Finite Group Databasebeta

← #10848840

Subgroups — #10848840

Order 768.

1094 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5731C211yesmaximal, characteristic, complemented
2C2×C3×#497C211yesmaximal, characteristic, complemented
3#83001C211yesmaximal, characteristic
4C2×#5056C211yesmaximal, characteristic, complemented
5#72071C212yesmaximal, complemented
6#72065C212yesmaximal, complemented
7#72054C212yesmaximal
8#72048C212yesmaximal, complemented
9#71785C211yesmaximal, characteristic, complemented
10#69830C211yesmaximal, characteristic, complemented
11#69827C211yesmaximal, characteristic, complemented
12#25612—31nomaximal, Sylow
13C22×C3×Q16C2212yes—
14C22×C3×SD16C2212yes—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#124C2211yescharacteristic
17C2×#946C2212yes—
18C2×#946C2211yescharacteristic
19C2×Q8×Dic3C2212yes—
20C2×#938C2211yescharacteristic
21#5731—24no—
22#5731—24no—
23C2×#898—22no—
24C2×#889—22no—
25C2×#872C2211yescharacteristic
26C2×#870C2211yescharacteristic
27C3×#497—24no—
28C2×C3×#97C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2212yes—
31C2×#835C2212yes—
32C2×#835C2212yes—
33#5221C2211yescharacteristic
34C2×#832C2212yes—
35C2×(C2×D3)⋊C8C2212yes—
36#5056—24no—
37#4961—22no—
38#4948—22no—
39#4935C2211yescharacteristic
40C2×C4×C3⋊C8C2211yescharacteristic
41#4551C2212yes—
42#4551C2212yes—
43#4550C2212yes—
44C2×#541—31no—
45C2×#497D311yescharacteristic, complemented
46C2×#488—31no—
47#2620—31no—
48#2001—32no—
49#2000—32no—
50#1992—31no—
51#1956—31no—
52#1880—32no—
53#1878—32no—
54#1750—31no—
55C22×C3×Q8C2311yescharacteristic
56C22×C3×Q8C2312yes—
57C22×C3×D4C2311yescharacteristic
58C22×#67—21no—
59C22×C4×D3C2311yescharacteristic
60C2×C3×Q16—24no—
61C2×C3×Q16—24no—
62C2×C3×Q16—42no—
63C2×C3×SD16—24no—
64C2×C3×SD16—24no—
65C2×C3×SD16—42no—
66C22×C3×C8C2312yes—
67C3×C4⋊Q8—22no—
68C3×C4⋊Q8D414yescomplemented
69C3×#124—22no—
70C3×#124—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C22⋊C4—22no—
73C2×C3×C42C2311yescharacteristic
74C2×#217—21no—
75C2×#217—22no—
76#946—24no—
77#946—24no—
78#946—42no—
79Q8×Dic3—24no—
80Q8×Dic3—24no—
81#938—42no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—21no—
84C2×(C2×D3)⋊C4D412yes—
85C2×C4⋊Dic3—22no—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3C2311yescharacteristic
88C2×C4⋊Dic3C2311yescharacteristic
89C2×C4⋊Dic3C2312yes—
90C2×Dic3⋊C4—22no—
91C2×Dic3⋊C4—22no—
92C2×Dic3⋊C4—21no—
93C2×Dic3⋊C4D412yes—
94C2×C4×Dic3—22no—
95C2×C4×Dic3C2311yescharacteristic
96C22×C3⋊C8C2312yes—
97#898—216no—
98#889—216no—
99#872—22no—
100#872—22no—
101#870—22no—
102#870D414yescomplemented
103C3×#97—22no—
104#835—24no—
105#835—42no—
106#835—42no—
107#832—24no—
108C3⋊#95—22no—
109(C2×D3)⋊C8—24no—
110C4×C3⋊C8—22no—
111C22×Q16C2×D312yes—
112C22×SD16C2×D312yes—
113#541—64no—
114#541—64no—
115C2×C4⋊Q8C2×D311yescharacteristic
116C2×#126—62no—
117C2×#125—31no—
118C2×#125—62no—
119C2×#124C2×D311yescharacteristic
120C2×C22⋊Q8—32no—
121C2×C22⋊Q8—31no—
122C2×#121—31no—
123C2×C4×Q8—32no—
124C2×#117—31no—
125#497—24no—
126#488—64no—
127C2×#103—32no—
128C2×#103—32no—
129C2×#103—32no—
130C2×#102—32no—
131C2×C22⋊C8—32no—
132C2×#97C2×D311yescharacteristic
133C2×C4×C8—31no—
134#396—62no—
135#394—62no—
136#391—31no—
137#390—31no—
138#384—31no—
139#340—32no—
140#340—32no—
141#336—32no—
142C24×C3—21no—
143C23×D3C2×D411yescharacteristic
144C2×C3×Q8—22no—
145C2×C3×Q8—21no—
146C2×C3×Q8—21no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8C2×D414yes—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2411yescharacteristic
162C2×#67—42no—
163C2×#67—82no—
164C22×Dic3—22no—
165C22×Dic3—21no—
166C22×Dic3—21no—
167C22×Dic3C2×D412yes—
168C22×Dic3C2×D411yescharacteristic
169C2×C4×D3—21no—
170C2×C4×D3—21no—
171C2×C4×D3—22no—
172C2×C4×D3—22no—
173C3×Q16—48no—
174C3×SD16—48no—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C2×C3×C8#4114yes—
178C3×C4⋊C4—28no—
179C3×C22⋊C4—44no—
180C3×C42C2×D412yes—
181C3×C42C2×D412yes—
182#217—42no—
183#217—82no—
184(C2×D3)⋊C4—28no—
185(C2×D3)⋊C4—24no—
186(C2×D3)⋊C4—42no—
187C4⋊Dic3—28no—
188C4⋊Dic3—22no—
189C4⋊Dic3—22no—
190C4⋊Dic3—22no—
191C4⋊Dic3—82no—
192C4⋊Dic3C2×D414yes—
193Dic3⋊C4—28no—
194Dic3⋊C4—28no—
195Dic3⋊C4—24no—
196Dic3⋊C4—42no—
197C4×Dic3—28no—
198C4×Dic3—21no—
199C4×Dic3—21no—
200C2×C3⋊C8—22no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8—22no—
203C22×Q8C22×D311yescharacteristic
204C22×Q8C22×D312yes—
205C22×D4—61no—
206C22×D4C22×D311yescharacteristic
207C23×C4—31no—
208C2×Q16—24no—
209C2×Q16—24no—
210C2×Q16—42no—
211C2×SD16—24no—
212C2×SD16—24no—
213C2×SD16—42no—
214C22×C8—32no—
215C22×C8C22×D312yes—
216C4⋊Q8—22no—
217C4⋊Q8#6714yescomplemented
218#126—616no—
219#125—34no—
220#125—616no—
221#125—62no—
222#124—22no—
223#124—22no—
224C22⋊Q8—64no—
225C22⋊Q8—64no—
226C22⋊Q8—122no—
227#121—122no—
228C4×Q8—64no—
229C4×Q8—64no—
230#117—62no—
231#117—62no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—32no—
234C2×C4⋊C4—32no—
235C2×C4⋊C4—31no—
236C2×C4⋊C4—31no—
237C2×C4⋊C4—62no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—62no—
240C2×C4⋊C4—61no—
241C2×C4⋊C4—62no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—32no—
244C2×C22⋊C4—62no—
245C2×C22⋊C4—61no—
246C2×C22⋊C4—61no—
247C2×C22⋊C4—62no—
248C2×C42—31no—
249C2×C42—62no—
250C2×C42C22×D311yescharacteristic
251#103—64no—
252#103—122no—
253#103—122no—
254#102—64no—
255C22⋊C8—64no—
256#97—22no—
257C4×C8—62no—
258#95—62no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3—42no—
263C23×C3—42no—
264C23×C3C22×D411yescharacteristic
265C22×D3—21no—
266C22×D3—21no—
267C22×D3—21no—
268C22×D3—22no—
269C22×D3—22no—
270C3×Q8—24no—
271C3×Q8—24no—
272C3×Q8—42no—
273C3×Q8—42no—
274C3×Q8—42no—
275C3×Q8—42no—
276C3×D4—42no—
277C3×D4—42no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—24no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4#12014yes—
287C2×C3×C4C22×D411yescharacteristic
288C2×C3×C4C22×D411yescharacteristic
289C2×C3×C4C2×#4112yes—
290C2×C3×C4H(5,𝔽2)12yes—
291#67—84no—
292C2×Dic3—22no—
293C2×Dic3—24no—
294C2×Dic3—21no—
295C2×Dic3—21no—
296C2×Dic3—21no—
297C2×Dic3—22no—
298C2×Dic3—22no—
299C2×Dic3—42no—
300C2×Dic3—42no—
301C2×Dic3—41no—
302C2×Dic3—41no—
303C2×Dic3—42no—
304C2×Dic3—42no—
305C4×D3—42no—
306C4×D3—42no—
307C3×C8—24no—
308C3⋊C8—42no—
309C24—21no—
310C24—31no—
311C2×Q8—22no—
312C2×Q8—21no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—21no—
320C2×Q8C2×#6714yes—
321C2×D4—21no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—21no—
325C2×D4—122no—
326C2×D4—242no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—32no—
330C22×C4—31no—
331C22×C4—62no—
332C22×C4—61no—
333C22×C4—62no—
334C22×C4—62no—
335C22×C4—61no—
336C22×C4—61no—
337C22×C4—61no—
338C22×C4D3×D412yes—
339C22×C4C23×D311yescharacteristic
340Q16—48no—
341SD16—48no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—62no—
345C2×C8—62no—
346C2×C8—62no—
347C2×C8#23714yes—
348C4⋊C4—28no—
349C4⋊C4—34no—
350C4⋊C4—68no—
351C4⋊C4—68no—
352C4⋊C4—64no—
353C4⋊C4—68no—
354C4⋊C4—62no—
355C4⋊C4—62no—
356C4⋊C4—62no—
357C4⋊C4—122no—
358C4⋊C4—242no—
359C22⋊C4—44no—
360C22⋊C4—68no—
361C22⋊C4—64no—
362C22⋊C4—122no—
363C22⋊C4—122no—
364C22⋊C4—242no—
365C42—68no—
366C42—61no—
367C42—61no—
368C42D3×D412yes—
369C42C2×#6712yes—
370C22×C3—82no—
371C22×C3—81no—
372C22×C3—81no—
373C22×C3—81no—
374C22×C3—82no—
375C22×C3C2×#12011yescharacteristic
376C22×C3#53211yescharacteristic
377C22×C3#53511yescharacteristic
378C22×C3#57712yes—
379C22×C3#57812yes—
380C2×D3—42no—
381C2×D3—41no—
382C2×D3—41no—
383C2×D3—41no—
384C2×D3—42no—
385C3×C4—24no—
386C3×C4—44no—
387C3×C4—81no—
388C3×C4#54612yes—
389C3×C4#55012yes—
390Dic3—41no—
391Dic3—42no—
392Dic3—44no—
393Dic3—81no—
394Dic3—81no—
395C23—41no—
396C23—41no—
397C23—42no—
398C23—42no—
399C23—41no—
400C23—61no—
401C23—61no—
402C23—62no—
403C23—62no—
404C23—61no—
405C23C2×D3×D411yescharacteristic
406Q8—24no—
407Q8—24no—
408Q8—42no—
409Q8—42no—
410Q8—42no—
411Q8—42no—
412D4—42no—
413D4—42no—
414D4—244no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—61no—
424C2×C4—61no—
425C2×C4—62no—
426C2×C4—62no—
427C2×C4—62no—
428C2×C4—64no—
429C2×C4—61no—
430C2×C4—122no—
431C2×C4—122no—
432C2×C4—122no—
433C2×C4—122no—
434C2×C4—121no—
435C2×C4—121no—
436C2×C4—122no—
437C2×C4—122no—
438C2×C4#93714yes—
439C2×C4C2×D3×D411yescharacteristic
440C2×C4C2×#23712yes—
441C2×C4#100412yes—
442C2×C4C22×#6711yescharacteristic
443C8—24no—
444C8—122no—
445C2×C3—161no—
446C2×C3#243111yescharacteristic
447C2×C3#301312yes—
448C2×C3#317812yes—
449C2×C3#322211yescharacteristic
450C2×C3#322311yescharacteristic
451D3—81no—
452C22—82no—
453C22—82no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22—122no—
458C22—122no—
459C22—121no—
460C22—121no—
461C22—121no—
462C22#571511yescharacteristic
463C22#571611yescharacteristic
464C22#577812yes—
465C22#577912yes—
466C22C2×#93711yescharacteristic
467C4—24no—
468C4—44no—
469C4—81no—
470C4—122no—
471C4—124no—
472C4—121no—
473C4—241no—
474C4—241no—
475C4#571812yes—
476C4#580612yes—
477C3#2561211yescharacteristic, complemented, Sylow
478C2—161no—
479C2—241no—
480C2#8186212yes—
481C2#8187811yescharacteristic
482C2#8187911yescharacteristic
483C2#8318711yescharacteristic
484C2#8398412yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#497C2
socleC23×C3C22×D4
prime radicalC2×#938C22
hypercenterC2×#497D3
hypocenterC3#25612
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#497

Complements

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

NH ≅ G/Ntype
1. C2×#5731C2semidirect
2. C2×C3×#497C2semidirect
4. C2×#5056C2semidirect
5. #72071C2semidirect
6. #72065C2semidirect
8. #72048C2semidirect
9. #71785C2semidirect
10. #69830C2semidirect
11. #69827C2semidirect
45. C2×#497D3semidirect
68. C3×C4⋊Q8D4semidirect
102. #870D4semidirect
217. C4⋊Q8#67semidirect
477. C3#25612semidirect

Maximal subgroups

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

subgroupindex
C2×#57312
C2×C3×#4972
#830012
C2×#50562
#720712
#720652
#720542
#720482
#717852
#698302
#698272
#256123