Finite Group Databasebeta

← #10778263

Subgroups — #10778263

Order 768.

714 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#494C211yesmaximal, characteristic, complemented
2#80969C212yesmaximal
3#79538C211yesmaximal, characteristic
4C3×#1689C211yesmaximal, characteristic, complemented
5#71306C211yesmaximal, characteristic, complemented
6#71101C212yesmaximal, complemented
7#70871C211yesmaximal, characteristic, complemented
8#68541C211yesmaximal, characteristic, complemented
9#68527C211yesmaximal, characteristic, complemented
10#68441C212yesmaximal, complemented
11#68420C212yesmaximal, complemented
12#19188—31nomaximal, Sylow
13C2×D3×Q16—21no—
14C2×D3×Q16—21no—
15C2×#907—21no—
16D3×C4⋊Q8C2212yes—
17#5548C2211yescharacteristic
18C3×#494C2211yescharacteristic
19C3×#474C2212yes—
20C3×#436C2211yescharacteristic
21#5183—21no—
22#5183—21no—
23#5128—21no—
24#5092C2212yes—
25#5087C2212yes—
26D4×C3⋊C8C2211yescharacteristic
27#5007C2212yes—
28#4882—22no—
29#4874—22no—
30#4846C2212yes—
31#4837C2211yescharacteristic
32#4834C2211yescharacteristic
33#4833C2212yes—
34#4774—22no—
35#4760—22no—
36#4727—21no—
37#67⋊C8—21no—
38#4699C2211yescharacteristic
39#4688C2211yescharacteristic
40C4×D3×C8C2211yescharacteristic
41C3×#334C2211yescharacteristic
42C3×#332C2212yes—
43#4547C2212yes—
44#4486C2212yes—
45#4482C2212yes—
46#4479C2212yes—
47#4457C2211yescharacteristic
48#3226—32no—
49C2×#494—31no—
50#2931—31no—
51#1689—31no—
52#1689—31no—
53#1689—31no—
54#1689D311yescharacteristic, complemented
55#1669—32no—
56#1669—32no—
57#1564—31no—
58#1546—32no—
59C2×D3×Q8—22no—
60C2×#235—21no—
61C2×C3×Q16—21no—
62C2×C3×Q16—21no—
63C2×C3×M16—21no—
64C3×C4⋊Q8C2312yes—
65C3×C4×D4C2311yescharacteristic
66#944—22no—
67C2×#216—22no—
68C2×#216—22no—
69#940D411yescharacteristic
70#938D411yescharacteristic
71C4×#67—21no—
72C22×C3⋊C8—21no—
73D3×Q16—44no—
74D3×Q16—44no—
75#907—44no—
76C2×Dic12—21no—
77C2×Dic12—21no—
78C2×ΓL(1,𝔽25)—21no—
79C2×ΓL(1,𝔽25)—21no—
80C2×D3×C8—21no—
81C2×D3×C8D411yescharacteristic
82C2×D3×C8D411yescharacteristic
83#895D411yescharacteristic
84D3×C4⋊C4—22no—
85D3×C4⋊C4—22no—
86#888—22no—
87#888—22no—
88#888D411yescharacteristic
89#885—21no—
90#884—21no—
91C4×D12C2311yescharacteristic
92C42×D3C2311yescharacteristic
93#869C2312yes—
94C3×#107—22no—
95C3×C4⋊C8C2312yes—
96C3×C4⋊C8C2311yescharacteristic
97C3×#102—22no—
98C3×C22⋊C8—21no—
99C3×C4×C8C2311yescharacteristic
100#832—22no—
101#830—21no—
102#821—22no—
103(C2×D3)⋊C8—22no—
104(C2×D3)⋊C8—21no—
105(C2×D3)⋊C8—21no—
106#818—22no—
107Dic3⋊C8—21no—
108C8×Dic3—22no—
109C8×Dic3D411yescharacteristic
110C8×Dic3D411yescharacteristic
111#809—22no—
112#807—22no—
113#807—22no—
114C4⋊(C3⋊C8)C2311yescharacteristic
115C4⋊(C3⋊C8)C2312yes—
116C4×C3⋊C8C2311yescharacteristic
117C22×Q16—61no—
118C22×Q16—61no—
119C2×#131—61no—
120#532—31no—
121C2×C4⋊Q8—32no—
122#500—62no—
123#494—32no—
124#494—31no—
125#494—61no—
126#494—61no—
127#494C2×D311yescharacteristic
128#486—62no—
129#480—62no—
130#480—62no—
131#474—32no—
132#474—32no—
133#474—32no—
134#474C2×D312yes—
135#436—31no—
136#436—61no—
137#436C2×D311yescharacteristic
138C8×D4—31no—
139C8×D4—31no—
140C8×D4—61no—
141#431—32no—
142#423—31no—
143#408—61no—
144C2×C4×C8—31no—
145#335—32no—
146#334—31no—
147#334C2×D311yescharacteristic
148#332—32no—
149#332—32no—
150#332—32no—
151#332C2×D312yes—
152C2×C3×Q8—22no—
153C2×C3×D4C2×D411yescharacteristic
154C22×C3×C4—21no—
155C2×#67—21no—
156C2×#67—21no—
157C2×#67—21no—
158D3×Q8—44no—
159D3×Q8—44no—
160#235—42no—
161#235—42no—
162C2×D12—21no—
163C2×D12C2×D411yescharacteristic
164C2×C4×D3—21no—
165C2×C4×D3—42no—
166C2×C4×D3C2×D411yescharacteristic
167C2×C4×D3C2×D411yescharacteristic
168C2×C4×D3#4111yescharacteristic
169C2×Dic6—21no—
170C2×Dic6—22no—
171C2×Dic6—42no—
172C3×Q16—42no—
173C3×Q16—42no—
174C3×M16—42no—
175C2×C3×C8—22no—
176C2×C3×C8—21no—
177C2×C3×C8C2×D411yescharacteristic
178C2×C3×C8C2×D411yescharacteristic
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4C2×D411yescharacteristic
182C3×C22⋊C4—21no—
183C3×C42C2411yescharacteristic
184#217—21no—
185#216—44no—
186#216—44no—
187(C2×D3)⋊C4—21no—
188(C2×D3)⋊C4—21no—
189C4⋊Dic3—22no—
190C4⋊Dic3—22no—
191C4⋊Dic3C2×D411yescharacteristic
192Dic3⋊C4—21no—
193Dic3⋊C4—42no—
194Dic3⋊C4—42no—
195C4×Dic3C2×D411yescharacteristic
196C4×Dic3C2×D411yescharacteristic
197C4×Dic3#4111yescharacteristic
198C2×C3⋊C8—22no—
199C2×C3⋊C8—21no—
200C2×C3⋊C8—42no—
201C2×C3⋊C8C2×D411yescharacteristic
202C2×C3⋊C8C2×D411yescharacteristic
203Dic12—42no—
204Dic12—42no—
205ΓL(1,𝔽25)—42no—
206ΓL(1,𝔽25)—42no—
207D3×C8—22no—
208D3×C8—42no—
209D3×C8D814yes—
210C2×#41—61no—
211C22×Q8—62no—
212C2×Q16—21no—
213C2×Q16—21no—
214C2×Q16—62no—
215C2×Q16—62no—
216C2×Q16—61no—
217C2×Q16—61no—
218C2×Q16—124no—
219C2×Q16—124no—
220#131—124no—
221C2×M16—21no—
222C2×M16—61no—
223C2×M16—61no—
224C22×C8—31no—
225C22×C8—31no—
226C22×C8—61no—
227C22×C8—61no—
228C4⋊Q8—31no—
229C4⋊Q8—32no—
230C4⋊Q8—62no—
231C4⋊Q8—62no—
232C4⋊Q8—62no—
233C4⋊Q8C22×D312yes—
234#127—31no—
235#124—61no—
236#121—31no—
237#121—31no—
238#121—61no—
239C4×D4—31no—
240C4×D4—61no—
241C4×D4C22×D311yescharacteristic
242C2×C4⋊C4—62no—
243C2×C4⋊C4—62no—
244C2×C42—31no—
245#107—22no—
246#107—62no—
247#107—62no—
248#107—62no—
249C4⋊C8—31no—
250C4⋊C8—32no—
251C4⋊C8—61no—
252C4⋊C8C22×D311yescharacteristic
253C4⋊C8C22×D312yes—
254#102—22no—
255#102—62no—
256#102—62no—
257#102—62no—
258C22⋊C8—21no—
259C22⋊C8—61no—
260C22⋊C8—62no—
261C22⋊C8—61no—
262C22⋊C8—61no—
263C4×C8—31no—
264C4×C8—31no—
265C4×C8—31no—
266C4×C8—62no—
267C4×C8C22×D311yescharacteristic
268C23×C3—21no—
269C22×D3—21no—
270C22×D3#12111yescharacteristic
271C3×Q8—42no—
272C3×Q8—42no—
273C3×D4—22no—
274C2×C3×C4—21no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—42no—
278C2×C3×C4C22×D411yescharacteristic
279C2×C3×C4C22×D411yescharacteristic
280C2×C3×C4C2×#4111yescharacteristic
281#67—42no—
282#67—42no—
283#67—42no—
284C2×Dic3—21no—
285C2×Dic3—42no—
286C2×Dic3#12111yescharacteristic
287C2×Dic3#12111yescharacteristic
288C2×Dic3#12111yescharacteristic
289D12—22no—
290D12—42no—
291C4×D3—22no—
292C4×D3—22no—
293C4×D3—41no—
294C4×D3—41no—
295C4×D3—84no—
296C4×D3C2×D812yes—
297C4×D3AGL(1,ℤ/8)12yes—
298Dic6—42no—
299Dic6—42no—
300Dic6—42no—
301Dic6—84no—
302C3×C8—21no—
303C3×C8—41no—
304C3×C8—42no—
305C3×C8C2×D812yes—
306C3⋊C8—21no—
307C3⋊C8—41no—
308C3⋊C8—42no—
309C3⋊C8C2×D812yes—
310#41—122no—
311#41—122no—
312C2×Q8—22no—
313C2×Q8—61no—
314C2×Q8—62no—
315C2×Q8—122no—
316C2×Q8—124no—
317C2×Q8—124no—
318C2×D4—31no—
319C2×D4—61no—
320C2×D4—61no—
321C2×D4—61no—
322C2×D4—61no—
323C2×D4D3×D411yescharacteristic
324C22×C4—21no—
325C22×C4—31no—
326C22×C4—31no—
327C22×C4—31no—
328C22×C4—61no—
329C22×C4—122no—
330Q16—42no—
331Q16—42no—
332Q16—122no—
333Q16—124no—
334Q16—124no—
335Q16—122no—
336M16—42no—
337M16—122no—
338M16—122no—
339C2×C8—22no—
340C2×C8—21no—
341C2×C8—31no—
342C2×C8—34no—
343C2×C8—31no—
344C2×C8—62no—
345C2×C8—62no—
346C2×C8—61no—
347C2×C8—122no—
348C2×C8—122no—
349C2×C8D3×D411yescharacteristic
350C2×C8D3×D411yescharacteristic
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—31no—
354C4⋊C4—61no—
355C4⋊C4—62no—
356C4⋊C4—62no—
357C4⋊C4—122no—
358C4⋊C4—122no—
359C4⋊C4D3×D411yescharacteristic
360C22⋊C4—21no—
361C22⋊C4—61no—
362C22⋊C4—61no—
363C22⋊C4—61no—
364C42—31no—
365C42—31no—
366C42—31no—
367C42C23×D311yescharacteristic
368C22×C3—41no—
369C22×C3—41no—
370C22×C3—41no—
371C22×C3C2×#12111yescharacteristic
372C2×D3—21no—
373C2×D3—21no—
374C2×D3—41no—
375C2×D3—41no—
376C2×D3—41no—
377C2×D3#45912yes—
378C3×C4—21no—
379C3×C4—41no—
380C3×C4—82no—
381C3×C4C22×D811yescharacteristic
382C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
383C3×C4#57812yes—
384Dic3—21no—
385Dic3—41no—
386Dic3—82no—
387Dic3#45912yes—
388Dic3#47312yes—
389C23—21no—
390C23—31no—
391C23—61no—
392Q8—42no—
393Q8—42no—
394Q8—122no—
395Q8—122no—
396Q8—122no—
397Q8—244no—
398D4—22no—
399D4—62no—
400D4—122no—
401D4—122no—
402D4—122no—
403D4—122no—
404C2×C4—21no—
405C2×C4—31no—
406C2×C4—31no—
407C2×C4—32no—
408C2×C4—32no—
409C2×C4—31no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—62no—
414C2×C4—62no—
415C2×C4—61no—
416C2×C4—121no—
417C2×C4—121no—
418C2×C4—122no—
419C2×C4—244no—
420C2×C4C2×D3×D411yescharacteristic
421C2×C4C2×D3×D411yescharacteristic
422C2×C4D3×#4111yescharacteristic
423C8—21no—
424C8—32no—
425C8—41no—
426C8—42no—
427C8—61no—
428C8—121no—
429C8—122no—
430C8D3×D812yes—
431C2×C3—81no—
432C2×C3C2×#45911yescharacteristic
433C2×C3#305911yescharacteristic
434C2×C3C2×#47311yescharacteristic
435D3—22no—
436D3—81no—
437C22—32no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—61no—
442C22—61no—
443C22—121no—
444C22—121no—
445C22—121no—
446C22D3×#12111yescharacteristic
447C4—21no—
448C4—32no—
449C4—32no—
450C4—41no—
451C4—61no—
452C4—82no—
453C4—121no—
454C4—242no—
455C4C2×D3×D811yescharacteristic
456C4D3×AGL(1,ℤ/8)11yescharacteristic
457C4#577912yes—
458C3#1918811yescharacteristic, complemented, Sylow
459C2—62no—
460C2—81no—
461C2—241no—
462C2D3×#45911yescharacteristic
463C2#8090511yescharacteristic
464C2D3×#47311yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1689C2
socleC22×C3C2×#121
prime radical#5548C22
hypercenter#1689D3
hypocenterC3#19188
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1689

Complements

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

NH ≅ G/Ntype
1. D3×#494C2semidirect
4. C3×#1689C2semidirect
5. #71306C2semidirect
6. #71101C2semidirect
7. #70871C2semidirect
8. #68541C2semidirect
9. #68527C2semidirect
10. #68441C2semidirect
11. #68420C2semidirect
54. #1689D3semidirect
458. C3#19188semidirect

Maximal subgroups

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

subgroupindex
D3×#4942
#809692
#795382
C3×#16892
#713062
#711012
#708712
#685412
#685272
#684412
#684202
#191883