Finite Group Databasebeta

← #10877269

Subgroups — #10877269

Order 768.

770 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2285C211yesmaximal, characteristic
2C4×#924C211yesmaximal, characteristic
3#71585C212yesmaximal
4#71576C212yesmaximal
5#71530C211yesmaximal, characteristic
6#69591C211yesmaximal, characteristic, complemented
7#69590C211yesmaximal, characteristic, complemented
8#69572C211yesmaximal, characteristic, complemented
9#69571C211yesmaximal, characteristic, complemented
10#69559C212yesmaximal, complemented
11#69540C212yesmaximal, complemented
12#27685—31nomaximal, Sylow
13C2×C3×#117C2212yes—
14C22×C3×C42C2211yescharacteristic
15C2×#924C2211yescharacteristic
16C2×C4×Dic6C2211yescharacteristic
17C3×#384C2211yescharacteristic
18C3×#384C2211yescharacteristic
19C3×#382C2212yes—
20C3×#380—21no—
21C3×#380—21no—
22C3×C4×C4⋊C4—21no—
23C3×C4×C4⋊C4—21no—
24#5212C2211yescharacteristic
25#5210—21no—
26C4×#217—21no—
27C2×#830C2212yes—
28C2×#810C2212yes—
29C2×#810C2212yes—
30C2×#808C2212yes—
31C2×#807C2212yes—
32C4×C4⋊Dic3C2211yescharacteristic
33#4933—21no—
34C4×Dic3⋊C4—21no—
35#4929C2211yescharacteristic
36#4928C2211yescharacteristic
37C2×C4⋊(C3⋊C8)C2212yes—
38#4529C2212yes—
39#4529C2212yes—
40#4529C2212yes—
41#4529C2212yes—
42#4524C2211yescharacteristic
43#4524C2211yescharacteristic
44C4×C22⋊Q8—31no—
45#2285D311yescharacteristic
46#2100—31no—
47#2099—31no—
48#2096—31no—
49#2095—31no—
50#1949—32no—
51#1873—32no—
52#1854—32no—
53#1805—32no—
54#1797—31no—
55C23×C3×C4D411yescharacteristic
56C23×C3×C4D411yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C22×Dic6C2311yescharacteristic
59C3×#117—42no—
60C3×#117—42no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C22⋊C4—22no—
68C2×C3×C42—24no—
69C2×C3×C42—22no—
70C2×C3×C42D411yescharacteristic
71C2×C3×C42D411yescharacteristic
72C2×C3×C42C2311yescharacteristic
73C2×C3×C42C2311yescharacteristic
74C2×#217—21no—
75C2×#217—21no—
76C2×C4⋊Dic3C2311yescharacteristic
77C2×C4⋊Dic3C2311yescharacteristic
78C2×C4⋊Dic3C2311yescharacteristic
79#924—22no—
80#924Q814yes—
81C2×Dic3⋊C4—21no—
82C2×Dic3⋊C4—21no—
83C2×Dic3⋊C4—21no—
84C2×C4×Dic3—21no—
85C2×C4×Dic3—21no—
86C22×C3⋊C8C2312yes—
87C22×C3⋊C8C2312yes—
88C4×Dic6—24no—
89C3×#95—22no—
90C3×#95—22no—
91C3⋊#95—21no—
92C3⋊#95—21no—
93C3⋊#95—21no—
94C3⋊#95—21no—
95C3⋊#95—21no—
96#830—24no—
97#810—42no—
98#810—42no—
99#808—42no—
100#807—42no—
101C4⋊(C3⋊C8)—24no—
102C2×C22⋊Q8—31no—
103C2×C4×Q8—31no—
104C2×#117C2×D312yes—
105C22×C42C2×D311yescharacteristic
106C2×#107—32no—
107C2×#106—32no—
108C2×C4⋊C8—32no—
109C2×#103—32no—
110C2×#103—32no—
111C2×C22⋊C8—32no—
112#391—31no—
113#389—31no—
114#387—61no—
115#385—61no—
116#384C2×D311yescharacteristic
117#384C2×D311yescharacteristic
118#382C2×D312yes—
119#380—21no—
120#380—21no—
121#380—31no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—31no—
125C4×C4⋊C4—61no—
126C4×C22⋊C4—61no—
127#340—32no—
128#340—32no—
129#340—32no—
130#340—32no—
131#336—31no—
132#336—31no—
133C24×C3C2×D411yescharacteristic
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—41no—
143C22×C3×C4—41no—
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×Q812yes—
150C22×C3×C4#4112yes—
151C22×C3×C4C2411yescharacteristic
152C22×Dic3—21no—
153C22×Dic3—21no—
154C22×Dic3—21no—
155C22×Dic3—21no—
156C2×Dic6—22no—
157C2×Dic6—22no—
158C2×Dic6—21no—
159C2×Dic6C2×Q812yes—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—42no—
165C3×C4⋊C4—42no—
166C3×C4⋊C4—42no—
167C3×C4⋊C4—42no—
168C3×C22⋊C4—44no—
169C3×C42—22no—
170C3×C42—22no—
171C3×C42—22no—
172C3×C42—44no—
173C3×C42#4114yes—
174#217—24no—
175#217—42no—
176C4⋊Dic3—21no—
177C4⋊Dic3—22no—
178C4⋊Dic3—22no—
179C4⋊Dic3C2×Q812yes—
180Dic3⋊C4—24no—
181Dic3⋊C4—42no—
182Dic3⋊C4—42no—
183C4×Dic3—42no—
184C4×Dic3—42no—
185C2×C3⋊C8—22no—
186C2×C3⋊C8—22no—
187C2×C3⋊C8—22no—
188C2×C3⋊C8—22no—
189C2×C3⋊C8—22no—
190C2×C3⋊C8—22no—
191C22×Q8—31no—
192C23×C4#6711yescharacteristic
193C23×C4#6711yescharacteristic
194C23×C4C22×D311yescharacteristic
195C22×C8—32no—
196C22×C8—32no—
197C22⋊Q8—34no—
198C22⋊Q8—62no—
199C4×Q8—64no—
200#117—42no—
201#117—42no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—31no—
207C2×C4⋊C4—31no—
208C2×C4⋊C4—31no—
209C2×C4⋊C4—61no—
210C2×C4⋊C4—61no—
211C2×C4⋊C4—61no—
212C2×C4⋊C4C22×D312yes—
213C2×C4⋊C4C22×D312yes—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—61no—
216C2×C22⋊C4—61no—
217C2×C42—24no—
218C2×C42—22no—
219C2×C42—61no—
220C2×C42—61no—
221C2×C42#6711yescharacteristic
222C2×C42#6711yescharacteristic
223C2×C42C22×D311yescharacteristic
224C2×C42C22×D311yescharacteristic
225#107—122no—
226#106—122no—
227C4⋊C8—64no—
228#103—122no—
229#103—122no—
230C22⋊C8—64no—
231#95—22no—
232#95—22no—
233#95—61no—
234#95—61no—
235#95—61no—
236#95—61no—
237#95—61no—
238C23×C3—21no—
239C23×C3—21no—
240C23×C3—21no—
241C23×C3—21no—
242C23×C3—21no—
243C23×C3—21no—
244C23×C3C22⋊Q812yes—
245C23×C3C22×D411yescharacteristic
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—24no—
257C2×C3×C4—24no—
258C2×C3×C4—44no—
259C2×C3×C4—44no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4C22⋊Q812yes—
267C2×C3×C4#12312yes—
268C2×C3×C4#12312yes—
269C2×C3×C4#13512yes—
270C2×C3×C4#13712yes—
271C2×C3×C4C22×Q811yescharacteristic
272C2×C3×C4C2×#4112yes—
273C2×C3×C4C2×#4111yescharacteristic
274C2×C3×C4H(5,𝔽2)12yes—
275C2×Dic3—22no—
276C2×Dic3—22no—
277C2×Dic3—41no—
278C2×Dic3—41no—
279C2×Dic3—41no—
280C2×Dic3—41no—
281C2×Dic3—41no—
282C2×Dic3—41no—
283C2×Dic3—41no—
284C2×Dic3—41no—
285C2×Dic3—41no—
286C2×Dic3—41no—
287Dic6—24no—
288Dic6—42no—
289C3⋊C8—42no—
290C3⋊C8—42no—
291C24C2×#6711yescharacteristic
292C2×Q8—32no—
293C2×Q8—61no—
294C2×Q8—62no—
295C2×Q8—62no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—61no—
307C22×C4—61no—
308C22×C4—61no—
309C22×C4—61no—
310C22×C4D3×Q812yes—
311C22×C4#23912yes—
312C22×C4C2×#6711yescharacteristic
313C22×C4C2×#6711yescharacteristic
314C22×C4C2×#6711yescharacteristic
315C22×C4C2×#6711yescharacteristic
316C22×C4C2×#6711yescharacteristic
317C22×C4C23×D311yescharacteristic
318C2×C8—62no—
319C2×C8—62no—
320C2×C8—62no—
321C2×C8—62no—
322C2×C8—62no—
323C2×C8—62no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—32no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—64no—
334C4⋊C4—61no—
335C4⋊C4—62no—
336C4⋊C4—62no—
337C4⋊C4—122no—
338C4⋊C4—122no—
339C22⋊C4—44no—
340C22⋊C4—64no—
341C22⋊C4—122no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—44no—
346C42—122no—
347C42—122no—
348C42#23514yes—
349C22×C3—22no—
350C22×C3—22no—
351C22×C3—22no—
352C22×C3—41no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3C2×C22⋊Q811yescharacteristic
357C22×C3C2×#12311yescharacteristic
358C22×C3#53511yescharacteristic
359C22×C3C2×#13511yescharacteristic
360C22×C3C2×#13711yescharacteristic
361C22×C3#57511yescharacteristic
362C22×C3#57511yescharacteristic
363C3×C4—22no—
364C3×C4—22no—
365C3×C4—22no—
366C3×C4—22no—
367C3×C4—22no—
368C3×C4—82no—
369C3×C4—82no—
370C3×C4#55312yes—
371C3×C4#55412yes—
372Dic3—42no—
373Dic3—42no—
374Dic3—81no—
375Dic3—81no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23#94612yes—
383C23C22×#6711yescharacteristic
384Q8—64no—
385Q8—122no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—62no—
407C2×C4—62no—
408C2×C4—121no—
409C2×C4—121no—
410C2×C4—121no—
411C2×C4—121no—
412C2×C4—121no—
413C2×C4—121no—
414C2×C4—121no—
415C2×C4—121no—
416C2×C4—121no—
417C2×C4—121no—
418C2×C4#92912yes—
419C2×C4#92912yes—
420C2×C4#94612yes—
421C2×C4#95012yes—
422C2×C4#95112yes—
423C2×C4C2×#23512yes—
424C2×C4#100412yes—
425C2×C4C2×D3×Q811yescharacteristic
426C2×C4C2×#23911yescharacteristic
427C8—122no—
428C8—122no—
429C2×C3—42no—
430C2×C3#248711yescharacteristic
431C2×C3#307611yescharacteristic
432C2×C3#307811yescharacteristic
433C2×C3#309111yescharacteristic
434C2×C3#309511yescharacteristic
435C2×C3#321011yescharacteristic
436C2×C3#321911yescharacteristic
437C22—22no—
438C22—22no—
439C22—22no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22C2×#92911yescharacteristic
445C22#580511yescharacteristic
446C22C2×#94611yescharacteristic
447C22C2×#95011yescharacteristic
448C22#582211yescharacteristic
449C22#582211yescharacteristic
450C22C2×#95111yescharacteristic
451C4—22no—
452C4—22no—
453C4—22no—
454C4—22no—
455C4—22no—
456C4—82no—
457C4—82no—
458C4—122no—
459C4—122no—
460C4—241no—
461C4—241no—
462C4#551112yes—
463C4#551412yes—
464C3#2768511yescharacteristic, complemented, Sylow
465C2—42no—
466C2#8219711yescharacteristic
467C2#8221511yescharacteristic
468C2#8222011yescharacteristic
469C2#8330911yescharacteristic
470C2#8331811yescharacteristic
471C2#8334911yescharacteristic
472C2#8336711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2285C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2285D3
hypocenterC3#27685
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2285

Complements

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

NH ≅ G/Ntype
6. #69591C2semidirect
7. #69590C2semidirect
8. #69572C2semidirect
9. #69571C2semidirect
10. #69559C2semidirect
11. #69540C2semidirect
464. C3#27685semidirect

Maximal subgroups

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

subgroupindex
C3×#22852
C4×#9242
#715852
#715762
#715302
#695912
#695902
#695722
#695712
#695592
#695402
#276853