Finite Group Databasebeta

← #10831720

Subgroups — #10831720

Order 768.

1112 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5579C211yesmaximal, characteristic, complemented
2#82501C211yesmaximal, characteristic, complemented
3C2×#4846C211yesmaximal, characteristic, complemented
4C2×#4711C211yesmaximal, characteristic
5C3×#1985C211yesmaximal, characteristic, complemented
6#70480C211yesmaximal, characteristic, complemented
7#69828C211yesmaximal, characteristic, complemented
8#68679C212yesmaximal, complemented
9#68673C212yesmaximal
10#68670C212yesmaximal, complemented
11#68634C212yesmaximal, complemented
12#24339—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#946—21no—
15C22×D24C2211yescharacteristic
16#5579—24no—
17#5579—24no—
18C2×#896—22no—
19C2×D3×C4⋊C4—21no—
20C2×C4×D12C2212yes—
21C2×C4×D12C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C2×#870C2212yes—
24C2×#869C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×#97C2211yescharacteristic
27#5231—21no—
28C2×#821C2211yescharacteristic
29C2×#821C2212yes—
30C2×#821C2212yes—
31C2×#821C2211yescharacteristic
32C2×#818C2212yes—
33C2×#817C2211yescharacteristic
34#4993—21no—
35#4989—22no—
36#4939C2212yes—
37#4936C2211yescharacteristic
38#4928C2212yes—
39#4846—24no—
40#4711—24no—
41C3×#336C2212yes—
42C3×#336C2212yes—
43#4551C2212yes—
44#4551C2212yes—
45#4551C2212yes—
46C2×#554—31no—
47C2×#474—31no—
48C2×#442—31no—
49#2580—31no—
50#2059—32no—
51#2025—32no—
52#1993—31no—
53#1985D311yescharacteristic, complemented
54#1879—32no—
55#1862—32no—
56#1804—31no—
57C22×C3×Q8D411yescharacteristic
58C22×D12C2311yescharacteristic
59C22×D12C2311yescharacteristic
60C22×C4×D3—21no—
61C22×C4×D3—22no—
62C22×C4×D3—21no—
63C22×Dic6—21no—
64C22×C3×C8C2311yescharacteristic
65C22×C3×C8C2312yes—
66C22×C3×C8C2311yescharacteristic
67C3×C4×Q8—22no—
68C3×C4×Q8—22no—
69C2×C3×C4⋊C4D412yes—
70C2×C3×C4⋊C4D411yescharacteristic
71C2×C3×C42C2312yes—
72C2×C3×C42C2311yescharacteristic
73#946—28no—
74C2×(C2×D3)⋊C4—21no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—21no—
78C2×(C2×D3)⋊C4—22no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3C2311yescharacteristic
81C2×C4⋊Dic3C2312yes—
82C2×C4⋊Dic3C2312yes—
83C2×C4⋊Dic3C2311yescharacteristic
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4—22no—
87C2×D24—22no—
88C2×D24—22no—
89C2×D24Q814yescomplemented
90#896—216no—
91D3×C4⋊C4—28no—
92C4×D12—24no—
93C4×D12—22no—
94C4×D12—22no—
95C4×D12—22no—
96C4×D12—24no—
97C4×D12D414yes—
98#870—24no—
99#870—24no—
100#869—22no—
101#869D414yes—
102C3×C4⋊C8—22no—
103C3×#97—22no—
104C3⋊#95—22no—
105C3⋊#95—21no—
106C3⋊#95—21no—
107#821—22no—
108#821—24no—
109#821—22no—
110#821—24no—
111#818—24no—
112#817—22no—
113C22×D8—31no—
114#554—64no—
115#554—64no—
116C2×C4⋊Q8—31no—
117C2×#125—32no—
118C2×C22⋊Q8—61no—
119C2×C22⋊Q8—62no—
120C2×C4×Q8C2×D311yescharacteristic
121C2×C4×D4—31no—
122C2×C4×D4—31no—
123C2×C4×D4—32no—
124C22×C4⋊C4—61no—
125#474—64no—
126#442—64no—
127C2×#107—32no—
128C2×#106—31no—
129C2×C4⋊C8C2×D311yescharacteristic
130C2×#102—32no—
131C2×#102—31no—
132C2×#102—32no—
133C2×#102—31no—
134C2×#97C2×D311yescharacteristic
135#399—61no—
136#393—61no—
137#390—32no—
138#389—32no—
139#385—62no—
140#382—31no—
141#340—32no—
142#340—32no—
143#340—32no—
144#336C2×D312yes—
145#336C2×D312yes—
146C23×D3—21no—
147C23×D3—21no—
148C2×C3×Q8—22no—
149C2×C3×Q8—42no—
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2411yescharacteristic
155C22×Dic3—21no—
156C22×Dic3—22no—
157C22×Dic3—22no—
158C22×Dic3—21no—
159C2×D12—21no—
160C2×D12—21no—
161C2×D12—21no—
162C2×D12—21no—
163C2×D12—21no—
164C2×D12—21no—
165C2×D12—21no—
166C2×D12—21no—
167C2×D12C2×D412yes—
168C2×D12C2×Q812yes—
169C2×D12C2×Q812yes—
170C2×D12#4112yes—
171C2×C4×D3—24no—
172C2×C4×D3—28no—
173C2×C4×D3—44no—
174C2×C4×D3—42no—
175C2×C4×D3—82no—
176C2×C4×D3—82no—
177C2×C4×D3—82no—
178C2×Dic6—24no—
179C2×Dic6—82no—
180C2×C3×C8—21no—
181C2×C3×C8—21no—
182C2×C3×C8—22no—
183C2×C3×C8—21no—
184C2×C3×C8—22no—
185C2×C3×C8—21no—
186C2×C3×C8—21no—
187C2×C3×C8—22no—
188C2×C3×C8C2×Q812yes—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—24no—
191C3×C42—22no—
192C3×C42—22no—
193C3×C42C2×D412yes—
194C3×C42C2×D412yes—
195(C2×D3)⋊C4—28no—
196(C2×D3)⋊C4—24no—
197(C2×D3)⋊C4—44no—
198(C2×D3)⋊C4—42no—
199(C2×D3)⋊C4—82no—
200C4⋊Dic3—24no—
201C4⋊Dic3—22no—
202C4⋊Dic3—21no—
203C4⋊Dic3—21no—
204C4⋊Dic3—21no—
205C4⋊Dic3—22no—
206C4⋊Dic3—22no—
207C4⋊Dic3C2×D412yes—
208C4⋊Dic3C2×D414yes—
209Dic3⋊C4—28no—
210Dic3⋊C4—28no—
211Dic3⋊C4—82no—
212D24—28no—
213C22×Q8—61no—
214C22×Q8D1211yescharacteristic
215C22×D4—31no—
216C22×D4—31no—
217C23×C4—61no—
218C23×C4—62no—
219C23×C4—61no—
220C2×D8—34no—
221C2×D8—62no—
222C2×D8—62no—
223C22×C8C22×D311yescharacteristic
224C22×C8C22×D312yes—
225C22×C8C22×D311yescharacteristic
226C4⋊Q8—34no—
227C4⋊Q8—62no—
228#125—64no—
229#125—64no—
230C22⋊Q8—616no—
231C22⋊Q8—68no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×D4—34no—
235C4×D4—64no—
236C4×D4—64no—
237C4×D4—62no—
238C4×D4—62no—
239C4×D4—62no—
240C2×C4⋊C4—31no—
241C2×C4⋊C4—32no—
242C2×C4⋊C4—31no—
243C2×C4⋊C4—32no—
244C2×C4⋊C4—68no—
245C2×C4⋊C4—62no—
246C2×C4⋊C4—61no—
247C2×C4⋊C4—62no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4D1211yescharacteristic
250C2×C4⋊C4D1212yes—
251C2×C22⋊C4—62no—
252C2×C22⋊C4—61no—
253C2×C22⋊C4—61no—
254C2×C22⋊C4—62no—
255C2×C22⋊C4—62no—
256C2×C42C22×D312yes—
257C2×C42C22×D311yescharacteristic
258#107—64no—
259#106—62no—
260C4⋊C8—22no—
261#102—62no—
262#102—64no—
263#102—64no—
264#102—62no—
265#97—22no—
266#95—61no—
267#95—61no—
268#95—62no—
269C23×C3C22×D411yescharacteristic
270C22×D3—22no—
271C22×D3—22no—
272C22×D3—22no—
273C22×D3—22no—
274C22×D3—42no—
275C22×D3—42no—
276C22×D3—41no—
277C22×D3—41no—
278C22×D3—42no—
279C22×D3—42no—
280C3×Q8—44no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4#12012yes—
290C2×C3×C4#12012yes—
291C2×C3×C4C22×D411yescharacteristic
292C2×C3×C4C22×D411yescharacteristic
293C2×C3×C4C22×Q811yescharacteristic
294C2×C3×C4C2×#4111yescharacteristic
295C2×C3×C42−1+411yescharacteristic
296C2×C3×C42−1+411yescharacteristic
297C2×Dic3—22no—
298C2×Dic3—24no—
299C2×Dic3—42no—
300C2×Dic3—42no—
301C2×Dic3—42no—
302C2×Dic3—41no—
303C2×Dic3—42no—
304C2×Dic3—42no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307D12—22no—
308D12—22no—
309D12—22no—
310D12—22no—
311D12—22no—
312D12—41no—
313D12—41no—
314D12C22⋊Q814yes—
315C4×D3—48no—
316C4×D3—88no—
317C4×D3—84no—
318Dic6—84no—
319C3×C8—22no—
320C3×C8—41no—
321C3×C8—42no—
322C24—61no—
323C24—61no—
324C2×Q8—22no—
325C2×Q8—42no—
326C2×Q8—64no—
327C2×Q8—242no—
328C2×D4—32no—
329C2×D4—32no—
330C2×D4—32no—
331C2×D4—32no—
332C2×D4—61no—
333C2×D4—61no—
334C2×D4—61no—
335C2×D4—61no—
336C2×D4—61no—
337C2×D4—61no—
338C2×D4—61no—
339C2×D4—61no—
340C22×C4—68no—
341C22×C4—61no—
342C22×C4—62no—
343C22×C4—64no—
344C22×C4—62no—
345C22×C4—61no—
346C22×C4—124no—
347C22×C4—122no—
348C22×C4—242no—
349C22×C4—242no—
350C22×C4—242no—
351C22×C4C2×D1212yes—
352C22×C4C2×D1212yes—
353C22×C4C2×D1211yescharacteristic
354C22×C4C2×D1211yescharacteristic
355C22×C4C23×D311yescharacteristic
356D8—68no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—22no—
361C2×C8—21no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—21no—
365C2×C8D3×Q812yes—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—32no—
369C4⋊C4—34no—
370C4⋊C4—68no—
371C4⋊C4—68no—
372C4⋊C4—64no—
373C4⋊C4—61no—
374C4⋊C4—62no—
375C4⋊C4—61no—
376C4⋊C4—61no—
377C4⋊C4—62no—
378C4⋊C4—62no—
379C4⋊C4—242no—
380C22⋊C4—68no—
381C22⋊C4—64no—
382C22⋊C4—124no—
383C22⋊C4—122no—
384C22⋊C4—242no—
385C42—22no—
386C42—22no—
387C42D3×D412yes—
388C42D3×D412yes—
389C22×C3C2×#12011yescharacteristic
390C22×C3#54811yescharacteristic
391C22×C3D4×Q811yescharacteristic
392C22×C3#57512yes—
393C22×C3#57612yes—
394C2×D3—24no—
395C2×D3—44no—
396C2×D3—44no—
397C2×D3—42no—
398C2×D3—42no—
399C2×D3—44no—
400C2×D3—42no—
401C2×D3—82no—
402C2×D3—81no—
403C2×D3—81no—
404C3×C4—22no—
405C3×C4—22no—
406C3×C4—42no—
407C3×C4—42no—
408C3×C4C2×C22⋊Q811yescharacteristic
409C3×C4#53611yescharacteristic
410C3×C4#54811yescharacteristic
411C3×C4D4×Q811yescharacteristic
412Dic3—42no—
413Dic3—44no—
414Dic3—82no—
415Dic3—81no—
416C23—62no—
417C23—62no—
418C23—62no—
419C23—62no—
420C23—122no—
421C23—122no—
422C23—121no—
423C23—122no—
424C23—122no—
425C23—121no—
426C23C22×D1211yescharacteristic
427Q8—44no—
428Q8—244no—
429D4—34no—
430D4—62no—
431D4—62no—
432D4—62no—
433D4—62no—
434D4—62no—
435D4—121no—
436D4—121no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—62no—
446C2×C4—64no—
447C2×C4—128no—
448C2×C4—122no—
449C2×C4—122no—
450C2×C4—122no—
451C2×C4—121no—
452C2×C4—122no—
453C2×C4—122no—
454C2×C4—122no—
455C2×C4—122no—
456C2×C4—244no—
457C2×C4—248no—
458C2×C4#88212yes—
459C2×C4#88212yes—
460C2×C4C2×D3×D411yescharacteristic
461C2×C4C2×D3×D411yescharacteristic
462C2×C4C2×D3×Q811yescharacteristic
463C2×C4#100711yescharacteristic
464C2×C4D3×#4111yescharacteristic
465C2×C4#101011yescharacteristic
466C8—22no—
467C8—42no—
468C8—41no—
469C2×C3#240911yescharacteristic
470C2×C3#300511yescharacteristic
471C2×C3#300811yescharacteristic
472C2×C3#337012yes—
473C2×C3#338912yes—
474D3—44no—
475D3—82no—
476C22—64no—
477C22—122no—
478C22—122no—
479C22—124no—
480C22—124no—
481C22—122no—
482C22—124no—
483C22—241no—
484C22—241no—
485C22—242no—
486C22C2×#88211yescharacteristic
487C22Q8×D1211yescharacteristic
488C22#557611yescharacteristic
489C22#574812yes—
490C22#575212yes—
491C4—22no—
492C4—22no—
493C4—42no—
494C4—42no—
495C4—122no—
496C4—124no—
497C4—241no—
498C4—242no—
499C4D3×C22⋊Q811yescharacteristic
500C4#562811yescharacteristic
501C4#567711yescharacteristic
502C4#572711yescharacteristic
503C3#2433911yescharacteristic, complemented, Sylow
504C2—124no—
505C2—242no—
506C2#7661911yescharacteristic
507C2#7863811yescharacteristic
508C2#7867011yescharacteristic
509C2#7997112yes—
510C2#7997912yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1985C2
socleC23×C3C22×D4
prime radicalC22×D24C22
hypercenter#1985D3
hypocenterC3#24339
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #1985

Complements

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

NH ≅ G/Ntype
1. C2×#5579C2semidirect
2. #82501C2semidirect
3. C2×#4846C2semidirect
5. C3×#1985C2semidirect
6. #70480C2semidirect
7. #69828C2semidirect
8. #68679C2semidirect
10. #68670C2semidirect
11. #68634C2semidirect
53. #1985D3semidirect
89. C2×D24Q8semidirect
503. C3#24339semidirect

Maximal subgroups

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

subgroupindex
C2×#55792
#825012
C2×#48462
C2×#47112
C3×#19852
#704802
#698282
#686792
#686732
#686702
#686342
#243393