Finite Group Databasebeta

← #10845924

Subgroups — #10845924

Order 768.

1018 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C8×D4C211yesmaximal, characteristic
2C2×#5116C212yesmaximal, complemented
3#82379C211yesmaximal, characteristic
4#82376C211yesmaximal, characteristic
5C2×#4849C211yesmaximal, characteristic, complemented
6#71972C212yesmaximal, complemented
7#70453C211yesmaximal, characteristic, complemented
8#69964C212yesmaximal, complemented
9#69937C212yesmaximal, complemented
10#69771C211yesmaximal, characteristic, complemented
11C4×#816C211yesmaximal, characteristic, complemented
12#25386—31nomaximal, Sylow
13C23×C3×C8C2212yes—
14C2×C3×C4×D4C2211yescharacteristic
15C2×#924C2212yes—
16C2×#924C2212yes—
17C2×#924C2212yes—
18C22×Dic12C2212yes—
19C2×#869C2211yescharacteristic
20C2×C4×Dic6C2211yescharacteristic
21C3×C8×D4—24no—
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C22⋊C8C2212yes—
24C2×C3×C4×C8C2211yescharacteristic
25#5221—21no—
26#5221—21no—
27#5212C2212yes—
28#5116—28no—
29C2×#818C2212yes—
30C2×#816C2212yes—
31C2×#816C2212yes—
32C2×#816C2211yescharacteristic
33C2×#816C2212yes—
34C2×#816C2211yescharacteristic
35#4975—21no—
36#4972—21no—
37#4949—22no—
38#4948—22no—
39#4935C2211yescharacteristic
40C4×C4⋊Dic3C2211yescharacteristic
41#4849—24no—
42#4551C2211yescharacteristic
43#4551C2212yes—
44#4551C2211yescharacteristic
45C2×#467—32no—
46C2×#462—31no—
47C2×C8×D4D311yescharacteristic
48#2542—31no—
49#2538—31no—
50#2036—32no—
51#2001—32no—
52#1956—31no—
53#1947—31no—
54#1892—32no—
55C4×#103—31no—
56C22×C3×D4D411yescharacteristic
57C23×C3×C4C2312yes—
58C22×Dic6C2312yes—
59C22×Dic6C2311yescharacteristic
60C22×C3×C8—24no—
61C22×C3×C8—24no—
62C22×C3×C8—42no—
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C22×C3×C8C2312yes—
66C3×C4×D4—22no—
67C3×C4×D4—22no—
68C2×C3×C4⋊C4D411yescharacteristic
69C2×C3×C22⋊C4D412yes—
70C2×C3×C42C2311yescharacteristic
71C2×#217—22no—
72C2×#217—22no—
73C2×#217—22no—
74C2×#217—22no—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3—21no—
77C2×C4⋊Dic3—21no—
78C2×C4⋊Dic3—21no—
79C2×C4⋊Dic3C2312yes—
80C2×C4⋊Dic3C2311yescharacteristic
81C2×C4⋊Dic3C2311yescharacteristic
82C2×C4⋊Dic3C2311yescharacteristic
83#924—24no—
84#924—24no—
85#924—24no—
86#924—42no—
87#924—42no—
88#924—42no—
89C2×Dic3⋊C4—21no—
90C2×Dic3⋊C4—22no—
91C2×Dic3⋊C4—22no—
92C2×Dic3⋊C4—22no—
93C2×C4×Dic3—21no—
94C2×C4×Dic3—21no—
95C2×Dic12—24no—
96C2×Dic12—24no—
97C2×Dic12—42no—
98#869—22no—
99#869—22no—
100C4×Dic6—22no—
101C4×Dic6—22no—
102C3×C4⋊C8D414yes—
103C3×C22⋊C8—24no—
104C3×C4×C8—22no—
105C3⋊#95—21no—
106C3⋊#95—22no—
107#818—24no—
108#816—24no—
109#816—24no—
110#816—22no—
111#816—42no—
112#816D414yescomplemented
113C22×Q16—32no—
114C23×C8C2×D312yes—
115C2×C4⋊Q8—31no—
116C2×C22⋊Q8—32no—
117C2×C22⋊Q8—32no—
118C2×C22⋊Q8—32no—
119C2×C4×Q8—31no—
120C2×C4×D4C2×D311yescharacteristic
121#467—68no—
122#462—64no—
123C8×D4—24no—
124C2×#107—32no—
125C2×C4⋊C8C2×D311yescharacteristic
126C2×#103—31no—
127C2×#103—32no—
128C2×#103—32no—
129C2×#103—31no—
130C2×#103—32no—
131C2×C22⋊C8C2×D312yes—
132C2×C4×C8C2×D311yescharacteristic
133#399—62no—
134#396—62no—
135#390—61no—
136#390—61no—
137#389—61no—
138#384—31no—
139#384—61no—
140#380—32no—
141C4×C4⋊C4—31no—
142#340—31no—
143#340—32no—
144#340—31no—
145C24×C3C2×D412yes—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4Q1614yescomplemented
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4C2×D412yes—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2411yescharacteristic
159C22×Dic3—21no—
160C22×Dic3—21no—
161C22×Dic3—21no—
162C22×Dic3—21no—
163C22×Dic3—22no—
164C22×Dic3—22no—
165C2×Dic6—21no—
166C2×Dic6—21no—
167C2×Dic6—21no—
168C2×Dic6—21no—
169C2×Dic6—21no—
170C2×Dic6—22no—
171C2×Dic6—22no—
172C2×Dic6—22no—
173C2×Dic6—22no—
174C2×Dic6—22no—
175C2×Dic6—22no—
176C2×Dic6C2×D412yes—
177C2×C3×C8—22no—
178C2×C3×C8—21no—
179C2×C3×C8—22no—
180C2×C3×C8—21no—
181C2×C3×C8—21no—
182C2×C3×C8—21no—
183C2×C3×C8—21no—
184C2×C3×C8—48no—
185C2×C3×C8C2×D412yes—
186C2×C3×C8C2×D414yes—
187C3×C4⋊C4—22no—
188C3×C22⋊C4—24no—
189C3×C42C2×D412yes—
190C3×C42#4112yes—
191#217—44no—
192#217—44no—
193#217—44no—
194#217—82no—
195C4⋊Dic3—21no—
196C4⋊Dic3—21no—
197C4⋊Dic3—21no—
198C4⋊Dic3—22no—
199C4⋊Dic3—21no—
200C4⋊Dic3—22no—
201C4⋊Dic3—42no—
202C4⋊Dic3—42no—
203C4⋊Dic3—42no—
204C4⋊Dic3—81no—
205C4⋊Dic3C2×D412yes—
206C4⋊Dic3#4112yes—
207Dic3⋊C4—44no—
208Dic3⋊C4—44no—
209Dic3⋊C4—44no—
210Dic3⋊C4—42no—
211C4×Dic3—42no—
212C4×Dic3—42no—
213Dic12—48no—
214C22×Q8—31no—
215C22×Q8—32no—
216C22×D4D1211yescharacteristic
217C23×C4C22×D312yes—
218C2×Q16—64no—
219C2×Q16—64no—
220C2×Q16—122no—
221C22×C8—24no—
222C22×C8—24no—
223C22×C8—42no—
224C22×C8C22×D311yescharacteristic
225C22×C8C22×D311yescharacteristic
226C22×C8C22×D312yes—
227C4⋊Q8—62no—
228C4⋊Q8—62no—
229C22⋊Q8—64no—
230C22⋊Q8—64no—
231C22⋊Q8—64no—
232C22⋊Q8—122no—
233C22⋊Q8—122no—
234C22⋊Q8—122no—
235C4×Q8—62no—
236C4×Q8—62no—
237C4×D4—22no—
238C4×D4—22no—
239C2×C4⋊C4—31no—
240C2×C4⋊C4—31no—
241C2×C4⋊C4—31no—
242C2×C4⋊C4—32no—
243C2×C4⋊C4—61no—
244C2×C4⋊C4—62no—
245C2×C4⋊C4—62no—
246C2×C4⋊C4—62no—
247C2×C4⋊C4—61no—
248C2×C4⋊C4—61no—
249C2×C4⋊C4—61no—
250C2×C4⋊C4—61no—
251C2×C4⋊C4D1211yescharacteristic
252C2×C22⋊C4—62no—
253C2×C22⋊C4—62no—
254C2×C22⋊C4—62no—
255C2×C22⋊C4—62no—
256C2×C22⋊C4D1212yes—
257C2×C42—61no—
258C2×C42—61no—
259C2×C42C22×D311yescharacteristic
260#107—64no—
261C4⋊C8#6714yes—
262#103—34no—
263#103—62no—
264#103—64no—
265#103—64no—
266#103—122no—
267C22⋊C8—24no—
268C4×C8—22no—
269#95—61no—
270#95—62no—
271C23×C3—22no—
272C23×C3—22no—
273C23×C3—22no—
274C23×C3—22no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3C2×Q1614yes—
278C23×C3C22×D411yescharacteristic
279C3×D4—28no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4#12112yes—
290C2×C3×C4#12112yes—
291C2×C3×C4C2×Q1612yes—
292C2×C3×C4#13512yes—
293C2×C3×C4C22×D411yescharacteristic
294C2×C3×C4C22×D411yescharacteristic
295C2×C3×C4C2×#4111yescharacteristic
296C2×C3×C4C2×#4111yescharacteristic
297C2×C3×C4H(5,𝔽2)11yescharacteristic
298C2×C3×C4H(5,𝔽2)11yescharacteristic
299C2×Dic3—22no—
300C2×Dic3—22no—
301C2×Dic3—22no—
302C2×Dic3—41no—
303C2×Dic3—41no—
304C2×Dic3—41no—
305C2×Dic3—41no—
306C2×Dic3—41no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317Dic6—22no—
318Dic6—22no—
319Dic6—41no—
320Dic6—41no—
321Dic6—42no—
322Dic6—42no—
323Dic6—42no—
324Dic6—42no—
325C3×C8—24no—
326C3×C8—22no—
327C3×C8—41no—
328C24C2×D1212yes—
329C2×Q8—32no—
330C2×Q8—61no—
331C2×Q8—61no—
332C2×Q8—62no—
333C2×Q8—62no—
334C2×Q8—61no—
335C2×Q8—62no—
336C2×Q8—61no—
337C2×Q8—62no—
338C2×Q8—62no—
339C2×Q8—61no—
340C2×Q8—62no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4Dic1214yescomplemented
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—61no—
351C22×C4—61no—
352C22×C4—61no—
353C22×C4—61no—
354C22×C4—62no—
355C22×C4—62no—
356C22×C4C2×D1212yes—
357C22×C4C2×D1211yescharacteristic
358C22×C4C2×D1211yescharacteristic
359C22×C4C23×D311yescharacteristic
360Q16—128no—
361C2×C8—22no—
362C2×C8—21no—
363C2×C8—22no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—48no—
369C2×C8D3×D412yes—
370C2×C8C2×#6714yes—
371C4⋊C4—22no—
372C4⋊C4—32no—
373C4⋊C4—32no—
374C4⋊C4—61no—
375C4⋊C4—61no—
376C4⋊C4—61no—
377C4⋊C4—62no—
378C4⋊C4—62no—
379C4⋊C4—61no—
380C4⋊C4—124no—
381C4⋊C4—124no—
382C4⋊C4—124no—
383C4⋊C4—122no—
384C4⋊C4—122no—
385C4⋊C4—122no—
386C4⋊C4—122no—
387C4⋊C4—241no—
388C22⋊C4—24no—
389C22⋊C4—124no—
390C22⋊C4—124no—
391C22⋊C4—124no—
392C22⋊C4—242no—
393C42—122no—
394C42—122no—
395C42#23512yes—
396C42C2×#6712yes—
397C22×C3—24no—
398C22×C3—24no—
399C22×C3—24no—
400C22×C3—42no—
401C22×C3—42no—
402C22×C3—42no—
403C22×C3—42no—
404C22×C3C2×#12111yescharacteristic
405C22×C3D4211yescharacteristic
406C22×C3#54611yescharacteristic
407C22×C3C22×Q1611yescharacteristic
408C22×C3C2×#13511yescharacteristic
409C22×C3#57711yescharacteristic
410C22×C3#57811yescharacteristic
411C3×C4—22no—
412C3×C4—42no—
413C3×C4#46714yescomplemented
414C3×C4#53211yescharacteristic
415C3×C4#54011yescharacteristic
416C3×C4D4211yescharacteristic
417C3×C4#55011yescharacteristic
418Dic3—42no—
419Dic3—42no—
420Dic3—42no—
421Dic3—82no—
422Dic3—82no—
423Dic3—81no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23C2×Dic1214yes—
431C23C22×D1211yescharacteristic
432Q8—62no—
433Q8—62no—
434Q8—121no—
435Q8—121no—
436Q8—122no—
437Q8—122no—
438Q8—122no—
439Q8—122no—
440D4—28no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—62no—
451C2×C4—62no—
452C2×C4—62no—
453C2×C4—121no—
454C2×C4—121no—
455C2×C4—121no—
456C2×C4—121no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4—122no—
460C2×C4—121no—
461C2×C4—122no—
462C2×C4—121no—
463C2×C4—121no—
464C2×C4—122no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4#90412yes—
469C2×C4C2×Dic1212yes—
470C2×C4#93012yes—
471C2×C4#93012yes—
472C2×C4C2×#23511yescharacteristic
473C2×C4C2×D3×D411yescharacteristic
474C2×C4#100411yescharacteristic
475C2×C4D3×#4111yescharacteristic
476C2×C4#100911yescharacteristic
477C2×C4C22×#6711yescharacteristic
478C8—22no—
479C8—24no—
480C8—41no—
481C2×C3—44no—
482C2×C3#242811yescharacteristic
483C2×C3C2×#46711yescharacteristic
484C2×C3#306311yescharacteristic
485C2×C3#328411yescharacteristic
486C2×C3D4×Q1611yescharacteristic
487C2×C3#329911yescharacteristic
488C2×C3#330211yescharacteristic
489C22—24no—
490C22—24no—
491C22—24no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22D4×D1211yescharacteristic
497C22#555411yescharacteristic
498C22C2×#90411yescharacteristic
499C22C22×Dic1211yescharacteristic
500C22#575111yescharacteristic
501C22#575311yescharacteristic
502C22C2×#93011yescharacteristic
503C4—22no—
504C4—42no—
505C4—122no—
506C4—122no—
507C4—122no—
508C4—242no—
509C4—241no—
510C4—242no—
511C4#511614yescomplemented
512C4#554811yescharacteristic
513C4#556211yescharacteristic
514C4D4×#6711yescharacteristic
515C4#583411yescharacteristic
516C3#2538611yescharacteristic, complemented, Sylow
517C2—44no—
518C2#7943711yescharacteristic
519C2#7944211yescharacteristic
520C2D4×Dic1211yescharacteristic
521C2#7944811yescharacteristic
522C2#8240911yescharacteristic
523C2C2×#511611yescharacteristic
524C2#8380811yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×C8×D4C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenterC2×C8×D4D3
hypocenterC3#25386
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
2. C2×#5116C2semidirect
5. C2×#4849C2semidirect
6. #71972C2semidirect
7. #70453C2semidirect
8. #69964C2semidirect
9. #69937C2semidirect
10. #69771C2semidirect
11. C4×#816C2semidirect
112. #816D4semidirect
148. C2×C3×D4Q16semidirect
343. C2×D4Dic12semidirect
413. C3×C4#467semidirect
511. C4#5116semidirect
516. C3#25386semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C8×D42
C2×#51162
#823792
#823762
C2×#48492
#719722
#704532
#699642
#699372
#697712
C4×#8162
#253863