Finite Group Databasebeta

← #10877242

Subgroups — #10877242

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×C4×C22⋊Q8C211yesmaximal, characteristic
2#82062C211yesmaximal, characteristic
3#71693C212yesmaximal
4#71626C211yesmaximal, characteristic, complemented
5#71626C211yesmaximal, characteristic, complemented
6#71530C211yesmaximal, characteristic
7#69686C212yesmaximal, complemented
8#69613C212yesmaximal, complemented
9#69506C212yesmaximal
10#69467C211yesmaximal, characteristic, complemented
11#69466C211yesmaximal, characteristic, complemented
12#27682—31nomaximal, Sylow
13C2×C3×C22⋊Q8C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16C2×#926C2212yes—
17C3×#391C2211yescharacteristic
18C3×#389C2211yescharacteristic
19C3×#387—21no—
20C3×#385—21no—
21C3×#380C2211yescharacteristic
22C3×C4×C4⋊C4—21no—
23C3×C4×C4⋊C4C2211yescharacteristic
24C3×C4×C22⋊C4—21no—
25C2×#835C2212yes—
26C2×#835C2212yes—
27#5212—21no—
28#5212—21no—
29C2×#830C2212yes—
30C2×#808C2212yes—
31C2×#807C2212yes—
32#4936C2212yes—
33#4935C2211yescharacteristic
34#4935C2211yescharacteristic
35C4×C4⋊Dic3—21no—
36C4×C4⋊Dic3—21no—
37C2×C4⋊(C3⋊C8)C2211yescharacteristic
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
39#4529C2212yes—
40#4529C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43#4524C2212yes—
44C4×C22⋊Q8D311yescharacteristic
45#2285—31no—
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—
55C22×C3×Q8C2311yescharacteristic
56C23×C3×C4D411yescharacteristic
57C23×C3×C4D411yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C3×C22⋊Q8—22no—
60C3×C22⋊Q8Q814yes—
61C3×C4×Q8—22no—
62C3×C4×Q8—22no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C22⋊C4—21no—
70C2×C3×C22⋊C4—21no—
71C2×C3×C42—22no—
72C2×C3×C42—22no—
73C2×C3×C42—21no—
74C2×C3×C42—21no—
75C2×C3×C42D411yescharacteristic
76C2×C3×C42D411yescharacteristic
77C2×C3×C42C2311yescharacteristic
78C2×C3×C42C2311yescharacteristic
79C2×#217—22no—
80#926—42no—
81#926—42no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3C2312yes—
87C2×C4⋊Dic3C2312yes—
88C2×C4×Dic3—22no—
89C22×C3⋊C8C2312yes—
90C22×C3⋊C8C2312yes—
91C3×#95—21no—
92C3×#95—21no—
93C3×#95—21no—
94C3×#95—21no—
95C3×#95—21no—
96#835—42no—
97#835—42no—
98C3⋊#95—22no—
99C3⋊#95—22no—
100#830—24no—
101#808—42no—
102#807—42no—
103C4⋊(C3⋊C8)—22no—
104C4⋊(C3⋊C8)—22no—
105C2×C22⋊Q8C2×D311yescharacteristic
106C2×C4×Q8C2×D311yescharacteristic
107C2×#117—32no—
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#391C2×D311yescharacteristic
117#389C2×D311yescharacteristic
118#387—21no—
119#385—21no—
120#384—31no—
121#384—31no—
122#382—32no—
123#380—61no—
124#380—61no—
125#380C2×D311yescharacteristic
126C4×C4⋊C4—21no—
127C4×C4⋊C4—61no—
128C4×C4⋊C4—61no—
129C4×C4⋊C4C2×D311yescharacteristic
130C4×C22⋊C4—21no—
131#340—32no—
132#340—32no—
133#340—32no—
134#340—32no—
135#336—32no—
136C24×C3C2×D411yescharacteristic
137C2×C3×Q8—21no—
138C2×C3×Q8—21no—
139C2×C3×Q8—21no—
140C2×C3×Q8—21no—
141C2×C3×Q8—21no—
142C2×C3×Q8C2×Q812yes—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—41no—
156C22×C3×C4—41no—
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×Q812yes—
163C22×C3×C4#4112yes—
164C22×C3×C4C2411yescharacteristic
165C22×Dic3—22no—
166C22×Dic3—22no—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—42no—
174C3×C4⋊C4—42no—
175C3×C4⋊C4C2×Q812yes—
176C3×C22⋊C4—24no—
177C3×C22⋊C4—42no—
178C3×C42—22no—
179C3×C42—22no—
180C3×C42—21no—
181C3×C42—21no—
182C3×C42—42no—
183C3×C42—42no—
184C3×C42#4112yes—
185C3×C42#4112yes—
186#217—44no—
187C4⋊Dic3—22no—
188C4⋊Dic3—22no—
189C4⋊Dic3—22no—
190C4⋊Dic3—22no—
191C4⋊Dic3—42no—
192C4⋊Dic3—42no—
193C4⋊Dic3—42no—
194C4⋊Dic3—42no—
195C4×Dic3—44no—
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×Q8C22×D311yescharacteristic
203C23×C4#6711yescharacteristic
204C23×C4#6711yescharacteristic
205C23×C4C22×D311yescharacteristic
206C22×C8—32no—
207C22×C8—32no—
208C22⋊Q8—22no—
209C22⋊Q8Dic614yes—
210C4×Q8—22no—
211C4×Q8—22no—
212#117—122no—
213#117—122no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—32no—
218C2×C4⋊C4—32no—
219C2×C4⋊C4—61no—
220C2×C4⋊C4—61no—
221C2×C4⋊C4—61no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4C22×D311yescharacteristic
224C2×C4⋊C4C22×D311yescharacteristic
225C2×C4⋊C4C22×D311yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—62no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—62no—
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—21no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—62no—
251#95—62no—
252C23×C3—21no—
253C23×C3—21no—
254C23×C3—21no—
255C23×C3—21no—
256C23×C3—21no—
257C23×C3—21no—
258C23×C3C22⋊Q812yes—
259C23×C3C22×D411yescharacteristic
260C3×Q8—22no—
261C3×Q8—22no—
262C3×Q8—41no—
263C3×Q8—41no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—44no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—44no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4C22⋊Q812yes—
293C2×C3×C4#12312yes—
294C2×C3×C4#12312yes—
295C2×C3×C4#13512yes—
296C2×C3×C4#13712yes—
297C2×C3×C4C22×Q811yescharacteristic
298C2×C3×C4C2×#4111yescharacteristic
299C2×C3×C4C2×#4111yescharacteristic
300C2×C3×C4C2×#4111yescharacteristic
301C2×C3×C4H(5,𝔽2)11yescharacteristic
302C2×C3×C4H(5,𝔽2)11yescharacteristic
303C2×Dic3—42no—
304C2×Dic3—42no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307C2×Dic3—42no—
308C2×Dic3—42no—
309C3⋊C8—42no—
310C3⋊C8—42no—
311C24C2×#6711yescharacteristic
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8C2×Dic612yes—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—62no—
333C22×C4—62no—
334C22×C4C2×Dic612yes—
335C22×C4#23512yes—
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—24no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—62no—
357C4⋊C4—62no—
358C4⋊C4—62no—
359C4⋊C4—62no—
360C4⋊C4—122no—
361C4⋊C4—122no—
362C4⋊C4—122no—
363C4⋊C4—122no—
364C4⋊C4C2×Dic612yes—
365C22⋊C4—24no—
366C22⋊C4—42no—
367C22⋊C4—124no—
368C42—22no—
369C42—22no—
370C42—21no—
371C42—21no—
372C42—42no—
373C42—42no—
374C42—124no—
375C42#23712yes—
376C42#23712yes—
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—42no—
394C3×C4—41no—
395C3×C4—41no—
396C3×C4—42no—
397C3×C4—81no—
398C3×C4—81no—
399C3×C4#48514yes—
400C3×C4#55311yescharacteristic
401C3×C4#55311yescharacteristic
402C3×C4#55411yescharacteristic
403C3×C4#55411yescharacteristic
404Dic3—82no—
405Dic3—82no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23#92412yes—
413C23C22×#6711yescharacteristic
414Q8—22no—
415Q8—22no—
416Q8—41no—
417Q8—41no—
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—22no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—44no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—44no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—122no—
447C2×C4—122no—
448C2×C4—122no—
449C2×C4—122no—
450C2×C4—122no—
451C2×C4—122no—
452C2×C4#92412yes—
453C2×C4#93512yes—
454C2×C4#93512yes—
455C2×C4#94212yes—
456C2×C4#95012yes—
457C2×C4C22×Dic611yescharacteristic
458C2×C4C2×#23511yescharacteristic
459C2×C4C2×#23711yescharacteristic
460C2×C4C2×#23711yescharacteristic
461C2×C4#100911yescharacteristic
462C2×C4#100911yescharacteristic
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×#92411yescharacteristic
481C22C2×#93511yescharacteristic
482C22C2×#94211yescharacteristic
483C22C2×#95011yescharacteristic
484C22#582211yescharacteristic
485C22#582211yescharacteristic
486C22#583011yescharacteristic
487C4—22no—
488C4—22no—
489C4—41no—
490C4—42no—
491C4—42no—
492C4—41no—
493C4—81no—
494C4—81no—
495C4—242no—
496C4—242no—
497C4#504514yes—
498C4#554311yescharacteristic
499C4#554311yescharacteristic
500C4#556511yescharacteristic
501C4#556511yescharacteristic
502C3#2768211yescharacteristic, complemented, Sylow
503C2—42no—
504C2#8237411yescharacteristic
505C2#8243111yescharacteristic
506C2#8250711yescharacteristic
507C2C2×#504511yescharacteristic
508C2#8291511yescharacteristic
509C2#8314511yescharacteristic
510C2#8335811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×C22⋊Q8C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC4×C22⋊Q8D3
hypocenterC3#27682
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
4. #71626C2semidirect
5. #71626C2semidirect
7. #69686C2semidirect
8. #69613C2semidirect
10. #69467C2semidirect
11. #69466C2semidirect
502. C3#27682semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C22⋊Q82
#820622
#716932
#716262
#715302
#696862
#696132
#695062
#694672
#694662
#276823