Finite Group Databasebeta

← #10822736

Subgroups — #10822736

Order 768.

1610 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5805C211yesmaximal, characteristic
2C22×C3×#102C211yesmaximal, characteristic, complemented
3C2×#5215C211yesmaximal, characteristic, complemented
4C2×#5110C211yesmaximal, characteristic, complemented
5C2×#5106C211yesmaximal, characteristic, complemented
6#82186C211yesmaximal, characteristic, complemented
7#71558C211yesmaximal, characteristic, complemented
8#70176C214yesmaximal, complemented
9#70175C214yesmaximal, complemented
10#23486—31nomaximal, Sylow
11C23×C3×D4C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C23×C3×C8C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C2×#953—22no—
16#5805—24no—
17#5805—24no—
18C2×#940—21no—
19C2×#938C2214yes—
20C2×#936—21no—
21C2×#926C2211yescharacteristic
22C2×#926C2211yescharacteristic
23C22×#208C2211yescharacteristic
24C2×C3×#102—24no—
25C2×C3×#102C2212yes—
26C2×C3×#102C2212yes—
27#5215—24no—
28C2×#832C2214yes—
29#5213—22no—
30#5110—24no—
31C2×(C2×D3)⋊C8C2212yescomplemented
32C2×(C2×D3)⋊C8C2212yescomplemented
33#5106—24no—
34C2×#818C2212yescomplemented
35C2×#817C2212yescomplemented
36#4988—21no—
37#4987C2214yescomplemented
38#4973—21no—
39#4529C2214yescomplemented
40C2×#535—31no—
41C2×#427—31no—
42C2×#418—31no—
43C22×#102D311yescharacteristic, complemented
44C2×#408—31no—
45#2311—31no—
46#2034—34no—
47#2033—34no—
48#1736—31no—
49C25×C3—21no—
50C22×C3×D4—22no—
51C22×C3×D4—21no—
52C22×C3×D4—22no—
53C22×C3×D4—22no—
54C22×C3×D4—22no—
55C22×C3×D4—22no—
56C22×C3×D4D414yes—
57C22×C3×D4C2312yes—
58C23×C3×C4—21no—
59C23×C3×C4C2311yescharacteristic
60C22×#67—24no—
61C22×#67D412yes—
62C2×#235—22no—
63C2×#235—22no—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235—22no—
67C2×#235—22no—
68C22×D12D411yescharacteristic
69C22×C4×D3C2312yes—
70C22×Dic6D411yescharacteristic
71C22×C3×C8—22no—
72C22×C3×C8—22no—
73C22×C3×C8D414yescomplemented
74C22×C3×C8C2311yescharacteristic
75C22×C3×C8C2311yescharacteristic
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4C2312yes—
80#953—216no—
81C2×#217—21no—
82C2×#217—24no—
83C2×#217—21no—
84#940—28no—
85#938—28no—
86#938—28no—
87#936—28no—
88C2×(C2×D3)⋊C4—24no—
89#926—22no—
90#926—22no—
91#926—22no—
92#926D414yes—
93C2×C4⋊Dic3C2312yes—
94C2×C4⋊Dic3C2312yes—
95C2×C4×Dic3—21no—
96C2×C4×Dic3—21no—
97C2×#208—22no—
98C2×#208—22no—
99C2×#208—22no—
100C22×C3⋊C8C2312yes—
101C3×#102—24no—
102C3×#102—24no—
103#832—28no—
104C3⋊#95—24no—
105(C2×D3)⋊C8—24no—
106(C2×D3)⋊C8—24no—
107#818—24no—
108#817—24no—
109C22×#41—31no—
110C23×D4C2×D311yescharacteristic
111C22×M16—31no—
112C23×C8C2×D311yescharacteristic
113#535—64no—
114#535—64no—
115C2×#127—61no—
116C2×#124—61no—
117C2×#121—34no—
118C2×#120—62no—
119C2×#117—31no—
120C2×#117—31no—
121C22×C4⋊C4C2×D311yescharacteristic
122#427—64no—
123C2×#107—32no—
124C2×#106—32no—
125#418—64no—
126C2×#102—24no—
127C2×#102—34no—
128C2×#102C2×D312yes—
129C2×#102C2×D312yes—
130#408—64no—
131C2×C22⋊C8—32no—
132C2×C22⋊C8—32no—
133#391—61no—
134#390—61no—
135#386—62no—
136#380—34no—
137#340—34no—
138C24×C3—22no—
139C24×C3—24no—
140C24×C3—42no—
141C24×C3—42no—
142C24×C3—82no—
143C24×C3—82no—
144C24×C3C2×D411yescharacteristic
145C23×D3C2×D412yes—
146C2×C3×D4—24no—
147C2×C3×D4—28no—
148C2×C3×D4—28no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—42no—
155C2×C3×D4—44no—
156C2×C3×D4—42no—
157C2×C3×D4—44no—
158C2×C3×D4C2×D414yes—
159C22×C3×C4—22no—
160C22×C3×C4—42no—
161C22×C3×C4—82no—
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D412yes—
166C22×C3×C4#4112yes—
167C22×C3×C4#4112yes—
168C22×C3×C4#4112yes—
169C22×C3×C4C2411yescharacteristic
170C2×#67—24no—
171C2×#67—24no—
172C2×#67—24no—
173C2×#67—216no—
174C2×#67—84no—
175C2×#67—84no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3C2×D412yes—
179#235—44no—
180#235—44no—
181#235—44no—
182#235—44no—
183C2×D12—22no—
184C2×D12—22no—
185C2×D12—22no—
186C2×C4×D3—22no—
187C2×C4×D3—22no—
188C2×C4×D3—22no—
189C2×C4×D3—22no—
190C2×C4×D3—22no—
191C2×C4×D3—22no—
192C2×Dic6—22no—
193C2×Dic6—22no—
194C2×Dic6—22no—
195C2×C3×C8—28no—
196C2×C3×C8—21no—
197C2×C3×C8—21no—
198C2×C3×C8—21no—
199C2×C3×C8—21no—
200C2×C3×C8C2×D412yes—
201C2×C3×C8C2×D412yes—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—42no—
205#217—216no—
206#217—24no—
207#217—42no—
208(C2×D3)⋊C4—84no—
209C4⋊Dic3—22no—
210C4⋊Dic3—22no—
211C4⋊Dic3—22no—
212C4⋊Dic3C2×D414yes—
213C4×Dic3—24no—
214C4×Dic3—42no—
215#208—44no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C25—21no—
220C2×#41—62no—
221C2×#41—62no—
222C2×#41—62no—
223C2×#41—62no—
224C2×#41—62no—
225C2×#41—62no—
226C22×Q8—31no—
227C22×D4—22no—
228C22×D4—21no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—32no—
234C22×D4—31no—
235C22×D4—64no—
236C22×D4D1214yes—
237C22×D4C22×D312yes—
238C23×C4—21no—
239C23×C4—32no—
240C23×C4C22×D311yescharacteristic
241C2×M16—62no—
242C2×M16—62no—
243C2×M16—62no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—32no—
247C22×C8#6714yescomplemented
248C22×C8C22×D311yescharacteristic
249C22×C8C22×D311yescharacteristic
250#127—68no—
251#124—68no—
252#121—68no—
253#121—68no—
254#120—616no—
255#117—34no—
256#117—62no—
257#117—62no—
258#117—62no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—32no—
263C2×C4⋊C4—32no—
264C2×C4⋊C4C22×D312yes—
265C2×C22⋊C4—64no—
266C2×C22⋊C4—61no—
267C2×C22⋊C4—61no—
268C2×C22⋊C4—64no—
269C2×C42—61no—
270C2×C42—61no—
271#107—64no—
272#106—64no—
273#102—24no—
274#102—24no—
275#102—68no—
276C22⋊C8—64no—
277C22⋊C8—64no—
278#95—64no—
279C23×C3—22no—
280C23×C3—22no—
281C23×C3—24no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—44no—
285C23×C3—44no—
286C23×C3—48no—
287C23×C3—88no—
288C23×C3—88no—
289C23×C3—84no—
290C23×C3—84no—
291C23×C3#12012yes—
292C23×C3#12112yes—
293C23×C3#12312yes—
294C23×C3C22×D411yescharacteristic
295C22×D3—22no—
296C22×D3—22no—
297C22×D3—22no—
298C22×D3—24no—
299C22×D3—24no—
300C3×D4—24no—
301C3×D4—24no—
302C3×D4—42no—
303C3×D4—42no—
304C3×D4—48no—
305C2×C3×C4—42no—
306C2×C3×C4—44no—
307C2×C3×C4—84no—
308C2×C3×C4#12012yes—
309C2×C3×C4#12114yes—
310C2×C3×C4#12112yes—
311C2×C3×C4#12114yes—
312C2×C3×C4#12312yes—
313C2×C3×C4#12314yes—
314C2×C3×C4#12414yes—
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C22×D411yescharacteristic
317C2×C3×C4C22×D411yescharacteristic
318C2×C3×C4C2×#4111yescharacteristic
319C2×C3×C4C2×#4111yescharacteristic
320C2×C3×C4C2×#4111yescharacteristic
321#67—48no—
322#67—816no—
323C2×Dic3—24no—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332D12—44no—
333C4×D3—42no—
334C4×D3—42no—
335C4×D3—42no—
336C4×D3—42no—
337Dic6—44no—
338C3×C8—22no—
339C3×C8—22no—
340C3⋊C8—42no—
341C24—22no—
342C24—24no—
343C24—32no—
344C24—42no—
345C24—42no—
346C24—82no—
347C24—82no—
348C24D3×D411yescharacteristic
349#41—124no—
350#41—124no—
351#41—124no—
352#41—124no—
353C2×Q8—62no—
354C2×Q8—62no—
355C2×Q8—62no—
356C2×D4—24no—
357C2×D4—28no—
358C2×D4—28no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—42no—
365C2×D4—44no—
366C2×D4—42no—
367C2×D4—44no—
368C2×D4—64no—
369C2×D4—64no—
370C2×D4—64no—
371C2×D4—616no—
372C2×D4—62no—
373C2×D4—62no—
374C2×D4—62no—
375C2×D4—244no—
376C2×D4—244no—
377C2×D4C2×D1214yes—
378C22×C4—22no—
379C22×C4—32no—
380C22×C4—42no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4—62no—
389C22×C4—82no—
390C22×C4C2×D1212yes—
391C22×C4#23512yes—
392C22×C4D3×D411yescharacteristic
393C22×C4D3×D412yes—
394C22×C4#23712yes—
395C22×C4#23712yes—
396C22×C4C2×#6712yes—
397C22×C4C23×D311yescharacteristic
398M16—124no—
399C2×C8—28no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—62no—
405C2×C8—62no—
406C2×C8—62no—
407C2×C8C2×#6712yes—
408C2×C8C2×#6712yes—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—34no—
412C4⋊C4—42no—
413C4⋊C4—62no—
414C4⋊C4—62no—
415C4⋊C4—62no—
416C22⋊C4—616no—
417C22⋊C4—64no—
418C22⋊C4—122no—
419C22⋊C4—244no—
420C42—64no—
421C42—122no—
422C22×C3—24no—
423C22×C3—24no—
424C22×C3—22no—
425C22×C3—22no—
426C22×C3—44no—
427C22×C3—48no—
428C22×C3—816no—
429C22×C3—84no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3#39414yes—
433C22×C3C2×#12011yescharacteristic
434C22×C3C2×#12111yescharacteristic
435C22×C3C2×#12311yescharacteristic
436C22×C3#57612yes—
437C22×C3#57712yes—
438C2×D3—44no—
439C2×D3—42no—
440C2×D3—44no—
441C2×D3—42no—
442C2×D3—42no—
443C3×C4—82no—
444C3×C4#39414yes—
445C3×C4C2×#12111yescharacteristic
446C3×C4C2×#12111yescharacteristic
447C3×C4C2×#12311yescharacteristic
448C3×C4C2×#12411yescharacteristic
449Dic3—42no—
450Dic3—44no—
451Dic3—82no—
452C23—22no—
453C23—22no—
454C23—24no—
455C23—48no—
456C23—42no—
457C23—44no—
458C23—44no—
459C23—42no—
460C23—62no—
461C23—62no—
462C23—64no—
463C23—64no—
464C23—62no—
465C23—88no—
466C23—84no—
467C23—84no—
468C23—88no—
469C23#88212yes—
470C23#88312yes—
471C23#93812yes—
472C23C2×D3×D411yescharacteristic
473Q8—124no—
474D4—24no—
475D4—24no—
476D4—48no—
477D4—42no—
478D4—42no—
479D4—128no—
480D4—124no—
481D4—2416no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—64no—
485C2×C4—62no—
486C2×C4—62no—
487C2×C4—62no—
488C2×C4—84no—
489C2×C4—122no—
490C2×C4—122no—
491C2×C4—122no—
492C2×C4—122no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4—122no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4#88212yes—
499C2×C4#88312yes—
500C2×C4#88514yes—
501C2×C4#88614yes—
502C2×C4#93014yes—
503C2×C4#93812yes—
504C2×C4#93914yes—
505C2×C4C22×D1211yescharacteristic
506C2×C4C2×#23511yescharacteristic
507C2×C4C2×D3×D411yescharacteristic
508C2×C4C2×#23711yescharacteristic
509C2×C4C2×#23711yescharacteristic
510C2×C4C22×#6711yescharacteristic
511C8—22no—
512C8—22no—
513C8—122no—
514C2×C3—24no—
515C2×C3—84no—
516C2×C3C2×#39411yescharacteristic
517C2×C3#301212yes—
518C2×C3#305711yescharacteristic
519C2×C3#305811yescharacteristic
520C2×C3#309612yes—
521D3—82no—
522C22—22no—
523C22—22no—
524C22—24no—
525C22—24no—
526C22—48no—
527C22—44no—
528C22—816no—
529C22—82no—
530C22—82no—
531C22—84no—
532C22—122no—
533C22—122no—
534C22—124no—
535C22—124no—
536C22—122no—
537C22#496114yes—
538C22C2×#88211yescharacteristic
539C22C2×#88311yescharacteristic
540C22#575712yes—
541C22#576212yes—
542C22C2×#93811yescharacteristic
543C4—82no—
544C4—124no—
545C4—122no—
546C4—242no—
547C4#496114yes—
548C4C2×#88511yescharacteristic
549C4C2×#88611yescharacteristic
550C4C2×#93011yescharacteristic
551C4C2×#93911yescharacteristic
552C3#2348611yescharacteristic, complemented, Sylow
553C2—24no—
554C2—84no—
555C2—242no—
556C2#7874712yes—
557C2#7875612yes—
558C2C2×#496111yescharacteristic
559C2#8383611yescharacteristic
560C2#8386311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×#102C2
socleC23×C3C22×D4
prime radicalC2×#5805C2
hypercenterC22×#102D3
hypocenterC3#23486
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#102

Complements

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

NH ≅ G/Ntype
2. C22×C3×#102C2semidirect
3. C2×#5215C2semidirect
4. C2×#5110C2semidirect
5. C2×#5106C2semidirect
6. #82186C2semidirect
7. #71558C2semidirect
8. #70176C2semidirect
9. #70175C2semidirect
31. C2×(C2×D3)⋊C8C22semidirect
32. C2×(C2×D3)⋊C8C22semidirect
34. C2×#818C22semidirect
35. C2×#817C22semidirect
37. #4987C22semidirect
39. #4529C22semidirect
43. C22×#102D3semidirect
73. C22×C3×C8D4semidirect
247. C22×C8#67semidirect
552. C3#23486semidirect

Maximal subgroups

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

subgroupindex
C2×#58052
C22×C3×#1022
C2×#52152
C2×#51102
C2×#51062
#821862
#715582
#701762
#701752
#234863