Finite Group Databasebeta

← #10796152

Subgroups — #10796152

Order 768.

1446 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×Q8C211yesmaximal, characteristic
2C2×#5225C211yesmaximal, characteristic
3C2×#5225C211yesmaximal, characteristic
4C2×#4964C211yesmaximal, characteristic
5#82069C211yesmaximal, characteristic
6#71629C214yesmaximal, complemented
7#71628C214yesmaximal, complemented
8#71558C211yesmaximal, characteristic
9#71527C211yesmaximal, characteristic
10#20874—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C2×C3×C4×Q8—24no—
13C2×C3×C4×Q8—24no—
14C2×C3×C4×Q8—22no—
15C2×C3×C4×Q8C2214yes—
16C22×C3×C4⋊C4—21no—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C42—21no—
19C22×C3×C42C2211yescharacteristic
20C2×#926C2211yescharacteristic
21C2×#926C2211yescharacteristic
22C22×#208C2211yescharacteristic
23C22×#208C2211yescharacteristic
24#5225—24no—
25#5225C418yescomplemented
26C2×#835C2212yes—
27C2×#835C2212yes—
28C2×#835C2214yes—
29#5210—21no—
30#5210—21no—
31#4964—24no—
32C2×#808C2212yes—
33C2×#807C2212yes—
34C4×Dic3⋊C4—22no—
35#4928C2212yes—
36#4928C2212yes—
37#4529C2214yes—
38#4524C2214yes—
39C22×C4×Q8D311yescharacteristic
40C2×#428—31no—
41C2×#419—31no—
42C2×#419—31no—
43#2296—31no—
44#1865—34no—
45#1863—34no—
46#1736—31no—
47#1728—31no—
48C22×C3×Q8—22no—
49C22×C3×Q8—21no—
50C22×C3×Q8—22no—
51C22×C3×Q8—22no—
52C22×C3×Q8C2×C414yes—
53C22×C3×Q8D414yes—
54C22×C3×Q8Q814yes—
55C22×C3×Q8C2312yes—
56C23×C3×C4—21no—
57C23×C3×C4—21no—
58C23×C3×C4D411yescharacteristic
59C23×C3×C4D411yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C3×C4×Q8—216no—
62C3×C4×Q8—28no—
63C3×C4×Q8—28no—
64C2×C3×C4⋊C4—28no—
65C2×C3×C4⋊C4—24no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4D414yes—
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C42—28no—
71C2×C3×C42—24no—
72C2×C3×C42—22no—
73C2×C3×C42—22no—
74C2×C3×C42D412yes—
75C2×C3×C42C2312yes—
76C2×#217—21no—
77C2×#217—21no—
78#926—22no—
79#926—22no—
80#926—22no—
81#926C2×C414yes—
82C2×C4⋊Dic3C2311yescharacteristic
83C2×C4⋊Dic3C2311yescharacteristic
84C2×C4⋊Dic3C2312yes—
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4—22no—
87C2×C4×Dic3—21no—
88C2×C4×Dic3—21no—
89C2×#208—22no—
90C2×#208—22no—
91C2×#208—22no—
92C2×#208—22no—
93C2×#208—22no—
94C2×#208C2×C414yes—
95C22×C3⋊C8C2312yes—
96C22×C3⋊C8C2312yes—
97#835—24no—
98#835—24no—
99#835C2×C4116yescomplemented
100C3⋊#95—22no—
101C3⋊#95—22no—
102#808—24no—
103#807—24no—
104C23×Q8C2×D311yescharacteristic
105C22×M16—31no—
106C22×M16—31no—
107C2×C4×Q8—24no—
108C2×C4×Q8—24no—
109C2×C4×Q8—22no—
110C2×C4×Q8C2×D314yes—
111C2×#117—31no—
112C2×#117—31no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4C2×D311yescharacteristic
115C22×C42—21no—
116C22×C42C2×D311yescharacteristic
117#428—64no—
118C2×#107—32no—
119C2×#106—32no—
120#419—38no—
121#419—64no—
122C2×#103—34no—
123C2×#103—32no—
124C2×#103—32no—
125#389—32no—
126#389—32no—
127#385—61no—
128#385—61no—
129C4×C4⋊C4—62no—
130#340—34no—
131#336—34no—
132C24×C3C2×D411yescharacteristic
133C2×C3×Q8—24no—
134C2×C3×Q8—28no—
135C2×C3×Q8—24no—
136C2×C3×Q8—24no—
137C2×C3×Q8—28no—
138C2×C3×Q8—28no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8C4⋊C4116yescomplemented
143C2×C3×Q8C22×C414yes—
144C2×C3×Q8C2×D414yes—
145C2×C3×Q8C2×Q814yes—
146C22×C3×C4—24no—
147C22×C3×C4—22no—
148C22×C3×C4—24no—
149C22×C3×C4—24no—
150C22×C3×C4—22no—
151C22×C3×C4—24no—
152C22×C3×C4—22no—
153C22×C3×C4—24no—
154C22×C3×C4—22no—
155C22×C3×C4—42no—
156C22×C3×C4—44no—
157C22×C3×C4C22×C412yes—
158C22×C3×C4C2×D412yes—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×Q812yes—
164C22×C3×C4#4112yes—
165C22×C3×C4#4112yes—
166C22×C3×C4C2411yescharacteristic
167C22×Dic3—22no—
168C22×Dic3—21no—
169C22×Dic3—21no—
170C3×C4⋊C4—216no—
171C3×C4⋊C4—24no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4C2×D414yes—
174C3×C42—216no—
175C3×C42—24no—
176C3×C42—22no—
177C3×C42—22no—
178#217—24no—
179#217—42no—
180C4⋊Dic3—22no—
181C4⋊Dic3—21no—
182C4⋊Dic3—21no—
183C4⋊Dic3—21no—
184C4⋊Dic3—21no—
185C4⋊Dic3C22×C414yes—
186Dic3⋊C4—44no—
187Dic3⋊C4—44no—
188C4×Dic3—24no—
189C4×Dic3—42no—
190#208—28no—
191#208—44no—
192C2×C3⋊C8—22no—
193C2×C3⋊C8—22no—
194C2×C3⋊C8—22no—
195C2×C3⋊C8—22no—
196C2×C3⋊C8—22no—
197C2×C3⋊C8C22×C414yes—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—21no—
202C22×Q8Dic614yes—
203C22×Q8C4×D314yes—
204C22×Q8#6714yes—
205C22×Q8C22×D312yes—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4#6711yescharacteristic
209C23×C4#6711yescharacteristic
210C23×C4C22×D311yescharacteristic
211C2×M16—34no—
212C2×M16—62no—
213C2×M16—62no—
214C2×M16—62no—
215C2×M16—62no—
216C2×M16—62no—
217C22×C8—32no—
218C22×C8—32no—
219C4×Q8—216no—
220C4×Q8—28no—
221C4×Q8—28no—
222#117—34no—
223#117—62no—
224#117—62no—
225#117—62no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—32no—
231C2×C4⋊C4—31no—
232C2×C4⋊C4—31no—
233C2×C4⋊C4—62no—
234C2×C4⋊C4—62no—
235C2×C4⋊C4#6714yes—
236C2×C4⋊C4C22×D312yes—
237C2×C22⋊C4—61no—
238C2×C22⋊C4—61no—
239C2×C42—28no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—24no—
243C2×C42—61no—
244C2×C42—61no—
245C2×C42#6712yes—
246C2×C42C22×D312yes—
247#107—64no—
248#106—64no—
249#103—316no—
250#103—64no—
251#103—64no—
252#95—62no—
253#95—62no—
254C23×C3—22no—
255C23×C3—22no—
256C23×C3C4×D412yes—
257C23×C3#12012yes—
258C23×C3C22⋊Q812yes—
259C23×C3C22×D411yescharacteristic
260C3×Q8—24no—
261C3×Q8—24no—
262C3×Q8—24no—
263C3×Q8—216no—
264C3×Q8C2×C4⋊C418yes—
265C2×C3×C4—28no—
266C2×C3×C4—24no—
267C2×C3×C4—22no—
268C2×C3×C4—24no—
269C2×C3×C4—24no—
270C2×C3×C4—22no—
271C2×C3×C4—216no—
272C2×C3×C4—22no—
273C2×C3×C4—24no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—44no—
277C2×C3×C4—48no—
278C2×C3×C4—44no—
279C2×C3×C4C2×C4⋊C414yes—
280C2×C3×C4C4×D414yes—
281C2×C3×C4C4×D412yes—
282C2×C3×C4#12012yes—
283C2×C3×C4C22⋊Q814yes—
284C2×C3×C4C22⋊Q812yes—
285C2×C3×C4#12314yes—
286C2×C3×C4#13714yes—
287C2×C3×C4C23×C411yescharacteristic
288C2×C3×C4C22×D411yescharacteristic
289C2×C3×C4C22×D411yescharacteristic
290C2×C3×C4C22×Q811yescharacteristic
291C2×C3×C4C2×#4111yescharacteristic
292C2×C3×C4C2×#4111yescharacteristic
293C2×Dic3—24no—
294C2×Dic3—42no—
295C2×Dic3—42no—
296C2×Dic3—41no—
297C2×Dic3—41no—
298C2×Dic3—42no—
299C2×Dic3—42no—
300C3⋊C8—24no—
301C3⋊C8—42no—
302C24C2×#6711yescharacteristic
303C2×Q8—28no—
304C2×Q8—24no—
305C2×Q8—28no—
306C2×Q8—24no—
307C2×Q8—28no—
308C2×Q8—24no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8Dic3⋊C4116yescomplemented
313C2×Q8C2×Dic614yes—
314C2×Q8C2×C4×D314yes—
315C2×Q8C2×#6714yes—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—42no—
326C22×C4—44no—
327C22×C4—62no—
328C22×C4—61no—
329C22×C4—61no—
330C22×C4C2×Dic612yes—
331C22×C4C2×C4×D312yes—
332C22×C4#23512yes—
333C22×C4#23512yes—
334C22×C4C2×#6712yes—
335C22×C4C2×#6712yes—
336C22×C4C2×#6712yes—
337C22×C4C2×#6711yescharacteristic
338C22×C4C2×#6712yes—
339C22×C4C23×D311yescharacteristic
340M16—68no—
341M16—124no—
342C2×C8—34no—
343C2×C8—62no—
344C2×C8—62no—
345C2×C8—62no—
346C2×C8—62no—
347C2×C8—62no—
348C4⋊C4—216no—
349C4⋊C4—24no—
350C4⋊C4—22no—
351C4⋊C4—34no—
352C4⋊C4—62no—
353C4⋊C4—61no—
354C4⋊C4—61no—
355C4⋊C4—61no—
356C4⋊C4—61no—
357C4⋊C4—124no—
358C4⋊C4—124no—
359C4⋊C4C2×#6714yes—
360C22⋊C4—64no—
361C22⋊C4—122no—
362C42—216no—
363C42—24no—
364C42—22no—
365C42—22no—
366C42—64no—
367C42—122no—
368C22×C3—24no—
369C22×C3—24no—
370C22×C3—22no—
371C22×C3—22no—
372C22×C3#38514yes—
373C22×C3C2×C4×D411yescharacteristic
374C22×C3C2×#12011yescharacteristic
375C22×C3C2×C22⋊Q811yescharacteristic
376C22×C3C2×#13712yes—
377C22×C3#57512yes—
378C3×C4—24no—
379C3×C4—28no—
380C3×C4—42no—
381C3×C4—44no—
382C3×C4#38514yes—
383C3×C4C22×C4⋊C411yescharacteristic
384C3×C4C2×C4×D411yescharacteristic
385C3×C4C2×C22⋊Q811yescharacteristic
386C3×C4C2×#12311yescharacteristic
387Dic3—44no—
388Dic3—81no—
389Dic3—81no—
390C23—22no—
391C23—22no—
392C23#92412yes—
393C23C4×#6712yes—
394C23#95312yes—
395C23C22×#6711yescharacteristic
396Q8—216no—
397Q8—24no—
398Q8—24no—
399Q8—24no—
400Q8C2×Dic3⋊C418yes—
401C2×C4—28no—
402C2×C4—216no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—44no—
413C2×C4—48no—
414C2×C4—44no—
415C2×C4—64no—
416C2×C4—122no—
417C2×C4—122no—
418C2×C4—122no—
419C2×C4—122no—
420C2×C4—121no—
421C2×C4—121no—
422C2×C4C2×Dic3⋊C414yes—
423C2×C4#92412yes—
424C2×C4#92414yes—
425C2×C4C4×#6712yes—
426C2×C4C4×#6714yes—
427C2×C4#92914yes—
428C2×C4#94214yes—
429C2×C4#95312yes—
430C2×C4C22×Dic611yescharacteristic
431C2×C4C22×C4×D311yescharacteristic
432C2×C4C2×#23511yescharacteristic
433C2×C4C2×#23511yescharacteristic
434C2×C4C22×#6711yescharacteristic
435C2×C4C22×#6711yescharacteristic
436C8—64no—
437C8—122no—
438C2×C3—24no—
439C2×C3C2×#38511yescharacteristic
440C2×C3C4×#13712yes—
441C2×C3#300411yescharacteristic
442C2×C3#300511yescharacteristic
443C2×C3#307812yes—
444C22—22no—
445C22—22no—
446C22—24no—
447C22—24no—
448C22#521014yes—
449C22C2×#92411yescharacteristic
450C22C2×C4×#6711yescharacteristic
451C22C2×#94212yes—
452C22#582212yes—
453C22C2×#95311yescharacteristic
454C4—28no—
455C4—24no—
456C4—42no—
457C4—44no—
458C4—124no—
459C4—241no—
460C4—241no—
461C4#521014yes—
462C4C22×Dic3⋊C411yescharacteristic
463C4C2×#92411yescharacteristic
464C4C2×C4×#6711yescharacteristic
465C4C2×#92911yescharacteristic
466C3#2087411yescharacteristic, complemented, Sylow
467C2—24no—
468C2C4×#94212yes—
469C2#8243112yes—
470C2C2×#521011yescharacteristic
471C2#8405911yescharacteristic
472C2#8408111yescharacteristic
subgroupG/H
centerC23C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC22×C3×C4×Q8C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
cosocleC22×C3×C4C24
hypercenterC22×C4×Q8D3
hypocenterC3#20874
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
6. #71629C2semidirect
7. #71628C2semidirect
25. #5225C4semidirect
99. #835C2×C4semidirect
142. C2×C3×Q8C4⋊C4semidirect
312. C2×Q8Dic3⋊C4semidirect
466. C3#20874semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×Q82
C2×#52252
C2×#49642
#820692
#716292
#716282
#715582
#715272
#208743