Finite Group Databasebeta

← #10710174

Subgroups — #10710174

Order 768.

794 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#413C211yesmaximal, characteristic
2#84087C211yesmaximal, characteristic
3C2×#5237C211yesmaximal, characteristic
4#67699C214yesmaximal, complemented
5#12587—31nomaximal, Sylow
6C22×C3×#41C2211yescharacteristic
7C22×C3×M16—21no—
8C2×#926C411yescharacteristic
9C22×C4⋊Dic3C411yescharacteristic
10C22×#208—21no—
11C3×#413—22no—
12C3×#413C414yescomplemented
13C2×C3×#101C2212yes—
14C2×C3×#100—21no—
15#5237—22no—
16#5237C414yescomplemented
17#5229C2212yes—
18C2×#836C2212yes—
19#5221C412yes—
20#5220—21no—
21#5220—21no—
22C2×#833—21no—
23#5212—21no—
24#5212—21no—
25C2×#413—31no—
26C2×#413D311yescharacteristic
27#2311—31no—
28#1394—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×Q8C2311yescharacteristic
37C22×C3×D4C2×C411yescharacteristic
38C22×C3×D4D411yescharacteristic
39C22×C3×D4D411yescharacteristic
40C23×C3×C4C2×C411yescharacteristic
41C23×C3×C4D411yescharacteristic
42C23×C3×C4D411yescharacteristic
43C23×Dic3—21no—
44C2×C3×M16—24no—
45C2×C3×M16—22no—
46C2×C3×M16—41no—
47C2×C3×M16—42no—
48C22×C3×C8—41no—
49C2×#217—21no—
50C2×#217—22no—
51C2×#217—22no—
52#926—42no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—41no—
56C2×C4⋊Dic3—41no—
57C2×C4⋊Dic3C2×C412yes—
58C2×C4×Dic3C2×C412yes—
59C2×#208—24no—
60C2×#208—22no—
61C2×#208—41no—
62C2×#208—42no—
63C22×C3⋊C8—41no—
64C3×#101—22no—
65C3×#101C2×C414yescomplemented
66C3×#100—24no—
67#836—22no—
68#836C2×C414yescomplemented
69#833—24no—
70C3⋊#95—22no—
71C3⋊#95—22no—
72C22×#41C2×D311yescharacteristic
73C22×M16—21no—
74C22×M16—61no—
75C2×#117—31no—
76C22×C4⋊C4—31no—
77#413—22no—
78#413—34no—
79#413—62no—
80#413Dic314yescomplemented
81C2×#101—32no—
82C2×#101C2×D312yes—
83C2×#100—21no—
84C2×#100—61no—
85#391—32no—
86#390—32no—
87#386—61no—
88#386—61no—
89#380—61no—
90#380—61no—
91C24×C3C22⋊C411yescharacteristic
92C24×C3C22⋊C411yescharacteristic
93C24×C3C22⋊C411yescharacteristic
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×C411yescharacteristic
105C2×C3×Q8C22×C411yescharacteristic
106C2×C3×Q8C2×D411yescharacteristic
107C2×C3×Q8C2×Q811yescharacteristic
108C2×C3×D4—41no—
109C2×C3×D4—41no—
110C2×C3×D4—41no—
111C2×C3×D4—41no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4C4212yes—
115C2×C3×D4C22⋊C412yes—
116C2×C3×D4C22⋊C412yes—
117C2×C3×D4C4⋊C412yes—
118C2×C3×D4C4⋊C412yes—
119C2×C3×D4C4⋊C412yes—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—22no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—22no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4C4212yes—
130C22×C3×C4C22⋊C412yes—
131C22×C3×C4C22⋊C411yescharacteristic
132C22×C3×C4C22⋊C412yes—
133C22×C3×C4C4⋊C412yes—
134C22×C3×C4C4⋊C412yes—
135C22×C3×C4C4⋊C412yes—
136C22×C3×C4C22×C411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×Dic3—22no—
140C22×Dic3—22no—
141C22×Dic3—42no—
142C22×Dic3—82no—
143C3×M16—24no—
144C3×M16—42no—
145C3×M16—44no—
146C2×C3×C8—42no—
147C2×C3×C8—44no—
148#217—42no—
149#217—82no—
150#217—82no—
151C4⋊Dic3—41no—
152C4⋊Dic3—41no—
153C4⋊Dic3—42no—
154C4⋊Dic3—42no—
155C4×Dic3—42no—
156#208—24no—
157#208—42no—
158#208—44no—
159C2×C3⋊C8—42no—
160C2×C3⋊C8—44no—
161C2×#41—41no—
162C2×#41—41no—
163C2×#41—42no—
164C2×#41Dic612yes—
165C2×#41C4×D312yes—
166C2×#41C2×Dic312yes—
167C2×#41#6712yes—
168C22×Q8C22×D311yescharacteristic
169C22×D4C4×D311yescharacteristic
170C22×D4D1211yescharacteristic
171C22×D4#6711yescharacteristic
172C23×C4—61no—
173C23×C4C4×D311yescharacteristic
174C23×C4D1211yescharacteristic
175C23×C4#6711yescharacteristic
176C2×M16—24no—
177C2×M16—22no—
178C2×M16—41no—
179C2×M16—42no—
180C2×M16—64no—
181C2×M16—62no—
182C2×M16—121no—
183C2×M16—122no—
184C22×C8—41no—
185C22×C8—121no—
186#117—122no—
187C2×C4⋊C4—32no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—121no—
191C2×C4⋊C4—121no—
192C2×C22⋊C4—61no—
193C2×C22⋊C4—62no—
194C2×C22⋊C4—62no—
195C2×C42—32no—
196#101—22no—
197#101—34no—
198#101—62no—
199#101C2×Dic314yescomplemented
200#100—24no—
201#100—64no—
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×C3#9512yes—
218C23×C3#9512yes—
219C23×C3C2×C22⋊C411yescharacteristic
220C3×Q8—41no—
221C3×Q8—41no—
222C3×Q8—41no—
223C3×Q8—41no—
224C3×D4—42no—
225C3×D4—42no—
226C3×D4—42no—
227C3×D4—42no—
228C3×D4—42no—
229C3×D4—42no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
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#9512yes—
249C2×C3×C4#9912yes—
250C2×C3×C4#9911yescharacteristic
251C2×C3×C4#9911yescharacteristic
252C2×C3×C4C2×C4211yescharacteristic
253C2×C3×C4C2×C22⋊C411yescharacteristic
254C2×C3×C4C2×C22⋊C411yescharacteristic
255C2×C3×C4C2×C4⋊C411yescharacteristic
256C2×C3×C4C2×C4⋊C411yescharacteristic
257C2×C3×C4C2×C4⋊C411yescharacteristic
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C2×Dic3—42no—
264C2×Dic3—84no—
265C3×C8—44no—
266C3⋊C8—44no—
267C24(C2×D3)⋊C411yescharacteristic
268C24(C2×D3)⋊C411yescharacteristic
269C24(C2×D3)⋊C411yescharacteristic
270#41—42no—
271#41—42no—
272#41—42no—
273#41—42no—
274#41—42no—
275#41—42no—
276#41—42no—
277#41—42no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8C2×Dic611yescharacteristic
281C2×Q8C2×C4×D311yescharacteristic
282C2×Q8C22×Dic311yescharacteristic
283C2×Q8C2×#6711yescharacteristic
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×D4Dic3⋊C412yes—
293C2×D4C4⋊Dic312yes—
294C2×D4(C2×D3)⋊C412yes—
295C2×D4#21712yes—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—62no—
306C22×C4—62no—
307C22×C4—122no—
308C22×C4—242no—
309C22×C4C4×Dic312yes—
310C22×C4Dic3⋊C412yes—
311C22×C4Dic3⋊C412yes—
312C22×C4C4⋊Dic312yes—
313C22×C4(C2×D3)⋊C412yes—
314C22×C4(C2×D3)⋊C411yescharacteristic
315C22×C4#21712yes—
316C22×C4C2×C4×D311yescharacteristic
317C22×C4C2×D1211yescharacteristic
318C22×C4C2×#6711yescharacteristic
319M16—24no—
320M16—42no—
321M16—44no—
322M16—64no—
323M16—122no—
324M16—124no—
325C2×C8—42no—
326C2×C8—44no—
327C2×C8—122no—
328C2×C8—124no—
329C4⋊C4—122no—
330C4⋊C4—121no—
331C4⋊C4—121no—
332C4⋊C4—122no—
333C22⋊C4—122no—
334C22⋊C4—242no—
335C22⋊C4—242no—
336C42—122no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—22no—
340C22×C3—22no—
341C22×C3—22no—
342C22×C3—42no—
343C22×C3—42no—
344C22×C3—42no—
345C22×C3—42no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—41no—
352C22×C3—81no—
353C22×C3—81no—
354C22×C3C2×#9511yescharacteristic
355C22×C3C2×#9911yescharacteristic
356C22×C3C2×#9911yescharacteristic
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4#34212yes—
364C3×C4#34212yes—
365Dic3—82no—
366Dic3—82no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23C3⋊#9512yes—
380C23C3⋊#9512yes—
381C23C3⋊#9512yes—
382C23C2×(C2×D3)⋊C411yescharacteristic
383Q8—41no—
384Q8—41no—
385Q8—41no—
386Q8—41no—
387D4—42no—
388D4—42no—
389D4—42no—
390D4—42no—
391D4—42no—
392D4—42no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—122no—
412C2×C4—122no—
413C2×C4—122no—
414C2×C4—122no—
415C2×C4—122no—
416C2×C4—122no—
417C2×C4—244no—
418C2×C4#80612yes—
419C2×C4#80611yescharacteristic
420C2×C4#80611yescharacteristic
421C2×C4C3⋊#9512yes—
422C2×C4C2×C4×Dic311yescharacteristic
423C2×C4C2×Dic3⋊C411yescharacteristic
424C2×C4C2×Dic3⋊C411yescharacteristic
425C2×C4C2×C4⋊Dic311yescharacteristic
426C2×C4C2×(C2×D3)⋊C411yescharacteristic
427C2×C4C2×#21711yescharacteristic
428C8—44no—
429C8—124no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—42no—
433C2×C3—42no—
434C2×C3—42no—
435C2×C3C2×#34211yescharacteristic
436C2×C3#213411yescharacteristic
437C2×C3#213511yescharacteristic
438C22—21no—
439C22—21no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—81no—
454C22—81no—
455C22C2×#80611yescharacteristic
456C22C2×#80611yescharacteristic
457C22C2×C3⋊#9511yescharacteristic
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—242no—
465C4—242no—
466C4#452712yes—
467C4#452712yes—
468C3#1258711yescharacteristic, complemented, Sylow
469C2—21no—
470C2—21no—
471C2—42no—
472C2—42no—
473C2—42no—
474C2#7014911yescharacteristic
475C2#7015011yescharacteristic
476C2C2×#452711yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC2×C3×C4C2×C42
FrattiniC22×Q8C22×D3
FittingC2×C3×#413C2
socleC22×C3C2×#99
prime radicalC22×C3×#41C22
cosocleC22×C3×Q8C23
hypercenterC2×#413D3
hypocenterC3#12587
LCSG ▷ C2×C3×C4 ▷ 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. #67699C2semidirect
12. C3×#413C4semidirect
16. #5237C4semidirect
65. C3×#101C2×C4semidirect
68. #836C2×C4semidirect
80. #413Dic3semidirect
199. #101C2×Dic3semidirect
468. C3#12587semidirect

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×#52372
#676992
#125873