Finite Group Databasebeta

← #10830092

Subgroups — #10830092

Order 768.

1180 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5713C211yesmaximal, characteristic, complemented
2C2×#5072C211yesmaximal, characteristic
3C2×#4710C211yesmaximal, characteristic, complemented
4#74190C211yesmaximal, characteristic, complemented
5C3×#1961C211yesmaximal, characteristic, complemented
6#70214C212yesmaximal, complemented
7#70178C212yesmaximal, complemented
8#70172C212yesmaximal, complemented
9#70168C212yesmaximal, complemented
10#69838C211yesmaximal, characteristic, complemented
11#69559C211yesmaximal, characteristic, complemented
12#24198—31nomaximal, Sylow
13C2×C3×#127C2211yescharacteristic
14C2×#936—21no—
15C2×#936C2212yes—
16C2×#935—22no—
17C2×D4×Dic3—21no—
18C2×D4×Dic3C2212yes—
19C22×#213C2212yes—
20C22×#204C2211yescharacteristic
21#5713—24no—
22#5713—24no—
23C2×C4×D12C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C2×C3×#102C2212yes—
27C2×C3×#102C2212yes—
28C2×C3×#97C2211yescharacteristic
29C3×#382C2211yescharacteristic
30C2×#832C2212yes—
31C2×#821C2211yescharacteristic
32C2×#818C2211yescharacteristic
33C2×#816C2211yescharacteristic
34#5072—24no—
35#4988C2212yes—
36#4984C2212yes—
37C2×#810C2212yes—
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
39#4710—24no—
40#4677—21no—
41#4673—21no—
42#4668—22no—
43#4550C2212yes—
44#4550C2212yes—
45#4529C2212yes—
46C2×#559—31no—
47C2×#459—31no—
48C2×#440—31no—
49#2580—31no—
50#2042—32no—
51#2012—32no—
52#1985—31no—
53#1961D311yescharacteristic, complemented
54#1879—32no—
55#1862—32no—
56#1805—31no—
57C22×C3×D4—21no—
58C22×C3×D4—21no—
59C22×C3×D4C2312yes—
60C23×Dic3—22no—
61C22×D12C2311yescharacteristic
62C22×C4×D3—21no—
63C22×Dic6D411yescharacteristic
64C22×Dic6C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C22×C3×C8C2311yescharacteristic
67C3×#127—22no—
68C3×#127D414yes—
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C42C2311yescharacteristic
71C2×#217—22no—
72C2×#217—22no—
73C2×#217—22no—
74#936—28no—
75#936—24no—
76#936—24no—
77#935—216no—
78D4×Dic3—28no—
79D4×Dic3—24no—
80D4×Dic3—24no—
81C2×#213—24no—
82C2×#213—24no—
83C2×#213—42no—
84C2×(C2×D3)⋊C4—21no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4—22no—
87C2×C4⋊Dic3—22no—
88C2×C4⋊Dic3D411yescharacteristic
89C2×C4⋊Dic3C2311yescharacteristic
90C2×C4⋊Dic3C2311yescharacteristic
91C2×Dic3⋊C4—21no—
92C2×Dic3⋊C4D412yes—
93C2×C4×Dic3—21no—
94C2×C4×Dic3C2312yes—
95C22×C3⋊C8C2312yes—
96C2×#204—22no—
97C2×#204—22no—
98C2×#204—22no—
99C4×D12—22no—
100C4×D12—22no—
101C4×Dic6—22no—
102C4×Dic6—22no—
103C4×Dic6—22no—
104C4×Dic6D414yes—
105C3×#102—24no—
106C3×#102—42no—
107C3×#97—22no—
108C3×#95—21no—
109C3×#95—21no—
110#832—24no—
111C3⋊#95—22no—
112#821—22no—
113#818—22no—
114#816—22no—
115#810—42no—
116C4⋊(C3⋊C8)—22no—
117C22×SD16—31no—
118C22×D8—32no—
119#559—64no—
120#559—64no—
121C2×#127C2×D311yescharacteristic
122C2×#124—32no—
123C2×#124—61no—
124C2×#123—62no—
125C2×C4×Q8—31no—
126C2×C4×Q8—31no—
127C2×C4×D4—32no—
128C2×C4×D4—31no—
129C2×C4×D4—61no—
130#459—64no—
131#440—64no—
132C2×#107—31no—
133C2×C4⋊C8—31no—
134C2×#103—31no—
135C2×#103—32no—
136C2×#102—32no—
137C2×#102—31no—
138C2×#102C2×D312yes—
139C2×#102C2×D312yes—
140C2×#97C2×D311yescharacteristic
141#399—61no—
142#393—61no—
143#390—32no—
144#389—32no—
145#385—62no—
146#382C2×D311yescharacteristic
147#340—32no—
148#336—32no—
149#336—32no—
150C24×C3—22no—
151C23×D3—21no—
152C2×C3×D4—24no—
153C2×C3×D4—24no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4—44no—
160C2×C3×D4—82no—
161C2×C3×D4C2×D414yes—
162C22×C3×C4—22no—
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2411yescharacteristic
166C22×Dic3—21no—
167C22×Dic3—21no—
168C22×Dic3—28no—
169C22×Dic3—21no—
170C22×Dic3—82no—
171C22×Dic3—82no—
172C22×Dic3C2×D412yes—
173C22×Dic3C2×D412yes—
174C2×D12—21no—
175C2×D12—21no—
176C2×D12—21no—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12#4112yes—
180C2×C4×D3—42no—
181C2×C4×D3—82no—
182C2×Dic6—21no—
183C2×Dic6—21no—
184C2×Dic6—21no—
185C2×Dic6—21no—
186C2×Dic6—22no—
187C2×Dic6—21no—
188C2×Dic6—42no—
189C2×Dic6C2×D412yes—
190C2×C3×C8—21no—
191C2×C3×C8—21no—
192C2×C3×C8—21no—
193C2×C3×C8—21no—
194C2×C3×C8—21no—
195C2×C3×C8#4112yes—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C42C2×D412yes—
199C3×C42C2×D412yes—
200#217—28no—
201#217—28no—
202#217—28no—
203#213—48no—
204(C2×D3)⋊C4—44no—
205(C2×D3)⋊C4—42no—
206(C2×D3)⋊C4—82no—
207C4⋊Dic3—22no—
208C4⋊Dic3—21no—
209C4⋊Dic3—21no—
210C4⋊Dic3—21no—
211C4⋊Dic3—82no—
212C4⋊Dic3C2×D412yes—
213Dic3⋊C4—24no—
214Dic3⋊C4—24no—
215C4×Dic3—24no—
216C4×Dic3—22no—
217C4×Dic3—22no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221#204—28no—
222C22×Q8—31no—
223C22×Q8—31no—
224C22×D4—21no—
225C22×D4—21no—
226C22×D4—31no—
227C22×D4C22×D312yes—
228C23×C4—62no—
229C23×C4—61no—
230C2×SD16—62no—
231C2×SD16—62no—
232C2×SD16—62no—
233C2×D8—64no—
234C2×D8—64no—
235C2×D8—122no—
236C22×C8—32no—
237C22×C8C22×D311yescharacteristic
238C22×C8C22×D311yescharacteristic
239#127—22no—
240#127D1214yes—
241#124—64no—
242#124—68no—
243#124—64no—
244#123—616no—
245C4×Q8—34no—
246C4×Q8—62no—
247C4×Q8—62no—
248C4×Q8—62no—
249C4×D4—64no—
250C4×D4—68no—
251C4×D4—64no—
252C4×D4—62no—
253C4×D4—62no—
254C2×C4⋊C4—31no—
255C2×C4⋊C4—31no—
256C2×C4⋊C4—32no—
257C2×C4⋊C4—31no—
258C2×C4⋊C4—61no—
259C2×C4⋊C4—62no—
260C2×C4⋊C4C22×D312yes—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4—62no—
264C2×C22⋊C4—62no—
265C2×C22⋊C4—61no—
266C2×C22⋊C4—62no—
267C2×C42—32no—
268C2×C42—61no—
269C2×C42C22×D311yescharacteristic
270#107—62no—
271C4⋊C8—62no—
272#103—62no—
273#103—122no—
274#102—24no—
275#102—42no—
276#102—62no—
277#102—64no—
278#97—22no—
279#95—21no—
280#95—21no—
281#95—62no—
282C23×C3—24no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3C22×D411yescharacteristic
290C22×D3—22no—
291C22×D3—41no—
292C22×D3—41no—
293C22×D3—42no—
294C22×D3—42no—
295C3×D4—24no—
296C3×D4—24no—
297C3×D4—48no—
298C3×D4—42no—
299C3×D4—42no—
300C3×D4—84no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4#12012yes—
307C2×C3×C4#12012yes—
308C2×C3×C4C22×D411yescharacteristic
309C2×C3×C4C22×D411yescharacteristic
310C2×C3×C4C2×#4111yescharacteristic
311C2×C3×C4C2×#4111yescharacteristic
312C2×C3×C4H(5,𝔽2)11yescharacteristic
313C2×C3×C42−1+411yescharacteristic
314C2×Dic3—22no—
315C2×Dic3—22no—
316C2×Dic3—22no—
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—22no—
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3—42no—
325C2×Dic3—41no—
326C2×Dic3—42no—
327C2×Dic3—88no—
328D12—22no—
329D12—22no—
330D12—41no—
331D12—41no—
332C4×D3—84no—
333Dic6—22no—
334Dic6—22no—
335Dic6—44no—
336Dic6—41no—
337Dic6—41no—
338C3×C8—22no—
339C3×C8—41no—
340C3⋊C8—42no—
341C24—22no—
342C24—61no—
343C2×Q8—32no—
344C2×Q8—62no—
345C2×Q8—61no—
346C2×Q8—61no—
347C2×Q8—61no—
348C2×Q8—61no—
349C2×Q8—61no—
350C2×Q8—122no—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—32no—
359C2×D4—44no—
360C2×D4—61no—
361C2×D4—61no—
362C2×D4—61no—
363C2×D4—61no—
364C2×D4—61no—
365C2×D4—82no—
366C2×D4C2×D1214yes—
367C22×C4—22no—
368C22×C4—32no—
369C22×C4—32no—
370C22×C4—61no—
371C22×C4—61no—
372C22×C4—68no—
373C22×C4—61no—
374C22×C4—122no—
375C22×C4—242no—
376C22×C4—242no—
377C22×C4—242no—
378C22×C4D3×D411yescharacteristic
379C22×C4D3×D411yescharacteristic
380C22×C4C23×D311yescharacteristic
381SD16—68no—
382D8—128no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—62no—
389C2×C8—62no—
390C2×C8—62no—
391C2×C8#23712yes—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—32no—
395C4⋊C4—62no—
396C4⋊C4—64no—
397C4⋊C4—64no—
398C4⋊C4—61no—
399C4⋊C4—61no—
400C4⋊C4—61no—
401C4⋊C4—242no—
402C22⋊C4—68no—
403C22⋊C4—68no—
404C22⋊C4—68no—
405C22⋊C4—124no—
406C22⋊C4—122no—
407C22⋊C4—242no—
408C42—64no—
409C42—62no—
410C42—62no—
411C42C2×D1212yes—
412C42D3×D412yes—
413C22×C3—44no—
414C22×C3—44no—
415C22×C3—44no—
416C22×C3—82no—
417C22×C3—82no—
418C22×C3—82no—
419C22×C3—82no—
420C22×C3C2×#12011yescharacteristic
421C22×C3#54811yescharacteristic
422C22×C3#55011yescharacteristic
423C22×C3#57512yes—
424C22×C3#57612yes—
425C2×D3—44no—
426C2×D3—42no—
427C2×D3—82no—
428C2×D3—81no—
429C2×D3—81no—
430C3×C4—22no—
431C3×C4—22no—
432C3×C4—82no—
433C3×C4#52511yescharacteristic
434C3×C4#53711yescharacteristic
435C3×C4#54611yescharacteristic
436C3×C4#54711yescharacteristic
437Dic3—42no—
438Dic3—42no—
439Dic3—42no—
440Dic3—42no—
441Dic3—81no—
442C23—24no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—62no—
450C23—121no—
451C23—121no—
452C23—122no—
453C23—122no—
454C23C2×D3×D411yescharacteristic
455Q8—62no—
456Q8—62no—
457Q8—124no—
458Q8—121no—
459Q8—121no—
460D4—24no—
461D4—24no—
462D4—48no—
463D4—42no—
464D4—42no—
465D4—62no—
466D4—62no—
467D4—84no—
468D4—121no—
469D4—121no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—62no—
476C2×C4—62no—
477C2×C4—62no—
478C2×C4—62no—
479C2×C4—62no—
480C2×C4—62no—
481C2×C4—62no—
482C2×C4—62no—
483C2×C4—121no—
484C2×C4—121no—
485C2×C4—122no—
486C2×C4—121no—
487C2×C4—122no—
488C2×C4—244no—
489C2×C4—248no—
490C2×C4#88212yes—
491C2×C4#88212yes—
492C2×C4C22×D1211yescharacteristic
493C2×C4C2×D3×D411yescharacteristic
494C2×C4C2×#23711yescharacteristic
495C2×C4#100411yescharacteristic
496C2×C4D3×#4111yescharacteristic
497C2×C4#101011yescharacteristic
498C8—22no—
499C8—41no—
500C8—122no—
501C2×C3—84no—
502C2×C3#240711yescharacteristic
503C2×C3#300511yescharacteristic
504C2×C3#300811yescharacteristic
505C2×C3#337012yes—
506C2×C3#340312yes—
507D3—82no—
508C22—44no—
509C22—44no—
510C22—44no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22—124no—
516C22—122no—
517C22—241no—
518C22—241no—
519C22—242no—
520C22C2×#88211yescharacteristic
521C22#567711yescharacteristic
522C22#571811yescharacteristic
523C22#577312yes—
524C22#577412yes—
525C4—22no—
526C4—22no—
527C4—82no—
528C4—122no—
529C4—122no—
530C4—122no—
531C4—122no—
532C4—241no—
533C4#555411yescharacteristic
534C4#555511yescharacteristic
535C4#560611yescharacteristic
536C4#560811yescharacteristic
537C3#2419811yescharacteristic, complemented, Sylow
538C2—84no—
539C2—242no—
540C2#7643511yescharacteristic
541C2#7892711yescharacteristic
542C2#7892811yescharacteristic
543C2#8171012yes—
544C2#8184212yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1961C2
socleC23×C3C22×D4
prime radicalC2×#5072C2
hypercenter#1961D3
hypocenterC3#24198
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1961

Complements

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

NH ≅ G/Ntype
1. C2×#5713C2semidirect
3. C2×#4710C2semidirect
4. #74190C2semidirect
5. C3×#1961C2semidirect
6. #70214C2semidirect
7. #70178C2semidirect
8. #70172C2semidirect
9. #70168C2semidirect
10. #69838C2semidirect
11. #69559C2semidirect
53. #1961D3semidirect
537. C3#24198semidirect

Maximal subgroups

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

subgroupindex
C2×#57132
C2×#50722
C2×#47102
#741902
C3×#19612
#702142
#701782
#701722
#701682
#698382
#695592
#241983