Finite Group Databasebeta

← #10877236

Subgroups — #10877236

Order 768.

770 conjugacy classes of subgroups.

The subgroup lattice has 512 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#69588C211yesmaximal, characteristic, complemented
7#69588C211yesmaximal, characteristic, complemented
8#69568C211yesmaximal, characteristic, complemented
9#69567C211yesmaximal, characteristic, complemented
10#69561C212yesmaximal, complemented
11#69542C212yesmaximal, complemented
12#27682—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)C2211yescharacteristic
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
39#4529C2212yes—
40#4529C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43#4524C2212yes—
44C4×C22⋊Q8—31no—
45#2285D311yescharacteristic
46#2101—32no—
47#2097—32no—
48#1948—31no—
49#1947—31no—
50#1873—32no—
51#1854—32no—
52#1803—31no—
53#1803—31no—
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—22no—
69C2×C3×C42—22no—
70C2×C3×C42—22no—
71C2×C3×C42D411yescharacteristic
72C2×C3×C42D411yescharacteristic
73C2×C3×C42C2311yescharacteristic
74C2×C3×C42C2311yescharacteristic
75C2×#217—21no—
76C2×#217—21no—
77C2×C4⋊Dic3C2311yescharacteristic
78C2×C4⋊Dic3C2311yescharacteristic
79C2×C4⋊Dic3C2311yescharacteristic
80#924—22no—
81#924Q814yes—
82C2×Dic3⋊C4—21no—
83C2×Dic3⋊C4—21no—
84C2×Dic3⋊C4—21no—
85C2×C4×Dic3—21no—
86C2×C4×Dic3—21no—
87C22×C3⋊C8C2312yes—
88C22×C3⋊C8C2312yes—
89C4×Dic6—22no—
90C4×Dic6—22no—
91C3×#95—22no—
92C3×#95—22no—
93C3⋊#95—21no—
94C3⋊#95—21no—
95C3⋊#95—21no—
96C3⋊#95—21no—
97C3⋊#95—21no—
98#830—24no—
99#810—42no—
100#810—42no—
101#808—42no—
102#807—42no—
103C4⋊(C3⋊C8)—22no—
104C4⋊(C3⋊C8)—22no—
105C2×C22⋊Q8—31no—
106C2×C4×Q8—31no—
107C2×#117C2×D312yes—
108C22×C42C2×D311yescharacteristic
109C2×#107—32no—
110C2×#106—32no—
111C2×C4⋊C8—31no—
112C2×C4⋊C8—31no—
113C2×#103—32no—
114C2×#103—32no—
115C2×C22⋊C8—32no—
116#391—31no—
117#389—31no—
118#387—61no—
119#385—61no—
120#384C2×D311yescharacteristic
121#384C2×D311yescharacteristic
122#382C2×D312yes—
123#380—21no—
124#380—21no—
125#380—31no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—31no—
129C4×C4⋊C4—61no—
130C4×C22⋊C4—61no—
131#340—32no—
132#340—32no—
133#340—32no—
134#340—32no—
135#336—32no—
136C24×C3C2×D411yescharacteristic
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—41no—
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×C4C2×Q812yes—
155C22×C3×C4#4112yes—
156C22×C3×C4C2411yescharacteristic
157C22×Dic3—21no—
158C22×Dic3—21no—
159C22×Dic3—21no—
160C22×Dic3—21no—
161C2×Dic6—21no—
162C2×Dic6—21no—
163C2×Dic6—21no—
164C2×Dic6—21no—
165C2×Dic6—21no—
166C2×Dic6C2×Q812yes—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—42no—
172C3×C4⋊C4—42no—
173C3×C4⋊C4—42no—
174C3×C4⋊C4—42no—
175C3×C22⋊C4—44no—
176C3×C42—22no—
177C3×C42—22no—
178C3×C42—21no—
179C3×C42—21no—
180C3×C42—44no—
181C3×C42#4112yes—
182C3×C42#4112yes—
183#217—24no—
184#217—42no—
185C4⋊Dic3—21no—
186C4⋊Dic3—21no—
187C4⋊Dic3—21no—
188C4⋊Dic3—21no—
189C4⋊Dic3—21no—
190C4⋊Dic3C2×Q812yes—
191Dic3⋊C4—24no—
192Dic3⋊C4—42no—
193Dic3⋊C4—42no—
194C4×Dic3—42no—
195C4×Dic3—42no—
196C2×C3⋊C8—22no—
197C2×C3⋊C8—22no—
198C2×C3⋊C8—22no—
199C2×C3⋊C8—22no—
200C2×C3⋊C8—22no—
201C2×C3⋊C8—22no—
202C22×Q8—31no—
203C23×C4#6711yescharacteristic
204C23×C4#6711yescharacteristic
205C23×C4C22×D311yescharacteristic
206C22×C8—32no—
207C22×C8—32no—
208C22⋊Q8—34no—
209C22⋊Q8—62no—
210C4×Q8—62no—
211C4×Q8—62no—
212#117—42no—
213#117—42no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—31no—
219C2×C4⋊C4—31no—
220C2×C4⋊C4—31no—
221C2×C4⋊C4—61no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4—61no—
224C2×C4⋊C4C22×D312yes—
225C2×C4⋊C4C22×D312yes—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—61no—
228C2×C22⋊C4—61no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—61no—
233C2×C42—61no—
234C2×C42#6711yescharacteristic
235C2×C42#6711yescharacteristic
236C2×C42C22×D311yescharacteristic
237C2×C42C22×D311yescharacteristic
238#107—122no—
239#106—122no—
240C4⋊C8—62no—
241C4⋊C8—62no—
242#103—122no—
243#103—122no—
244C22⋊C8—64no—
245#95—22no—
246#95—22no—
247#95—61no—
248#95—61no—
249#95—61no—
250#95—61no—
251#95—61no—
252C23×C3—21no—
253C23×C3—21no—
254C23×C3—21no—
255C23×C3—21no—
256C23×C3—21no—
257C23×C3—21no—
258C23×C3C22⋊Q812yes—
259C23×C3C22×D411yescharacteristic
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—44no—
275C2×C3×C4—44no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4C22⋊Q812yes—
283C2×C3×C4#12312yes—
284C2×C3×C4#12312yes—
285C2×C3×C4#13512yes—
286C2×C3×C4#13712yes—
287C2×C3×C4C22×Q811yescharacteristic
288C2×C3×C4C2×#4111yescharacteristic
289C2×C3×C4C2×#4111yescharacteristic
290C2×C3×C4C2×#4111yescharacteristic
291C2×C3×C4H(5,𝔽2)11yescharacteristic
292C2×C3×C4H(5,𝔽2)11yescharacteristic
293C2×Dic3—22no—
294C2×Dic3—22no—
295C2×Dic3—41no—
296C2×Dic3—41no—
297C2×Dic3—41no—
298C2×Dic3—41no—
299C2×Dic3—41no—
300C2×Dic3—41no—
301C2×Dic3—41no—
302C2×Dic3—41no—
303C2×Dic3—41no—
304C2×Dic3—41no—
305Dic6—22no—
306Dic6—22no—
307Dic6—41no—
308Dic6—41no—
309C3⋊C8—42no—
310C3⋊C8—42no—
311C24C2×#6711yescharacteristic
312C2×Q8—32no—
313C2×Q8—61no—
314C2×Q8—61no—
315C2×Q8—61no—
316C2×Q8—61no—
317C2×Q8—61no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—61no—
331C22×C4—61no—
332C22×C4—61no—
333C22×C4—61no—
334C22×C4D3×Q812yes—
335C22×C4#23912yes—
336C22×C4C2×#6711yescharacteristic
337C22×C4C2×#6711yescharacteristic
338C22×C4C2×#6711yescharacteristic
339C22×C4C2×#6711yescharacteristic
340C22×C4C2×#6711yescharacteristic
341C22×C4C23×D311yescharacteristic
342C2×C8—62no—
343C2×C8—62no—
344C2×C8—62no—
345C2×C8—62no—
346C2×C8—62no—
347C2×C8—62no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—32no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—64no—
358C4⋊C4—61no—
359C4⋊C4—61no—
360C4⋊C4—61no—
361C4⋊C4—61no—
362C4⋊C4—61no—
363C4⋊C4—122no—
364C4⋊C4—122no—
365C22⋊C4—44no—
366C22⋊C4—64no—
367C22⋊C4—122no—
368C42—22no—
369C42—22no—
370C42—21no—
371C42—21no—
372C42—44no—
373C42—122no—
374C42—122no—
375C42#23512yes—
376C42#23512yes—
377C22×C3—22no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—41no—
381C22×C3—41no—
382C22×C3—41no—
383C22×C3—41no—
384C22×C3C2×C22⋊Q811yescharacteristic
385C22×C3C2×#12311yescharacteristic
386C22×C3#53511yescharacteristic
387C22×C3C2×#13511yescharacteristic
388C22×C3C2×#13711yescharacteristic
389C22×C3#57511yescharacteristic
390C22×C3#57511yescharacteristic
391C3×C4—22no—
392C3×C4—22no—
393C3×C4—41no—
394C3×C4—41no—
395C3×C4—82no—
396C3×C4—82no—
397C3×C4#48514yes—
398C3×C4#55311yescharacteristic
399C3×C4#55311yescharacteristic
400C3×C4#55411yescharacteristic
401C3×C4#55411yescharacteristic
402Dic3—42no—
403Dic3—42no—
404Dic3—81no—
405Dic3—81no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23#94612yes—
413C23C22×#6711yescharacteristic
414Q8—62no—
415Q8—62no—
416Q8—121no—
417Q8—121no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—62no—
441C2×C4—62no—
442C2×C4—121no—
443C2×C4—121no—
444C2×C4—121no—
445C2×C4—121no—
446C2×C4—121no—
447C2×C4—121no—
448C2×C4—121no—
449C2×C4—121no—
450C2×C4—121no—
451C2×C4—121no—
452C2×C4#92912yes—
453C2×C4#92912yes—
454C2×C4#94612yes—
455C2×C4#95012yes—
456C2×C4#95112yes—
457C2×C4C2×#23511yescharacteristic
458C2×C4C2×#23511yescharacteristic
459C2×C4#100411yescharacteristic
460C2×C4#100411yescharacteristic
461C2×C4C2×D3×Q811yescharacteristic
462C2×C4C2×#23911yescharacteristic
463C8—122no—
464C8—122no—
465C2×C3—42no—
466C2×C3#248711yescharacteristic
467C2×C3#307611yescharacteristic
468C2×C3#307811yescharacteristic
469C2×C3C2×#48511yescharacteristic
470C2×C3#309311yescharacteristic
471C2×C3#321211yescharacteristic
472C2×C3#321611yescharacteristic
473C22—22no—
474C22—22no—
475C22—22no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22C2×#92911yescharacteristic
481C22#580511yescharacteristic
482C22C2×#94611yescharacteristic
483C22C2×#95011yescharacteristic
484C22#582211yescharacteristic
485C22#582211yescharacteristic
486C22C2×#95111yescharacteristic
487C4—22no—
488C4—22no—
489C4—41no—
490C4—41no—
491C4—82no—
492C4—82no—
493C4—122no—
494C4—122no—
495C4—241no—
496C4—241no—
497C4#499614yes—
498C4#551111yescharacteristic
499C4#551111yescharacteristic
500C4#551411yescharacteristic
501C4#551411yescharacteristic
502C3#2768211yescharacteristic, complemented, Sylow
503C2—42no—
504C2#8219711yescharacteristic
505C2#8221611yescharacteristic
506C2C2×#499611yescharacteristic
507C2#8330511yescharacteristic
508C2#8331911yescharacteristic
509C2#8334911yescharacteristic
510C2#8336711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2285C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2285D3
hypocenterC3#27682
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. #69588C2semidirect
7. #69588C2semidirect
8. #69568C2semidirect
9. #69567C2semidirect
10. #69561C2semidirect
11. #69542C2semidirect
502. C3#27682semidirect

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
#695882
#695682
#695672
#695612
#695422
#276823