Finite Group Databasebeta

← #10690896

Subgroups — #10690896

Order 768.

708 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
1C2×#4835C211yesmaximal, characteristic, complemented
2#74375C211yesmaximal, characteristic
3#70475C211yesmaximal, characteristic, complemented
4C2×#4485C211yesmaximal, characteristic, complemented
5C2×#4485C211yesmaximal, characteristic, complemented
6C3×#1296C211yesmaximal, characteristic, complemented
7#67634C211yesmaximal, characteristic, complemented
8#9142—31nomaximal, Sylow
9C2×#895—21no—
10C2×#895—21no—
11C2×#895C412yes—
12C2×#874C2211yescharacteristic
13C2×C42×D3C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×C3×C4⋊C8C2211yescharacteristic
16C3×#384C2211yescharacteristic
17C2×(C2×D3)⋊C8—21no—
18C2×(C2×D3)⋊C8—21no—
19C2×#815—21no—
20#4988—21no—
21#4988—21no—
22#4988C412yes—
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25#4835—22no—
26#4835—22no—
27#4550—21no—
28#4485—22no—
29#4485—22no—
30C2×#432—31no—
31#2856—31no—
32#1797—31no—
33C2×#331—31no—
34C2×#331—31no—
35#1296—31no—
36#1296D311yescharacteristic, complemented
37C22×D12—22no—
38C22×D12—21no—
39C22×D12—21no—
40C22×D12C2×C412yes—
41C22×C4×D3C2×C411yescharacteristic
42C22×C4×D3D411yescharacteristic
43C22×C4×D3D411yescharacteristic
44C22×C3×C8—21no—
45C22×C3×C8—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4C2×C412yes—
49C2×C3×C42C2311yescharacteristic
50C2×(C2×D3)⋊C4—22no—
51C2×(C2×D3)⋊C4—22no—
52C2×(C2×D3)⋊C4—22no—
53C2×C4×Dic3C2×C411yescharacteristic
54C2×C4×Dic3D411yescharacteristic
55C2×C4×Dic3D411yescharacteristic
56C22×C3⋊C8—21no—
57C22×C3⋊C8—21no—
58#895—44no—
59#895—42no—
60#895—42no—
61#895—42no—
62#895—42no—
63#874—21no—
64#874—21no—
65#874—21no—
66#874—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—
77#815—24no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)D412yescomplemented
82C2×#127—31no—
83C2×#121—32no—
84C2×#121—61no—
85C2×#121—61no—
86C22×C42—31no—
87#432—62no—
88#432—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×#97—61no—
96#390—32no—
97#390—61no—
98#390—61no—
99#384C2×D311yescharacteristic
100#336—61no—
101#331—62no—
102#331—62no—
103C23×D3—22no—
104C23×D3C22⋊C411yescharacteristic
105C22×C3×C4—22no—
106C22×C3×C4C22×C411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×Dic3C22⋊C411yescharacteristic
110C22×Dic3C22⋊C411yescharacteristic
111C22×Dic3C22⋊C411yescharacteristic
112C2×D12—22no—
113C2×D12—22no—
114C2×D12—42no—
115C2×D12—42no—
116C2×D12—44no—
117C2×D12—44no—
118C2×D12—44no—
119C2×D12—41no—
120C2×D12—41no—
121C2×D12—41no—
122C2×D12—41no—
123C2×D12—41no—
124C2×D12—41no—
125C2×D12—41no—
126C2×D12—41no—
127C2×D12—41no—
128C2×D12—41no—
129C2×D12—41no—
130C2×D12—41no—
131C2×C4×D3—22no—
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—21no—
145C2×C4×D3—21no—
146C2×C4×D3—41no—
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×C4⋊C4—42no—
154C3×C4⋊C4—41no—
155C3×C4⋊C4—41no—
156C3×C4⋊C4—41no—
157C3×C4⋊C4—41no—
158C3×C42C2×D411yescharacteristic
159C3×C42C2×D411yescharacteristic
160C3×C42#4111yescharacteristic
161C3×C42#4111yescharacteristic
162(C2×D3)⋊C4—44no—
163(C2×D3)⋊C4—44no—
164(C2×D3)⋊C4—44no—
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—32no—
178C22×D4—62no—
179C22×D4—61no—
180C22×D4—61no—
181C23×C4—31no—
182C23×C4—31no—
183C23×C4—31no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—61no—
187C22×C8—61no—
188#127—61no—
189#127—61no—
190#127—61no—
191#127—61no—
192#121—124no—
193#121—122no—
194#121—122no—
195#121—122no—
196#121—122no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4C4×D312yes—
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⋊C8D1212yescomplemented
219C22⋊C8—64no—
220C22⋊C8—122no—
221#97—64no—
222C23×C3C2×C22⋊C411yescharacteristic
223C22×D3—21no—
224C22×D3—21no—
225C22×D3—21no—
226C22×D3—21no—
227C22×D3—21no—
228C22×D3—21no—
229C22×D3—42no—
230C22×D3—42no—
231C22×D3—42no—
232C22×D3—42no—
233C22×D3—42no—
234C22×D3—42no—
235C22×D3—42no—
236C22×D3#10012yes—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4C4×D411yescharacteristic
243C2×C3×C4C4×D411yescharacteristic
244C2×C3×C4#12011yescharacteristic
245C2×C3×C4#12111yescharacteristic
246C2×C3×C4#12111yescharacteristic
247C2×C3×C4#12311yescharacteristic
248C2×C3×C4#13511yescharacteristic
249C2×C3×C4#13511yescharacteristic
250C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
251C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
252C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
253C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
254C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
255C2×C3×C4#13711yescharacteristic
256C2×Dic3—21no—
257C2×Dic3—21no—
258C2×Dic3—21no—
259C2×Dic3—21no—
260C2×Dic3—21no—
261C2×Dic3—21no—
262C2×Dic3—21no—
263C2×Dic3—21no—
264C2×Dic3—21no—
265D12—82no—
266D12—82no—
267D12—88no—
268D12—81no—
269D12—81no—
270D12—81no—
271D12—81no—
272D12—81no—
273D12—81no—
274D12—81no—
275D12—81no—
276C4×D3—42no—
277C4×D3—42no—
278C4×D3—41no—
279C4×D3—41no—
280C4×D3—41no—
281C4×D3—41no—
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—31no—
291C24—62no—
292C2×D4—62no—
293C2×D4—62no—
294C2×D4—122no—
295C2×D4—122no—
296C2×D4—124no—
297C2×D4—124no—
298C2×D4—124no—
299C2×D4—121no—
300C2×D4—121no—
301C2×D4—121no—
302C2×D4—121no—
303C2×D4—121no—
304C2×D4—121no—
305C2×D4—121no—
306C2×D4—121no—
307C2×D4—121no—
308C2×D4—121no—
309C2×D4—121no—
310C2×D4—121no—
311C22×C4—22no—
312C22×C4—31no—
313C22×C4—31no—
314C22×C4—31no—
315C22×C4—62no—
316C22×C4—61no—
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—61no—
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—42no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C22⋊C4—124no—
352C22⋊C4—124no—
353C22⋊C4—124no—
354C42—61no—
355C42—61no—
356C42—61no—
357C42—61no—
358C42—61no—
359C42—61no—
360C42C2×D1211yescharacteristic
361C42#23511yescharacteristic
362C42D3×D411yescharacteristic
363C42#23911yescharacteristic
364C22×C3#38611yescharacteristic
365C22×C3C2×#10011yescharacteristic
366C22×C3#41311yescharacteristic
367C22×C3#41611yescharacteristic
368C22×C3#41711yescharacteristic
369C22×C3#41811yescharacteristic
370C22×C3#41811yescharacteristic
371C2×D3—22no—
372C2×D3—22no—
373C2×D3—22no—
374C2×D3—41no—
375C2×D3—41no—
376C2×D3—41no—
377C2×D3—41no—
378C2×D3—82no—
379C2×D3—82no—
380C2×D3—82no—
381C2×D3—82no—
382C2×D3—82no—
383C2×D3—82no—
384C2×D3—82no—
385C3×C4—21no—
386C3×C4—21no—
387C3×C4—82no—
388C3×C4#44911yescharacteristic
389C3×C4#44911yescharacteristic
390C3×C4#46811yescharacteristic
391C3×C4#46811yescharacteristic
392C3×C4#46911yescharacteristic
393C3×C4#46911yescharacteristic
394C3×C4#48211yescharacteristic
395C3×C4#48211yescharacteristic
396Dic3—41no—
397Dic3—41no—
398Dic3—41no—
399C23—32no—
400C23—61no—
401C23—61no—
402C23—61no—
403C23—61no—
404C23—61no—
405C23—61no—
406C23—122no—
407C23—122no—
408C23—122no—
409C23—122no—
410C23—122no—
411C23—122no—
412C23—122no—
413C23D3×C22⋊C411yescharacteristic
414D4—242no—
415D4—242no—
416D4—248no—
417D4—241no—
418D4—241no—
419D4—241no—
420D4—241no—
421D4—241no—
422D4—241no—
423D4—241no—
424D4—241no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—61no—
431C2×C4—61no—
432C2×C4—61no—
433C2×C4—61no—
434C2×C4—61no—
435C2×C4—61no—
436C2×C4—61no—
437C2×C4—61no—
438C2×C4—61no—
439C2×C4—122no—
440C2×C4—122no—
441C2×C4—121no—
442C2×C4—121no—
443C2×C4—121no—
444C2×C4—121no—
445C2×C4—121no—
446C2×C4—121no—
447C2×C4—121no—
448C2×C4—121no—
449C2×C4C4×D1211yescharacteristic
450C2×C4#88211yescharacteristic
451C2×C4#88411yescharacteristic
452C2×C4#89311yescharacteristic
453C2×C4#89411yescharacteristic
454C2×C4#89511yescharacteristic
455C2×C4#91111yescharacteristic
456C2×C4#91111yescharacteristic
457C2×C4#91411yescharacteristic
458C2×C4#91411yescharacteristic
459C2×C4#91411yescharacteristic
460C2×C4#91611yescharacteristic
461C2×C4#91811yescharacteristic
462C2×C4#91911yescharacteristic
463C8—42no—
464C8—81no—
465C8—122no—
466C8—241no—
467C2×C3#155111yescharacteristic
468C2×C3#155311yescharacteristic
469C2×C3#155511yescharacteristic
470C2×C3#155711yescharacteristic
471C2×C3#187511yescharacteristic
472C2×C3#189511yescharacteristic
473C2×C3#190011yescharacteristic
474D3—42no—
475D3—162no—
476C22—62no—
477C22—62no—
478C22—62no—
479C22—121no—
480C22—121no—
481C22—121no—
482C22—121no—
483C22—242no—
484C22—242no—
485C22—242no—
486C22—242no—
487C22—242no—
488C22—242no—
489C22—242no—
490C22#466511yescharacteristic
491C22D3×#10011yescharacteristic
492C22#475111yescharacteristic
493C22#477011yescharacteristic
494C22#477011yescharacteristic
495C22#480311yescharacteristic
496C22#480411yescharacteristic
497C4—21no—
498C4—21no—
499C4—82no—
500C4—121no—
501C4—121no—
502C4—121no—
503C4#478111yescharacteristic
504C4#478211yescharacteristic
505C4#481011yescharacteristic
506C4#481211yescharacteristic
507C4#486511yescharacteristic
508C4#486611yescharacteristic
509C4#488511yescharacteristic
510C4#488611yescharacteristic
511C3#914211yescharacteristic, complemented, Sylow
512C2—122no—
513C2—482no—
514C2#6838711yescharacteristic
515C2#6842611yescharacteristic
516C2#6842911yescharacteristic
517C2#6848511yescharacteristic
518C2#6888111yescharacteristic
519C2#6888511yescharacteristic
520C2#6899211yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1296C2
socleC23×C3C2×C22⋊C4
prime radical#74375C2
cosocleC2×C3×C42C23
hypercenter#1296D3
hypocenterC3#9142
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
1. C2×#4835C2semidirect
3. #70475C2semidirect
4. C2×#4485C2semidirect
5. C2×#4485C2semidirect
6. C3×#1296C2semidirect
7. #67634C2semidirect
36. #1296D3semidirect
74. C3×C4⋊C8D4semidirect
81. C4⋊(C3⋊C8)D4semidirect
218. C4⋊C8D12semidirect
511. C3#9142semidirect

Maximal subgroups

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

subgroupindex
C2×#48352
#743752
#704752
C2×#44852
C3×#12962
#676342
#91423