Finite Group Databasebeta

← #10710139

Subgroups — #10710139

Order 768.

810 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
1C2×C3×#413C211yesmaximal, characteristic
2#84087C211yesmaximal, characteristic
3C2×#5240C211yesmaximal, characteristic
4#67698C214yesmaximal, complemented
5#12575—31nomaximal, Sylow
6C22×C3×#41C2211yescharacteristic
7C22×C3×M16—21no—
8C2×#926—21no—
9C2×#926C411yescharacteristic, complemented
10C22×C4⋊Dic3C411yescharacteristic, complemented
11C3×#413—22no—
12C3×#413C414yescomplemented
13C2×C3×#101—21no—
14C2×C3×#100C2212yes—
15#5240—22no—
16#5240C414yescomplemented
17#5229—21no—
18#5221—21no—
19#5220C412yescomplemented
20#5220C2212yes—
21C2×#834—21no—
22C2×#834C2212yes—
23#5212—21no—
24#5212—21no—
25C2×#413D311yescharacteristic
26C2×#410—31no—
27#2311—31no—
28#1390—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×Q8D411yescharacteristic
37C22×C3×D4C2×C411yescharacteristic
38C22×C3×D4D411yescharacteristic
39C22×C3×D4C2311yescharacteristic
40C23×C3×C4C2×C411yescharacteristic
41C23×C3×C4D411yescharacteristic
42C23×C3×C4D411yescharacteristic
43C23×Dic3C2×C412yes—
44C2×C3×M16—24no—
45C2×C3×M16—22no—
46C2×C3×M16—41no—
47C2×C3×M16—42no—
48C22×C3×C8—41no—
49C2×#217—22no—
50C2×#217—22no—
51C2×#217—22no—
52C2×#217C2×C412yes—
53#926—24no—
54#926—42no—
55#926—42no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3—41no—
60C2×C4⋊Dic3—41no—
61C2×C4⋊Dic3—41no—
62C2×C4×Dic3—21no—
63C2×C4×Dic3—41no—
64C3×#101—24no—
65C3×#100—22no—
66C3×#100C2×C414yes—
67#834—24no—
68#834—22no—
69#834C2×C414yes—
70C3⋊#95—22no—
71C3⋊#95—22no—
72C22×#41C2×D311yescharacteristic
73C22×M16—21no—
74C2×#117—31no—
75C2×#117—61no—
76C22×C4⋊C4—31no—
77#413—22no—
78#413Dic314yescomplemented
79C2×#101—21no—
80C2×#100C2×D312yes—
81#410—34no—
82#410—62no—
83C2×#99—32no—
84C2×#99—61no—
85#391—61no—
86#390—61no—
87#386—32no—
88#386—32no—
89#380—61no—
90#380—61no—
91C24×C3C22⋊C411yescharacteristic
92C24×C3C22×C411yescharacteristic
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×Q8C22⋊C412yes—
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×D4C4212yes—
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×C4C4212yes—
130C22×C3×C4C22⋊C411yescharacteristic
131C22×C3×C4C22⋊C411yescharacteristic
132C22×C3×C4C22⋊C412yes—
133C22×C3×C4C22⋊C411yescharacteristic
134C22×C3×C4C22⋊C412yes—
135C22×C3×C4C4⋊C412yes—
136C22×C3×C4C4⋊C412yes—
137C22×C3×C4C4⋊C412yes—
138C22×C3×C4C2×D411yescharacteristic
139C22×Dic3—22no—
140C22×Dic3—22no—
141C22×Dic3—42no—
142C22×Dic3—42no—
143C22×Dic3—82no—
144C3×M16—24no—
145C3×M16—42no—
146C3×M16—44no—
147C2×C3×C8—42no—
148C2×C3×C8—44no—
149#217—24no—
150#217—42no—
151#217—42no—
152#217—82no—
153#217—82no—
154C4⋊Dic3—42no—
155C4⋊Dic3—44no—
156C4⋊Dic3—41no—
157C4⋊Dic3—41no—
158C4⋊Dic3—42no—
159C4×Dic3—42no—
160C4×Dic3—44no—
161C2×#41—41no—
162C2×#41—41no—
163C2×#41—42no—
164C2×#41Dic612yes—
165C2×#41C4×D312yes—
166C2×#41C2×Dic312yes—
167C2×#41#6712yes—
168C22×Q8#6711yescharacteristic
169C22×D4C4×D311yescharacteristic
170C22×D4D1211yescharacteristic
171C22×D4C22×D311yescharacteristic
172C23×C4—32no—
173C23×C4C4×D311yescharacteristic
174C23×C4D1211yescharacteristic
175C23×C4#6711yescharacteristic
176C2×M16—24no—
177C2×M16—22no—
178C2×M16—41no—
179C2×M16—42no—
180C22×C8—41no—
181#117—64no—
182#117—122no—
183#117—122no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4—121no—
188C2×C4⋊C4—121no—
189C2×C4⋊C4—121no—
190C2×C22⋊C4—32no—
191C2×C22⋊C4—62no—
192C2×C22⋊C4—62no—
193C2×C22⋊C4—62no—
194C2×C42—61no—
195C2×C42—121no—
196#101—24no—
197#100—22no—
198#100C2×Dic314yes—
199#99—34no—
200#99—64no—
201#99—62no—
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×C4211yescharacteristic
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#9512yes—
251C2×C3×C4#9512yes—
252C2×C3×C4#9512yes—
253C2×C3×C4#9912yes—
254C2×C3×C4#9911yescharacteristic
255C2×C3×C4#9911yescharacteristic
256C2×C3×C4C2×C22⋊C411yescharacteristic
257C2×C3×C4C2×C4⋊C411yescharacteristic
258C2×Dic3—44no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265C2×Dic3—82no—
266C2×Dic3—82no—
267C2×Dic3—84no—
268C3×C8—44no—
269C24(C2×D3)⋊C411yescharacteristic
270C24C2×C4×D311yescharacteristic
271C24C2×D1211yescharacteristic
272#41—42no—
273#41—42no—
274#41—42no—
275#41—42no—
276#41—42no—
277#41—42no—
278#41—42no—
279#41—42no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8Dic3⋊C412yes—
283C2×Q8#21712yes—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4C4×Dic312yes—
291C2×D4Dic3⋊C412yes—
292C2×D4C4⋊Dic312yes—
293C2×D4(C2×D3)⋊C412yes—
294C2×D4C2×Dic611yescharacteristic
295C2×D4C2×C4×D311yescharacteristic
296C2×D4C22×Dic311yescharacteristic
297C2×D4C2×#6711yescharacteristic
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—62no—
308C22×C4—62no—
309C22×C4—122no—
310C22×C4—122no—
311C22×C4—242no—
312C22×C4C4×Dic312yes—
313C22×C4Dic3⋊C412yes—
314C22×C4Dic3⋊C412yes—
315C22×C4C4⋊Dic312yes—
316C22×C4(C2×D3)⋊C411yescharacteristic
317C22×C4(C2×D3)⋊C411yescharacteristic
318C22×C4(C2×D3)⋊C412yes—
319C22×C4(C2×D3)⋊C411yescharacteristic
320C22×C4#21712yes—
321C22×C4C2×#6711yescharacteristic
322M16—24no—
323M16—42no—
324M16—44no—
325C2×C8—42no—
326C2×C8—44no—
327C4⋊C4—122no—
328C4⋊C4—124no—
329C4⋊C4—121no—
330C4⋊C4—121no—
331C4⋊C4—122no—
332C22⋊C4—64no—
333C22⋊C4—122no—
334C22⋊C4—122no—
335C22⋊C4—242no—
336C22⋊C4—242no—
337C42—122no—
338C42—124no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—22no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—42no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—42no—
352C22×C3—41no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3—41no—
357C22×C3—41no—
358C22×C3—81no—
359C22×C3—81no—
360C22×C3C2×#9511yescharacteristic
361C22×C3C2×#9911yescharacteristic
362C22×C3C2×#9911yescharacteristic
363C3×C4—21no—
364C3×C4—22no—
365C3×C4—22no—
366C3×C4—21no—
367C3×C4#34212yes—
368C3×C4#34212yes—
369Dic3—82no—
370Dic3—84no—
371Dic3—82no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23C3⋊#9512yes—
385C23C2×C4×Dic311yescharacteristic
386C23C2×Dic3⋊C411yescharacteristic
387C23C2×C4⋊Dic311yescharacteristic
388C23C2×(C2×D3)⋊C411yescharacteristic
389C23C2×(C2×D3)⋊C411yescharacteristic
390Q8—42no—
391Q8—42no—
392D4—42no—
393D4—42no—
394D4—42no—
395D4—42no—
396D4—41no—
397D4—41no—
398D4—41no—
399D4—41no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
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—124no—
419C2×C4—122no—
420C2×C4—122no—
421C2×C4—122no—
422C2×C4—122no—
423C2×C4—122no—
424C2×C4—122no—
425C2×C4—242no—
426C2×C4—242no—
427C2×C4—244no—
428C2×C4#80612yes—
429C2×C4#80611yescharacteristic
430C2×C4#80611yescharacteristic
431C2×C4C3⋊#9512yes—
432C2×C4C3⋊#9512yes—
433C2×C4C3⋊#9512yes—
434C2×C4C2×Dic3⋊C411yescharacteristic
435C2×C4C2×#21711yescharacteristic
436C8—44no—
437C2×C3—21no—
438C2×C3—21no—
439C2×C3—41no—
440C2×C3—41no—
441C2×C3—41no—
442C2×C3—42no—
443C2×C3—41no—
444C2×C3C2×#34211yescharacteristic
445C2×C3#212211yescharacteristic
446C2×C3#212311yescharacteristic
447C22—21no—
448C22—21no—
449C22—21no—
450C22—22no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—42no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—42no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—81no—
467C22—81no—
468C22C2×#80611yescharacteristic
469C22C2×#80611yescharacteristic
470C22C2×C3⋊#9511yescharacteristic
471C4—21no—
472C4—22no—
473C4—22no—
474C4—21no—
475C4—242no—
476C4—244no—
477C4—242no—
478C4#452712yes—
479C4#452712yes—
480C3#1257511yescharacteristic, complemented, Sylow
481C2—21no—
482C2—21no—
483C2—41no—
484C2—41no—
485C2—41no—
486C2—42no—
487C2—41no—
488C2#7014611yescharacteristic
489C2#7014711yescharacteristic
490C2C2×#452711yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC23×C3C2×C42
FrattiniC22×D4C22×D3
FittingC2×C3×#413C2
socleC22×C3C2×#99
prime radicalC22×C3×#41C22
cosocleC22×C3×D4C23
hypercenterC2×#413D3
hypocenterC3#12575
LCSG ▷ C23×C3 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#413

Complements

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

NH ≅ G/Ntype
4. #67698C2semidirect
9. C2×#926C4semidirect
10. C22×C4⋊Dic3C4semidirect
12. C3×#413C4semidirect
16. #5240C4semidirect
19. #5220C4semidirect
78. #413Dic3semidirect
480. C3#12575semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4132
#840872
C2×#52402
#676982
#125753