Finite Group Databasebeta

← #10690542

Subgroups — #10690542

Order 768.

690 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4685C211yesmaximal, characteristic, complemented
2C4×#895C211yesmaximal, characteristic
3#70475C211yesmaximal, characteristic, complemented
4C2×#4483C211yesmaximal, characteristic, complemented
5C2×#4459C211yesmaximal, characteristic, complemented
6C3×#1279C211yesmaximal, characteristic, complemented
7#67619C211yesmaximal, characteristic, complemented
8#9076—31nomaximal, Sylow
9C2×#895C411yescharacteristic
10C2×C4×D12—21no—
11C2×C4×D12C2211yescharacteristic
12C2×C42×D3C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×C3×C4×C8C2211yescharacteristic
15C3×C4×C4⋊C4C2211yescharacteristic
16C2×(C2×D3)⋊C8—21no—
17C2×(C2×D3)⋊C8—21no—
18C2×Dic3⋊C8—21no—
19#4988C411yescharacteristic
20#4987C411yescharacteristic
21#4984C411yescharacteristic
22C4×(C2×D3)⋊C4—21no—
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24C2×C4×C3⋊C8C2211yescharacteristic
25#4685—22no—
26#4685—22no—
27#4670—21no—
28#4665—21no—
29#4550—21no—
30#4483—22no—
31#4459—22no—
32C2×#431—31no—
33C4×#121—31no—
34#1797—31no—
35C2×D4⋊C8—31no—
36C2×D4⋊C8—31no—
37#1279—31no—
38#1279D311yescharacteristic, complemented
39C22×D12—21no—
40C22×D12C2×C411yescharacteristic
41C22×C4×D3—21no—
42C22×C4×D3C2×C411yescharacteristic
43C22×C4×D3D411yescharacteristic
44C22×C4×D3D411yescharacteristic
45C22×C3×C8—21no—
46C22×C3×C8—21no—
47C2×C3×C4⋊C4C2×C411yescharacteristic
48C2×C3×C4⋊C4C2×C411yescharacteristic
49C2×C3×C42—21no—
50C2×C3×C42C2311yescharacteristic
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—21no—
53C2×(C2×D3)⋊C4—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3C2×C411yescharacteristic
58C2×C4×Dic3C2×C411yescharacteristic
59C2×C4×Dic3D411yescharacteristic
60C2×C4×Dic3D411yescharacteristic
61C22×C3⋊C8—21no—
62C22×C3⋊C8—21no—
63#895—22no—
64#895—22no—
65C4×D12—21no—
66C4×D12—21no—
67C4×D12—21no—
68C4×D12—21no—
69C4×D12—42no—
70C4×D12—42no—
71C42×D3—21no—
72C42×D3—21no—
73C42×D3—21no—
74C42×D3—21no—
75C3×C4⋊C8—21no—
76C3×C4⋊C8—21no—
77C3×C4×C8—21no—
78C3×C4×C8D412yescomplemented
79C3×#95—21no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82(C2×D3)⋊C8—24no—
83(C2×D3)⋊C8—42no—
84Dic3⋊C8—24no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87C4×C3⋊C8—21no—
88C4×C3⋊C8D412yescomplemented
89C2×#121—31no—
90C2×C4×D4—31no—
91C2×C4×D4—61no—
92C22×C42—31no—
93#431—62no—
94#431—62no—
95C2×C4⋊C8—31no—
96C2×C4⋊C8—61no—
97C2×C4⋊C8C2×D311yescharacteristic
98C2×C22⋊C8—61no—
99C2×C22⋊C8—61no—
100C2×C4×C8—31no—
101C2×C4×C8C2×D311yescharacteristic
102#390—31no—
103#389—31no—
104#388—61no—
105#386—61no—
106#380—31no—
107C4×C4⋊C4C2×D311yescharacteristic
108C4×C22⋊C4—61no—
109#336—61no—
110D4⋊C8—62no—
111D4⋊C8—62no—
112C23×D3—21no—
113C23×D3C22⋊C411yescharacteristic
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4C22×C411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×Dic3—21no—
120C22×Dic3C22⋊C411yescharacteristic
121C22×Dic3C22⋊C411yescharacteristic
122C22×Dic3C22⋊C411yescharacteristic
123C2×D12—21no—
124C2×D12—21no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—24no—
128C2×D12—21no—
129C2×D12—42no—
130C2×D12—42no—
131C2×D12M1612yes—
132C2×C4×D3—21no—
133C2×C4×D3—21no—
134C2×C4×D3—21no—
135C2×C4×D3—21no—
136C2×C4×D3—21no—
137C2×C4×D3—21no—
138C2×C4×D3—21no—
139C2×C4×D3—21no—
140C2×C4×D3—21no—
141C2×C4×D3—21no—
142C2×C4×D3—21no—
143C2×C4×D3—21no—
144C2×C4×D3—41no—
145C2×C4×D3—41no—
146C2×C4×D3—41no—
147C2×C4×D3—41no—
148C2×C4×D3—41no—
149C2×C4×D3—41no—
150C2×C4×D3—41no—
151C2×C4×D3—41no—
152C2×C4×D3D812yes—
153C2×C4×D3SD1612yes—
154C2×C3×C8—22no—
155C2×C3×C8—41no—
156C2×C3×C8—41no—
157C2×C3×C8—41no—
158C2×C3×C8—41no—
159C2×C3×C8—41no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—21no—
162C3×C4⋊C4—21no—
163C3×C4⋊C4M1612yes—
164C3×C42—41no—
165C3×C42—41no—
166C3×C42C2×D411yescharacteristic
167C3×C42C2×D411yescharacteristic
168C3×C42#4111yescharacteristic
169C3×C42#4111yescharacteristic
170(C2×D3)⋊C4—24no—
171(C2×D3)⋊C4—42no—
172(C2×D3)⋊C4—42no—
173(C2×D3)⋊C4—41no—
174(C2×D3)⋊C4—41no—
175(C2×D3)⋊C4—42no—
176C4⋊Dic3—21no—
177C4⋊Dic3—21no—
178C4⋊Dic3—42no—
179C4×Dic3—21no—
180C4×Dic3—21no—
181C4×Dic3—21no—
182C4×Dic3—21no—
183C4×Dic3—21no—
184C4×Dic3—21no—
185C2×C3⋊C8—22no—
186C2×C3⋊C8—41no—
187C2×C3⋊C8—41no—
188C2×C3⋊C8—41no—
189C2×C3⋊C8—41no—
190C2×C3⋊C8—41no—
191C22×D4—31no—
192C22×D4—61no—
193C23×C4—31no—
194C23×C4—31no—
195C23×C4—31no—
196C23×C4—61no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—61no—
200C22×C8—61no—
201#121—62no—
202#121—62no—
203C4×D4—61no—
204C4×D4—61no—
205C4×D4—61no—
206C4×D4—61no—
207C4×D4—122no—
208C4×D4—122no—
209C2×C4⋊C4—31no—
210C2×C4⋊C4—61no—
211C2×C4⋊C4C4×D311yescharacteristic
212C2×C4⋊C4C4×D311yescharacteristic
213C2×C22⋊C4—61no—
214C2×C22⋊C4—61no—
215C2×C22⋊C4—61no—
216C2×C22⋊C4—61no—
217C2×C22⋊C4—61no—
218C2×C42—21no—
219C2×C42—31no—
220C2×C42—31no—
221C2×C42—31no—
222C2×C42—61no—
223C2×C42—61no—
224C2×C42—61no—
225C2×C42—61no—
226C2×C42C22×D311yescharacteristic
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—64no—
230C4⋊C8—61no—
231C4⋊C8—61no—
232C22⋊C8—64no—
233C22⋊C8—122no—
234C4×C8—21no—
235C4×C8—32no—
236C4×C8—61no—
237C4×C8D1212yescomplemented
238#95—21no—
239#95—61no—
240#95—61no—
241C23×C3C2×C22⋊C411yescharacteristic
242C22×D3—21no—
243C22×D3—21no—
244C22×D3—21no—
245C22×D3—21no—
246C22×D3—21no—
247C22×D3—21no—
248C22×D3—22no—
249C22×D3—41no—
250C22×D3—41no—
251C22×D3—41no—
252C22×D3—41no—
253C22×D3—41no—
254C22×D3—41no—
255C22×D3#10212yes—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—22no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4C4×D411yescharacteristic
265C2×C3×C4C4×D411yescharacteristic
266C2×C3×C4#12011yescharacteristic
267C2×C3×C4#12111yescharacteristic
268C2×C3×C4#12111yescharacteristic
269C2×C3×C4#12311yescharacteristic
270C2×C3×C4C2×M1611yescharacteristic
271C2×C3×C4C2×M1611yescharacteristic
272C2×C3×C4#13111yescharacteristic
273C2×C3×C4#13111yescharacteristic
274C2×C3×C4C2×D811yescharacteristic
275C2×C3×C4C2×SD1611yescharacteristic
276C2×C3×C4#13511yescharacteristic
277C2×C3×C4#13511yescharacteristic
278C2×Dic3—21no—
279C2×Dic3—21no—
280C2×Dic3—21no—
281C2×Dic3—21no—
282C2×Dic3—21no—
283C2×Dic3—21no—
284C2×Dic3—21no—
285C2×Dic3—21no—
286C2×Dic3—41no—
287C2×Dic3—41no—
288C2×Dic3—41no—
289C2×Dic3#10212yes—
290D12—22no—
291D12—22no—
292D12—48no—
293D12—41no—
294D12—41no—
295C4×D3—22no—
296C4×D3—22no—
297C4×D3—22no—
298C4×D3—22no—
299C4×D3—42no—
300C4×D3—42no—
301C4×D3—41no—
302C4×D3—41no—
303C4×D3—41no—
304C4×D3—41no—
305C4×D3—81no—
306C4×D3—81no—
307C4×D3—81no—
308C4×D3—81no—
309C3×C8—42no—
310C3×C8—81no—
311C3⋊C8—42no—
312C3⋊C8—81no—
313C24—31no—
314C24—61no—
315C2×D4—32no—
316C2×D4—64no—
317C2×D4—61no—
318C2×D4—61no—
319C2×D4—61no—
320C2×D4—61no—
321C2×D4—61no—
322C2×D4—122no—
323C2×D4—122no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—31no—
327C22×C4—32no—
328C22×C4—32no—
329C22×C4—31no—
330C22×C4—31no—
331C22×C4—61no—
332C22×C4—61no—
333C22×C4—61no—
334C22×C4—61no—
335C22×C4—61no—
336C22×C4—61no—
337C22×C4—61no—
338C22×C4—61no—
339C22×C4—61no—
340C22×C4—61no—
341C22×C4—61no—
342C22×C4—61no—
343C22×C4—61no—
344C22×C4—121no—
345C22×C4—121no—
346C22×C4—121no—
347C22×C4—121no—
348C22×C4—121no—
349C22×C4—121no—
350C22×C4—121no—
351C22×C4—121no—
352C22×C4C2×C4×D311yescharacteristic
353C22×C4D3×D411yescharacteristic
354C22×C4D3×D411yescharacteristic
355C2×C8—22no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—62no—
362C2×C8—121no—
363C2×C8—121no—
364C2×C8—121no—
365C2×C8—121no—
366C2×C8—121no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—61no—
371C4⋊C4—61no—
372C4⋊C4—122no—
373C4⋊C4ΓL(1,𝔽25)12yes—
374C22⋊C4—64no—
375C22⋊C4—122no—
376C22⋊C4—122no—
377C22⋊C4—122no—
378C22⋊C4—121no—
379C22⋊C4—121no—
380C42—41no—
381C42—41no—
382C42—61no—
383C42—61no—
384C42—61no—
385C42—61no—
386C42—61no—
387C42—61no—
388C42C2×D1211yescharacteristic
389C42#23511yescharacteristic
390C42D3×D411yescharacteristic
391C42#23911yescharacteristic
392C22×C3#38611yescharacteristic
393C22×C3#40711yescharacteristic
394C22×C3#40811yescharacteristic
395C22×C3C2×#10211yescharacteristic
396C22×C3#41611yescharacteristic
397C22×C3#42111yescharacteristic
398C22×C3#42111yescharacteristic
399C2×D3—22no—
400C2×D3—22no—
401C2×D3—22no—
402C2×D3—41no—
403C2×D3—41no—
404C2×D3—41no—
405C2×D3—41no—
406C2×D3—42no—
407C2×D3—42no—
408C2×D3—42no—
409C2×D3—81no—
410C2×D3—81no—
411C2×D3—81no—
412C2×D3—81no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—42no—
416C3×C4—81no—
417C3×C4#43511yescharacteristic
418C3×C4#43511yescharacteristic
419C3×C4#43611yescharacteristic
420C3×C4#43611yescharacteristic
421C3×C4#46511yescharacteristic
422C3×C4#46511yescharacteristic
423C3×C4#46611yescharacteristic
424C3×C4#46611yescharacteristic
425Dic3—22no—
426Dic3—41no—
427Dic3—41no—
428Dic3—81no—
429C23—32no—
430C23—61no—
431C23—61no—
432C23—61no—
433C23—61no—
434C23—61no—
435C23—61no—
436C23—62no—
437C23—121no—
438C23—121no—
439C23—121no—
440C23—121no—
441C23—121no—
442C23—121no—
443C23D3×C22⋊C411yescharacteristic
444D4—62no—
445D4—62no—
446D4—128no—
447D4—121no—
448D4—121no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—32no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—61no—
459C2×C4—61no—
460C2×C4—62no—
461C2×C4—62no—
462C2×C4—62no—
463C2×C4—62no—
464C2×C4—61no—
465C2×C4—61no—
466C2×C4—61no—
467C2×C4—61no—
468C2×C4—61no—
469C2×C4—61no—
470C2×C4—122no—
471C2×C4—122no—
472C2×C4—121no—
473C2×C4—121no—
474C2×C4—121no—
475C2×C4—121no—
476C2×C4—121no—
477C2×C4—121no—
478C2×C4—121no—
479C2×C4—241no—
480C2×C4—241no—
481C2×C4—241no—
482C2×C4—241no—
483C2×C4C4×D1211yescharacteristic
484C2×C4#88211yescharacteristic
485C2×C4#88411yescharacteristic
486C2×C4#89311yescharacteristic
487C2×C4#89411yescharacteristic
488C2×C4#89511yescharacteristic
489C2×C4C2×ΓL(1,𝔽25)11yescharacteristic
490C2×C4#90111yescharacteristic
491C2×C4D3×M1611yescharacteristic
492C2×C4#90711yescharacteristic
493C2×C4D3×D811yescharacteristic
494C2×C4D3×SD1611yescharacteristic
495C2×C4#91611yescharacteristic
496C2×C4#91911yescharacteristic
497C8—42no—
498C8—81no—
499C8—122no—
500C8—241no—
501C2×C3#148011yescharacteristic
502C2×C3#148211yescharacteristic
503C2×C3#148611yescharacteristic
504C2×C3#148811yescharacteristic
505C2×C3#187011yescharacteristic
506C2×C3#189311yescharacteristic
507C2×C3#189711yescharacteristic
508D3—42no—
509D3—82no—
510C22—62no—
511C22—62no—
512C22—62no—
513C22—121no—
514C22—121no—
515C22—121no—
516C22—121no—
517C22—122no—
518C22—122no—
519C22—122no—
520C22—241no—
521C22—241no—
522C22—241no—
523C22—241no—
524C22#466511yescharacteristic
525C22#472611yescharacteristic
526C22#472711yescharacteristic
527C22D3×#10211yescharacteristic
528C22#480411yescharacteristic
529C22#482111yescharacteristic
530C22#482111yescharacteristic
531C4—21no—
532C4—21no—
533C4—42no—
534C4—62no—
535C4—81no—
536C4—121no—
537C4—121no—
538C4—241no—
539C4#468811yescharacteristic
540C4#470611yescharacteristic
541C4#477511yescharacteristic
542C4#480711yescharacteristic
543C4#483611yescharacteristic
544C4#483811yescharacteristic
545C4#486411yescharacteristic
546C4#488311yescharacteristic
547C3#907611yescharacteristic, complemented, Sylow
548C2—122no—
549C2—242no—
550C2#6816811yescharacteristic
551C2#6817211yescharacteristic
552C2#6823311yescharacteristic
553C2#6823711yescharacteristic
554C2#6866911yescharacteristic
555C2#6875711yescharacteristic
556C2#6888311yescharacteristic
subgroupG/H
centerC22×C4D3×D4
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1279C2
socleC23×C3C2×C22⋊C4
prime radicalC2×#895C4
cosocleC2×C3×C42C23
hypercenter#1279D3
hypocenterC3#9076
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
1. C2×#4685C2semidirect
3. #70475C2semidirect
4. C2×#4483C2semidirect
5. C2×#4459C2semidirect
6. C3×#1279C2semidirect
7. #67619C2semidirect
38. #1279D3semidirect
78. C3×C4×C8D4semidirect
88. C4×C3⋊C8D4semidirect
237. C4×C8D12semidirect
547. C3#9076semidirect

Maximal subgroups

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

subgroupindex
C2×#46852
C4×#8952
#704752
C2×#44832
C2×#44592
C3×#12792
#676192
#90763