Finite Group Databasebeta

← #10848833

Subgroups — #10848833

Order 768.

1222 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×#5535C211yesmaximal, characteristic, complemented
2#82287C211yesmaximal, characteristic, complemented
3C2×#4712C211yesmaximal, characteristic, complemented
4C2×#4697C211yesmaximal, characteristic, complemented
5C3×#1750C211yesmaximal, characteristic, complemented
6#69962C212yesmaximal, complemented
7#69961C212yesmaximal, complemented
8#69827C211yesmaximal, characteristic, complemented
9#69772C211yesmaximal, characteristic, complemented
10#68673C212yesmaximal, complemented
11#68672C212yesmaximal, complemented
12#25611—31nomaximal, Sylow
13C2×C3×#117C2211yescharacteristic
14C2×#930C2212yes—
15C2×#930C2211yescharacteristic
16C2×#924C2211yescharacteristic
17C22×D24C2212yes—
18C22×#204C2212yes—
19#5535—24no—
20#5535—24no—
21C2×#894—22no—
22C2×#884—22no—
23C2×#875C2211yescharacteristic
24C2×#874C2211yescharacteristic
25C2×C4×D12C2212yes—
26C2×#870C2211yescharacteristic
27C2×C3×C22⋊C8C2212yes—
28C2×C3×#97C2211yescharacteristic
29C2×C3×C4×C8C2211yescharacteristic
30C2×#821C2212yes—
31C2×#821C2212yes—
32C2×#821C2212yes—
33C2×#816C2212yes—
34#4994—22no—
35#4961—22no—
36#4939C2211yescharacteristic
37#4935C2211yescharacteristic
38#4929C2211yescharacteristic
39#4712—24no—
40#4697—24no—
41C3×#336C2212yes—
42#4551C2212yes—
43#4551C2212yes—
44C2×#540—31no—
45C2×#496—31no—
46C2×#488—31no—
47#2620—31no—
48#2000—32no—
49#1999—32no—
50#1992—31no—
51#1957—31no—
52#1879—32no—
53#1877—32no—
54#1750D311yescharacteristic, complemented
55C23×C3×C4C2311yescharacteristic
56C22×#67—22no—
57C22×#67—21no—
58C22×D12—22no—
59C22×D12C2311yescharacteristic
60C22×D12C2312yes—
61C22×C4×D3—22no—
62C22×Dic6C2311yescharacteristic
63C22×C3×C8C2312yes—
64C22×C3×C8C2312yes—
65C3×#117—22no—
66C3×#117—22no—
67C2×C3×C4⋊C4D412yes—
68C2×C3×C22⋊C4D412yes—
69C2×C3×C42C2311yescharacteristic
70C2×C3×C42C2311yescharacteristic
71C2×#217—21no—
72#930—24no—
73#930—24no—
74#930—42no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—21no—
79C2×(C2×D3)⋊C4—22no—
80C2×C4⋊Dic3—22no—
81C2×C4⋊Dic3C2311yescharacteristic
82C2×C4⋊Dic3C2311yescharacteristic
83C2×C4⋊Dic3C2312yes—
84#924—42no—
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4—21no—
87C2×D24—24no—
88C2×D24—24no—
89C2×D24—42no—
90C2×#204—24no—
91C2×#204—24no—
92C2×#204—42no—
93#894—216no—
94#884—216no—
95#875—22no—
96#875—22no—
97#874—22no—
98#874D414yescomplemented
99C4×D12—24no—
100C4×D12—24no—
101#870—22no—
102#870D414yescomplemented
103C3×C22⋊C8—24no—
104C3×#97—22no—
105C3×C4×C8—22no—
106C3⋊#95—22no—
107#821—24no—
108#821—42no—
109#821—42no—
110#816—24no—
111C22×SD16—32no—
112C22×D8—32no—
113#540—64no—
114#540—64no—
115C2×#127—31no—
116C2×#125—31no—
117C2×#124—31no—
118C2×#123—62no—
119C2×C22⋊Q8—31no—
120C2×#121—32no—
121C2×#121—31no—
122C2×#121—62no—
123C2×C4×D4—32no—
124C2×#117C2×D311yescharacteristic
125#496—64no—
126#488—64no—
127C2×#103—32no—
128C2×#102—32no—
129C2×#102—32no—
130C2×#102—32no—
131C2×C22⋊C8C2×D312yes—
132C2×#97C2×D311yescharacteristic
133C2×C4×C8C2×D311yescharacteristic
134#396—62no—
135#394—62no—
136#391—31no—
137#390—31no—
138#384—31no—
139#340—32no—
140#340—32no—
141#336C2×D312yes—
142C24×C3C2×D411yescharacteristic
143C23×D3—22no—
144C23×D3—21no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2411yescharacteristic
153C2×#67—28no—
154C2×#67—42no—
155C2×#67—82no—
156C2×#67—82no—
157C2×#67—82no—
158C22×Dic3—22no—
159C22×Dic3—21no—
160C22×Dic3—21no—
161C22×Dic3—21no—
162C2×D12—22no—
163C2×D12—22no—
164C2×D12—22no—
165C2×D12—22no—
166C2×D12—22no—
167C2×D12—21no—
168C2×D12—21no—
169C2×D12—22no—
170C2×D12—22no—
171C2×D12—28no—
172C2×D12—82no—
173C2×D12—82no—
174C2×D12C2×D414yes—
175C2×C4×D3—28no—
176C2×C4×D3—82no—
177C2×C4×D3—82no—
178C2×Dic6—21no—
179C2×Dic6—21no—
180C2×Dic6—22no—
181C2×Dic6—22no—
182C2×C3×C8—22no—
183C2×C3×C8—22no—
184C2×C3×C8—22no—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C2×C3×C8#4114yes—
188C3×C4⋊C4—24no—
189C3×C22⋊C4—24no—
190C3×C42—21no—
191C3×C42—21no—
192C3×C42C2×D412yes—
193C3×C42C2×D412yes—
194#217—42no—
195(C2×D3)⋊C4—28no—
196(C2×D3)⋊C4—28no—
197(C2×D3)⋊C4—42no—
198(C2×D3)⋊C4—44no—
199(C2×D3)⋊C4—82no—
200C4⋊Dic3—22no—
201C4⋊Dic3—22no—
202C4⋊Dic3—22no—
203C4⋊Dic3—82no—
204C4⋊Dic3C2×D414yes—
205Dic3⋊C4—28no—
206Dic3⋊C4—42no—
207D24—48no—
208#204—48no—
209C22×Q8—31no—
210C22×D4—32no—
211C22×D4—31no—
212C22×D4—62no—
213C22×D4—61no—
214C22×D4—62no—
215C23×C4—62no—
216C23×C4C22×D311yescharacteristic
217C2×SD16—64no—
218C2×SD16—64no—
219C2×SD16—122no—
220C2×D8—64no—
221C2×D8—64no—
222C2×D8—122no—
223C22×C8C22×D312yes—
224C22×C8C22×D312yes—
225#127—34no—
226#127—62no—
227#125—34no—
228#125—62no—
229#124—62no—
230#124—62no—
231#123—616no—
232C22⋊Q8—122no—
233#121—616no—
234#121—64no—
235#121—64no—
236#121—122no—
237C4×D4—64no—
238C4×D4—64no—
239#117—22no—
240#117—22no—
241C2×C4⋊C4—32no—
242C2×C4⋊C4—31no—
243C2×C4⋊C4—31no—
244C2×C4⋊C4—62no—
245C2×C4⋊C4—61no—
246C2×C4⋊C4—62no—
247C2×C4⋊C4D1212yes—
248C2×C22⋊C4—62no—
249C2×C22⋊C4—62no—
250C2×C22⋊C4—61no—
251C2×C22⋊C4—61no—
252C2×C22⋊C4—62no—
253C2×C22⋊C4—62no—
254C2×C22⋊C4D1212yes—
255C2×C42C22×D311yescharacteristic
256C2×C42C22×D311yescharacteristic
257#103—64no—
258#102—64no—
259#102—122no—
260#102—122no—
261C22⋊C8—24no—
262#97—22no—
263C4×C8—22no—
264#95—62no—
265C23×C3—21no—
266C23×C3—21no—
267C23×C3—21no—
268C23×C3—22no—
269C23×C3—22no—
270C23×C3C22×D411yescharacteristic
271C22×D3—24no—
272C22×D3—42no—
273C22×D3—42no—
274C22×D3—42no—
275C22×D3—42no—
276C22×D3—42no—
277C22×D3—42no—
278C22×D3—41no—
279C22×D3—41no—
280C22×D3—41no—
281C22×D3—42no—
282C22×D3—42no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—21no—
286C2×C3×C4—22no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4#12014yes—
294C2×C3×C4C22×D411yescharacteristic
295C2×C3×C4C22×D411yescharacteristic
296C2×C3×C4C2×#4112yes—
297C2×C3×C4H(5,𝔽2)12yes—
298#67—88no—
299#67—84no—
300C2×Dic3—24no—
301C2×Dic3—42no—
302C2×Dic3—42no—
303C2×Dic3—41no—
304C2×Dic3—41no—
305C2×Dic3—41no—
306C2×Dic3—41no—
307C2×Dic3—41no—
308C2×Dic3—42no—
309C2×Dic3—42no—
310D12—24no—
311D12—24no—
312D12—42no—
313D12—42no—
314D12—42no—
315D12—42no—
316D12—88no—
317C4×D3—88no—
318Dic6—42no—
319Dic6—42no—
320C3×C8—24no—
321C3×C8—42no—
322C24—62no—
323C24—61no—
324C24C2×D1211yescharacteristic
325C2×Q8—62no—
326C2×Q8—61no—
327C2×Q8—61no—
328C2×Q8—62no—
329C2×D4—34no—
330C2×D4—68no—
331C2×D4—68no—
332C2×D4—62no—
333C2×D4—62no—
334C2×D4—62no—
335C2×D4—62no—
336C2×D4—61no—
337C2×D4—61no—
338C2×D4—62no—
339C2×D4—62no—
340C2×D4—62no—
341C2×D4—122no—
342C2×D4—242no—
343C2×D4—242no—
344C2×D4—242no—
345C2×D4—242no—
346C2×D4—242no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—68no—
352C22×C4—62no—
353C22×C4—61no—
354C22×C4—61no—
355C22×C4—61no—
356C22×C4—242no—
357C22×C4—242no—
358C22×C4C2×D1212yes—
359C22×C4C2×D1211yescharacteristic
360C22×C4C2×D1212yes—
361C22×C4C23×D311yescharacteristic
362SD16—128no—
363D8—128no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8#23514yes—
370C4⋊C4—24no—
371C4⋊C4—34no—
372C4⋊C4—68no—
373C4⋊C4—62no—
374C4⋊C4—62no—
375C4⋊C4—62no—
376C4⋊C4—122no—
377C4⋊C4—242no—
378C22⋊C4—24no—
379C22⋊C4—68no—
380C22⋊C4—68no—
381C22⋊C4—122no—
382C22⋊C4—122no—
383C22⋊C4—124no—
384C22⋊C4—242no—
385C42—21no—
386C42—21no—
387C42D3×D412yes—
388C42D3×D412yes—
389C22×C3—42no—
390C22×C3—41no—
391C22×C3—41no—
392C22×C3—41no—
393C22×C3—42no—
394C22×C3C2×#12011yescharacteristic
395C22×C3#53211yescharacteristic
396C22×C3#53511yescharacteristic
397C22×C3#57612yes—
398C22×C3#57712yes—
399C2×D3—44no—
400C2×D3—44no—
401C2×D3—44no—
402C2×D3—82no—
403C2×D3—82no—
404C2×D3—82no—
405C2×D3—82no—
406C2×D3—82no—
407C2×D3—81no—
408C2×D3—81no—
409C2×D3—81no—
410C2×D3—82no—
411C3×C4—24no—
412C3×C4—41no—
413C3×C4—42no—
414C3×C4#54612yes—
415C3×C4#55012yes—
416Dic3—44no—
417Dic3—81no—
418Dic3—81no—
419Dic3—81no—
420C23—21no—
421C23—21no—
422C23—22no—
423C23—22no—
424C23—21no—
425C23—64no—
426C23—122no—
427C23—122no—
428C23—122no—
429C23—122no—
430C23—122no—
431C23—122no—
432C23—121no—
433C23—121no—
434C23—122no—
435C23—122no—
436C23—121no—
437C23C22×D1211yescharacteristic
438Q8—122no—
439Q8—122no—
440D4—64no—
441D4—64no—
442D4—122no—
443D4—122no—
444D4—122no—
445D4—122no—
446D4—248no—
447D4—248no—
448D4—244no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—64no—
460C2×C4—122no—
461C2×C4—122no—
462C2×C4—121no—
463C2×C4—121no—
464C2×C4—122no—
465C2×C4—122no—
466C2×C4—121no—
467C2×C4—121no—
468C2×C4—121no—
469C2×C4—248no—
470C2×C4#88214yes—
471C2×C4C2×#23512yes—
472C2×C4C2×D3×D411yescharacteristic
473C2×C4C2×D3×D411yescharacteristic
474C2×C4#100912yes—
475C8—24no—
476C8—42no—
477C2×C3—81no—
478C2×C3#243111yescharacteristic
479C2×C3#301212yes—
480C2×C3#317912yes—
481C2×C3#322011yescharacteristic
482C2×C3#322111yescharacteristic
483D3—84no—
484D3—161no—
485C22—41no—
486C22—41no—
487C22—42no—
488C22—41no—
489C22—42no—
490C22—124no—
491C22—124no—
492C22—124no—
493C22—242no—
494C22—242no—
495C22—242no—
496C22—242no—
497C22—241no—
498C22—241no—
499C22—242no—
500C22—241no—
501C22—242no—
502C22#547711yescharacteristic
503C22C2×#88211yescharacteristic
504C22#552511yescharacteristic
505C22#575112yes—
506C22#575212yes—
507C4—24no—
508C4—42no—
509C4—41no—
510C4—124no—
511C4—241no—
512C4—241no—
513C4—241no—
514C4#555012yes—
515C4#555312yes—
516C3#2561111yescharacteristic, complemented, Sylow
517C2—81no—
518C2—244no—
519C2—481no—
520C2#7358611yescharacteristic
521C2#7857812yes—
522C2#7867212yes—
523C2#7939011yescharacteristic
524C2#7939711yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1750C2
socleC23×C3C22×D4
hypercenter#1750D3
hypocenterC3#25611
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1750

Complements

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

NH ≅ G/Ntype
1. C2×#5535C2semidirect
2. #82287C2semidirect
3. C2×#4712C2semidirect
4. C2×#4697C2semidirect
5. C3×#1750C2semidirect
6. #69962C2semidirect
7. #69961C2semidirect
8. #69827C2semidirect
9. #69772C2semidirect
10. #68673C2semidirect
11. #68672C2semidirect
54. #1750D3semidirect
98. #874D4semidirect
102. #870D4semidirect
516. C3#25611semidirect

Maximal subgroups

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

subgroupindex
C2×#55352
#822872
C2×#47122
C2×#46972
C3×#17502
#699622
#699612
#698272
#697722
#686732
#686722
#256113