Finite Group Databasebeta

← #10690530

Subgroups — #10690530

Order 768.

694 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
1C4×#938C211yesmaximal, characteristic
2C2×#4685C211yesmaximal, characteristic, complemented
3C2×C3×D4⋊C8C211yesmaximal, characteristic, complemented
4C2×D4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
5#70475C211yesmaximal, characteristic, complemented
6#67704C211yesmaximal, characteristic, complemented
7#67655C211yesmaximal, characteristic, complemented
8#9074—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×#938C411yescharacteristic
11C2×C4×#67—21no—
12C2×C42×D3C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×C3×C4×C8C2211yescharacteristic
15#5221C411yescharacteristic
16C4×#217—21no—
17C2×(C2×D3)⋊C8—21no—
18C2×(C2×D3)⋊C8—21no—
19C2×Dic3⋊C8—21no—
20#4987C411yescharacteristic
21#4972C411yescharacteristic
22#4956—21no—
23#4944—21no—
24C4×C4⋊Dic3C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26C2×C4×C3⋊C8C2211yescharacteristic
27#4685—22no—
28#4685—22no—
29C3×D4⋊C8—22no—
30#4550—21no—
31D4⋊(C3⋊C8)—22no—
32C2×#431—31no—
33C4×#121—31no—
34#1797—31no—
35C2×D4⋊C8—31no—
36C2×D4⋊C8D311yescharacteristic, complemented
37#1279—31no—
38#1279—31no—
39C22×C3×D4C2×C411yescharacteristic
40C23×C3×C4—21no—
41C22×#67—21no—
42C22×C4×D3C2×C411yescharacteristic
43C22×C4×D3D411yescharacteristic
44C22×C4×D3D411yescharacteristic
45C22×C3×C8—21no—
46C22×C3×C8—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C2×C3×C4⋊C4C2×C411yescharacteristic
52C2×C3×C22⋊C4—21no—
53C2×C3×C42C2311yescharacteristic
54C2×#217—21no—
55C2×#217—21no—
56C2×#217—21no—
57#938—22no—
58#938C814yescomplemented
59C4×#67—42no—
60C4×#67—42no—
61C2×(C2×D3)⋊C4—21no—
62C2×C4⋊Dic3C2×C411yescharacteristic
63C2×C4⋊Dic3C2×C411yescharacteristic
64C2×Dic3⋊C4—21no—
65C2×C4×Dic3—21no—
66C2×C4×Dic3C2×C411yescharacteristic
67C2×C4×Dic3D411yescharacteristic
68C2×C4×Dic3D411yescharacteristic
69C22×C3⋊C8—21no—
70C22×C3⋊C8—21no—
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—61no—
91C2×C4×D4C2×D311yescharacteristic
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⋊C4—31no—
108C4×C22⋊C4—61no—
109#336—61no—
110D4⋊C8—22no—
111D4⋊C8—62no—
112C24×C3—21no—
113C23×D3C22⋊C411yescharacteristic
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4C2×C812yes—
120C22×C3×C4—21no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4—41no—
127C22×C3×C4C22×C411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C2×#67—24no—
131C2×#67—42no—
132C2×#67—42no—
133C22×Dic3—21no—
134C22×Dic3—21no—
135C22×Dic3C22⋊C411yescharacteristic
136C22×Dic3C22⋊C411yescharacteristic
137C22×Dic3C22⋊C411yescharacteristic
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—21no—
145C2×C4×D3—21no—
146C2×C4×D3—21no—
147C2×C4×D3—21no—
148C2×C4×D3—21no—
149C2×C4×D3—21no—
150C2×C4×D3—41no—
151C2×C4×D3—41no—
152C2×C4×D3C2×C812yes—
153C2×C4×D3M1612yes—
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×C22⋊C4—41no—
163C3×C22⋊C4—41no—
164C3×C42C2×D411yescharacteristic
165C3×C42C2×D411yescharacteristic
166C3×C42#4111yescharacteristic
167C3×C42#4111yescharacteristic
168#217—24no—
169#217—42no—
170#217—42no—
171(C2×D3)⋊C4—42no—
172C4⋊Dic3—21no—
173C4⋊Dic3—21no—
174C4⋊Dic3—21no—
175C4⋊Dic3C2×C812yes—
176Dic3⋊C4—42no—
177C4×Dic3—21no—
178C4×Dic3—21no—
179C4×Dic3—21no—
180C4×Dic3—21no—
181C4×Dic3—21no—
182C4×Dic3—21no—
183C4×Dic3—41no—
184C4×Dic3—41no—
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—61no—
192C22×D4C4×D311yescharacteristic
193C23×C4—21no—
194C23×C4—31no—
195C23×C4—31no—
196C23×C4—31no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—61no—
200C22×C8—61no—
201#121—34no—
202#121—62no—
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—122no—
208C4×D4—122no—
209C2×C4⋊C4—31no—
210C2×C4⋊C4—31no—
211C2×C4⋊C4—61no—
212C2×C4⋊C4C4×D311yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—61no—
215C2×C22⋊C4—61no—
216C2×C22⋊C4—61no—
217C2×C22⋊C4—61no—
218C2×C42—31no—
219C2×C42—31no—
220C2×C42—31no—
221C2×C42—61no—
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×C8#6712yescomplemented
238#95—61no—
239#95—61no—
240#95—61no—
241C23×C3—22no—
242C23×C3—41no—
243C23×C3—41no—
244C23×C3—41no—
245C23×C3—41no—
246C23×C3—41no—
247C23×C3—41no—
248C23×C3C2×C22⋊C411yescharacteristic
249C22×D3—21no—
250C22×D3—21no—
251C22×D3—21no—
252C22×D3—21no—
253C22×D3—21no—
254C22×D3—21no—
255C22×D3C22⋊C812yes—
256C3×D4—22no—
257C3×D4—22no—
258C3×D4—41no—
259C3×D4—41no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—81no—
266C2×C3×C4—81no—
267C2×C3×C4—81no—
268C2×C3×C4—81no—
269C2×C3×C4C4×D411yescharacteristic
270C2×C3×C4C4×D411yescharacteristic
271C2×C3×C4#12011yescharacteristic
272C2×C3×C4#12111yescharacteristic
273C2×C3×C4#12111yescharacteristic
274C2×C3×C4#12311yescharacteristic
275C2×C3×C4C22×C811yescharacteristic
276C2×C3×C4C2×M1611yescharacteristic
277C2×C3×C4#13111yescharacteristic
278C2×C3×C4#13111yescharacteristic
279C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
280C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
281C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
282C2×C3×C4#13711yescharacteristic
283#67—48no—
284C2×Dic3—21no—
285C2×Dic3—21no—
286C2×Dic3—21no—
287C2×Dic3—21no—
288C2×Dic3—22no—
289C2×Dic3—21no—
290C2×Dic3—21no—
291C2×Dic3—21no—
292C2×Dic3—21no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—41no—
297C2×Dic3—41no—
298C2×Dic3C22⋊C812yes—
299C4×D3—22no—
300C4×D3—22no—
301C4×D3—22no—
302C4×D3—22no—
303C4×D3—42no—
304C4×D3—42no—
305C4×D3—41no—
306C4×D3—41no—
307C4×D3—41no—
308C4×D3—41no—
309C3×C8—42no—
310C3×C8—81no—
311C3⋊C8—42no—
312C3⋊C8—81no—
313C24—21no—
314C24—31no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—64no—
321C2×D4—122no—
322C2×D4—122no—
323C2×D4D3×C812yes—
324C22×C4—21no—
325C22×C4—31no—
326C22×C4—32no—
327C22×C4—32no—
328C22×C4—31no—
329C22×C4—31no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
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—61no—
345C22×C4—61no—
346C22×C4—61no—
347C22×C4—61no—
348C22×C4—61no—
349C22×C4—61no—
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—32no—
370C4⋊C4—61no—
371C4⋊C4—61no—
372C4⋊C4—61no—
373C4⋊C4—122no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—64no—
377C22⋊C4—122no—
378C22⋊C4—122no—
379C22⋊C4—122no—
380C42—61no—
381C42—61no—
382C42—61no—
383C42—61no—
384C42—61no—
385C42—61no—
386C42—121no—
387C42—121no—
388C42#23511yescharacteristic
389C42D3×D411yescharacteristic
390C42#23711yescharacteristic
391C42C2×#6711yescharacteristic
392C22×C3—42no—
393C22×C3—42no—
394C22×C3—42no—
395C22×C3—81no—
396C22×C3—81no—
397C22×C3—81no—
398C22×C3—81no—
399C22×C3#38611yescharacteristic
400C22×C3C2×C22⋊C811yescharacteristic
401C22×C3#40811yescharacteristic
402C22×C3#41711yescharacteristic
403C22×C3#41811yescharacteristic
404C22×C3#42111yescharacteristic
405C22×C3#42111yescharacteristic
406C2×D3—22no—
407C2×D3—22no—
408C2×D3—22no—
409C2×D3—41no—
410C2×D3—41no—
411C2×D3—41no—
412C2×D3—41no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—81no—
416C3×C4C8×D411yescharacteristic
417C3×C4C8×D411yescharacteristic
418C3×C4#43511yescharacteristic
419C3×C4#43511yescharacteristic
420C3×C4#46811yescharacteristic
421C3×C4#46811yescharacteristic
422C3×C4#46911yescharacteristic
423C3×C4#46911yescharacteristic
424Dic3—22no—
425Dic3—41no—
426Dic3—41no—
427Dic3—42no—
428Dic3—81no—
429C23—22no—
430C23—32no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—61no—
438C23—61no—
439C23—61no—
440C23—61no—
441C23—61no—
442C23—61no—
443C23D3×C22⋊C411yescharacteristic
444D4—22no—
445D4—22no—
446D4—41no—
447D4—41no—
448D4—128no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—32no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—61no—
456C2×C4—61no—
457C2×C4—62no—
458C2×C4—62no—
459C2×C4—62no—
460C2×C4—62no—
461C2×C4—61no—
462C2×C4—61no—
463C2×C4—62no—
464C2×C4—61no—
465C2×C4—61no—
466C2×C4—61no—
467C2×C4—61no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—122no—
473C2×C4—122no—
474C2×C4—121no—
475C2×C4—121no—
476C2×C4—121no—
477C2×C4—121no—
478C2×C4—121no—
479C2×C4—121no—
480C2×C4—121no—
481C2×C4—121no—
482C2×C4—121no—
483C2×C4#67⋊C411yescharacteristic
484C2×C4#88311yescharacteristic
485C2×C4#88411yescharacteristic
486C2×C4C2×D3×C811yescharacteristic
487C2×C4#90111yescharacteristic
488C2×C4D3×M1611yescharacteristic
489C2×C4#90711yescharacteristic
490C2×C4#91111yescharacteristic
491C2×C4#91111yescharacteristic
492C2×C4#91411yescharacteristic
493C2×C4#91511yescharacteristic
494C2×C4C4×#6711yescharacteristic
495C2×C4#93711yescharacteristic
496C2×C4#93811yescharacteristic
497C8—42no—
498C8—81no—
499C8—122no—
500C8—241no—
501C2×C3—82no—
502C2×C3#147711yescharacteristic
503C2×C3#147811yescharacteristic
504C2×C3#149511yescharacteristic
505C2×C3#149611yescharacteristic
506C2×C3#186911yescharacteristic
507C2×C3#189511yescharacteristic
508C2×C3#189711yescharacteristic
509D3—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—62no—
514C22—62no—
515C22—62no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22—81no—
520C22—121no—
521C22—121no—
522C22—121no—
523C22—121no—
524C22D3×C22⋊C811yescharacteristic
525C22#472711yescharacteristic
526C22#477011yescharacteristic
527C22#477111yescharacteristic
528C22#482111yescharacteristic
529C22#482111yescharacteristic
530C22#495611yescharacteristic
531C4—21no—
532C4—21no—
533C4—62no—
534C4—81no—
535C4—121no—
536C4—122no—
537C4—121no—
538C4—241no—
539C4#67⋊C811yescharacteristic
540C4#472811yescharacteristic
541C4#478011yescharacteristic
542C4#478211yescharacteristic
543C4C8×#6711yescharacteristic
544C4#512711yescharacteristic
545C4#515911yescharacteristic
546C4#517411yescharacteristic
547C3#907411yescharacteristic, complemented, Sylow
548C2—82no—
549C2—122no—
550C2#6816611yescharacteristic
551C2#6816711yescharacteristic
552C2#6817811yescharacteristic
553C2#6817911yescharacteristic
554C2#6995711yescharacteristic
555C2#7020311yescharacteristic
556C2#7038311yescharacteristic
subgroupG/H
centerC22×C4D3×D4
commutatorC2×C3×C4C22×C8
FrattiniC2×C42C22×D3
FittingC2×C3×D4⋊C8C2
socleC23×C3C2×C22⋊C4
prime radicalC2×#938C4
cosocleC2×C3×C42C23
hypercenterC2×D4⋊C8D3
hypocenterC3#9074
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×D4⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#4685C2semidirect
3. C2×C3×D4⋊C8C2semidirect
4. C2×D4⋊(C3⋊C8)C2semidirect
5. #70475C2semidirect
6. #67704C2semidirect
7. #67655C2semidirect
36. C2×D4⋊C8D3semidirect
58. #938C8semidirect
78. C3×C4×C8D4semidirect
88. C4×C3⋊C8D4semidirect
237. C4×C8#67semidirect
547. C3#9074semidirect

Maximal subgroups

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

subgroupindex
C4×#9382
C2×#46852
C2×C3×D4⋊C82
C2×D4⋊(C3⋊C8)2
#704752
#677042
#676552
#90743