Finite Group Databasebeta

← #10883527

Subgroups — #10883527

Order 768.

782 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83966C212yesmaximal
2C2×#5184C211yesmaximal, characteristic, complemented
3#78658C211yesmaximal, characteristic
4C3×#2020C211yesmaximal, characteristic, complemented
5#72103C212yesmaximal, complemented
6#72074C211yesmaximal, characteristic, complemented
7#70123C212yesmaximal, complemented
8#70097C212yesmaximal, complemented
9#69995C211yesmaximal, characteristic, complemented
10#68929C211yesmaximal, characteristic, complemented
11#68922C211yesmaximal, characteristic, complemented
12#28179—31nomaximal, Sylow
13C2×C3×#137C2212yes—
14C22×C3×Q16C2211yescharacteristic
15#5832C2211yescharacteristic
16C2×#948C2211yescharacteristic
17C2×Q8×Dic3C2212yes—
18C2×#942C2212yes—
19C2×#907C2211yescharacteristic
20C2×C3×#101C2212yes—
21C3×#408C2211yescharacteristic
22#5232C2212yes—
23C2×#836C2212yes—
24C2×#835C2212yes—
25#5185—22no—
26#5184—22no—
27#5184—22no—
28#5182—22no—
29#5166—22no—
30#5163—22no—
31#5133C2212yes—
32#5118C2212yes—
33#5118C2211yescharacteristic
34#5110C2211yescharacteristic
35C2×#817C2211yescharacteristic
36C2×#815C2211yescharacteristic
37#4730—22no—
38#4728—22no—
39C3×#341C2211yescharacteristic
40#4558C2212yes—
41#4554C2212yes—
42#4554C2212yes—
43#4552C2211yescharacteristic
44#3037—32no—
45C2×#441—31no—
46#2932—31no—
47#2079—31no—
48#2079—31no—
49#2020—31no—
50#2020D311yescharacteristic, complemented
51#1908—32no—
52#1908—32no—
53#1838—31no—
54#1801—32no—
55C2×C3×#41C2311yescharacteristic
56C22×C3×Q8C2312yes—
57C2×#235C2311yescharacteristic
58C3×#137—24no—
59C3×#137—42no—
60C2×C3×Q16—24no—
61C2×C3×Q16—22no—
62C2×C3×Q16—22no—
63C2×C3×SD16—22no—
64C2×C3×M16C2312yes—
65C2×C3×M16C2311yescharacteristic
66C22×C3×C8C2311yescharacteristic
67#948—42no—
68#947D411yescharacteristic
69Q8×Dic3—22no—
70Q8×Dic3—22no—
71Q8×Dic3—22no—
72Q8×Dic3—22no—
73#944D411yescharacteristic
74C2×#216—22no—
75#942—24no—
76#942—42no—
77C2×#215—22no—
78#940D411yescharacteristic
79#939—22no—
80#936D411yescharacteristic
81C4×#67—22no—
82C2×C4⋊Dic3—22no—
83C2×C4⋊Dic3C2312yes—
84C2×C4×Dic3—22no—
85C2×C4×Dic3C2311yescharacteristic
86C2×#208—21no—
87C2×#208C2311yescharacteristic
88C2×#208C2312yes—
89C22×C3⋊C8—21no—
90C22×C3⋊C8C2311yescharacteristic
91#907—42no—
92C2×ΓL(1,𝔽25)—21no—
93C2×D3×C8—21no—
94C3×#101—42no—
95C3×C22⋊C8—22no—
96#836—42no—
97#835—22no—
98#835—22no—
99#835—22no—
100#832—22no—
101#821—22no—
102(C2×D3)⋊C8—22no—
103#817—22no—
104#816—22no—
105#815D412yes—
106#815D412yes—
107Dic3⋊C8—22no—
108Dic3⋊C8—22no—
109Dic3⋊C8—22no—
110C2×#137—32no—
111C2×#137C2×D312yes—
112C22×Q16C2×D311yescharacteristic
113C2×#131—31no—
114C2×#131—31no—
115#532—31no—
116C2×C4×Q8—32no—
117#464—62no—
118#464—62no—
119#462—62no—
120#460—62no—
121#441—62no—
122#441—62no—
123#435—62no—
124#435—62no—
125C2×#106—31no—
126#423—31no—
127#423—32no—
128#417—32no—
129#417—32no—
130C2×#103—32no—
131C2×#101—32no—
132C2×#101C2×D312yes—
133#408—31no—
134#408C2×D311yescharacteristic
135C2×#97—31no—
136#343—32no—
137#343—32no—
138#341—32no—
139#341—31no—
140#341C2×D311yescharacteristic
141C3×#41—22no—
142C3×#41—22no—
143C2×C3×Q8—22no—
144C2×C3×Q8—22no—
145C2×C3×Q8—22no—
146C2×C3×Q8—22no—
147C2×C3×Q8—22no—
148C2×C3×Q8C2×D411yescharacteristic
149C2×C3×Q8#4114yes—
150C2×C3×D4—21no—
151C2×C3×D4C2×D411yescharacteristic
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4C2411yescharacteristic
155C2×#67—21no—
156C2×#67—22no—
157C22×Dic3—22no—
158C22×Dic3#4112yes—
159#235—22no—
160#235—22no—
161C2×D12C2×D411yescharacteristic
162C2×C4×D3—21no—
163C2×Dic6C2×D411yescharacteristic
164C3×Q16—28no—
165C3×Q16—44no—
166C3×SD16—44no—
167C3×M16—22no—
168C3×M16—22no—
169C3×M16—22no—
170C2×C3×C8—22no—
171C2×C3×C8—22no—
172C2×C3×C8—21no—
173C2×C3×C8C2×D412yes—
174#217—22no—
175#216—44no—
176#215—44no—
177(C2×D3)⋊C4—22no—
178C4⋊Dic3—22no—
179C4⋊Dic3—22no—
180C4⋊Dic3—42no—
181C4⋊Dic3—42no—
182Dic3⋊C4—22no—
183C4×Dic3—42no—
184C4×Dic3—42no—
185C4×Dic3C2×D411yescharacteristic
186C4×Dic3C2×D411yescharacteristic
187C4×Dic3C2×D412yes—
188#208—22no—
189#208—22no—
190#208—22no—
191#208—42no—
192C2×C3⋊C8—22no—
193C2×C3⋊C8—21no—
194C2×C3⋊C8—22no—
195C2×C3⋊C8—42no—
196C2×C3⋊C8C2×D411yescharacteristic
197C2×C3⋊C8C2×D411yescharacteristic
198ΓL(1,𝔽25)—42no—
199D3×C8—42no—
200C2×#41—31no—
201C2×#41C22×D311yescharacteristic
202C22×Q8C22×D312yes—
203#137—24no—
204#137—42no—
205#137—64no—
206#137—122no—
207C2×Q16—24no—
208C2×Q16—22no—
209C2×Q16—22no—
210C2×Q16—62no—
211C2×SD16—22no—
212C2×SD16—62no—
213#131—122no—
214#131—122no—
215C2×M16—31no—
216C2×M16—32no—
217C2×M16—61no—
218C2×M16—61no—
219C2×M16C22×D311yescharacteristic
220C2×M16C22×D312yes—
221C22×C8—31no—
222C22×C8—61no—
223C22×C8—61no—
224C22×C8C22×D311yescharacteristic
225C4⋊Q8—31no—
226#127—31no—
227#124—31no—
228#124—31no—
229#121—62no—
230C4×Q8—62no—
231C4×Q8—62no—
232C4×Q8—62no—
233C4×Q8—62no—
234C4×D4—62no—
235C2×C4⋊C4—32no—
236C2×C4⋊C4—62no—
237C2×C42—31no—
238C2×C42—62no—
239#106—62no—
240C4⋊C8—62no—
241C4⋊C8—62no—
242C4⋊C8—62no—
243#103—62no—
244#103—62no—
245#103—62no—
246#103—62no—
247#102—62no—
248#102—62no—
249#101—42no—
250#101—122no—
251C22⋊C8—22no—
252C22⋊C8—62no—
253#97—32no—
254#97—32no—
255C23×C3—21no—
256C23×C3C2×#4111yescharacteristic
257C22×D3—21no—
258C3×Q8—24no—
259C3×Q8—24no—
260C3×Q8—22no—
261C3×Q8—42no—
262C3×Q8—42no—
263C3×D4—22no—
264C3×D4—42no—
265C2×C3×C4—21no—
266C2×C3×C4—24no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4C22×D411yescharacteristic
272C2×C3×C4C22×D411yescharacteristic
273C2×C3×C4C2×#4112yes—
274C2×C3×C4H(5,𝔽2)12yes—
275#67—42no—
276#67—82no—
277C2×Dic3—21no—
278C2×Dic3—42no—
279C2×Dic3—42no—
280C2×Dic3—82no—
281C2×Dic3#12112yes—
282C2×Dic3#12112yes—
283D12—22no—
284C4×D3—41no—
285C4×D3—41no—
286Dic6—22no—
287C3×C8—22no—
288C3×C8—41no—
289C3×C8—42no—
290C3⋊C8—21no—
291C3⋊C8—21no—
292C3⋊C8—41no—
293C3⋊C8—42no—
294#41—22no—
295#41—22no—
296#41—62no—
297#41—62no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—31no—
304C2×Q8D3×D411yescharacteristic
305C2×Q8#23714yes—
306C2×D4—21no—
307C2×D4—31no—
308C2×D4—62no—
309C2×D4—61no—
310C2×D4D3×D411yescharacteristic
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—32no—
314C22×C4—61no—
315C22×C4—62no—
316C22×C4C23×D311yescharacteristic
317Q16—28no—
318Q16—44no—
319Q16—124no—
320SD16—44no—
321SD16—124no—
322M16—22no—
323M16—22no—
324M16—22no—
325M16—62no—
326M16—62no—
327M16—62no—
328M16—122no—
329M16—122no—
330C2×C8—22no—
331C2×C8—21no—
332C2×C8—22no—
333C2×C8—31no—
334C2×C8—31no—
335C2×C8—62no—
336C2×C8—61no—
337C2×C8—62no—
338C2×C8—122no—
339C2×C8—122no—
340C2×C8D3×D412yes—
341C4⋊C4—62no—
342C4⋊C4—62no—
343C4⋊C4—62no—
344C4⋊C4—122no—
345C4⋊C4—122no—
346C22⋊C4—62no—
347C22⋊C4—62no—
348C42—32no—
349C42—31no—
350C42—31no—
351C42—122no—
352C42—122no—
353C22×C3—22no—
354C22×C3—41no—
355C22×C3—42no—
356C22×C3C2×#12111yescharacteristic
357C22×C3#55312yes—
358C2×D3—41no—
359C2×D3—42no—
360C3×C4—41no—
361C3×C4—44no—
362C3×C4#52511yescharacteristic
363C3×C4#53511yescharacteristic
364C3×C4#54612yes—
365Dic3—22no—
366Dic3—22no—
367Dic3—41no—
368Dic3—82no—
369C23—21no—
370C23—61no—
371C23D3×#4111yescharacteristic
372Q8—24no—
373Q8—24no—
374Q8—22no—
375Q8—42no—
376Q8—42no—
377Q8—62no—
378D4—22no—
379D4—42no—
380D4—62no—
381D4—122no—
382D4—242no—
383C2×C4—24no—
384C2×C4—21no—
385C2×C4—32no—
386C2×C4—32no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—61no—
392C2×C4—122no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—122no—
396C2×C4—242no—
397C2×C4C2×D3×D411yescharacteristic
398C2×C4C2×D3×D411yescharacteristic
399C2×C4C2×#23712yes—
400C2×C4#100412yes—
401C8—22no—
402C8—41no—
403C8—42no—
404C8—61no—
405C8—61no—
406C8—121no—
407C8—122no—
408C2×C3—22no—
409C2×C3—81no—
410C2×C3#253511yescharacteristic
411C2×C3#306912yes—
412D3—81no—
413C22—22no—
414C22—42no—
415C22—41no—
416C22—122no—
417C22—121no—
418C22#560112yes—
419C22D3×#12111yescharacteristic
420C4—41no—
421C4—44no—
422C4—62no—
423C4—62no—
424C4—121no—
425C4—242no—
426C4#558611yescharacteristic
427C4#559112yes—
428C4#571511yescharacteristic
429C3#2817911yescharacteristic, complemented, Sylow
430C2—22no—
431C2—81no—
432C2—241no—
433C2#7668611yescharacteristic
434C2#8094412yes—
subgroupG/H
centerC22D3×#121
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2020C2
socleC22×C3C2×#121
prime radical#5832C22
hypercenter#2020D3
hypocenterC3#28179
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2020

Complements

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

NH ≅ G/Ntype
2. C2×#5184C2semidirect
4. C3×#2020C2semidirect
5. #72103C2semidirect
6. #72074C2semidirect
7. #70123C2semidirect
8. #70097C2semidirect
9. #69995C2semidirect
10. #68929C2semidirect
11. #68922C2semidirect
50. #2020D3semidirect
429. C3#28179semidirect

Maximal subgroups

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

subgroupindex
#839662
C2×#51842
#786582
C3×#20202
#721032
#720742
#701232
#700972
#699952
#689292
#689222
#281793