Finite Group Databasebeta

← #10844715

Subgroups — #10844715

Order 768.

886 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C8×Q8C211yesmaximal, characteristic, complemented
2#82482C211yesmaximal, characteristic, complemented
3#82476C211yesmaximal, characteristic
4C2×#4848C211yesmaximal, characteristic
5#70483C212yesmaximal, complemented
6#70482C212yesmaximal, complemented
7#70454C211yesmaximal, characteristic, complemented
8#69774C212yesmaximal, complemented
9#69772C211yesmaximal, characteristic
10#69768C212yesmaximal, complemented
11C4×#816C211yesmaximal, characteristic, complemented
12#25295—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#947—21no—
15C22×#204C2212yes—
16C2×#874C2211yescharacteristic
17C2×C4×Dic6C2211yescharacteristic
18C3×C8×Q8—24no—
19C2×C3×C4⋊C8C2212yes—
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×C4×C8C2211yescharacteristic
22C2×C3×C4×C8C2212yes—
23#5229—21no—
24C2×#821C2212yes—
25C2×#821C2212yes—
26C2×#821C2212yes—
27C2×#817C2212yes—
28C2×#816C2211yescharacteristic
29C2×#816C2211yescharacteristic
30#4991—22no—
31#4988—21no—
32#4982—22no—
33#4972—21no—
34#4939C2212yes—
35#4939C2212yes—
36#4935C2211yescharacteristic
37C4×C4⋊Dic3C2211yescharacteristic
38#4929C2212yes—
39#4928C2212yes—
40#4848—24no—
41#4551C2212yes—
42#4551C2211yescharacteristic
43#4551C2211yescharacteristic
44C2×#460—31no—
45C2×C8×Q8D311yescharacteristic, complemented
46#2582—31no—
47#2548—31no—
48#2065—32no—
49#2055—32no—
50#1968—32no—
51#1957—31no—
52#1948—31no—
53#1934—32no—
54C4×#103—31no—
55C22×C3×Q8D411yescharacteristic
56C22×D12—21no—
57C22×D12—21no—
58C22×D12C2312yes—
59C22×Dic6C2311yescharacteristic
60C22×C3×C8C2311yescharacteristic
61C22×C3×C8C2312yes—
62C22×C3×C8C2311yescharacteristic
63C3×C4×Q8—22no—
64C3×C4×Q8—22no—
65C2×C3×C4⋊C4D411yescharacteristic
66C2×C3×C4⋊C4D412yes—
67C2×C3×C42C2312yes—
68C2×C3×C42C2311yescharacteristic
69#947—44no—
70C2×(C2×D3)⋊C4—22no—
71C2×(C2×D3)⋊C4—22no—
72C2×(C2×D3)⋊C4—22no—
73C2×(C2×D3)⋊C4—22no—
74C2×C4⋊Dic3—21no—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3C2311yescharacteristic
77C2×C4⋊Dic3C2312yes—
78C2×C4⋊Dic3C2311yescharacteristic
79C2×C4⋊Dic3C2311yescharacteristic
80C2×Dic3⋊C4—21no—
81C2×Dic3⋊C4—22no—
82C2×C4×Dic3—21no—
83C2×C4×Dic3—21no—
84C2×#204—24no—
85C2×#204—24no—
86C2×#204—42no—
87#874—22no—
88#874—22no—
89C4×Dic6—22no—
90C4×Dic6—22no—
91C3×C4⋊C8—24no—
92C3×C4⋊C8D414yes—
93C3×C4×C8—24no—
94C3×C4×C8—22no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97C3⋊#95—22no—
98C3⋊#95—21no—
99#821—24no—
100#821—24no—
101#821—42no—
102#817—42no—
103#816—22no—
104#816—22no—
105C22×SD16—32no—
106C2×#127—31no—
107C2×#124—61no—
108C2×C4×Q8—31no—
109C2×C4×Q8C2×D311yescharacteristic
110#460—64no—
111C8×Q8—24no—
112C2×#106—32no—
113C2×C4⋊C8C2×D312yes—
114C2×C4⋊C8C2×D311yescharacteristic
115C2×#103—31no—
116C2×#103—31no—
117C2×#102—32no—
118C2×#102—32no—
119C2×#102—32no—
120C2×C4×C8C2×D312yes—
121C2×C4×C8C2×D311yescharacteristic
122#398—62no—
123#394—62no—
124#391—32no—
125#391—61no—
126#390—32no—
127#390—32no—
128#390—61no—
129#389—32no—
130#389—61no—
131#384—31no—
132C4×C4⋊C4—31no—
133#340—32no—
134#340—31no—
135#340—31no—
136C23×D3—22no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8SD1614yescomplemented
140C22×C3×C4C2×D412yes—
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D412yes—
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2411yescharacteristic
145C22×Dic3—21no—
146C22×Dic3—22no—
147C22×Dic3—21no—
148C22×Dic3—21no—
149C22×Dic3—21no—
150C2×D12—22no—
151C2×D12—22no—
152C2×D12—22no—
153C2×D12—22no—
154C2×D12—22no—
155C2×D12—22no—
156C2×D12—42no—
157C2×D12—42no—
158C2×D12—42no—
159C2×D12—42no—
160C2×D12—81no—
161C2×D12—81no—
162C2×Dic6—21no—
163C2×Dic6—21no—
164C2×Dic6—21no—
165C2×Dic6—21no—
166C2×Dic6—21no—
167C2×Dic6#4112yes—
168C2×C3×C8—21no—
169C2×C3×C8—22no—
170C2×C3×C8—22no—
171C2×C3×C8—21no—
172C2×C3×C8—21no—
173C2×C3×C8—21no—
174C2×C3×C8—21no—
175C2×C3×C8C2×D414yes—
176C2×C3×C8#4112yes—
177C3×C4⋊C4—24no—
178C3×C4⋊C4—22no—
179C3×C42—22no—
180C3×C42—22no—
181C3×C42C2×D412yes—
182C3×C42#4112yes—
183(C2×D3)⋊C4—44no—
184(C2×D3)⋊C4—44no—
185(C2×D3)⋊C4—82no—
186(C2×D3)⋊C4—82no—
187C4⋊Dic3—21no—
188C4⋊Dic3—21no—
189C4⋊Dic3—21no—
190C4⋊Dic3—22no—
191C4⋊Dic3—21no—
192C4⋊Dic3—21no—
193C4⋊Dic3—22no—
194C4⋊Dic3—21no—
195C4⋊Dic3—42no—
196C4⋊Dic3—81no—
197Dic3⋊C4—42no—
198Dic3⋊C4—82no—
199C4×Dic3—42no—
200C4×Dic3—42no—
201#204—48no—
202C22×Q8—31no—
203C22×Q8D1211yescharacteristic
204C22×D4—32no—
205C22×D4—61no—
206C22×D4—61no—
207C2×SD16—64no—
208C2×SD16—64no—
209C2×SD16—122no—
210C22×C8C22×D311yescharacteristic
211C22×C8C22×D311yescharacteristic
212C22×C8C22×D312yes—
213#127—62no—
214#127—62no—
215#124—124no—
216C4×Q8—22no—
217C4×Q8—22no—
218C4×Q8—62no—
219C4×Q8—62no—
220C2×C4⋊C4—31no—
221C2×C4⋊C4—31no—
222C2×C4⋊C4—31no—
223C2×C4⋊C4—32no—
224C2×C4⋊C4—61no—
225C2×C4⋊C4—61no—
226C2×C4⋊C4—61no—
227C2×C4⋊C4—62no—
228C2×C4⋊C4D1211yescharacteristic
229C2×C4⋊C4D1212yes—
230C2×C22⋊C4—62no—
231C2×C22⋊C4—62no—
232C2×C22⋊C4—62no—
233C2×C22⋊C4—62no—
234C2×C42—61no—
235C2×C42—61no—
236C2×C42C22×D312yes—
237C2×C42C22×D311yescharacteristic
238#106—122no—
239C4⋊C8—24no—
240C4⋊C8#6714yes—
241#103—62no—
242#103—62no—
243#102—64no—
244#102—64no—
245#102—122no—
246C4×C8—24no—
247C4×C8—22no—
248#95—61no—
249#95—62no—
250#95—62no—
251#95—62no—
252C23×C3C22×D411yescharacteristic
253C22×D3—42no—
254C22×D3—42no—
255C22×D3—42no—
256C22×D3—42no—
257C22×D3—42no—
258C22×D3—42no—
259C22×D3—42no—
260C3×Q8—28no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4#12112yes—
269C2×C3×C4#12112yes—
270C2×C3×C4C2×SD1614yes—
271C2×C3×C4C2×SD1612yes—
272C2×C3×C4#13512yes—
273C2×C3×C4C22×D411yescharacteristic
274C2×C3×C4C2×#4111yescharacteristic
275C2×C3×C4C2×#4111yescharacteristic
276C2×C3×C4C2×#4111yescharacteristic
277C2×C3×C4H(5,𝔽2)11yescharacteristic
278C2×C3×C42−1+411yescharacteristic
279C2×Dic3—22no—
280C2×Dic3—41no—
281C2×Dic3—41no—
282C2×Dic3—41no—
283C2×Dic3—41no—
284C2×Dic3—41no—
285C2×Dic3—42no—
286C2×Dic3—41no—
287C2×Dic3—41no—
288C2×Dic3—42no—
289C2×Dic3—41no—
290C2×Dic3—42no—
291C2×Dic3—41no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294D12—42no—
295D12—42no—
296D12—42no—
297D12—42no—
298D12—84no—
299D12—84no—
300Dic6—22no—
301Dic6—22no—
302Dic6—41no—
303Dic6—41no—
304C3×C8—24no—
305C3×C8—22no—
306C3×C8—41no—
307C24—62no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—32no—
311C2×Q8—61no—
312C2×Q8—61no—
313C2×Q8—61no—
314C2×Q8—61no—
315C2×Q8—61no—
316C2×Q8#20414yescomplemented
317C2×D4—62no—
318C2×D4—62no—
319C2×D4—62no—
320C2×D4—62no—
321C2×D4—62no—
322C2×D4—62no—
323C2×D4—122no—
324C2×D4—122no—
325C2×D4—122no—
326C2×D4—122no—
327C2×D4—241no—
328C2×D4—241no—
329C22×C4—61no—
330C22×C4—62no—
331C22×C4—61no—
332C22×C4—61no—
333C22×C4—61no—
334C22×C4C2×D1212yes—
335C22×C4C2×D1212yes—
336C22×C4C2×D1211yescharacteristic
337C22×C4C2×D1211yescharacteristic
338C22×C4C23×D311yescharacteristic
339SD16—128no—
340C2×C8—21no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8#23912yes—
348C2×C8C2×#6714yes—
349C4⋊C4—24no—
350C4⋊C4—22no—
351C4⋊C4—61no—
352C4⋊C4—62no—
353C4⋊C4—61no—
354C4⋊C4—61no—
355C4⋊C4—61no—
356C4⋊C4—62no—
357C4⋊C4—61no—
358C4⋊C4—61no—
359C4⋊C4—122no—
360C4⋊C4—122no—
361C4⋊C4—242no—
362C4⋊C4—241no—
363C22⋊C4—124no—
364C22⋊C4—124no—
365C22⋊C4—242no—
366C22⋊C4—242no—
367C42—22no—
368C42—22no—
369C42—122no—
370C42—122no—
371C42#23512yes—
372C42C2×#6712yes—
373C22×C3C2×#12111yescharacteristic
374C22×C3#54811yescharacteristic
375C22×C3#55011yescharacteristic
376C22×C3C22×SD1611yescharacteristic
377C22×C3C2×#13511yescharacteristic
378C22×C3#57611yescharacteristic
379C22×C3#57711yescharacteristic
380C2×D3—82no—
381C2×D3—82no—
382C2×D3—82no—
383C2×D3—82no—
384C2×D3—82no—
385C2×D3—82no—
386C2×D3—82no—
387C3×C4—24no—
388C3×C4—22no—
389C3×C4—42no—
390C3×C4#46514yes—
391C3×C4#52911yescharacteristic
392C3×C4#54211yescharacteristic
393C3×C4#54611yescharacteristic
394C3×C4#54811yescharacteristic
395Dic3—42no—
396Dic3—81no—
397Dic3—81no—
398Dic3—82no—
399Dic3—81no—
400C23—122no—
401C23—122no—
402C23—122no—
403C23—122no—
404C23—122no—
405C23—122no—
406C23—122no—
407C23C22×D1211yescharacteristic
408Q8—28no—
409Q8—62no—
410Q8—62no—
411Q8—121no—
412Q8—121no—
413D4—122no—
414D4—122no—
415D4—122no—
416D4—122no—
417D4—244no—
418D4—244no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—62no—
427C2×C4—121no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—122no—
432C2×C4—121no—
433C2×C4—121no—
434C2×C4—121no—
435C2×C4—122no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4—121no—
440C2×C4—122no—
441C2×C4C2×#20414yes—
442C2×C4C2×#20412yes—
443C2×C4#90412yes—
444C2×C4#93012yes—
445C2×C4#93012yes—
446C2×C4C2×#23511yescharacteristic
447C2×C4C2×#23911yescharacteristic
448C2×C4#100711yescharacteristic
449C2×C4D3×#4111yescharacteristic
450C2×C4#100911yescharacteristic
451C2×C4C22×#6711yescharacteristic
452C8—22no—
453C8—24no—
454C8—41no—
455C2×C3#242611yescharacteristic
456C2×C3C2×#46511yescharacteristic
457C2×C3#306211yescharacteristic
458C2×C3#336211yescharacteristic
459C2×C3#336511yescharacteristic
460C2×C3#339211yescharacteristic
461C2×C3#339611yescharacteristic
462D3—162no—
463C22—242no—
464C22—242no—
465C22—242no—
466C22—242no—
467C22—242no—
468C22—242no—
469C22—242no—
470C22#557611yescharacteristic
471C22#557711yescharacteristic
472C22C22×#20411yescharacteristic
473C22C2×#90411yescharacteristic
474C22#575111yescharacteristic
475C22#575211yescharacteristic
476C22C2×#93011yescharacteristic
477C4—24no—
478C4—22no—
479C4—42no—
480C4—122no—
481C4—241no—
482C4—242no—
483C4—241no—
484C4—241no—
485C4#511414yes—
486C4#558011yescharacteristic
487C4#558211yescharacteristic
488C4#581711yescharacteristic
489C4#582911yescharacteristic
490C3#2529511yescharacteristic, complemented, Sylow
491C2—482no—
492C2#7991111yescharacteristic
493C2#7991511yescharacteristic
494C2#7993211yescharacteristic
495C2#7993511yescharacteristic
496C2#8249511yescharacteristic
497C2C2×#511411yescharacteristic
498C2#8380211yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×C8×Q8C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenterC2×C8×Q8D3
hypocenterC3#25295
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C8×Q8C2semidirect
2. #82482C2semidirect
5. #70483C2semidirect
6. #70482C2semidirect
7. #70454C2semidirect
8. #69774C2semidirect
10. #69768C2semidirect
11. C4×#816C2semidirect
45. C2×C8×Q8D3semidirect
139. C2×C3×Q8SD16semidirect
316. C2×Q8#204semidirect
490. C3#25295semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C8×Q82
#824822
#824762
C2×#48482
#704832
#704822
#704542
#697742
#697722
#697682
C4×#8162
#252953