Finite Group Databasebeta

← #10848831

Subgroups — #10848831

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×#5719C211yesmaximal, characteristic, complemented
2C2×#5074C211yesmaximal, characteristic, complemented
3C2×#4697C211yesmaximal, characteristic, complemented
4#74370C211yesmaximal, characteristic, complemented
5C3×#1957C211yesmaximal, characteristic, complemented
6#70205C212yesmaximal, complemented
7#70204C212yesmaximal, complemented
8#70172C212yesmaximal, complemented
9#70171C212yesmaximal, complemented
10#69753C211yesmaximal, characteristic, complemented
11#69583C211yesmaximal, characteristic, complemented
12#25611—31nomaximal, Sylow
13C2×C3×#127C2211yescharacteristic
14C2×#939—22no—
15C2×#938C2212yes—
16C2×#935—22no—
17C2×D4×Dic3C2212yes—
18C22×#214C2212yes—
19C22×#213C2212yes—
20#5719—24no—
21#5719—24no—
22C2×#897C2211yescharacteristic
23C2×#895C2211yescharacteristic
24C2×#875C2211yescharacteristic
25C2×#872C2211yescharacteristic
26C2×#870C2211yescharacteristic
27C2×C3×#102C2212yes—
28C2×C3×#102C2212yes—
29C2×C3×C4×C8C2211yescharacteristic
30C3×#384C2211yescharacteristic
31C2×#821C2212yes—
32C2×(C2×D3)⋊C8C2212yes—
33C2×#816C2212yes—
34#5074—24no—
35#4994—22no—
36#4991—22no—
37#4988C2211yescharacteristic
38#4973C2211yescharacteristic
39C2×#803C2211yescharacteristic
40#4697—24no—
41#4550C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44C2×#540—31no—
45C2×#496—31no—
46C2×#488—31no—
47#2620—31no—
48#2000—32no—
49#1999—32no—
50#1992—31no—
51#1957D311yescharacteristic, complemented
52#1879—32no—
53#1877—32no—
54#1750—31no—
55C22×C3×D4—22no—
56C22×C3×D4C2312yes—
57C22×#67—22no—
58C23×Dic3—22no—
59C22×D12—21no—
60C22×D12C2311yescharacteristic
61C22×C4×D3C2311yescharacteristic
62C22×Dic6C2311yescharacteristic
63C22×C3×C8C2312yes—
64C3×#127—22no—
65C3×#127D414yescomplemented
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C42C2311yescharacteristic
70C2×#217—22no—
71C2×#217—22no—
72#939—216no—
73#938—24no—
74#938—24no—
75#935—216no—
76D4×Dic3—24no—
77D4×Dic3—24no—
78C2×#214—24no—
79C2×#214—24no—
80C2×#214—42no—
81C2×#213—24no—
82C2×#213—24no—
83C2×#213—42no—
84C2×(C2×D3)⋊C4—21no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4—21no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4D412yes—
89C2×C4⋊Dic3—21no—
90C2×C4⋊Dic3C2312yes—
91C2×Dic3⋊C4—22no—
92C2×Dic3⋊C4D412yes—
93C2×C4×Dic3C2311yescharacteristic
94C22×C3⋊C8C2312yes—
95#897—42no—
96#895—42no—
97#875—22no—
98#875—22no—
99#872—22no—
100#872—22no—
101#870—22no—
102#870D414yescomplemented
103C3×#102—42no—
104C3×#102—42no—
105C3×C4×C8—22no—
106C3⋊#95—22no—
107#821—24no—
108(C2×D3)⋊C8—24no—
109#816—24no—
110#803—22no—
111C22×SD16—32no—
112C22×D8—32no—
113#540—64no—
114#540—64no—
115C2×#127C2×D311yescharacteristic
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×#117—31no—
125#496—64no—
126#488—64no—
127C2×#103—32no—
128C2×#102—32no—
129C2×#102C2×D312yes—
130C2×#102C2×D312yes—
131C2×C22⋊C8—32no—
132C2×#97—31no—
133C2×C4×C8C2×D311yescharacteristic
134#396—62no—
135#394—62no—
136#391—31no—
137#390—31no—
138#384C2×D311yescharacteristic
139#340—32no—
140#340—32no—
141#336—32no—
142C24×C3—22no—
143C23×D3—21no—
144C23×D3C2×D411yescharacteristic
145C2×C3×D4—28no—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4—82no—
152C2×C3×D4—82no—
153C2×C3×D4C2×D414yes—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2411yescharacteristic
158C2×#67—28no—
159C2×#67—82no—
160C2×#67—82no—
161C22×Dic3—22no—
162C22×Dic3—28no—
163C22×Dic3—21no—
164C22×Dic3—82no—
165C22×Dic3—82no—
166C22×Dic3C2×D412yes—
167C22×Dic3C2×D411yescharacteristic
168C2×D12—21no—
169C2×D12—21no—
170C2×D12—22no—
171C2×D12—22no—
172C2×D12—42no—
173C2×D12—82no—
174C2×C4×D3—21no—
175C2×C4×D3—21no—
176C2×C4×D3—22no—
177C2×C4×D3—22no—
178C2×Dic6—21no—
179C2×Dic6—21no—
180C2×Dic6—22no—
181C2×Dic6—22no—
182C2×C3×C8—22no—
183C2×C3×C8—22no—
184C2×C3×C8—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—82no—
188C3×C42C2×D412yes—
189C3×C42C2×D412yes—
190#217—28no—
191#217—28no—
192#214—48no—
193#213—48no—
194(C2×D3)⋊C4—24no—
195(C2×D3)⋊C4—42no—
196(C2×D3)⋊C4—42no—
197(C2×D3)⋊C4—44no—
198(C2×D3)⋊C4—82no—
199C4⋊Dic3—22no—
200C4⋊Dic3—42no—
201C4⋊Dic3C2×D414yes—
202Dic3⋊C4—28no—
203Dic3⋊C4—24no—
204C4×Dic3—21no—
205C4×Dic3—21no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8#4114yes—
209C22×Q8—31no—
210C22×D4—22no—
211C22×D4—31no—
212C22×D4—61no—
213C22×D4—62no—
214C22×D4C22×D312yes—
215C23×C4—31no—
216C23×C4—62no—
217C2×SD16—64no—
218C2×SD16—64no—
219C2×SD16—122no—
220C2×D8—64no—
221C2×D8—64no—
222C2×D8—122no—
223C22×C8—32no—
224C22×C8C22×D312yes—
225#127—22no—
226#127D1214yescomplemented
227#125—34no—
228#125—62no—
229#124—62no—
230#124—62no—
231#123—616no—
232C22⋊Q8—122no—
233#121—64no—
234#121—616no—
235#121—64no—
236#121—122no—
237C4×D4—64no—
238C4×D4—64no—
239#117—62no—
240#117—62no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—32no—
243C2×C4⋊C4—32no—
244C2×C4⋊C4—62no—
245C2×C4⋊C4—61no—
246C2×C4⋊C4C22×D311yescharacteristic
247C2×C4⋊C4C22×D311yescharacteristic
248C2×C22⋊C4—32no—
249C2×C22⋊C4—62no—
250C2×C22⋊C4—62no—
251C2×C22⋊C4—61no—
252C2×C22⋊C4—61no—
253C2×C22⋊C4—62no—
254C2×C22⋊C4—62no—
255C2×C42—31no—
256C2×C42C22×D311yescharacteristic
257#103—64no—
258#102—42no—
259#102—42no—
260#102—64no—
261C22⋊C8—64no—
262#97—62no—
263C4×C8—22no—
264#95—62no—
265C23×C3—24no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3C22×D411yescharacteristic
273C22×D3—21no—
274C22×D3—21no—
275C22×D3—21no—
276C22×D3—22no—
277C22×D3—22no—
278C22×D3—41no—
279C22×D3—41no—
280C22×D3—41no—
281C22×D3—42no—
282C22×D3—42no—
283C3×D4—24no—
284C3×D4—24no—
285C3×D4—42no—
286C3×D4—42no—
287C3×D4—88no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4#12014yes—
295C2×C3×C4C22×D411yescharacteristic
296C2×C3×C4C22×D411yescharacteristic
297C2×C3×C4C2×#4112yes—
298C2×C3×C4H(5,𝔽2)12yes—
299#67—88no—
300C2×Dic3—22no—
301C2×Dic3—24no—
302C2×Dic3—21no—
303C2×Dic3—21no—
304C2×Dic3—21no—
305C2×Dic3—22no—
306C2×Dic3—22no—
307C2×Dic3—42no—
308C2×Dic3—42no—
309C2×Dic3—41no—
310C2×Dic3—41no—
311C2×Dic3—41no—
312C2×Dic3—88no—
313D12—42no—
314D12—42no—
315D12—84no—
316C4×D3—42no—
317C4×D3—42no—
318Dic6—42no—
319Dic6—42no—
320C3×C8—42no—
321C3⋊C8—24no—
322C24—22no—
323C24—31no—
324C24—61no—
325C2×Q8—61no—
326C2×Q8—61no—
327C2×Q8—62no—
328C2×Q8—62no—
329C2×D4—28no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—68no—
336C2×D4—61no—
337C2×D4—61no—
338C2×D4—62no—
339C2×D4—62no—
340C2×D4—82no—
341C2×D4—82no—
342C2×D4—122no—
343C2×D4—242no—
344C2×D4—242no—
345C2×D4—242no—
346C2×D4C2×D1214yes—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—32no—
350C22×C4—31no—
351C22×C4—62no—
352C22×C4—68no—
353C22×C4—61no—
354C22×C4—61no—
355C22×C4—62no—
356C22×C4—62no—
357C22×C4—61no—
358C22×C4—242no—
359C22×C4—242no—
360C22×C4D3×D412yes—
361C22×C4C23×D311yescharacteristic
362SD16—128no—
363D8—128no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—34no—
368C2×C8—62no—
369C2×C8—62no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—34no—
373C4⋊C4—64no—
374C4⋊C4—68no—
375C4⋊C4—62no—
376C4⋊C4—82no—
377C4⋊C4—122no—
378C22⋊C4—68no—
379C22⋊C4—68no—
380C22⋊C4—64no—
381C22⋊C4—122no—
382C22⋊C4—122no—
383C22⋊C4—124no—
384C22⋊C4—242no—
385C42—61no—
386C42—61no—
387C42C2×D1212yes—
388C42D3×D412yes—
389C22×C3—44no—
390C22×C3—44no—
391C22×C3—44no—
392C22×C3—82no—
393C22×C3—82no—
394C22×C3—82no—
395C22×C3—82no—
396C22×C3C2×#12011yescharacteristic
397C22×C3#53211yescharacteristic
398C22×C3#53511yescharacteristic
399C22×C3#57612yes—
400C22×C3#57712yes—
401C2×D3—42no—
402C2×D3—41no—
403C2×D3—41no—
404C2×D3—41no—
405C2×D3—42no—
406C2×D3—82no—
407C2×D3—81no—
408C2×D3—81no—
409C2×D3—81no—
410C2×D3—82no—
411C3×C4—24no—
412C3×C4—81no—
413C3×C4—81no—
414C3×C4#54612yes—
415C3×C4#55012yes—
416Dic3—41no—
417Dic3—42no—
418Dic3—44no—
419Dic3—81no—
420C23—24no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—61no—
428C23—61no—
429C23—61no—
430C23—62no—
431C23—62no—
432C23—121no—
433C23—121no—
434C23—121no—
435C23—122no—
436C23—122no—
437C23C2×D3×D411yescharacteristic
438Q8—122no—
439Q8—122no—
440D4—24no—
441D4—24no—
442D4—42no—
443D4—42no—
444D4—88no—
445D4—122no—
446D4—122no—
447D4—244no—
448D4—248no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—61no—
456C2×C4—61no—
457C2×C4—62no—
458C2×C4—61no—
459C2×C4—64no—
460C2×C4—62no—
461C2×C4—62no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4—122no—
466C2×C4—121no—
467C2×C4—121no—
468C2×C4—121no—
469C2×C4—248no—
470C2×C4#88214yes—
471C2×C4C22×D1211yescharacteristic
472C2×C4C2×D3×D411yescharacteristic
473C2×C4#100412yes—
474C2×C4D3×#4112yes—
475C8—42no—
476C8—64no—
477C2×C3—84no—
478C2×C3#243111yescharacteristic
479C2×C3#301212yes—
480C2×C3#317912yes—
481C2×C3#322011yescharacteristic
482C2×C3#322111yescharacteristic
483D3—81no—
484D3—161no—
485C22—44no—
486C22—44no—
487C22—44no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22—82no—
492C22—121no—
493C22—121no—
494C22—121no—
495C22—122no—
496C22—122no—
497C22—241no—
498C22—241no—
499C22—241no—
500C22—242no—
501C22—242no—
502C22C2×#88211yescharacteristic
503C22#566811yescharacteristic
504C22#571511yescharacteristic
505C22#575712yes—
506C22#576212yes—
507C4—24no—
508C4—81no—
509C4—81no—
510C4—121no—
511C4—122no—
512C4—124no—
513C4—241no—
514C4#555412yes—
515C4#559912yes—
516C3#2561111yescharacteristic, complemented, Sylow
517C2—84no—
518C2—241no—
519C2—481no—
520C2#7631311yescharacteristic
521C2#7874712yes—
522C2#8155812yes—
523C2#8176811yescharacteristic
524C2#8178711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1957C2
socleC23×C3C22×D4
hypercenter#1957D3
hypocenterC3#25611
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1957

Complements

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

NH ≅ G/Ntype
1. C2×#5719C2semidirect
2. C2×#5074C2semidirect
3. C2×#4697C2semidirect
4. #74370C2semidirect
5. C3×#1957C2semidirect
6. #70205C2semidirect
7. #70204C2semidirect
8. #70172C2semidirect
9. #70171C2semidirect
10. #69753C2semidirect
11. #69583C2semidirect
51. #1957D3semidirect
65. C3×#127D4semidirect
102. #870D4semidirect
226. #127D12semidirect
516. C3#25611semidirect

Maximal subgroups

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

subgroupindex
C2×#57192
C2×#50742
C2×#46972
#743702
C3×#19572
#702052
#702042
#701722
#701712
#697532
#695832
#256113