Finite Group Databasebeta

← #10829977

Subgroups — #10829977

Order 768.

1082 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×#5732C211yesmaximal, characteristic
2C2×#5090C211yesmaximal, characteristic, complemented
3C2×C4×#204C211yesmaximal, characteristic
4#74343C211yesmaximal, characteristic, complemented
5C3×#1956C211yesmaximal, characteristic, complemented
6#70278C212yesmaximal
7#70245C212yesmaximal
8#70239C212yesmaximal, complemented
9#70238C212yesmaximal, complemented
10#69788C211yesmaximal, characteristic, complemented
11#69584C211yesmaximal, characteristic
12#24189—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×#947C2212yes—
15C2×Q8×Dic3C2212yes—
16C22×#216C2212yes—
17C22×#204C2211yescharacteristic
18#5732—24no—
19#5732—24no—
20C2×#898—22no—
21C2×#897—21no—
22C2×#892—21no—
23C2×C4×D12C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C2×C3×#103C2212yes—
27C2×C3×#103C2212yes—
28C2×C3×C4×C8C2211yescharacteristic
29C3×#384C2211yescharacteristic
30C2×#835C2212yes—
31C2×#821C2211yescharacteristic
32C2×#817C2211yescharacteristic
33C2×#816C2211yescharacteristic
34#5090—24no—
35#4984—21no—
36#4984C2212yes—
37#4981—22no—
38#4973—21no—
39#4973C2212yes—
40C2×#809C2212yes—
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C4×#204—24no—
43#4550C2212yes—
44#4550C2212yes—
45#4529C2212yes—
46C2×#556—31no—
47C2×#462—31no—
48C2×C4×SD16—31no—
49#2582—31no—
50#2042—32no—
51#2015—32no—
52#1984—31no—
53#1956D311yescharacteristic, complemented
54#1878—32no—
55#1865—32no—
56#1802—31no—
57C22×C3×Q8C2312yes—
58C22×D12C2311yescharacteristic
59C22×C4×D3—21no—
60C22×Dic6D411yescharacteristic
61C22×Dic6C2311yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C3×C4⋊Q8—22no—
65C3×C4⋊Q8D414yes—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C42C2311yescharacteristic
72#947—24no—
73#947—24no—
74Q8×Dic3—24no—
75Q8×Dic3—24no—
76C2×#216—24no—
77C2×#216—24no—
78C2×#216—42no—
79C2×(C2×D3)⋊C4—21no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—22no—
82C2×C4⋊Dic3—22no—
83C2×C4⋊Dic3—22no—
84C2×C4⋊Dic3D411yescharacteristic
85C2×C4⋊Dic3C2311yescharacteristic
86C2×C4⋊Dic3C2311yescharacteristic
87C2×Dic3⋊C4—21no—
88C2×Dic3⋊C4D412yes—
89C2×C4×Dic3—22no—
90C2×C4×Dic3—21no—
91C2×C4×Dic3C2312yes—
92C22×C3⋊C8C2312yes—
93C2×#204—22no—
94C2×#204—22no—
95C2×#204—22no—
96#898—216no—
97#897—28no—
98#892—28no—
99C4×D12—22no—
100C4×D12D414yes—
101C4×Dic6—22no—
102C4×Dic6—22no—
103C4×Dic6—22no—
104C4×Dic6—22no—
105C3×#103—24no—
106C3×#103—42no—
107C3×C4×C8—22no—
108#835—24no—
109C3⋊#95—22no—
110C3⋊#95—22no—
111C3⋊#95—22no—
112#821—22no—
113#817—22no—
114#816—22no—
115#809—42no—
116C4⋊(C3⋊C8)—22no—
117C22×Q16—32no—
118C22×SD16—31no—
119#556—64no—
120#556—64no—
121C2×C4⋊Q8C2×D311yescharacteristic
122C2×#126—62no—
123C2×#124—32no—
124C2×C22⋊Q8—61no—
125C2×C4×Q8—31no—
126C2×C4×Q8—32no—
127C2×C4×Q8—31no—
128C2×C4×D4—31no—
129C2×#117—61no—
130#462—64no—
131C4×SD16—64no—
132C2×#106—31no—
133C2×C4⋊C8—31no—
134C2×#103—32no—
135C2×#103—31no—
136C2×#103C2×D312yes—
137C2×#103C2×D312yes—
138C2×#102—31no—
139C2×#102—32no—
140C2×C4×C8C2×D311yescharacteristic
141#398—62no—
142#391—32no—
143#391—61no—
144#389—32no—
145#389—61no—
146#384C2×D311yescharacteristic
147#340—32no—
148#336—32no—
149#336—32no—
150C23×D3—21no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8C2×D414yes—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2411yescharacteristic
162C22×Dic3—21no—
163C22×Dic3—21no—
164C22×Dic3—21no—
165C22×Dic3C2×D412yes—
166C22×Dic3C2×D412yes—
167C2×D12—21no—
168C2×D12—21no—
169C2×D12—21no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12C2×D412yes—
173C2×C4×D3—24no—
174C2×C4×D3—42no—
175C2×C4×D3—42no—
176C2×Dic6—21no—
177C2×Dic6—21no—
178C2×Dic6—21no—
179C2×Dic6—21no—
180C2×Dic6—22no—
181C2×Dic6—22no—
182C2×Dic6—22no—
183C2×Dic6—21no—
184C2×Dic6#4112yes—
185C2×C3×C8—21no—
186C2×C3×C8—21no—
187C2×C3×C8—21no—
188C2×C3×C8—21no—
189C2×C3×C8—21no—
190C2×C3×C8#4112yes—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—42no—
196C3×C4⋊C4—81no—
197C3×C42C2×D412yes—
198C3×C42C2×D412yes—
199#216—48no—
200(C2×D3)⋊C4—28no—
201(C2×D3)⋊C4—28no—
202(C2×D3)⋊C4—24no—
203C4⋊Dic3—28no—
204C4⋊Dic3—22no—
205C4⋊Dic3—21no—
206C4⋊Dic3—21no—
207C4⋊Dic3—21no—
208C4⋊Dic3—82no—
209C4⋊Dic3C2×D412yes—
210Dic3⋊C4—24no—
211Dic3⋊C4—42no—
212C4×Dic3—28no—
213C4×Dic3—22no—
214C4×Dic3—22no—
215C4×Dic3—42no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219#204—28no—
220C22×Q8—31no—
221C22×Q8—31no—
222C22×Q8C22×D312yes—
223C22×D4—31no—
224C23×C4—61no—
225C2×Q16—64no—
226C2×Q16—64no—
227C2×Q16—122no—
228C2×SD16—62no—
229C2×SD16—62no—
230C2×SD16—62no—
231C22×C8—32no—
232C22×C8C22×D311yescharacteristic
233C22×C8C22×D311yescharacteristic
234C4⋊Q8—22no—
235C4⋊Q8D1214yes—
236#126—616no—
237#124—64no—
238#124—64no—
239C22⋊Q8—68no—
240C4×Q8—64no—
241C4×Q8—64no—
242C4×Q8—62no—
243C4×Q8—62no—
244C4×Q8—62no—
245C4×Q8—62no—
246C4×D4—34no—
247C4×D4—62no—
248#117—68no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—31no—
254C2×C4⋊C4—31no—
255C2×C4⋊C4—32no—
256C2×C4⋊C4—31no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4—61no—
259C2×C4⋊C4—62no—
260C2×C4⋊C4C22×D312yes—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4—61no—
264C2×C42—32no—
265C2×C42—62no—
266C2×C42—61no—
267C2×C42C22×D311yescharacteristic
268#106—62no—
269C4⋊C8—62no—
270#103—24no—
271#103—42no—
272#103—62no—
273#103—64no—
274#102—62no—
275#102—122no—
276C4×C8—22no—
277#95—62no—
278#95—62no—
279#95—62no—
280C23×C3C22×D411yescharacteristic
281C22×D3—22no—
282C22×D3—41no—
283C22×D3—41no—
284C22×D3—41no—
285C22×D3—41no—
286C22×D3—41no—
287C22×D3—41no—
288C3×Q8—24no—
289C3×Q8—24no—
290C3×Q8—42no—
291C3×Q8—42no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—24no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4#12012yes—
301C2×C3×C4#12012yes—
302C2×C3×C4#13512yes—
303C2×C3×C4#13712yes—
304C2×C3×C4C22×D411yescharacteristic
305C2×C3×C4C22×D411yescharacteristic
306C2×C3×C4C2×#4111yescharacteristic
307C2×C3×C4C2×#4111yescharacteristic
308C2×C3×C4H(5,𝔽2)11yescharacteristic
309C2×C3×C42−1+411yescharacteristic
310C2×Dic3—22no—
311C2×Dic3—22no—
312C2×Dic3—22no—
313C2×Dic3—22no—
314C2×Dic3—22no—
315C2×Dic3—22no—
316C2×Dic3—22no—
317C2×Dic3—22no—
318C2×Dic3—41no—
319C2×Dic3—41no—
320C2×Dic3—41no—
321C2×Dic3—41no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325D12—22no—
326D12—22no—
327D12—41no—
328D12—41no—
329C4×D3—48no—
330Dic6—22no—
331Dic6—22no—
332Dic6—41no—
333Dic6—41no—
334Dic6—44no—
335C3×C8—22no—
336C3×C8—41no—
337C3⋊C8—42no—
338C24—61no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—32no—
345C2×Q8—62no—
346C2×Q8—62no—
347C2×Q8—62no—
348C2×Q8—61no—
349C2×Q8—61no—
350C2×Q8—61no—
351C2×Q8—61no—
352C2×Q8—61no—
353C2×Q8C2×D1214yes—
354C2×D4—32no—
355C2×D4—61no—
356C2×D4—61no—
357C2×D4—61no—
358C2×D4—61no—
359C2×D4—61no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—32no—
363C22×C4—32no—
364C22×C4—61no—
365C22×C4—61no—
366C22×C4—64no—
367C22×C4—61no—
368C22×C4—122no—
369C22×C4—122no—
370C22×C4D3×D411yescharacteristic
371C22×C4D3×D411yescharacteristic
372C22×C4C23×D311yescharacteristic
373Q16—128no—
374SD16—68no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—62no—
381C2×C8—62no—
382C2×C8—62no—
383C2×C8#23712yes—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—32no—
389C4⋊C4—42no—
390C4⋊C4—68no—
391C4⋊C4—62no—
392C4⋊C4—64no—
393C4⋊C4—61no—
394C4⋊C4—61no—
395C4⋊C4—61no—
396C4⋊C4—81no—
397C4⋊C4—122no—
398C4⋊C4—242no—
399C22⋊C4—68no—
400C22⋊C4—68no—
401C22⋊C4—64no—
402C42—68no—
403C42—62no—
404C42—62no—
405C42—122no—
406C42C2×D1212yes—
407C42D3×D412yes—
408C22×C3C2×#12011yescharacteristic
409C22×C3#54811yescharacteristic
410C22×C3#55011yescharacteristic
411C22×C3C2×#13511yescharacteristic
412C22×C3C2×#13711yescharacteristic
413C22×C3#57711yescharacteristic
414C22×C3#57811yescharacteristic
415C2×D3—42no—
416C2×D3—42no—
417C2×D3—42no—
418C2×D3—81no—
419C2×D3—81no—
420C2×D3—81no—
421C2×D3—81no—
422C3×C4—22no—
423C3×C4—44no—
424C3×C4—82no—
425C3×C4#45214yes—
426C3×C4#52511yescharacteristic
427C3×C4#53711yescharacteristic
428C3×C4#54611yescharacteristic
429C3×C4#54711yescharacteristic
430Dic3—42no—
431Dic3—42no—
432Dic3—42no—
433Dic3—42no—
434Dic3—81no—
435C23—62no—
436C23—121no—
437C23—121no—
438C23—121no—
439C23—121no—
440C23—121no—
441C23—121no—
442C23C2×D3×D411yescharacteristic
443Q8—24no—
444Q8—24no—
445Q8—42no—
446Q8—42no—
447Q8—62no—
448Q8—62no—
449Q8—121no—
450Q8—124no—
451Q8—121no—
452D4—62no—
453D4—62no—
454D4—121no—
455D4—121no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—24no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—62no—
465C2×C4—62no—
466C2×C4—62no—
467C2×C4—62no—
468C2×C4—62no—
469C2×C4—62no—
470C2×C4—62no—
471C2×C4—62no—
472C2×C4—121no—
473C2×C4—121no—
474C2×C4—121no—
475C2×C4—121no—
476C2×C4—128no—
477C2×C4—121no—
478C2×C4—121no—
479C2×C4—121no—
480C2×C4#88212yes—
481C2×C4#88212yes—
482C2×C4#91612yes—
483C2×C4#91812yes—
484C2×C4C22×D1211yescharacteristic
485C2×C4C2×D3×D411yescharacteristic
486C2×C4C2×#23711yescharacteristic
487C2×C4#100411yescharacteristic
488C2×C4D3×#4111yescharacteristic
489C2×C4#101011yescharacteristic
490C8—22no—
491C8—41no—
492C8—122no—
493C2×C3#240711yescharacteristic
494C2×C3C2×#45211yescharacteristic
495C2×C3#301311yescharacteristic
496C2×C3#336411yescharacteristic
497C2×C3#336911yescharacteristic
498C2×C3#339411yescharacteristic
499C2×C3#339811yescharacteristic
500D3—82no—
501C22—122no—
502C22—122no—
503C22—122no—
504C22—241no—
505C22—241no—
506C22—241no—
507C22—241no—
508C22C2×#88211yescharacteristic
509C22#567711yescharacteristic
510C22#571811yescharacteristic
511C22C2×#91611yescharacteristic
512C22#576211yescharacteristic
513C22C2×#91811yescharacteristic
514C22#576611yescharacteristic
515C4—22no—
516C4—44no—
517C4—82no—
518C4—122no—
519C4—122no—
520C4—122no—
521C4—122no—
522C4—241no—
523C4#480814yes—
524C4#555411yescharacteristic
525C4#555511yescharacteristic
526C4#560611yescharacteristic
527C4#560811yescharacteristic
528C3#2418911yescharacteristic, complemented, Sylow
529C2—242no—
530C2#7643511yescharacteristic
531C2C2×#480811yescharacteristic
532C2#7879511yescharacteristic
533C2#8162511yescharacteristic
534C2#8162611yescharacteristic
535C2#8175611yescharacteristic
536C2#8179711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1956C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1956D3
hypocenterC3#24189
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
2. C2×#5090C2semidirect
4. #74343C2semidirect
5. C3×#1956C2semidirect
8. #70239C2semidirect
9. #70238C2semidirect
10. #69788C2semidirect
53. #1956D3semidirect
528. C3#24189semidirect

Maximal subgroups

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

subgroupindex
C2×#57322
C2×#50902
C2×C4×#2042
#743432
C3×#19562
#702782
#702452
#702392
#702382
#697882
#695842
#241893