Finite Group Databasebeta

← #10845322

Subgroups — #10845322

Order 768.

874 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2538C211yesmaximal, characteristic
2C2×#5021C211yesmaximal, characteristic, complemented
3#82384C211yesmaximal, characteristic
4C2×D4×C3⋊C8C211yesmaximal, characteristic
5#71696C212yesmaximal, complemented
6#71695C212yesmaximal, complemented
7C4×#835C211yesmaximal, characteristic, complemented
8#69582C211yesmaximal, characteristic, complemented
9#69567C211yesmaximal, characteristic, complemented
10#69519C212yesmaximal, complemented
11#69512C212yesmaximal, complemented
12#25340—31nomaximal, Sylow
13C2×C3×C22⋊Q8C2212yes—
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C23×C3⋊C8C2212yes—
17C2×#869C2211yescharacteristic
18C3×#396—22no—
19C3×#390—21no—
20C3×#389—21no—
21C3×#384C2211yescharacteristic
22C3×#380C2212yes—
23C2×#835C2211yescharacteristic
24C2×#835C2212yes—
25C2×#835C2211yescharacteristic
26#5221—21no—
27#5212C2212yes—
28#5212C2212yes—
29C2×#830C2212yes—
30#5021—24no—
31D4×C3⋊C8—24no—
32#4977—21no—
33C2×#807C2212yes—
34C2×#807C2212yes—
35#4947—22no—
36C4×C4⋊Dic3C2211yescharacteristic
37C2×C4⋊(C3⋊C8)C2211yescharacteristic
38C2×C4×C3⋊C8C2211yescharacteristic
39#4529C2212yes—
40#4529C2212yes—
41#4529C2212yes—
42#4529C2211yescharacteristic
43#4529C2211yescharacteristic
44C2×#477—31no—
45C2×C8×D4—31no—
46#2563—31no—
47#2538D311yescharacteristic
48#2076—32no—
49#2036—32no—
50#1990—31no—
51#1947—31no—
52#1892—32no—
53#1808—32no—
54C4×#103—31no—
55C22×C3×Q8C2311yescharacteristic
56C22×C3×D4D411yescharacteristic
57C23×C3×C4C2312yes—
58C22×Dic6—21no—
59C3×C22⋊Q8—24no—
60C3×C22⋊Q8—42no—
61C3×C4×Q8—22no—
62C3×C4×Q8—22no—
63C3×C4×D4—22no—
64C3×C4×D4—22no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4D411yescharacteristic
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C22⋊C4—22no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4D412yes—
75C2×C3×C42—21no—
76C2×C3×C42C2311yescharacteristic
77C2×#217—22no—
78C2×#217—22no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3C2311yescharacteristic
81C2×C4⋊Dic3C2311yescharacteristic
82C2×C4⋊Dic3C2312yes—
83C2×C4×Dic3—21no—
84C22×C3⋊C8—24no—
85C22×C3⋊C8—24no—
86C22×C3⋊C8—42no—
87C22×C3⋊C8C2311yescharacteristic
88C22×C3⋊C8C2312yes—
89C22×C3⋊C8C2311yescharacteristic
90#869—22no—
91#869—22no—
92C3×#95—22no—
93#835—22no—
94#835—24no—
95#835—22no—
96C3⋊#95—21no—
97C3⋊#95—22no—
98C3⋊#95—22no—
99#830—24no—
100#807—24no—
101#807—42no—
102C4⋊(C3⋊C8)D414yes—
103C4×C3⋊C8—22no—
104C23×C8—32no—
105C2×C4⋊Q8—31no—
106C2×C22⋊Q8C2×D312yes—
107C2×C4×Q8C2×D311yescharacteristic
108C2×C4×D4C2×D311yescharacteristic
109#477—64no—
110C8×D4—64no—
111C2×#107—32no—
112C2×#107—32no—
113C2×C4⋊C8—31no—
114C2×#103—32no—
115C2×#103—31no—
116C2×#103—31no—
117C2×C22⋊C8—32no—
118C2×C4×C8—31no—
119#396—22no—
120#393—62no—
121#390—21no—
122#390—61no—
123#389—21no—
124#384C2×D311yescharacteristic
125#382—61no—
126#380—32no—
127#380—32no—
128#380C2×D312yes—
129C4×C4⋊C4—31no—
130#340—31no—
131#340—32no—
132#340—31no—
133#340—32no—
134#340—32no—
135C24×C3C2×D412yes—
136C2×C3×Q8—21no—
137C2×C3×Q8—21no—
138C2×C3×Q8—21no—
139C2×C3×Q8—21no—
140C2×C3×Q8—21no—
141C2×C3×Q8#4112yes—
142C2×C3×D4—22no—
143C2×C3×D4—22no—
144C2×C3×D4Q1614yes—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2411yescharacteristic
158C22×Dic3—21no—
159C22×Dic3—21no—
160C22×Dic3—22no—
161C2×Dic6—42no—
162C2×Dic6—81no—
163C2×Dic6—81no—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—21no—
166C3×C4⋊C4—21no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—44no—
170C3×C4⋊C4—42no—
171C3×C4⋊C4—42no—
172C3×C4⋊C4—81no—
173C3×C22⋊C4—24no—
174C3×C22⋊C4—44no—
175C3×C22⋊C4—82no—
176C3×C42—42no—
177C3×C42C2×D412yes—
178C3×C42#4112yes—
179#217—82no—
180#217—82no—
181C4⋊Dic3—22no—
182C4⋊Dic3—21no—
183C4⋊Dic3—21no—
184C4⋊Dic3—22no—
185C4⋊Dic3—21no—
186C4⋊Dic3—21no—
187C4⋊Dic3—42no—
188C4×Dic3—42no—
189C2×C3⋊C8—21no—
190C2×C3⋊C8—22no—
191C2×C3⋊C8—22no—
192C2×C3⋊C8—21no—
193C2×C3⋊C8—21no—
194C2×C3⋊C8—21no—
195C2×C3⋊C8—21no—
196C2×C3⋊C8—48no—
197C2×C3⋊C8C2×D414yes—
198C2×C3⋊C8#4112yes—
199C22×Q8—61no—
200C22×Q8C22×D311yescharacteristic
201C22×D4#6711yescharacteristic
202C23×C4C22×D312yes—
203C22×C8—31no—
204C22×C8—32no—
205C22×C8—31no—
206C22×C8—64no—
207C22×C8—64no—
208C22×C8—122no—
209C4⋊Q8—62no—
210C4⋊Q8—62no—
211C22⋊Q8—24no—
212C22⋊Q8—42no—
213C4×Q8—22no—
214C4×Q8—22no—
215C4×D4—22no—
216C4×D4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—32no—
222C2×C4⋊C4—31no—
223C2×C4⋊C4—31no—
224C2×C4⋊C4—61no—
225C2×C4⋊C4#6711yescharacteristic
226C2×C4⋊C4C22×D311yescharacteristic
227C2×C4⋊C4C22×D312yes—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—62no—
231C2×C22⋊C4—62no—
232C2×C22⋊C4#6712yes—
233C2×C42—21no—
234C2×C42—61no—
235C2×C42C22×D311yescharacteristic
236#107—64no—
237#107—122no—
238C4⋊C8—34no—
239#103—62no—
240#103—64no—
241#103—62no—
242C22⋊C8—64no—
243C4×C8—62no—
244#95—22no—
245#95—61no—
246#95—62no—
247#95—62no—
248C23×C3—22no—
249C23×C3—22no—
250C23×C3—22no—
251C23×C3—22no—
252C23×C3—22no—
253C23×C3—22no—
254C23×C3C2×Q1614yes—
255C23×C3C22×D411yescharacteristic
256C3×Q8—22no—
257C3×Q8—22no—
258C3×Q8—41no—
259C3×Q8—41no—
260C3×D4—28no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—41no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—41no—
277C2×C3×C4—42no—
278C2×C3×C4—41no—
279C2×C3×C4#12112yes—
280C2×C3×C4#12112yes—
281C2×C3×C4C2×Q1612yes—
282C2×C3×C4#13512yes—
283C2×C3×C4C22×D411yescharacteristic
284C2×C3×C4C2×#4111yescharacteristic
285C2×C3×C4C2×#4111yescharacteristic
286C2×C3×C4C2×#4111yescharacteristic
287C2×C3×C4H(5,𝔽2)11yescharacteristic
288C2×C3×C42−1+411yescharacteristic
289C2×Dic3—42no—
290C2×Dic3—41no—
291C2×Dic3—41no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—42no—
296C2×Dic3—41no—
297C2×Dic3—42no—
298Dic6—84no—
299C3⋊C8—22no—
300C3⋊C8—24no—
301C3⋊C8—41no—
302C24C2×#6712yes—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—122no—
309C2×Q8—241no—
310C2×Q8—241no—
311C2×Q8#23712yes—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4#21614yes—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—61no—
325C22×C4—61no—
326C22×C4—62no—
327C22×C4C2×#6712yes—
328C22×C4C2×#6711yescharacteristic
329C22×C4C2×#6711yescharacteristic
330C22×C4C23×D311yescharacteristic
331C2×C8—34no—
332C2×C8—32no—
333C2×C8—61no—
334C2×C8—62no—
335C2×C8—61no—
336C2×C8—62no—
337C2×C8—61no—
338C2×C8—61no—
339C2×C8—61no—
340C2×C8—128no—
341C4⋊C4—22no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—44no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—62no—
350C4⋊C4—61no—
351C4⋊C4—62no—
352C4⋊C4—61no—
353C4⋊C4—61no—
354C4⋊C4—61no—
355C4⋊C4—81no—
356C4⋊C4—122no—
357C22⋊C4—24no—
358C22⋊C4—44no—
359C22⋊C4—82no—
360C22⋊C4—242no—
361C22⋊C4—242no—
362C42—42no—
363C42—122no—
364C42D3×D412yes—
365C42#23712yes—
366C22×C3—24no—
367C22×C3—24no—
368C22×C3—24no—
369C22×C3—42no—
370C22×C3—42no—
371C22×C3—42no—
372C22×C3—42no—
373C22×C3C2×#12111yescharacteristic
374C22×C3#54611yescharacteristic
375C22×C3#54711yescharacteristic
376C22×C3C22×Q1611yescharacteristic
377C22×C3C2×#13511yescharacteristic
378C22×C3#57611yescharacteristic
379C22×C3#57711yescharacteristic
380C3×C4—22no—
381C3×C4—42no—
382C3×C4—42no—
383C3×C4—82no—
384C3×C4—81no—
385C3×C4#46714yes—
386C3×C4#52911yescharacteristic
387C3×C4#54211yescharacteristic
388C3×C4#54611yescharacteristic
389C3×C4#54811yescharacteristic
390Dic3—81no—
391Dic3—81no—
392Dic3—82no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23C2×#21614yes—
400C23C22×#6711yescharacteristic
401Q8—22no—
402Q8—22no—
403Q8—41no—
404Q8—41no—
405Q8—244no—
406D4—28no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—122no—
426C2×C4—121no—
427C2×C4—121no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—122no—
432C2×C4—121no—
433C2×C4—122no—
434C2×C4#93912yes—
435C2×C4#93912yes—
436C2×C4C2×#21612yes—
437C2×C4#95012yes—
438C2×C4C2×D3×D411yescharacteristic
439C2×C4C2×#23711yescharacteristic
440C2×C4C2×#23711yescharacteristic
441C2×C4#100411yescharacteristic
442C2×C4D3×#4111yescharacteristic
443C2×C4#101011yescharacteristic
444C8—62no—
445C8—64no—
446C8—121no—
447C2×C3—44no—
448C2×C3#242711yescharacteristic
449C2×C3C2×#46711yescharacteristic
450C2×C3#306211yescharacteristic
451C2×C3#329911yescharacteristic
452C2×C3#330511yescharacteristic
453C2×C3#333811yescharacteristic
454C2×C3#334011yescharacteristic
455C22—24no—
456C22—24no—
457C22—24no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22C2×#93911yescharacteristic
463C22#580611yescharacteristic
464C22C22×#21611yescharacteristic
465C22C2×#95011yescharacteristic
466C22#582411yescharacteristic
467C22#583511yescharacteristic
468C22#583711yescharacteristic
469C4—22no—
470C4—42no—
471C4—42no—
472C4—82no—
473C4—81no—
474C4—241no—
475C4—241no—
476C4—242no—
477C4#505114yes—
478C4#559811yescharacteristic
479C4#560011yescharacteristic
480C4#560311yescharacteristic
481C4#561211yescharacteristic
482C3#2534011yescharacteristic, complemented, Sylow
483C2—44no—
484C2#8264711yescharacteristic
485C2C2×#505111yescharacteristic
486C2#8342211yescharacteristic
487C2#8355111yescharacteristic
488C2#8357111yescharacteristic
489C2#8362311yescharacteristic
490C2#8365811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2538C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenter#2538D3
hypocenterC3#25340
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2538

Complements

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

NH ≅ G/Ntype
2. C2×#5021C2semidirect
5. #71696C2semidirect
6. #71695C2semidirect
7. C4×#835C2semidirect
8. #69582C2semidirect
9. #69567C2semidirect
10. #69519C2semidirect
11. #69512C2semidirect
482. C3#25340semidirect

Maximal subgroups

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

subgroupindex
C3×#25382
C2×#50212
#823842
C2×D4×C3⋊C82
#716962
#716952
C4×#8352
#695822
#695672
#695192
#695122
#253403