—
166C4⋊Dic3—22no—
167C4⋊Dic3—22no—
168C4⋊Dic3—22no—
169Dic3⋊C4—44no—
170Dic3⋊C4—42no—
171Dic3⋊C4—44no—
172C4×Dic3—22no—
173C4×Dic3—24no—
174C4×Dic3—44no—
175C2×C3⋊C8—22no—
176C2×C3⋊C8—22no—
177C2×C3⋊C8—22no—
178ΓL(1,𝔽25)—48no—
179C22×Q8—34no—
180C22×Q8—31no—
181C22×Q8—61no—
182C22×Q8—62no—
183C22×Q8—64no—
184C22×Q8—64no—
185C22×Q8C22×D311yescharacteristic
186C23×C4—31no—
187C23×C4—32no—
188C23×C4—61no—
189C23×C4—61no—
190C2×M16—64no—
191C2×M16—64no—
192C2×M16—64no—
193C22×C8—32no—
194C22×C8C22×D312yes—
195#117—64no—
196#117—64no—
197C2×C4⋊C4—31no—
198C2×C4⋊C4—32no—
199C2×C4⋊C4—64no—
200C2×C4⋊C4—61no—
201C2×C4⋊C4—62no—
202C2×C4⋊C4—62no—
203C2×C4⋊C4C22×D311yescharacteristic
204C2×C4⋊C4C22×D312yes—
205C2×C22⋊C4—62no—
206C2×C42—32no—
207C2×C42—31no—
208C2×C42—64no—
209C2×C42—62no—
210C2×C42C22×D311yescharacteristic
211#103—24no—
212#103—64no—
213#103—64no—
214#103—64no—
215#95—22no—
216#95—22no—
217#95—62no—
218#95—62no—
219#95—62no—
220#95—62no—
221C23×C3C22×D411yescharacteristic
222C22×D3—22no—
223C22×D3—22no—
224C22×D3#12112yes—
225C22×D3#12312yes—
226C22×D3#12412yes—
227C3×Q8—24no—
228C3×Q8—42no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—42no—
236C2×C3×C4—41no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4C22×D411yescharacteristic
241C2×C3×C4C2×#4112yes—
242C2×C3×C4C2×#4112yes—
243C2×C3×C4C2×#4111yescharacteristic
244C2×Dic3—22no—
245C2×Dic3—22no—
246C2×Dic3—22no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—42no—
250C2×Dic3—41no—
251C2×Dic3—42no—
252C2×Dic3—42no—
253C2×Dic3—42no—
254C2×Dic3#12112yes—
255C2×Dic3#12312yes—
256C2×Dic3#12412yes—
257C2×Dic3#13714yes—
258C4×D3—48no—
259C4×D3—48no—
260C4×D3—84no—
261C4×D3#12318yes—
262C4×D3#12618yes—
263Dic6—24no—
264Dic6—42no—
265Dic6—48no—
266C3×C8—42no—
267C3⋊C8—42no—
268C24—31no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—32no—
273C2×Q8—64no—
274C2×Q8—616no—
275C2×Q8—61no—
276C2×Q8—62no—
277C2×Q8—62no—
278C2×Q8—122no—
279C2×Q8—122no—
280C2×Q8—128no—
281C2×Q8D3×D412yes—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—31no—
286C22×C4—32no—
287C22×C4—34no—
288C22×C4—34no—
289C22×C4—32no—
290C22×C4—32no—
291C22×C4—64no—
292C22×C4—64no—
293C22×C4—64no—
294C22×C4—61no—
295C22×C4—61no—
296C22×C4—62no—
297C22×C4—122no—
298C22×C4—122no—
299C22×C4—122no—
300C22×C4—122no—
301C22×C4—124no—
302C22×C4D3×D412yes—
303C22×C4C23×D311yescharacteristic
304M16—128no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—62no—
309C2×C8—62no—
310C2×C8—62no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—62no—
315C4⋊C4—62no—
316C4⋊C4—62no—
317C4⋊C4—122no—
318C4⋊C4—124no—
319C4⋊C4—124no—
320C22⋊C4—124no—
321C42—22no—
322C42—64no—
323C42—62no—
324C42—124no—
325C22×C3C2×#12111yescharacteristic
326C22×C3C2×#12311yescharacteristic
327C22×C3C2×#12411yescharacteristic
328C22×C3C2×#13712yes—
329C22×C3#57512yes—
330C2×D3—24no—
331C2×D3—22no—
332C2×D3—22no—
333C2×D3—24no—
334C2×D3#39714yes—
335C3×C4—42no—
336C3×C4—42no—
337C3×C4—81no—
338C3×C4—82no—
339C3×C4C2×#12312yes—
340C3×C4C2×#12612yes—
341Dic3—24no—
342Dic3—42no—
343Dic3—81no—
344Dic3—82no—
345Dic3#39714yes—
346C23—32no—
347C23—32no—
348C23—32no—
349C23—62no—
350C23—62no—
351C23C2×D3×D411yescharacteristic
352Q8—24no—
353Q8—42no—
354Q8—64no—
355Q8—128no—
356Q8—122no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—32no—
362C2×C4—32no—
363C2×C4—32no—
364C2×C4—38no—
365C2×C4—38no—
366C2×C4—34no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—62no—
375C2×C4—62no—
376C2×C4—62no—
377C2×C4—128no—
378C2×C4—128no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—122no—
383C2×C4—122no—
384C2×C4—122no—
385C2×C4—122no—
386C2×C4—244no—
387C2×C4C2×D3×D411yescharacteristic
388C2×C4D3×#4112yes—
389C2×C4D3×#4112yes—
390C2×C4D3×#4111yescharacteristic
391C8—42no—
392C8—122no—
393C2×C3C2×#39711yescharacteristic
394C2×C3#303712yes—
395C2×C3#309512yes—
396C2×C3#313911yescharacteristic
397C2×C3#314011yescharacteristic
398D3—24no—
399C22—34no—
400C22—62no—
401C22—64no—
402C22—64no—
403C22—62no—
404C22D3×#12111yescharacteristic
405C22D3×#12311yescharacteristic
406C22D3×#12411yescharacteristic
407C22#576812yes—
408C22D3×#13712yes—
409C4—34no—
410C4—42no—
411C4—42no—
412C4—64no—
413C4—82no—
414C4—81no—
415C4—122no—
416C4—242no—
417C4—241no—
418C4D3×#12312yes—
419C4D3×#12612yes—
420C3#2362911yescharacteristic, complemented, Sylow
421C2—64no—
422C2D3×#39711yescharacteristic
423C2#8059812yes—
424C2#8151712yes—
425C2#8164011yescharacteristic
426C2#8173911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2074C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2074D3
hypocenterC3#23629
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2074

Complements

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

NH ≅ G/Ntype
4. C3×#2074C2semidirect
5. #70233C2semidirect
6. #69671C2semidirect
7. #69560C2semidirect
8. #68914C2semidirect
11. #68638C2semidirect
47. #2074D3semidirect
420. C3#23629semidirect

Maximal subgroups

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

subgroupindex
C2×#48022
D3×#3912
#741482
C3×#20742
#702332
#696712
#695602
#689142
#688672
#686602
#686382
#236293