Finite Group Databasebeta

← #10814345

Subgroups — #10814345

Order 768.

978 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D24C211yesmaximal, characteristic
2C4⋊C4×D12C211yesmaximal, characteristic, complemented
3#73638C211yesmaximal, characteristic, complemented
4C3×#1770C211yesmaximal, characteristic, complemented
5#70479C211yesmaximal, characteristic, complemented
6#70479C211yesmaximal, characteristic, complemented
7#69818C212yesmaximal, complemented
8#69812C212yesmaximal, complemented
9#69762C211yesmaximal, characteristic, complemented
10#68673C212yesmaximal
11#68624C212yesmaximal, complemented
12#22758—31nomaximal, Sylow
13C22×D24C2211yescharacteristic
14C2×#892—21no—
15C2×D3×C4⋊C4—21no—
16C2×C4×D12C2212yes—
17C2×C4×D12C2211yescharacteristic
18C2×C4×D12C2211yescharacteristic
19C2×C3×C4⋊C8C2211yescharacteristic
20C2×C3×#97C2212yes—
21C2×C3×C4×C8C2211yescharacteristic
22C3×C4×C4⋊C4C2211yescharacteristic
23C2×#821C2212yes—
24C2×#821C2212yes—
25C2×#821C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#818C2211yescharacteristic
28#4987—21no—
29#4987—21no—
30#4939C2212yes—
31#4935C2211yescharacteristic
32#4935C2211yescharacteristic
33C4×C4⋊Dic3C2212yes—
34#4928C2212yes—
35C4×D24C418yescomplemented
36#4670—22no—
37#4668—22no—
38C3×#336C2212yes—
39#4551C2212yes—
40#4551C2212yes—
41#4551C2212yes—
42#4551C2212yes—
43C2×C4×D8—31no—
44#2354—31no—
45D4×C4⋊C4—31no—
46#1936—32no—
47#1921—32no—
48#1879—32no—
49#1834—31no—
50#1802—31no—
51#1802—31no—
52#1770D311yescharacteristic, complemented
53#1762—32no—
54C22×D12C2311yescharacteristic
55C22×D12C2311yescharacteristic
56C22×C4×D3—21no—
57C22×C4×D3—21no—
58C22×C4×D3—22no—
59C22×C3×C8C2312yes—
60C22×C3×C8C2311yescharacteristic
61C22×C3×C8C2311yescharacteristic
62C2×C3×C4⋊C4D411yescharacteristic
63C2×C3×C4⋊C4D411yescharacteristic
64C2×C3×C42C2312yes—
65C2×C3×C42C2311yescharacteristic
66C2×(C2×D3)⋊C4—21no—
67C2×(C2×D3)⋊C4—21no—
68C2×(C2×D3)⋊C4—22no—
69C2×(C2×D3)⋊C4—22no—
70C2×(C2×D3)⋊C4—22no—
71C2×C4⋊Dic3—21no—
72C2×C4⋊Dic3—21no—
73C2×C4⋊Dic3—21no—
74C2×C4⋊Dic3—21no—
75C2×C4⋊Dic3C2312yes—
76C2×C4⋊Dic3C2312yes—
77C2×C4⋊Dic3C2311yescharacteristic
78C2×C4⋊Dic3C2311yescharacteristic
79C2×Dic3⋊C4—22no—
80C2×C4×Dic3—22no—
81C2×D24—22no—
82C2×D24—22no—
83C2×D24C2×C414yes—
84#892—44no—
85D3×C4⋊C4—44no—
86C4×D12—22no—
87C4×D12—22no—
88C4×D12—24no—
89C4×D12—42no—
90C4×D12C2×C414yes—
91C4×D12C2×C414yescomplemented
92C3×C4⋊C8—22no—
93C3×#97—42no—
94C3×C4×C8C2×C414yescomplemented
95C3×#95D412yes—
96C3⋊#95—22no—
97C3⋊#95—22no—
98#821—24no—
99#821—42no—
100#821C2×C414yes—
101#821C2×C414yescomplemented
102#818C2×C414yescomplemented
103C22×D8—31no—
104C2×C4×D4—31no—
105C2×C4×D4—31no—
106C2×C4×D4—32no—
107C2×#117—61no—
108C22×C4⋊C4—61no—
109C4×D8—38no—
110C2×#107—31no—
111C2×C4⋊C8C2×D311yescharacteristic
112C2×#102—31no—
113C2×#102—31no—
114C2×#102—32no—
115C2×#102—32no—
116C2×#97C2×D312yes—
117C2×C4×C8C2×D311yescharacteristic
118#390—32no—
119#389—32no—
120#388—62no—
121#385—62no—
122#384—31no—
123#384—31no—
124#380—61no—
125#380—61no—
126C4×C4⋊C4—32no—
127C4×C4⋊C4C2×D311yescharacteristic
128#340—32no—
129#340—32no—
130#340—32no—
131#340—32no—
132#336C2×D312yes—
133C23×D3—21no—
134C23×D3—21no—
135C22×C3×C4C2×D412yes—
136C22×C3×C4C2×D412yes—
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2411yescharacteristic
140C22×Dic3—21no—
141C22×Dic3—21no—
142C22×Dic3—22no—
143C22×Dic3—22no—
144C2×D12—21no—
145C2×D12—21no—
146C2×D12—21no—
147C2×D12—21no—
148C2×D12—21no—
149C2×D12—21no—
150C2×D12—21no—
151C2×D12—21no—
152C2×D12C22×C412yes—
153C2×D12C22×C412yes—
154C2×D12#4112yes—
155C2×D12#4112yes—
156C2×C4×D3—24no—
157C2×C4×D3—24no—
158C2×C4×D3—42no—
159C2×C4×D3—42no—
160C2×C4×D3—42no—
161C2×C4×D3—42no—
162C2×C4×D3—44no—
163C2×C4×D3—44no—
164C2×C4×D3—82no—
165C2×C3×C8—21no—
166C2×C3×C8—21no—
167C2×C3×C8—21no—
168C2×C3×C8—21no—
169C2×C3×C8—22no—
170C2×C3×C8—22no—
171C2×C3×C8—22no—
172C2×C3×C8C22×C412yes—
173C2×C3×C8C22×C412yes—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C42—22no—
177C3×C42—22no—
178C3×C42C22×C412yes—
179C3×C42#4112yes—
180(C2×D3)⋊C4—24no—
181(C2×D3)⋊C4—24no—
182(C2×D3)⋊C4—44no—
183(C2×D3)⋊C4—44no—
184(C2×D3)⋊C4—82no—
185C4⋊Dic3—21no—
186C4⋊Dic3—21no—
187C4⋊Dic3—22no—
188C4⋊Dic3—22no—
189C4⋊Dic3—22no—
190C4⋊Dic3—22no—
191C4⋊Dic3—42no—
192C4⋊Dic3—42no—
193C4⋊Dic3—81no—
194C4⋊Dic3—81no—
195C4⋊Dic3C22×C412yes—
196C4⋊Dic3C22×C412yes—
197Dic3⋊C4—44no—
198C4×Dic3—44no—
199D24—28no—
200C22×D4—31no—
201C22×D4—31no—
202C23×C4—61no—
203C23×C4—61no—
204C23×C4—62no—
205C2×D8—34no—
206C2×D8—62no—
207C2×D8—62no—
208C22×C8C22×D312yes—
209C22×C8C22×D311yescharacteristic
210C22×C8C22×D311yescharacteristic
211C4×D4—34no—
212C4×D4—34no—
213C4×D4—64no—
214C4×D4—62no—
215C4×D4—62no—
216C4×D4—122no—
217#117—124no—
218C2×C4⋊C4—31no—
219C2×C4⋊C4—31no—
220C2×C4⋊C4—32no—
221C2×C4⋊C4—32no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4—62no—
224C2×C4⋊C4—61no—
225C2×C4⋊C4—61no—
226C2×C4⋊C4—61no—
227C2×C4⋊C4—124no—
228C2×C4⋊C4D1211yescharacteristic
229C2×C4⋊C4D1211yescharacteristic
230C2×C22⋊C4—61no—
231C2×C22⋊C4—61no—
232C2×C22⋊C4—62no—
233C2×C22⋊C4—62no—
234C2×C22⋊C4—62no—
235C2×C42—62no—
236C2×C42C22×D312yes—
237C2×C42C22×D311yescharacteristic
238#107—34no—
239C4⋊C8—22no—
240#102—34no—
241#102—34no—
242#102—64no—
243#102—122no—
244#97—42no—
245C4×C8C4×D314yescomplemented
246#95—62no—
247#95—62no—
248#95D1212yes—
249C23×C3C22×D411yescharacteristic
250C22×D3—22no—
251C22×D3—22no—
252C22×D3—22no—
253C22×D3—22no—
254C22×D3—41no—
255C22×D3—41no—
256C22×D3—41no—
257C22×D3—41no—
258C22×D3—41no—
259C22×D3—41no—
260C22×D3—41no—
261C22×D3—41no—
262C22×D3—41no—
263C22×D3—41no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4C4×D412yes—
273C2×C3×C4C4×D412yes—
274C2×C3×C4AGL(1,ℤ/8)12yes—
275C2×C3×C4AGL(1,ℤ/8)12yes—
276C2×C3×C4C23×C411yescharacteristic
277C2×C3×C4C2×#4111yescharacteristic
278C2×C3×C4C2×#4111yescharacteristic
279C2×C3×C4C2×#4111yescharacteristic
280C2×C3×C4H(5,𝔽2)11yescharacteristic
281C2×C3×C42−1+411yescharacteristic
282C2×Dic3—22no—
283C2×Dic3—22no—
284C2×Dic3—41no—
285C2×Dic3—41no—
286C2×Dic3—41no—
287C2×Dic3—41no—
288C2×Dic3—42no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—42no—
292C2×Dic3—42no—
293C2×Dic3—42no—
294D12—22no—
295D12—22no—
296D12—22no—
297D12—22no—
298D12—22no—
299D12—41no—
300D12—41no—
301D12#11714yescomplemented
302C4×D3—48no—
303C4×D3—48no—
304C4×D3—88no—
305C3×C8—22no—
306C3×C8—22no—
307C3×C8—42no—
308C24—61no—
309C24—61no—
310C2×D4—32no—
311C2×D4—32no—
312C2×D4—32no—
313C2×D4—32no—
314C2×D4—61no—
315C2×D4—61no—
316C2×D4—61no—
317C2×D4—61no—
318C2×D4—61no—
319C2×D4—61no—
320C2×D4—61no—
321C2×D4—61no—
322C22×C4—61no—
323C22×C4—61no—
324C22×C4—64no—
325C22×C4—64no—
326C22×C4—62no—
327C22×C4—62no—
328C22×C4—124no—
329C22×C4—124no—
330C22×C4—122no—
331C22×C4—122no—
332C22×C4—122no—
333C22×C4—122no—
334C22×C4—242no—
335C22×C4C2×D1212yes—
336C22×C4C2×D1212yes—
337C22×C4C2×D1211yescharacteristic
338C22×C4C2×D1211yescharacteristic
339C22×C4C23×D311yescharacteristic
340D8—68no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8C2×C4×D312yes—
349C2×C8C2×C4×D312yes—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—32no—
353C4⋊C4—32no—
354C4⋊C4—61no—
355C4⋊C4—61no—
356C4⋊C4—62no—
357C4⋊C4—62no—
358C4⋊C4—62no—
359C4⋊C4—62no—
360C4⋊C4—124no—
361C4⋊C4—122no—
362C4⋊C4—122no—
363C4⋊C4—241no—
364C4⋊C4—241no—
365C22⋊C4—64no—
366C22⋊C4—64no—
367C22⋊C4—124no—
368C22⋊C4—124no—
369C22⋊C4—242no—
370C42—22no—
371C42—22no—
372C42—124no—
373C42C2×C4×D312yes—
374C42#23512yes—
375C22×C3C2×C4×D411yescharacteristic
376C22×C3#54611yescharacteristic
377C22×C3#54811yescharacteristic
378C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
379C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
380C22×C3#57611yescharacteristic
381C22×C3#57811yescharacteristic
382C2×D3—24no—
383C2×D3—42no—
384C2×D3—42no—
385C2×D3—42no—
386C2×D3—42no—
387C2×D3—42no—
388C2×D3—42no—
389C2×D3—42no—
390C2×D3—42no—
391C2×D3—42no—
392C2×D3—81no—
393C2×D3—81no—
394C2×D3—81no—
395C2×D3—81no—
396C3×C4—22no—
397C3×C4—42no—
398C3×C4—42no—
399C3×C4#44214yes—
400C3×C4C2×#11711yescharacteristic
401C3×C4#52011yescharacteristic
402C3×C4#55211yescharacteristic
403C3×C4#55311yescharacteristic
404Dic3—42no—
405Dic3—42no—
406Dic3—82no—
407Dic3—82no—
408C23—62no—
409C23—62no—
410C23—62no—
411C23—62no—
412C23—121no—
413C23—121no—
414C23—121no—
415C23—121no—
416C23—121no—
417C23—121no—
418C23—121no—
419C23—121no—
420C23—121no—
421C23—121no—
422C23C22×D1211yescharacteristic
423D4—34no—
424D4—62no—
425D4—62no—
426D4—62no—
427D4—62no—
428D4—62no—
429D4—121no—
430D4—121no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—62no—
440C2×C4—62no—
441C2×C4—121no—
442C2×C4—121no—
443C2×C4—121no—
444C2×C4—121no—
445C2×C4—128no—
446C2×C4—128no—
447C2×C4—122no—
448C2×C4—122no—
449C2×C4—122no—
450C2×C4—122no—
451C2×C4—122no—
452C2×C4—122no—
453C2×C4—248no—
454C2×C4C4×D1212yes—
455C2×C4C4×D1212yes—
456C2×C4#90812yes—
457C2×C4#90812yes—
458C2×C4C22×C4×D311yescharacteristic
459C2×C4C2×#23511yescharacteristic
460C2×C4#100411yescharacteristic
461C2×C4#100711yescharacteristic
462C2×C4D3×#4111yescharacteristic
463C2×C4D3×#4111yescharacteristic
464C8—22no—
465C8—22no—
466C8—42no—
467C2×C3#235211yescharacteristic
468C2×C3C2×#44211yescharacteristic
469C2×C3#295111yescharacteristic
470C2×C3#332511yescharacteristic
471C2×C3#333011yescharacteristic
472C2×C3#336611yescharacteristic
473C2×C3#336711yescharacteristic
474D3—44no—
475D3—82no—
476C22—64no—
477C22—122no—
478C22—122no—
479C22—122no—
480C22—122no—
481C22—122no—
482C22—122no—
483C22—122no—
484C22—122no—
485C22—122no—
486C22—241no—
487C22—241no—
488C22—241no—
489C22—241no—
490C22C2×C4×D1211yescharacteristic
491C22#555411yescharacteristic
492C22#557611yescharacteristic
493C22C2×#90811yescharacteristic
494C22C2×#90811yescharacteristic
495C22#575211yescharacteristic
496C22#575311yescharacteristic
497C4—22no—
498C4—42no—
499C4—42no—
500C4—122no—
501C4—122no—
502C4—242no—
503C4—242no—
504C4#471114yes—
505C4#550411yescharacteristic
506C4D3×#11711yescharacteristic
507C4#555811yescharacteristic
508C4#558111yescharacteristic
509C3#2275811yescharacteristic, complemented, Sylow
510C2—124no—
511C2—242no—
512C2#7377111yescharacteristic
513C2C2×#471111yescharacteristic
514C2#7856211yescharacteristic
515C2#7955611yescharacteristic
516C2#7955711yescharacteristic
517C2#7996911yescharacteristic
518C2#7997011yescharacteristic
subgroupG/H
centerC23C22×D12
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1770C2
socleC23×C3C22×D4
prime radicalC22×D24C22
cosocleC22×C3×C4C24
hypercenter#1770D3
hypocenterC3#22758
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1770

Complements

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

NH ≅ G/Ntype
2. C4⋊C4×D12C2semidirect
3. #73638C2semidirect
4. C3×#1770C2semidirect
5. #70479C2semidirect
6. #70479C2semidirect
7. #69818C2semidirect
8. #69812C2semidirect
9. #69762C2semidirect
11. #68624C2semidirect
35. C4×D24C4semidirect
52. #1770D3semidirect
91. C4×D12C2×C4semidirect
94. C3×C4×C8C2×C4semidirect
101. #821C2×C4semidirect
102. #818C2×C4semidirect
245. C4×C8C4×D3semidirect
301. D12#117semidirect
509. C3#22758semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D242
C4⋊C4×D122
#736382
C3×#17702
#704792
#698182
#698122
#697622
#686732
#686242
#227583