Finite Group Databasebeta

← #10690854

Subgroups — #10690854

Order 768.

728 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83192C211yesmaximal, characteristic
2C2×#4833C211yesmaximal, characteristic, complemented
3C2×C3×#331C211yesmaximal, characteristic, complemented
4C2×#4544C211yesmaximal, characteristic, complemented
5#70475C211yesmaximal, characteristic, complemented
6#67702C211yesmaximal, characteristic, complemented
7#67702C211yesmaximal, characteristic, complemented
8#9134—31nomaximal, Sylow
9C2×C3×#127C2211yescharacteristic
10C2×#938—21no—
11C2×#938—21no—
12C2×#938C412yes—
13C2×C42×D3C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×C3×C4⋊C8C2211yescharacteristic
16#5221—21no—
17#5221—21no—
18#5221C412yes—
19C2×(C2×D3)⋊C8—21no—
20C2×(C2×D3)⋊C8—21no—
21C2×C8×Dic3—21no—
22#4935C2211yescharacteristic
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25#4833—22no—
26#4833—22no—
27C3×#331—22no—
28#4550—21no—
29#4544—22no—
30C2×#431—31no—
31#2856—31no—
32#1797—31no—
33C2×#331—31no—
34C2×#331D311yescharacteristic, complemented
35#1296—31no—
36#1296—31no—
37C22×C3×D4—21no—
38C22×C3×D4—21no—
39C22×C3×D4C2×C412yes—
40C22×#67—22no—
41C22×C4×D3C2×C411yescharacteristic
42C22×C4×D3D411yescharacteristic
43C22×C4×D3D411yescharacteristic
44C22×C3×C8—21no—
45C22×C3×C8—21no—
46C3×#127—21no—
47C3×#127—21no—
48C3×#127—21no—
49C3×#127—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#217—22no—
52C2×#217—22no—
53C2×#217—22no—
54#938—42no—
55#938—42no—
56#938—42no—
57#938—42no—
58#938—44no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3C2×C412yes—
62C2×C4×Dic3C2×C411yescharacteristic
63C2×C4×Dic3D411yescharacteristic
64C2×C4×Dic3D411yescharacteristic
65C22×C3⋊C8—21no—
66C22×C3⋊C8—21no—
67C42×D3—21no—
68C42×D3—21no—
69C42×D3—21no—
70C42×D3—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8—21no—
74C3×C4⋊C8D412yescomplemented
75(C2×D3)⋊C8—24no—
76(C2×D3)⋊C8—42no—
77C8×Dic3—24no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)D412yescomplemented
82C2×#127C2×D311yescharacteristic
83C2×#121—32no—
84C2×#121—61no—
85C2×#121—61no—
86C22×C42—31no—
87#431—62no—
88#431—62no—
89C2×C4⋊C8—31no—
90C2×C4⋊C8—31no—
91C2×C4⋊C8C2×D311yescharacteristic
92C2×C4⋊C8C2×D311yescharacteristic
93C2×C22⋊C8—61no—
94C2×C22⋊C8—61no—
95C2×C4×C8—61no—
96#390—32no—
97#390—61no—
98#390—61no—
99#384—31no—
100#336—61no—
101#331—22no—
102#331—62no—
103C24×C3—22no—
104C23×D3C22⋊C411yescharacteristic
105C2×C3×D4—22no—
106C2×C3×D4—22no—
107C2×C3×D4—42no—
108C2×C3×D4—42no—
109C2×C3×D4—41no—
110C2×C3×D4—41no—
111C2×C3×D4—41no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C22×C3×C4C22×C411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C2×#67—44no—
125C2×#67—44no—
126C2×#67—44no—
127C22×Dic3—22no—
128C22×Dic3C22⋊C411yescharacteristic
129C22×Dic3C22⋊C411yescharacteristic
130C22×Dic3C22⋊C411yescharacteristic
131C2×C4×D3—21no—
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—22no—
144C2×C4×D3—41no—
145C2×C4×D3D812yes—
146C2×C4×D3SD1612yes—
147C2×C3×C8—22no—
148C2×C3×C8—41no—
149C2×C3×C8—41no—
150C2×C3×C8—41no—
151C2×C3×C8—41no—
152C2×C3×C8—41no—
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42#4111yescharacteristic
156C3×C42#4111yescharacteristic
157#217—44no—
158#217—44no—
159#217—44no—
160C4⋊Dic3—41no—
161C4⋊Dic3—41no—
162C4⋊Dic3—41no—
163C4⋊Dic3—41no—
164C4⋊Dic3—42no—
165C4×Dic3—21no—
166C4×Dic3—21no—
167C4×Dic3—21no—
168C4×Dic3—21no—
169C4×Dic3—21no—
170C4×Dic3—21no—
171C2×C3⋊C8—22no—
172C2×C3⋊C8—41no—
173C2×C3⋊C8—41no—
174C2×C3⋊C8—41no—
175C2×C3⋊C8—41no—
176C2×C3⋊C8—41no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—62no—
180C22×D4C4×D312yes—
181C23×C4—31no—
182C23×C4—31no—
183C23×C4—31no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—61no—
187C22×C8—61no—
188#127—21no—
189#127—21no—
190#127—21no—
191#127—21no—
192#121—122no—
193#121—122no—
194#121—122no—
195#121—122no—
196#121—124no—
197C2×C4⋊C4—32no—
198C2×C4⋊C4—61no—
199C2×C4⋊C4—61no—
200C2×C22⋊C4—62no—
201C2×C22⋊C4—62no—
202C2×C22⋊C4—62no—
203C2×C42—31no—
204C2×C42—31no—
205C2×C42—31no—
206C2×C42—61no—
207C2×C42—61no—
208C2×C42—61no—
209C2×C42—61no—
210C2×C42C22×D311yescharacteristic
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—32no—
215C4⋊C8—61no—
216C4⋊C8—61no—
217C4⋊C8—61no—
218C4⋊C8#6712yescomplemented
219C22⋊C8—64no—
220C22⋊C8—122no—
221C4×C8—64no—
222C23×C3—42no—
223C23×C3—42no—
224C23×C3—42no—
225C23×C3—42no—
226C23×C3—42no—
227C23×C3—42no—
228C23×C3—42no—
229C23×C3C2×C22⋊C411yescharacteristic
230C22×D3—21no—
231C22×D3—21no—
232C22×D3—21no—
233C22×D3—21no—
234C22×D3—21no—
235C22×D3—21no—
236C22×D3#10212yes—
237C3×D4—82no—
238C3×D4—82no—
239C3×D4—81no—
240C3×D4—81no—
241C3×D4—81no—
242C3×D4—81no—
243C3×D4—81no—
244C3×D4—81no—
245C3×D4—81no—
246C3×D4—81no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4C4×D411yescharacteristic
250C2×C3×C4C4×D411yescharacteristic
251C2×C3×C4#12011yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×C3×C4#12111yescharacteristic
254C2×C3×C4#12311yescharacteristic
255C2×C3×C4C2×D811yescharacteristic
256C2×C3×C4C2×SD1611yescharacteristic
257C2×C3×C4#13511yescharacteristic
258C2×C3×C4#13511yescharacteristic
259C2×C3×C4#13511yescharacteristic
260C2×C3×C4#13511yescharacteristic
261C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
262C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
263#67—88no—
264C2×Dic3—21no—
265C2×Dic3—21no—
266C2×Dic3—21no—
267C2×Dic3—21no—
268C2×Dic3—21no—
269C2×Dic3—21no—
270C2×Dic3—21no—
271C2×Dic3—42no—
272C2×Dic3—42no—
273C2×Dic3—42no—
274C2×Dic3#10012yes—
275C2×Dic3#10212yes—
276C4×D3—22no—
277C4×D3—22no—
278C4×D3—22no—
279C4×D3—22no—
280C4×D3—42no—
281C4×D3—42no—
282C4×D3—41no—
283C4×D3—41no—
284C4×D3—41no—
285C4×D3—41no—
286C3×C8—42no—
287C3×C8—81no—
288C3⋊C8—42no—
289C3⋊C8—81no—
290C24—22no—
291C24—31no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—124no—
309C2×D4—124no—
310C2×D4—124no—
311C22×C4—31no—
312C22×C4—31no—
313C22×C4—32no—
314C22×C4—32no—
315C22×C4—31no—
316C22×C4—62no—
317C22×C4—61no—
318C22×C4—61no—
319C22×C4—61no—
320C22×C4—61no—
321C22×C4—61no—
322C22×C4—61no—
323C22×C4—61no—
324C22×C4—61no—
325C22×C4—61no—
326C22×C4—61no—
327C22×C4—61no—
328C22×C4—61no—
329C22×C4—62no—
330C22×C4—121no—
331C22×C4C2×C4×D311yescharacteristic
332C22×C4D3×D411yescharacteristic
333C22×C4D3×D411yescharacteristic
334C2×C8—22no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—62no—
341C2×C8—121no—
342C2×C8—121no—
343C2×C8—121no—
344C2×C8—121no—
345C2×C8—121no—
346C4⋊C4—122no—
347C4⋊C4—121no—
348C4⋊C4—121no—
349C4⋊C4—121no—
350C4⋊C4—121no—
351C22⋊C4—124no—
352C22⋊C4—124no—
353C22⋊C4—124no—
354C42—61no—
355C42—61no—
356C42—61no—
357C42—61no—
358C42—61no—
359C42—61no—
360C42#23511yescharacteristic
361C42D3×D411yescharacteristic
362C42#23711yescharacteristic
363C42C2×#6711yescharacteristic
364C22×C3—82no—
365C22×C3—82no—
366C22×C3—82no—
367C22×C3—82no—
368C22×C3—82no—
369C22×C3—82no—
370C22×C3—82no—
371C22×C3#38611yescharacteristic
372C22×C3C2×#10011yescharacteristic
373C22×C3#41311yescharacteristic
374C22×C3C2×#10211yescharacteristic
375C22×C3#41611yescharacteristic
376C22×C3#41611yescharacteristic
377C22×C3#41811yescharacteristic
378C2×D3—22no—
379C2×D3—22no—
380C2×D3—22no—
381C2×D3—41no—
382C2×D3—41no—
383C2×D3—41no—
384C2×D3—41no—
385C3×C4—21no—
386C3×C4—21no—
387C3×C4#44911yescharacteristic
388C3×C4#44911yescharacteristic
389C3×C4#46511yescharacteristic
390C3×C4#46511yescharacteristic
391C3×C4#46611yescharacteristic
392C3×C4#46611yescharacteristic
393C3×C4#48211yescharacteristic
394C3×C4#48211yescharacteristic
395Dic3—22no—
396Dic3—22no—
397Dic3—41no—
398Dic3—82no—
399C23—32no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—61no—
408C23—61no—
409C23—61no—
410C23—61no—
411C23—61no—
412C23—61no—
413C23D3×C22⋊C411yescharacteristic
414D4—82no—
415D4—82no—
416D4—81no—
417D4—81no—
418D4—81no—
419D4—81no—
420D4—81no—
421D4—81no—
422D4—81no—
423D4—81no—
424D4—248no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—32no—
428C2×C4—32no—
429C2×C4—61no—
430C2×C4—61no—
431C2×C4—61no—
432C2×C4—61no—
433C2×C4—62no—
434C2×C4—62no—
435C2×C4—62no—
436C2×C4—62no—
437C2×C4—61no—
438C2×C4—61no—
439C2×C4—61no—
440C2×C4—122no—
441C2×C4—122no—
442C2×C4—121no—
443C2×C4—121no—
444C2×C4—121no—
445C2×C4—121no—
446C2×C4—122no—
447C2×C4—122no—
448C2×C4—122no—
449C2×C4#67⋊C411yescharacteristic
450C2×C4#88311yescharacteristic
451C2×C4#88411yescharacteristic
452C2×C4D3×D811yescharacteristic
453C2×C4#91111yescharacteristic
454C2×C4#91211yescharacteristic
455C2×C4#91211yescharacteristic
456C2×C4D3×SD1611yescharacteristic
457C2×C4#91411yescharacteristic
458C2×C4#91611yescharacteristic
459C2×C4#91611yescharacteristic
460C2×C4C4×#6711yescharacteristic
461C2×C4#93711yescharacteristic
462C2×C4#93811yescharacteristic
463C8—42no—
464C8—81no—
465C8—122no—
466C8—241no—
467C2×C3—162no—
468C2×C3#154011yescharacteristic
469C2×C3#154311yescharacteristic
470C2×C3#154511yescharacteristic
471C2×C3#155311yescharacteristic
472C2×C3#187511yescharacteristic
473C2×C3#189311yescharacteristic
474C2×C3#190011yescharacteristic
475D3—42no—
476C22—62no—
477C22—62no—
478C22—62no—
479C22—82no—
480C22—82no—
481C22—82no—
482C22—82no—
483C22—82no—
484C22—82no—
485C22—82no—
486C22—121no—
487C22—121no—
488C22—121no—
489C22—121no—
490C22D3×#10011yescharacteristic
491C22#474511yescharacteristic
492C22D3×#10211yescharacteristic
493C22#477011yescharacteristic
494C22#477211yescharacteristic
495C22#477211yescharacteristic
496C22#495611yescharacteristic
497C4—21no—
498C4—21no—
499C4—62no—
500C4—62no—
501C4—121no—
502C4—242no—
503C4#477511yescharacteristic
504C4#477811yescharacteristic
505C4#477911yescharacteristic
506C4#478411yescharacteristic
507C4#515711yescharacteristic
508C4#515811yescharacteristic
509C4#517111yescharacteristic
510C4#517211yescharacteristic
511C3#913411yescharacteristic, complemented, Sylow
512C2—122no—
513C2—162no—
514C2#6838111yescharacteristic
515C2#6838211yescharacteristic
516C2#6838311yescharacteristic
517C2#6838611yescharacteristic
518C2#7008111yescharacteristic
519C2#7020111yescharacteristic
520C2#7020211yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×#331C2
socleC23×C3C2×C22⋊C4
prime radical#83192C2
cosocleC2×C3×C42C23
hypercenterC2×#331D3
hypocenterC3#9134
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#331

Complements

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

NH ≅ G/Ntype
2. C2×#4833C2semidirect
3. C2×C3×#331C2semidirect
4. C2×#4544C2semidirect
5. #70475C2semidirect
6. #67702C2semidirect
7. #67702C2semidirect
34. C2×#331D3semidirect
74. C3×C4⋊C8D4semidirect
81. C4⋊(C3⋊C8)D4semidirect
218. C4⋊C8#67semidirect
511. C3#9134semidirect

Maximal subgroups

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

subgroupindex
#831922
C2×#48332
C2×C3×#3312
C2×#45442
#704752
#677022
#91343