Finite Group Databasebeta

← #10876183

Subgroups — #10876183

Order 768.

868 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2283C211yesmaximal, characteristic
2C2×#4929C211yesmaximal, characteristic
3#71531C211yesmaximal, characteristic
4#69561C212yesmaximal, complemented
5#69560C212yesmaximal, complemented
6#69525C211yesmaximal, characteristic
7#69525C211yesmaximal, characteristic
8#69505C211yesmaximal, characteristic
9#69504C211yesmaximal, characteristic
10#69465C212yesmaximal, complemented
11#69464C212yesmaximal, complemented
12#27589—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C22×C3×C42C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic
16C3×#383—21no—
17C3×#382C2211yescharacteristic
18C3×#382C2211yescharacteristic
19C3×#381—21no—
20C3×#380C2211yescharacteristic
21C3×#380C2211yescharacteristic
22C3×C4×C4⋊C4C2211yescharacteristic
23C3×C4×C4⋊C4C2211yescharacteristic
24C3×C4×C22⋊C4—21no—
25C3×#376—21no—
26C2×C3⋊#95—21no—
27C2×C3⋊#95—21no—
28C2×#830C2211yescharacteristic
29C2×#830C2211yescharacteristic
30C2×#810C2212yes—
31C2×#810C2212yes—
32C2×#810C2212yes—
33C2×#810C2212yes—
34C2×#808C2212yes—
35C2×#807C2212yes—
36#4929—21no—
37#4929—21no—
38#4929C2212yes—
39#4929C2212yes—
40#4529C2212yes—
41#4529C2212yes—
42#4524C2212yes—
43#4524C2212yes—
44C2×#391—31no—
45#2283D311yescharacteristic
46#2097—32no—
47#2074—32no—
48#1928—32no—
49#1927—32no—
50#1853—31no—
51#1852—31no—
52#1817—31no—
53#1787—31no—
54#1787—31no—
55C23×C3×C4D411yescharacteristic
56C23×C3×C4D411yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×Dic3—21no—
59C23×Dic3—21no—
60C22×Dic6—21no—
61C22×Dic6—24no—
62C22×Dic6—22no—
63C22×Dic6—22no—
64C22×Dic6—42no—
65C22×Dic6C2312yes—
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C22⋊C4—21no—
71C2×C3×C22⋊C4—21no—
72C2×C3×C42—22no—
73C2×C3×C42—22no—
74C2×C3×C42—21no—
75C2×C3×C42—21no—
76C2×C3×C42D411yescharacteristic
77C2×C3×C42D411yescharacteristic
78C2×C3×C42C2311yescharacteristic
79C2×C3×C42C2311yescharacteristic
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3—42no—
82C2×C4⋊Dic3C2312yes—
83C22×C3⋊C8C2312yes—
84C22×C3⋊C8C2312yes—
85C3×#95—21no—
86C3×#95—21no—
87C3×#95—21no—
88C3×#95—21no—
89C3×#95—21no—
90C3×#95—21no—
91C3×#95—21no—
92C3×#95—21no—
93C3⋊#95—22no—
94C3⋊#95—22no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97#830—22no—
98#830—22no—
99#810—24no—
100#810—24no—
101#810—42no—
102#810—42no—
103#808—42no—
104#807—42no—
105C23×Q8—31no—
106C22×C4⋊C4—31no—
107C22×C42C2×D311yescharacteristic
108C2×#107—32no—
109C2×#106—32no—
110C2×#103—32no—
111C2×#103—32no—
112C2×#103—32no—
113C2×#103—32no—
114C2×C22⋊C8—31no—
115C2×C22⋊C8—31no—
116#391—32no—
117#391—32no—
118#391—61no—
119#391—61no—
120#383—21no—
121#382C2×D311yescharacteristic
122#382C2×D311yescharacteristic
123#381—21no—
124#380C2×D311yescharacteristic
125#380C2×D311yescharacteristic
126C4×C4⋊C4C2×D311yescharacteristic
127C4×C4⋊C4C2×D311yescharacteristic
128C4×C22⋊C4—21no—
129#376—21no—
130C2×#95—61no—
131C2×#95—61no—
132#340—32no—
133#340—32no—
134#336—32no—
135#336—32no—
136C24×C3C2×D411yescharacteristic
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—41no—
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4#4112yes—
155C22×C3×C4#4112yes—
156C22×C3×C4C2411yescharacteristic
157C22×Dic3—24no—
158C22×Dic3—22no—
159C22×Dic3—22no—
160C22×Dic3—42no—
161C22×Dic3—42no—
162C22×Dic3—42no—
163C22×Dic3—42no—
164C22×Dic3—42no—
165C2×Dic6—22no—
166C2×Dic6—22no—
167C2×Dic6—22no—
168C2×Dic6—22no—
169C2×Dic6—24no—
170C2×Dic6—22no—
171C2×Dic6—48no—
172C2×Dic6—48no—
173C2×Dic6—42no—
174C2×Dic6—42no—
175C2×Dic6#4114yes—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180C3×C22⋊C4—42no—
181C3×C22⋊C4—42no—
182C3×C42—22no—
183C3×C42—22no—
184C3×C42—21no—
185C3×C42—21no—
186C3×C42—21no—
187C3×C42—21no—
188C3×C42—42no—
189C3×C42—42no—
190C4⋊Dic3—22no—
191C4⋊Dic3—22no—
192C4⋊Dic3—42no—
193C2×C3⋊C8—22no—
194C2×C3⋊C8—22no—
195C2×C3⋊C8—22no—
196C2×C3⋊C8—22no—
197C2×C3⋊C8—22no—
198C2×C3⋊C8—22no—
199C22×Q8—32no—
200C22×Q8—64no—
201C22×Q8—61no—
202C22×Q8—62no—
203C22×Q8—62no—
204C22×Q8—122no—
205C23×C4—61no—
206C23×C4—61no—
207C23×C4#6711yescharacteristic
208C23×C4#6711yescharacteristic
209C23×C4C22×D311yescharacteristic
210C22×C8—32no—
211C22×C8—32no—
212C2×C4⋊C4—32no—
213C2×C4⋊C4—61no—
214C2×C4⋊C4—122no—
215C2×C4⋊C4C22×D311yescharacteristic
216C2×C4⋊C4C22×D311yescharacteristic
217C2×C4⋊C4C22×D311yescharacteristic
218C2×C4⋊C4C22×D311yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42#6711yescharacteristic
226C2×C42#6711yescharacteristic
227C2×C42C22×D311yescharacteristic
228C2×C42C22×D311yescharacteristic
229#107—122no—
230#106—122no—
231#103—64no—
232#103—64no—
233#103—122no—
234#103—122no—
235C22⋊C8—62no—
236C22⋊C8—62no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—62no—
246#95—62no—
247#95—62no—
248#95—62no—
249C23×C3—21no—
250C23×C3—21no—
251C23×C3—21no—
252C23×C3—21no—
253C23×C3#12312yes—
254C23×C3#13512yes—
255C23×C3#13712yes—
256C23×C3C22×D411yescharacteristic
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—24no—
262C2×C3×C4—24no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—44no—
273C2×C3×C4—44no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4#12312yes—
287C2×C3×C4#13712yes—
288C2×C3×C4C2×#4111yescharacteristic
289C2×C3×C4C2×#4111yescharacteristic
290C2×C3×C4C2×#4112yes—
291C2×C3×C4H(5,𝔽2)12yes—
292C2×Dic3—24no—
293C2×Dic3—48no—
294C2×Dic3—42no—
295C2×Dic3—42no—
296C2×Dic3—42no—
297C2×Dic3—42no—
298C2×Dic3—42no—
299C2×Dic3—84no—
300Dic6—24no—
301Dic6—24no—
302Dic6—48no—
303Dic6—42no—
304Dic6—42no—
305C3⋊C8—42no—
306C3⋊C8—42no—
307C24C2×#6711yescharacteristic
308C2×Q8—34no—
309C2×Q8—64no—
310C2×Q8—62no—
311C2×Q8—62no—
312C2×Q8—62no—
313C2×Q8—62no—
314C2×Q8—62no—
315C2×Q8—128no—
316C2×Q8—128no—
317C2×Q8—122no—
318C2×Q8—122no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—41no—
331C22×C4—64no—
332C22×C4—62no—
333C22×C4—62no—
334C22×C4—122no—
335C22×C4—122no—
336C22×C4—122no—
337C22×C4—122no—
338C22×C4—122no—
339C22×C4#23512yes—
340C22×C4#23512yes—
341C22×C4C2×#6711yescharacteristic
342C22×C4C2×#6711yescharacteristic
343C22×C4C2×#6711yescharacteristic
344C22×C4C2×#6711yescharacteristic
345C22×C4C2×#6711yescharacteristic
346C22×C4C23×D311yescharacteristic
347C2×C8—62no—
348C2×C8—62no—
349C2×C8—62no—
350C2×C8—62no—
351C2×C8—62no—
352C2×C8—62no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—62no—
358C4⋊C4—62no—
359C4⋊C4—122no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C42—22no—
363C42—22no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—42no—
369C42—42no—
370C22×C3—22no—
371C22×C3—22no—
372C22×C3—22no—
373C22×C3—22no—
374C22×C3—22no—
375C22×C3—22no—
376C22×C3#48514yes—
377C22×C3C2×#12311yescharacteristic
378C22×C3#52511yescharacteristic
379C22×C3#53511yescharacteristic
380C22×C3C2×#13511yescharacteristic
381C22×C3C2×#13711yescharacteristic
382C22×C3C2×#13711yescharacteristic
383C22×C3#57511yescharacteristic
384C3×C4—22no—
385C3×C4—22no—
386C3×C4—41no—
387C3×C4—41no—
388C3×C4—41no—
389C3×C4—81no—
390C3×C4—81no—
391C3×C4—81no—
392C3×C4—81no—
393C3×C4#55112yes—
394C3×C4#55112yes—
395Dic3—44no—
396Dic3—82no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23#92912yes—
402C23#95012yes—
403C23#95112yes—
404C23C22×#6711yescharacteristic
405Q8—64no—
406Q8—64no—
407Q8—128no—
408Q8—122no—
409Q8—122no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—64no—
440C2×C4—128no—
441C2×C4—122no—
442C2×C4—122no—
443C2×C4—122no—
444C2×C4—122no—
445C2×C4—122no—
446C2×C4—244no—
447C2×C4#92912yes—
448C2×C4#95112yes—
449C2×C4C2×#23511yescharacteristic
450C2×C4C2×#23511yescharacteristic
451C2×C4D3×#4112yes—
452C2×C4#100912yes—
453C8—122no—
454C8—122no—
455C2×C3—24no—
456C2×C3#248211yescharacteristic
457C2×C3C2×#48511yescharacteristic
458C2×C3#309511yescharacteristic
459C2×C3#310811yescharacteristic
460C2×C3#311711yescharacteristic
461C2×C3#321111yescharacteristic
462C2×C3#321811yescharacteristic
463C22—22no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22#499614yes—
470C22C2×#92911yescharacteristic
471C22C2×#95011yescharacteristic
472C22#582211yescharacteristic
473C22C2×#95111yescharacteristic
474C22C2×#95111yescharacteristic
475C22#582811yescharacteristic
476C22#583011yescharacteristic
477C4—22no—
478C4—22no—
479C4—41no—
480C4—41no—
481C4—41no—
482C4—81no—
483C4—81no—
484C4—81no—
485C4—81no—
486C4—124no—
487C4—242no—
488C4#552712yes—
489C4#552712yes—
490C3#2758911yescharacteristic, complemented, Sylow
491C2—24no—
492C2C2×#499611yescharacteristic
493C2#8222011yescharacteristic
494C2#8228211yescharacteristic
495C2#8314411yescharacteristic
496C2#8315211yescharacteristic
497C2#8315311yescharacteristic
498C2#8315411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2283C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2283D3
hypocenterC3#27589
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2283

Complements

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

NH ≅ G/Ntype
4. #69561C2semidirect
5. #69560C2semidirect
10. #69465C2semidirect
11. #69464C2semidirect
490. C3#27589semidirect

Maximal subgroups

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

subgroupindex
C3×#22832
C2×#49292
#715312
#695612
#695602
#695252
#695052
#695042
#694652
#694642
#275893