Finite Group Databasebeta

← #10829259

Subgroups — #10829259

Order 768.

1140 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5542C211yesmaximal, characteristic, complemented
2#82415C211yesmaximal, characteristic
3C2×#4842C211yesmaximal, characteristic, complemented
4C2×#4710C211yesmaximal, characteristic, complemented
5C3×#1955C211yesmaximal, characteristic, complemented
6#70486C211yesmaximal, characteristic, complemented
7#69962C212yesmaximal, complemented
8#69936C212yesmaximal, complemented
9#69933C212yesmaximal, complemented
10#69929C212yesmaximal, complemented
11#69828C211yesmaximal, characteristic, complemented
12#24131—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C2×#936—21no—
15C2×#930C2212yes—
16C2×#926C2212yes—
17C2×#924C2212yes—
18C22×#204C2211yescharacteristic
19#5542—24no—
20#5542—24no—
21C2×#887—21no—
22C2×#879—22no—
23C2×C4×D12C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×C22⋊C8C2212yes—
28C2×C3×C22⋊C8C2212yes—
29C2×C3×#97C2211yescharacteristic
30#5223—21no—
31#5212C2212yes—
32C2×#821C2211yescharacteristic
33C2×#821C2212yes—
34C2×#818C2211yescharacteristic
35C2×#817C2212yes—
36C2×#816C2211yescharacteristic
37C2×#816C2212yes—
38#4992—21no—
39#4956—22no—
40#4936C2211yescharacteristic
41#4842—24no—
42#4710—24no—
43#4551C2212yes—
44#4551C2212yes—
45#4551C2212yes—
46C2×#556—31no—
47C2×#476—31no—
48C2×#440—31no—
49#2535—31no—
50#2035—32no—
51#2000—32no—
52#1993—31no—
53#1955D311yescharacteristic, complemented
54#1875—32no—
55#1854—32no—
56#1805—31no—
57C22×C3×D4D411yescharacteristic
58C23×C3×C4C2312yes—
59C22×#67—22no—
60C22×D12C2311yescharacteristic
61C22×C4×D3—21no—
62C22×Dic6—21no—
63C22×Dic6C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C22×C3×C8C2312yes—
66C22×C3×C8C2311yescharacteristic
67C3×C4×D4—22no—
68C3×C4×D4—22no—
69C2×C3×C4⋊C4D411yescharacteristic
70C2×C3×C22⋊C4D412yes—
71C2×C3×C42C2311yescharacteristic
72C2×#217—22no—
73C2×#217—22no—
74C2×#217—22no—
75#936—28no—
76#930—24no—
77#930—42no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—21no—
80#926—24no—
81#926—24no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3C2312yes—
84C2×C4⋊Dic3C2311yescharacteristic
85C2×C4⋊Dic3C2311yescharacteristic
86C2×C4⋊Dic3C2312yes—
87#924—24no—
88#924—24no—
89C2×Dic3⋊C4—21no—
90C2×Dic3⋊C4—22no—
91C2×C4×Dic3—21no—
92C2×C4×Dic3—22no—
93C2×#204—22no—
94C2×#204—22no—
95C2×#204—22no—
96#887—28no—
97#879—216no—
98C4×D12—22no—
99C4×D12—22no—
100#869—22no—
101#869D414yescomplemented
102C4×Dic6—22no—
103C4×Dic6D414yescomplemented
104C3×C4⋊C8—22no—
105C3×C22⋊C8—24no—
106C3×C22⋊C8—42no—
107C3×#97—22no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110C3⋊#95—22no—
111#821—22no—
112#821—24no—
113#818—22no—
114#817—24no—
115#816—22no—
116#816—42no—
117C22×SD16—31no—
118#556—64no—
119#556—64no—
120C2×C4⋊Q8—31no—
121C2×#126—62no—
122C2×#124—61no—
123C2×C22⋊Q8—32no—
124C2×#121—32no—
125C2×C4×Q8—31no—
126C2×C4×Q8—61no—
127C2×C4×D4—31no—
128C2×C4×D4C2×D311yescharacteristic
129C2×#117—32no—
130#476—64no—
131#440—64no—
132C2×#107—31no—
133C2×#106—32no—
134C2×C4⋊C8C2×D311yescharacteristic
135C2×#103—31no—
136C2×#103—32no—
137C2×#102—31no—
138C2×#102—32no—
139C2×C22⋊C8C2×D312yes—
140C2×C22⋊C8C2×D312yes—
141C2×#97C2×D311yescharacteristic
142#397—61no—
143#392—61no—
144#386—62no—
145#382—31no—
146#380—32no—
147#340—32no—
148#340—32no—
149#340—32no—
150C24×C3C2×D412yes—
151C23×D3—21no—
152C2×C3×D4—22no—
153C2×C3×D4—42no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2411yescharacteristic
164C2×#67—44no—
165C2×#67—44no—
166C2×#67—82no—
167C22×Dic3—21no—
168C22×Dic3—21no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—21no—
172C2×D12—21no—
173C2×D12—21no—
174C2×D12—21no—
175C2×D12—21no—
176C2×D12—21no—
177C2×D12#4112yes—
178C2×C4×D3—42no—
179C2×C4×D3—82no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6—21no—
183C2×Dic6—21no—
184C2×Dic6—24no—
185C2×Dic6—21no—
186C2×Dic6—44no—
187C2×Dic6C2×D412yes—
188C2×C3×C8—21no—
189C2×C3×C8—21no—
190C2×C3×C8—21no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—21no—
195C2×C3×C8—21no—
196C2×C3×C8#4112yes—
197C3×C4⋊C4—22no—
198C3×C22⋊C4—24no—
199C3×C42C2×D412yes—
200C3×C42C2×D412yes—
201#217—28no—
202#217—28no—
203#217—82no—
204(C2×D3)⋊C4—44no—
205(C2×D3)⋊C4—42no—
206C4⋊Dic3—24no—
207C4⋊Dic3—21no—
208C4⋊Dic3—22no—
209C4⋊Dic3—21no—
210C4⋊Dic3—21no—
211C4⋊Dic3—22no—
212C4⋊Dic3—22no—
213C4⋊Dic3C2×D414yes—
214C4⋊Dic3C2×D412yes—
215Dic3⋊C4—28no—
216Dic3⋊C4—24no—
217C4×Dic3—28no—
218C4×Dic3—24no—
219#204—28no—
220C22×Q8—31no—
221C22×Q8—61no—
222C22×D4—31no—
223C22×D4—62no—
224C22×D4D1211yescharacteristic
225C23×C4—61no—
226C23×C4C22×D312yes—
227C2×SD16—62no—
228C2×SD16—62no—
229C2×SD16—62no—
230C22×C8C22×D311yescharacteristic
231C22×C8C22×D312yes—
232C22×C8C22×D311yescharacteristic
233C4⋊Q8—34no—
234C4⋊Q8—62no—
235#126—616no—
236#124—68no—
237C22⋊Q8—64no—
238C22⋊Q8—64no—
239#121—64no—
240#121—122no—
241C4×Q8—34no—
242C4×Q8—68no—
243C4×Q8—62no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—62no—
247C4×D4—62no—
248#117—64no—
249#117—64no—
250C2×C4⋊C4—31no—
251C2×C4⋊C4—32no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—32no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4D1211yescharacteristic
258C2×C22⋊C4—62no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—61no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4D1212yes—
264C2×C42—62no—
265C2×C42—61no—
266C2×C42C22×D311yescharacteristic
267#107—62no—
268#106—64no—
269C4⋊C8—22no—
270#103—62no—
271#103—122no—
272#102—62no—
273#102—64no—
274C22⋊C8—24no—
275C22⋊C8—42no—
276#97—22no—
277#95—62no—
278#95—61no—
279#95—61no—
280C23×C3—22no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3C22×D411yescharacteristic
288C22×D3—22no—
289C22×D3—41no—
290C22×D3—41no—
291C22×D3—42no—
292C22×D3—42no—
293C3×D4—44no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4#12012yes—
304C2×C3×C4#12012yes—
305C2×C3×C4C22×D411yescharacteristic
306C2×C3×C4C22×D411yescharacteristic
307C2×C3×C4C2×#4111yescharacteristic
308C2×C3×C4C2×#4111yescharacteristic
309C2×C3×C4H(5,𝔽2)11yescharacteristic
310C2×C3×C42−1+411yescharacteristic
311#67—88no—
312C2×Dic3—22no—
313C2×Dic3—22no—
314C2×Dic3—24no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317C2×Dic3—44no—
318C2×Dic3—42no—
319C2×Dic3—41no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324D12—22no—
325D12—22no—
326D12—41no—
327D12—41no—
328C4×D3—84no—
329Dic6—22no—
330Dic6—22no—
331Dic6—48no—
332Dic6—41no—
333Dic6—41no—
334C3×C8—22no—
335C3×C8—41no—
336C3×C8—42no—
337C24—61no—
338C24C2×D1212yes—
339C2×Q8—32no—
340C2×Q8—64no—
341C2×Q8—61no—
342C2×Q8—61no—
343C2×Q8—61no—
344C2×Q8—61no—
345C2×Q8—61no—
346C2×Q8—124no—
347C2×D4—22no—
348C2×D4—32no—
349C2×D4—42no—
350C2×D4—61no—
351C2×D4—61no—
352C2×D4—61no—
353C2×D4—61no—
354C2×D4—61no—
355C2×D4—124no—
356C2×D4—124no—
357C2×D4—242no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—61no—
365C22×C4—62no—
366C22×C4—61no—
367C22×C4—62no—
368C22×C4—61no—
369C22×C4—122no—
370C22×C4—242no—
371C22×C4C2×D1212yes—
372C22×C4C2×D1211yescharacteristic
373C22×C4C2×D1211yescharacteristic
374C22×C4C23×D311yescharacteristic
375SD16—68no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—21no—
382C2×C8—22no—
383C2×C8—21no—
384C2×C8#23712yes—
385C4⋊C4—22no—
386C4⋊C4—34no—
387C4⋊C4—32no—
388C4⋊C4—68no—
389C4⋊C4—64no—
390C4⋊C4—64no—
391C4⋊C4—61no—
392C4⋊C4—62no—
393C4⋊C4—61no—
394C4⋊C4—61no—
395C4⋊C4—62no—
396C4⋊C4—62no—
397C22⋊C4—24no—
398C22⋊C4—68no—
399C22⋊C4—68no—
400C22⋊C4—124no—
401C22⋊C4—122no—
402C22⋊C4—242no—
403C42—68no—
404C42—64no—
405C42D3×D412yes—
406C42D3×D412yes—
407C22×C3—42no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3—42no—
413C22×C3—42no—
414C22×C3C2×#12011yescharacteristic
415C22×C3#54611yescharacteristic
416C22×C3#54711yescharacteristic
417C22×C3#57512yes—
418C22×C3#57712yes—
419C2×D3—42no—
420C2×D3—44no—
421C2×D3—82no—
422C2×D3—81no—
423C2×D3—81no—
424C3×C4—22no—
425C3×C4—22no—
426C3×C4—42no—
427C3×C4#52511yescharacteristic
428C3×C4#53711yescharacteristic
429C3×C4#54811yescharacteristic
430C3×C4#55011yescharacteristic
431Dic3—42no—
432Dic3—44no—
433Dic3—42no—
434Dic3—81no—
435Dic3—82no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—62no—
444C23—121no—
445C23—122no—
446C23—122no—
447C23—121no—
448C23C22×D1211yescharacteristic
449Q8—62no—
450Q8—62no—
451Q8—128no—
452Q8—121no—
453Q8—121no—
454D4—44no—
455D4—62no—
456D4—62no—
457D4—121no—
458D4—121no—
459D4—248no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—62no—
470C2×C4—64no—
471C2×C4—62no—
472C2×C4—124no—
473C2×C4—122no—
474C2×C4—121no—
475C2×C4—121no—
476C2×C4—121no—
477C2×C4—122no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4—244no—
482C2×C4#88212yes—
483C2×C4#88212yes—
484C2×C4C2×D3×D411yescharacteristic
485C2×C4C2×D3×D411yescharacteristic
486C2×C4C2×#23711yescharacteristic
487C2×C4#100411yescharacteristic
488C2×C4D3×#4111yescharacteristic
489C2×C4#101011yescharacteristic
490C8—22no—
491C8—42no—
492C8—41no—
493C2×C3—82no—
494C2×C3#240711yescharacteristic
495C2×C3#300511yescharacteristic
496C2×C3#301011yescharacteristic
497C2×C3#331112yes—
498C2×C3#335012yes—
499D3—82no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—124no—
508C22—122no—
509C22—241no—
510C22—241no—
511C22—242no—
512C22C2×#88211yescharacteristic
513C22#555411yescharacteristic
514C22#555511yescharacteristic
515C22#574812yes—
516C22#575112yes—
517C4—22no—
518C4—22no—
519C4—42no—
520C4—124no—
521C4—122no—
522C4—122no—
523C4—241no—
524C4—242no—
525C4#560611yescharacteristic
526C4#560811yescharacteristic
527C4#567711yescharacteristic
528C4#571811yescharacteristic
529C3#2413111yescharacteristic, complemented, Sylow
530C2—82no—
531C2—242no—
532C2#7643511yescharacteristic
533C2#7863811yescharacteristic
534C2#7867811yescharacteristic
535C2#7947612yes—
536C2#7950212yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1955C2
socleC23×C3C22×D4
prime radical#82415C2
hypercenter#1955D3
hypocenterC3#24131
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1955

Complements

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

NH ≅ G/Ntype
1. C2×#5542C2semidirect
3. C2×#4842C2semidirect
4. C2×#4710C2semidirect
5. C3×#1955C2semidirect
6. #70486C2semidirect
7. #69962C2semidirect
8. #69936C2semidirect
9. #69933C2semidirect
10. #69929C2semidirect
11. #69828C2semidirect
53. #1955D3semidirect
101. #869D4semidirect
103. C4×Dic6D4semidirect
529. C3#24131semidirect

Maximal subgroups

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

subgroupindex
C2×#55422
#824152
C2×#48422
C2×#47102
C3×#19552
#704862
#699622
#699362
#699332
#699292
#698282
#241313