Finite Group Databasebeta

← #10845310

Subgroups — #10845310

Order 768.

958 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83260C211yesmaximal, characteristic, complemented
2C2×#5084C211yesmaximal, characteristic, complemented
3C2×C8×D12C211yesmaximal, characteristic, complemented
4#74368C211yesmaximal, characteristic
5C3×#1990C211yesmaximal, characteristic, complemented
6C4×#821C211yesmaximal, characteristic, complemented
7#69568C211yesmaximal, characteristic, complemented
8#68883C212yesmaximal, complemented
9#68868C212yesmaximal, complemented
10#68833C212yesmaximal, complemented
11#68823C212yesmaximal, complemented
12#25339—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C22×D3×C8C2212yes—
15C2×#895—21no—
16C2×#895C2212yes—
17C2×C4×D12C2211yescharacteristic
18C2×C4×D12C2211yescharacteristic
19C2×C3×C4×C8C2211yescharacteristic
20C3×#384C2211yescharacteristic
21#5231—22no—
22C2×#821C2211yescharacteristic
23C2×#821C2211yescharacteristic
24C2×(C2×D3)⋊C8C2212yes—
25#5084—24no—
26#4991—22no—
27#4988—21no—
28#4987—21no—
29#4987C2212yes—
30#4987C2212yes—
31#4987C2212yes—
32#4977—21no—
33C2×#809C2212yes—
34C2×#808C2212yes—
35C2×#808C2212yes—
36C4×C4⋊Dic3C2211yescharacteristic
37C2×C4⋊(C3⋊C8)C2211yescharacteristic
38C8×D12—24no—
39C3×#340C2212yes—
40C3×#340C2212yes—
41#4551C2211yescharacteristic
42#4551C2211yescharacteristic
43#4529C2212yes—
44C2×#476—31no—
45C2×C8×D4—31no—
46#2563—31no—
47#2537—31no—
48#2077—32no—
49#2034—32no—
50#1990D311yescharacteristic, complemented
51#1948—31no—
52#1893—32no—
53#1807—32no—
54C4×#102—31no—
55C22×C3×Q8—21no—
56C22×D12—22no—
57C22×D12D411yescharacteristic
58C22×D12C2311yescharacteristic
59C22×C4×D3—21no—
60C22×C4×D3C2312yes—
61C22×C3×C8C2311yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C3×C4⋊Q8—22no—
64C3×C4⋊Q8—22no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4C2312yes—
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C42C2311yescharacteristic
71C2×(C2×D3)⋊C4—22no—
72C2×(C2×D3)⋊C4—21no—
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4—22no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4D412yes—
77C2×C4⋊Dic3D411yescharacteristic
78C2×C4⋊Dic3C2311yescharacteristic
79C2×C4⋊Dic3C2311yescharacteristic
80C2×C4⋊Dic3C2311yescharacteristic
81C2×C4×Dic3—21no—
82C22×C3⋊C8C2312yes—
83C2×D3×C8—24no—
84C2×D3×C8—24no—
85C2×D3×C8—42no—
86#895—24no—
87#895—42no—
88#895—44no—
89C4×D12—22no—
90C4×D12—22no—
91C4×D12—22no—
92C4×D12—22no—
93C3×C4×C8—22no—
94C3⋊#95—21no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97C3⋊#95—22no—
98#821—22no—
99#821—22no—
100(C2×D3)⋊C8—24no—
101#809—24no—
102#808—24no—
103#808—42no—
104C4⋊(C3⋊C8)D414yescomplemented
105C23×C8—32no—
106C2×C4⋊Q8C2×D311yescharacteristic
107C2×#121—32no—
108C2×#121—61no—
109C2×C4×D4—31no—
110C2×C4×D4—31no—
111#476—64no—
112C8×D4—64no—
113C2×#106—32no—
114C2×#106—32no—
115C2×C4⋊C8—31no—
116C2×#102—31no—
117C2×#102—31no—
118C2×#102—32no—
119C2×C22⋊C8—32no—
120C2×C4×C8C2×D311yescharacteristic
121#394—62no—
122#393—62no—
123#390—61no—
124#384C2×D311yescharacteristic
125#382—61no—
126#380—32no—
127#380—32no—
128#380—32no—
129#380—61no—
130C4×C4⋊C4—31no—
131#340—31no—
132#340—31no—
133#340—32no—
134#340C2×D312yes—
135#340C2×D312yes—
136C23×D3—21no—
137C23×D3C2×D412yes—
138C2×C3×Q8—42no—
139C2×C3×Q8—81no—
140C2×C3×Q8—81no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2411yescharacteristic
146C22×Dic3—21no—
147C22×Dic3—21no—
148C22×Dic3—21no—
149C22×Dic3C2×D412yes—
150C2×D12—22no—
151C2×D12—22no—
152C2×D12—21no—
153C2×D12—21no—
154C2×D12—21no—
155C2×D12—21no—
156C2×D12—21no—
157C2×D12—44no—
158C2×D12—44no—
159C2×D12—82no—
160C2×D12SD1614yes—
161C2×D12#4112yes—
162C2×C4×D3—22no—
163C2×C4×D3—22no—
164C2×C4×D3—22no—
165C2×C4×D3—22no—
166C2×C4×D3—22no—
167C2×C4×D3—22no—
168C2×C4×D3—42no—
169C2×C4×D3—42no—
170C2×C4×D3—42no—
171C2×C3×C8—21no—
172C2×C3×C8—21no—
173C2×C3×C8—21no—
174C2×C3×C8—21no—
175C2×C3×C8—21no—
176C2×C3×C8#4112yes—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—42no—
182C3×C4⋊C4—42no—
183C3×C4⋊C4—81no—
184C3×C42C2×D412yes—
185C3×C42#4112yes—
186(C2×D3)⋊C4—24no—
187(C2×D3)⋊C4—44no—
188(C2×D3)⋊C4—44no—
189(C2×D3)⋊C4—42no—
190(C2×D3)⋊C4—82no—
191(C2×D3)⋊C4—82no—
192C4⋊Dic3—22no—
193C4⋊Dic3—21no—
194C4⋊Dic3—21no—
195C4⋊Dic3—21no—
196C4⋊Dic3—21no—
197C4⋊Dic3—21no—
198C4⋊Dic3—21no—
199C4×Dic3—42no—
200C2×C3⋊C8—22no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8C2×D414yes—
203D3×C8—48no—
204C22×Q8—21no—
205C22×D4—31no—
206C22×D4—31no—
207C22×D4—62no—
208C23×C4—32no—
209C23×C4—61no—
210C22×C8—32no—
211C22×C8—64no—
212C22×C8—64no—
213C22×C8—122no—
214C22×C8C22×D311yescharacteristic
215C22×C8C22×D311yescharacteristic
216C4⋊Q8—22no—
217C4⋊Q8—22no—
218#121—64no—
219#121—122no—
220#121—124no—
221C4×D4—62no—
222C4×D4—62no—
223C4×D4—62no—
224C4×D4—62no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—31no—
229C2×C4⋊C4—31no—
230C2×C4⋊C4—31no—
231C2×C4⋊C4—31no—
232C2×C4⋊C4C22×D312yes—
233C2×C4⋊C4C22×D312yes—
234C2×C22⋊C4—32no—
235C2×C22⋊C4—62no—
236C2×C22⋊C4—62no—
237C2×C22⋊C4—61no—
238C2×C22⋊C4—62no—
239C2×C22⋊C4—62no—
240C2×C42—61no—
241C2×C42C22×D311yescharacteristic
242#106—64no—
243#106—122no—
244C4⋊C8—34no—
245#102—64no—
246#102—62no—
247#102—62no—
248C22⋊C8—64no—
249C4×C8—22no—
250#95—62no—
251#95—62no—
252#95—62no—
253#95—61no—
254C23×C3C22×D411yescharacteristic
255C22×D3—22no—
256C22×D3—22no—
257C22×D3—22no—
258C22×D3—22no—
259C22×D3—22no—
260C22×D3—22no—
261C22×D3—22no—
262C22×D3—41no—
263C22×D3—41no—
264C22×D3—41no—
265C22×D3—41no—
266C22×D3—41no—
267C22×D3—41no—
268C22×D3C2×SD1614yes—
269C3×Q8—84no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4#12112yes—
279C2×C3×C4#12112yes—
280C2×C3×C4C2×SD1612yes—
281C2×C3×C4#13512yes—
282C2×C3×C4C22×D411yescharacteristic
283C2×C3×C4C2×#4111yescharacteristic
284C2×C3×C4C2×#4111yescharacteristic
285C2×C3×C4C2×#4111yescharacteristic
286C2×C3×C4H(5,𝔽2)11yescharacteristic
287C2×C3×C42−1+411yescharacteristic
288C2×Dic3—22no—
289C2×Dic3—22no—
290C2×Dic3—22no—
291C2×Dic3—41no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—41no—
297C2×Dic3—41no—
298C2×Dic3—41no—
299C2×Dic3—41no—
300D12—22no—
301D12—22no—
302D12—28no—
303D12—41no—
304D12—41no—
305D12—88no—
306C4×D3—42no—
307C4×D3—42no—
308C4×D3—42no—
309C4×D3—42no—
310C4×D3—84no—
311C3×C8—22no—
312C3×C8—41no—
313C3⋊C8—24no—
314C24—32no—
315C24—61no—
316C2×Q8—42no—
317C2×Q8—81no—
318C2×Q8—81no—
319C2×D4—34no—
320C2×D4—32no—
321C2×D4—62no—
322C2×D4—62no—
323C2×D4—61no—
324C2×D4—61no—
325C2×D4—61no—
326C2×D4—61no—
327C2×D4—61no—
328C2×D4—124no—
329C2×D4—124no—
330C2×D4—242no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—32no—
334C22×C4—62no—
335C22×C4—62no—
336C22×C4—62no—
337C22×C4—62no—
338C22×C4—62no—
339C22×C4—62no—
340C22×C4—61no—
341C22×C4—61no—
342C22×C4—61no—
343C22×C4—122no—
344C22×C4—122no—
345C22×C4—122no—
346C22×C4D3×D411yescharacteristic
347C22×C4D3×D411yescharacteristic
348C22×C4C23×D311yescharacteristic
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—34no—
355C2×C8—62no—
356C2×C8—62no—
357C2×C8—128no—
358C2×C8#23912yes—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—62no—
366C4⋊C4—61no—
367C4⋊C4—61no—
368C4⋊C4—61no—
369C4⋊C4—61no—
370C4⋊C4—61no—
371C4⋊C4—61no—
372C4⋊C4—81no—
373C22⋊C4—64no—
374C22⋊C4—124no—
375C22⋊C4—124no—
376C22⋊C4—122no—
377C22⋊C4—242no—
378C22⋊C4—242no—
379C42—122no—
380C42#23512yes—
381C42D3×D412yes—
382C22×C3C2×#12111yescharacteristic
383C22×C3#54611yescharacteristic
384C22×C3#54711yescharacteristic
385C22×C3C22×SD1611yescharacteristic
386C22×C3C2×#13511yescharacteristic
387C22×C3#57611yescharacteristic
388C22×C3#57711yescharacteristic
389C2×D3—24no—
390C2×D3—24no—
391C2×D3—24no—
392C2×D3—42no—
393C2×D3—42no—
394C2×D3—42no—
395C2×D3—42no—
396C2×D3—42no—
397C2×D3—42no—
398C2×D3—42no—
399C2×D3—81no—
400C2×D3—81no—
401C2×D3—81no—
402C2×D3—81no—
403C3×C4—22no—
404C3×C4—82no—
405C3×C4—82no—
406C3×C4#46514yes—
407C3×C4#52911yescharacteristic
408C3×C4#54211yescharacteristic
409C3×C4#54611yescharacteristic
410C3×C4#54811yescharacteristic
411Dic3—42no—
412Dic3—81no—
413Dic3—81no—
414Dic3—81no—
415C23—34no—
416C23—62no—
417C23—62no—
418C23—62no—
419C23—62no—
420C23—62no—
421C23—62no—
422C23—62no—
423C23—121no—
424C23—121no—
425C23—121no—
426C23—121no—
427C23—121no—
428C23—121no—
429C23C2×D3×D411yescharacteristic
430Q8—84no—
431D4—62no—
432D4—68no—
433D4—62no—
434D4—121no—
435D4—121no—
436D4—248no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—62no—
446C2×C4—62no—
447C2×C4—62no—
448C2×C4—122no—
449C2×C4—122no—
450C2×C4—122no—
451C2×C4—122no—
452C2×C4—121no—
453C2×C4—121no—
454C2×C4—121no—
455C2×C4—121no—
456C2×C4—121no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4—121no—
460C2×C4—121no—
461C2×C4—244no—
462C2×C4#88412yes—
463C2×C4#88412yes—
464C2×C4D3×SD1612yes—
465C2×C4#91912yes—
466C2×C4C2×#23511yescharacteristic
467C2×C4C2×D3×D411yescharacteristic
468C2×C4C2×#23911yescharacteristic
469C2×C4#100711yescharacteristic
470C2×C4D3×#4111yescharacteristic
471C2×C4#100911yescharacteristic
472C8—22no—
473C8—41no—
474C8—64no—
475C2×C3#242711yescharacteristic
476C2×C3C2×#46511yescharacteristic
477C2×C3#306211yescharacteristic
478C2×C3#329511yescharacteristic
479C2×C3#330311yescharacteristic
480C2×C3#333511yescharacteristic
481C2×C3#334111yescharacteristic
482D3—44no—
483D3—82no—
484C22—64no—
485C22—64no—
486C22—64no—
487C22—122no—
488C22—122no—
489C22—122no—
490C22—122no—
491C22—122no—
492C22—122no—
493C22—122no—
494C22—241no—
495C22—241no—
496C22—241no—
497C22—241no—
498C22C2×#88411yescharacteristic
499C22#567611yescharacteristic
500C22#572611yescharacteristic
501C22#575711yescharacteristic
502C22C2×D3×SD1611yescharacteristic
503C22#576211yescharacteristic
504C22C2×#91911yescharacteristic
505C4—22no—
506C4—82no—
507C4—82no—
508C4—122no—
509C4—241no—
510C4—241no—
511C4—241no—
512C4#480714yes—
513C4#558011yescharacteristic
514C4#558211yescharacteristic
515C4#563011yescharacteristic
516C4#563311yescharacteristic
517C3#2533911yescharacteristic, complemented, Sylow
518C2—124no—
519C2—242no—
520C2#7662411yescharacteristic
521C2#7875211yescharacteristic
522C2C2×#480711yescharacteristic
523C2#8157711yescharacteristic
524C2#8161511yescharacteristic
525C2#8189911yescharacteristic
526C2#8194811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1990C2
socleC23×C3C22×D4
prime radical#74368C2
hypercenter#1990D3
hypocenterC3#25339
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
1. #83260C2semidirect
2. C2×#5084C2semidirect
3. C2×C8×D12C2semidirect
5. C3×#1990C2semidirect
6. C4×#821C2semidirect
7. #69568C2semidirect
8. #68883C2semidirect
9. #68868C2semidirect
10. #68833C2semidirect
11. #68823C2semidirect
50. #1990D3semidirect
104. C4⋊(C3⋊C8)D4semidirect
517. C3#25339semidirect

Maximal subgroups

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

subgroupindex
#832602
C2×#50842
C2×C8×D122
#743682
C3×#19902
C4×#8212
#695682
#688832
#688682
#688332
#688232
#253393