Finite Group Databasebeta

← #10710162

Subgroups — #10710162

Order 768.

826 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#410C211yesmaximal, characteristic
2#84087C211yesmaximal, characteristic
3C2×#5240C211yesmaximal, characteristic
4#67695C214yesmaximal, complemented
5#12583—31nomaximal, Sylow
6C22×C3×#41C2211yescharacteristic
7C2×C3×#117—21no—
8C2×#926—21no—
9C2×#926C411yescharacteristic, complemented
10C22×C4⋊Dic3C411yescharacteristic, complemented
11C3×#410—22no—
12C3×#410C414yescomplemented
13C2×C3×#99—21no—
14C2×C3×#99C2212yes—
15#5240—22no—
16#5240C414yescomplemented
17#5229C412yescomplemented
18#5221C2212yes—
19#5220—21no—
20#5220—21no—
21C2×#834—21no—
22C2×#834C2212yes—
23#5212—21no—
24#5212—21no—
25C2×#410—31no—
26C2×#410D311yescharacteristic
27#2311—31no—
28#1393—34no—
29C2×C3×#41—41no—
30C2×C3×#41—41no—
31C2×C3×#41—42no—
32C2×C3×#41C2×C412yes—
33C2×C3×#41C2×C412yes—
34C2×C3×#41D412yes—
35C2×C3×#41Q812yes—
36C22×C3×Q8C2×C411yescharacteristic
37C22×C3×D4D411yescharacteristic
38C22×C3×D4D411yescharacteristic
39C22×C3×D4C2311yescharacteristic
40C23×C3×C4C2×C411yescharacteristic
41C23×C3×C4D411yescharacteristic
42C23×C3×C4D411yescharacteristic
43C23×Dic3—21no—
44C3×#117—24no—
45C3×#117—42no—
46C2×C3×C4⋊C4—41no—
47C2×C3×C22⋊C4—22no—
48C2×C3×C42—41no—
49C2×#217—22no—
50C2×#217—21no—
51C2×#217—22no—
52C2×#217—22no—
53#926—24no—
54#926—42no—
55#926—42no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—41no—
59C2×C4⋊Dic3—41no—
60C2×C4⋊Dic3—41no—
61C2×C4⋊Dic3C2×C412yes—
62C2×C4×Dic3—41no—
63C2×C4×Dic3C2×C412yes—
64C3×#99—24no—
65C3×#99—22no—
66C3×#99C2×C414yescomplemented
67#834—24no—
68#834—22no—
69#834C2×C414yescomplemented
70C3⋊#95—22no—
71C3⋊#95—22no—
72C22×#41C2×D311yescharacteristic
73C2×#117—21no—
74C2×#117—31no—
75C2×#117—61no—
76C22×C4⋊C4—31no—
77#410—22no—
78#410—34no—
79#410—62no—
80#410Dic314yescomplemented
81C2×#99—21no—
82C2×#99—32no—
83C2×#99—61no—
84C2×#99C2×D312yes—
85#391—32no—
86#390—32no—
87#386—61no—
88#386—61no—
89#380—61no—
90#380—61no—
91C24×C3C22⋊C411yescharacteristic
92C24×C3C2×D411yescharacteristic
93C24×C3C2×D411yescharacteristic
94C3×#41—42no—
95C3×#41—42no—
96C3×#41—42no—
97C3×#41—42no—
98C3×#41—42no—
99C3×#41—42no—
100C3×#41—42no—
101C3×#41—42no—
102C2×C3×Q8—41no—
103C2×C3×Q8—41no—
104C2×C3×Q8C4212yes—
105C2×C3×Q8C4⋊C412yes—
106C2×C3×D4—41no—
107C2×C3×D4—41no—
108C2×C3×D4—41no—
109C2×C3×D4—41no—
110C2×C3×D4—41no—
111C2×C3×D4—41no—
112C2×C3×D4C22⋊C412yes—
113C2×C3×D4C22⋊C412yes—
114C2×C3×D4C4⋊C412yes—
115C2×C3×D4C4⋊C412yes—
116C2×C3×D4C22×C411yescharacteristic
117C2×C3×D4C22×C411yescharacteristic
118C2×C3×D4C2×D411yescharacteristic
119C2×C3×D4C2×Q811yescharacteristic
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4—42no—
130C22×C3×C4C4212yes—
131C22×C3×C4C22⋊C411yescharacteristic
132C22×C3×C4C22⋊C411yescharacteristic
133C22×C3×C4C22⋊C412yes—
134C22×C3×C4C22⋊C411yescharacteristic
135C22×C3×C4C22⋊C412yes—
136C22×C3×C4C4⋊C412yes—
137C22×C3×C4C4⋊C412yes—
138C22×C3×C4C4⋊C412yes—
139C22×C3×C4C22×C411yescharacteristic
140C22×Dic3—22no—
141C22×Dic3—22no—
142C22×Dic3—42no—
143C22×Dic3—42no—
144C22×Dic3—82no—
145C3×C4⋊C4—44no—
146C3×C22⋊C4—24no—
147C3×C22⋊C4—42no—
148C3×C42—44no—
149#217—24no—
150#217—42no—
151#217—42no—
152#217—82no—
153#217—82no—
154C4⋊Dic3—44no—
155C4⋊Dic3—41no—
156C4⋊Dic3—41no—
157C4⋊Dic3—42no—
158C4⋊Dic3—42no—
159C4×Dic3—44no—
160C4×Dic3—42no—
161C2×#41—41no—
162C2×#41—41no—
163C2×#41—42no—
164C2×#41Dic612yes—
165C2×#41C4×D312yes—
166C2×#41C2×Dic312yes—
167C2×#41#6712yes—
168C22×Q8C4×D311yescharacteristic
169C22×D4D1211yescharacteristic
170C22×D4#6711yescharacteristic
171C22×D4C22×D311yescharacteristic
172C23×C4—61no—
173C23×C4C4×D311yescharacteristic
174C23×C4D1211yescharacteristic
175C23×C4#6711yescharacteristic
176#117—24no—
177#117—42no—
178#117—64no—
179#117—122no—
180#117—122no—
181C2×C4⋊C4—32no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4—121no—
186C2×C4⋊C4—121no—
187C2×C4⋊C4—121no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—62no—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—62no—
192C2×C22⋊C4—62no—
193C2×C42—32no—
194C2×C42—41no—
195C2×C42—121no—
196#99—24no—
197#99—22no—
198#99—34no—
199#99—64no—
200#99—62no—
201#99C2×Dic314yescomplemented
202#95—62no—
203#95—62no—
204C23×C3—21no—
205C23×C3—21no—
206C23×C3—21no—
207C23×C3—21no—
208C23×C3—21no—
209C23×C3—21no—
210C23×C3—41no—
211C23×C3—41no—
212C23×C3—41no—
213C23×C3—41no—
214C23×C3—41no—
215C23×C3—41no—
216C23×C3#9512yes—
217C23×C3C2×C22⋊C411yescharacteristic
218C23×C3C2×C22⋊C411yescharacteristic
219C23×C3C2×C22⋊C411yescharacteristic
220C23×C3C2×C4⋊C411yescharacteristic
221C23×C3C2×C4⋊C411yescharacteristic
222C3×Q8—42no—
223C3×Q8—42no—
224C3×D4—42no—
225C3×D4—42no—
226C3×D4—42no—
227C3×D4—42no—
228C3×D4—41no—
229C3×D4—41no—
230C3×D4—41no—
231C3×D4—41no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—44no—
251C2×C3×C4—82no—
252C2×C3×C4—82no—
253C2×C3×C4#9512yes—
254C2×C3×C4#9512yes—
255C2×C3×C4#9512yes—
256C2×C3×C4#9912yes—
257C2×C3×C4#9911yescharacteristic
258C2×C3×C4#9911yescharacteristic
259C2×C3×C4C2×C4211yescharacteristic
260C2×C3×C4C2×C4⋊C411yescharacteristic
261C2×Dic3—44no—
262C2×Dic3—42no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3—42no—
267C2×Dic3—42no—
268C2×Dic3—82no—
269C2×Dic3—82no—
270C2×Dic3—84no—
271C24(C2×D3)⋊C411yescharacteristic
272C24C2×D1211yescharacteristic
273C24C2×#6711yescharacteristic
274#41—42no—
275#41—42no—
276#41—42no—
277#41—42no—
278#41—42no—
279#41—42no—
280#41—42no—
281#41—42no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8C4×Dic312yes—
285C2×Q8Dic3⋊C412yes—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4Dic3⋊C412yes—
293C2×D4C4⋊Dic312yes—
294C2×D4(C2×D3)⋊C412yes—
295C2×D4#21712yes—
296C2×D4C2×Dic611yescharacteristic
297C2×D4C2×C4×D311yescharacteristic
298C2×D4C22×Dic311yescharacteristic
299C2×D4C2×#6711yescharacteristic
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—42no—
310C22×C4—62no—
311C22×C4—62no—
312C22×C4—122no—
313C22×C4—122no—
314C22×C4—242no—
315C22×C4C4×Dic312yes—
316C22×C4Dic3⋊C412yes—
317C22×C4Dic3⋊C412yes—
318C22×C4C4⋊Dic312yes—
319C22×C4(C2×D3)⋊C411yescharacteristic
320C22×C4(C2×D3)⋊C411yescharacteristic
321C22×C4(C2×D3)⋊C412yes—
322C22×C4(C2×D3)⋊C411yescharacteristic
323C22×C4#21712yes—
324C22×C4C2×C4×D311yescharacteristic
325C4⋊C4—44no—
326C4⋊C4—122no—
327C4⋊C4—124no—
328C4⋊C4—121no—
329C4⋊C4—121no—
330C4⋊C4—122no—
331C22⋊C4—24no—
332C22⋊C4—42no—
333C22⋊C4—64no—
334C22⋊C4—122no—
335C22⋊C4—122no—
336C22⋊C4—242no—
337C22⋊C4—242no—
338C42—44no—
339C42—122no—
340C42—124no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—22no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—42no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3—42no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3—41no—
357C22×C3—41no—
358C22×C3—41no—
359C22×C3—41no—
360C22×C3—81no—
361C22×C3—81no—
362C22×C3C2×#9511yescharacteristic
363C22×C3C2×#9911yescharacteristic
364C22×C3C2×#9911yescharacteristic
365C3×C4—22no—
366C3×C4—22no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—84no—
370C3×C4#34212yes—
371C3×C4#34212yes—
372Dic3—82no—
373Dic3—84no—
374Dic3—82no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23C3⋊#9512yes—
388C23C2×Dic3⋊C411yescharacteristic
389C23C2×C4⋊Dic311yescharacteristic
390C23C2×(C2×D3)⋊C411yescharacteristic
391C23C2×(C2×D3)⋊C411yescharacteristic
392C23C2×#21711yescharacteristic
393Q8—42no—
394Q8—42no—
395D4—42no—
396D4—42no—
397D4—42no—
398D4—42no—
399D4—41no—
400D4—41no—
401D4—41no—
402D4—41no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—44no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—124no—
425C2×C4—122no—
426C2×C4—122no—
427C2×C4—122no—
428C2×C4—122no—
429C2×C4—122no—
430C2×C4—122no—
431C2×C4—244no—
432C2×C4—242no—
433C2×C4—242no—
434C2×C4#80612yes—
435C2×C4#80611yescharacteristic
436C2×C4#80611yescharacteristic
437C2×C4C3⋊#9512yes—
438C2×C4C3⋊#9512yes—
439C2×C4C3⋊#9512yes—
440C2×C4C2×C4×Dic311yescharacteristic
441C2×C4C2×Dic3⋊C411yescharacteristic
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—41no—
445C2×C3—41no—
446C2×C3—42no—
447C2×C3—41no—
448C2×C3—41no—
449C2×C3C2×#34211yescharacteristic
450C2×C3#212811yescharacteristic
451C2×C3#212911yescharacteristic
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—22no—
457C22—21no—
458C22—21no—
459C22—42no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—42no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—81no—
472C22—81no—
473C22C2×#80611yescharacteristic
474C22C2×#80611yescharacteristic
475C22C2×C3⋊#9511yescharacteristic
476C4—22no—
477C4—22no—
478C4—21no—
479C4—21no—
480C4—84no—
481C4—242no—
482C4—244no—
483C4—242no—
484C4#452712yes—
485C4#452712yes—
486C3#1258311yescharacteristic, complemented, Sylow
487C2—21no—
488C2—21no—
489C2—41no—
490C2—41no—
491C2—42no—
492C2—41no—
493C2—41no—
494C2#7005911yescharacteristic
495C2#7006111yescharacteristic
496C2C2×#452711yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC2×C3×C4C2×C42
FrattiniC22×D4C22×D3
FittingC2×C3×#410C2
socleC22×C3C2×#99
prime radicalC22×C3×#41C22
cosocleC22×C3×D4C23
hypercenterC2×#410D3
hypocenterC3#12583
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
4. #67695C2semidirect
9. C2×#926C4semidirect
10. C22×C4⋊Dic3C4semidirect
12. C3×#410C4semidirect
16. #5240C4semidirect
17. #5229C4semidirect
66. C3×#99C2×C4semidirect
69. #834C2×C4semidirect
80. #410Dic3semidirect
201. #99C2×Dic3semidirect
486. C3#12583semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4102
#840872
C2×#52402
#676952
#125833