Finite Group Databasebeta

← #10804132

Subgroups — #10804132

Order 768.

1554 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5791C211yesmaximal, characteristic
2C3×#2280C211yesmaximal, characteristic, complemented
3C2×#5211C211yesmaximal, characteristic, complemented
4#71595C214yesmaximal, complemented
5#69537C212yesmaximal, complemented
6#69534C212yesmaximal, complemented
7#69496C214yesmaximal, complemented
8#21748—31nomaximal, Sylow
9C24×C3×C4C2211yescharacteristic
10C22×#235C2211yescharacteristic
11C2×C3×#117C2212yes—
12C2×#953C2212yes—
13C22×#217C2212yes—
14#5791—22no—
15#5791—22no—
16#5791—22no—
17#5791—22no—
18C2×#930C412yescomplemented
19C2×#929C2212yes—
20C2×C4×#67C414yescomplemented
21C2×#926C2211yescharacteristic
22C2×#924C412yescomplemented
23C3×#381C2212yes—
24C3×#380—22no—
25C3×#379C2212yes—
26C3×C4×C22⋊C4—22no—
27#5211—22no—
28#5211C2212yes—
29#5211C2212yes—
30C2×C3⋊#95—22no—
31C2×#806C2218yes—
32#4527C2218yes—
33C2×#525—31no—
34#2280D311yescharacteristic, complemented
35C2×#381—31no—
36#1899—34no—
37#1898—34no—
38#1820—32no—
39#1782—32no—
40C25×C3C2311yescharacteristic
41C23×C3×C4—44no—
42C23×C3×C4C2×C412yes—
43C23×C3×C4C2×C412yes—
44C23×C3×C4D412yes—
45C23×C3×C4D412yes—
46C23×C3×C4D412yes—
47C23×C3×C4D412yes—
48C23×C3×C4C2311yescharacteristic
49C23×C3×C4C2311yescharacteristic
50C22×#67C2×C414yes—
51C22×#67C2312yes—
52C23×Dic3—22no—
53C2×#235—22no—
54C2×#235—22no—
55C2×#235—44no—
56C22×D12C2×C411yescharacteristic, complemented
57C22×C4×D3C2×C412yescomplemented
58C22×Dic6C2×C411yescharacteristic, complemented
59C3×#117—44no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C22⋊C4—24no—
62C2×C3×C22⋊C4C2314yes—
63C2×C3×C42—22no—
64#953—22no—
65#953—22no—
66#953—24no—
67C2×#217—22no—
68C2×#217—44no—
69C2×#217C2×C414yes—
70C2×#217C2314yes—
71C2×#217C2312yes—
72#930—22no—
73#930—22no—
74#930—22no—
75#930—22no—
76#929—24no—
77#929—22no—
78#929—22no—
79C4×#67—24no—
80C4×#67—24no—
81C4×#67—24no—
82C4×#67—24no—
83C2×(C2×D3)⋊C4C2×C414yescomplemented
84#926—22no—
85#926—22no—
86C2×C4⋊Dic3C2×C412yescomplemented
87#924—22no—
88#924—22no—
89#924—22no—
90#924—22no—
91C2×Dic3⋊C4C2×C414yescomplemented
92C2×C4×Dic3C2×C412yescomplemented
93C3×#95—24no—
94C3⋊#95—24no—
95C3⋊#95—24no—
96#806—48no—
97C22×#41—31no—
98C24×C4C2×D311yescharacteristic
99#525—62no—
100#525—62no—
101#525—62no—
102#525—62no—
103C2×#123—32no—
104C2×C22⋊Q8—32no—
105C2×#121—32no—
106C2×#120—32no—
107C2×C4×D4—34no—
108C2×#117—31no—
109C2×#117C2×D312yes—
110C22×C22⋊C4—32no—
111C2×#99—38no—
112#381—32no—
113#381—32no—
114#381—62no—
115#381C2×D312yes—
116#380—22no—
117#379C2×D312yes—
118C4×C22⋊C4—22no—
119C2×#95—62no—
120#342—38no—
121C24×C3—24no—
122C24×C3—44no—
123C24×C3C22×C412yes—
124C24×C3C2×D411yescharacteristic
125C24×C3C2×D411yescharacteristic
126C24×C3C2×D412yes—
127C24×C3C2411yescharacteristic
128C23×D3C22×C412yes—
129C22×C3×C4—24no—
130C22×C3×C4—24no—
131C22×C3×C4—24no—
132C22×C3×C4—44no—
133C22×C3×C4—44no—
134C22×C3×C4—44no—
135C22×C3×C4—44no—
136C22×C3×C4—44no—
137C22×C3×C4—44no—
138C22×C3×C4C22⋊C412yes—
139C22×C3×C4C22⋊C412yes—
140C22×C3×C4C22⋊C412yes—
141C22×C3×C4C22⋊C412yes—
142C22×C3×C4C22⋊C412yes—
143C22×C3×C4C22⋊C412yes—
144C22×C3×C4C22⋊C412yes—
145C22×C3×C4C22⋊C412yes—
146C22×C3×C4C22×C411yescharacteristic
147C22×C3×C4C22×C412yes—
148C22×C3×C4C22×C411yescharacteristic
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C2×#67—24no—
155C2×#67—24no—
156C2×#67—22no—
157C2×#67—22no—
158C2×#67—28no—
159C2×#67—48no—
160C22×Dic3—24no—
161C22×Dic3—44no—
162C22×Dic3—48no—
163C22×Dic3C22×C412yes—
164C22×Dic3C22×C414yes—
165#235—48no—
166#235—48no—
167C2×D12—22no—
168C2×D12—42no—
169C2×C4×D3—24no—
170C2×C4×D3—44no—
171C2×Dic6—22no—
172C2×Dic6—42no—
173C3×C4⋊C4—44no—
174C3×C22⋊C4—28no—
175C3×C22⋊C4—48no—
176C3×C42—44no—
177#217—24no—
178#217—24no—
179#217—22no—
180#217—22no—
181#217—28no—
182#217—44no—
183(C2×D3)⋊C4—24no—
184(C2×D3)⋊C4—24no—
185C4⋊Dic3—22no—
186C4⋊Dic3—22no—
187Dic3⋊C4—24no—
188Dic3⋊C4—24no—
189C4×Dic3—24no—
190C25C22×D311yescharacteristic
191C2×#41—62no—
192C2×#41—62no—
193C2×#41—124no—
194C22×Q8—31no—
195C22×D4—32no—
196C22×D4—34no—
197C22×D4—31no—
198C23×C4—32no—
199C23×C4—44no—
200C23×C4—62no—
201C23×C4C4×D312yes—
202C23×C4C4×D312yes—
203C23×C4D1212yes—
204C23×C4D1212yes—
205C23×C4#6712yes—
206C23×C4#6712yes—
207C23×C4C22×D311yescharacteristic
208C23×C4C22×D311yescharacteristic
209#123—64no—
210#123—62no—
211#123—62no—
212C22⋊Q8—62no—
213C22⋊Q8—62no—
214C22⋊Q8—62no—
215C22⋊Q8—62no—
216#121—62no—
217#121—62no—
218#121—62no—
219#121—62no—
220#120—62no—
221#120—62no—
222#120—64no—
223C4×D4—64no—
224C4×D4—64no—
225C4×D4—64no—
226C4×D4—64no—
227#117—44no—
228#117—62no—
229#117—62no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—32no—
232C2×C4⋊C4—34no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—34no—
235C2×C22⋊C4—32no—
236C2×C22⋊C4—34no—
237C2×C22⋊C4—34no—
238C2×C22⋊C4—62no—
239C2×C22⋊C4—124no—
240C2×C22⋊C4C22×D314yes—
241C2×C42—22no—
242C2×C42—32no—
243#99—128no—
244#95—24no—
245#95—64no—
246#95—64no—
247C23×C3—22no—
248C23×C3—24no—
249C23×C3—24no—
250C23×C3—22no—
251C23×C3—24no—
252C23×C3—24no—
253C23×C3—24no—
254C23×C3—42no—
255C23×C3—42no—
256C23×C3—42no—
257C23×C3—42no—
258C23×C3—416no—
259C23×C3C2×C22⋊C412yes—
260C23×C3C2×C22⋊C412yes—
261C23×C3C23×C411yescharacteristic
262C23×C3C22×D411yescharacteristic
263C23×C3C22×D411yescharacteristic
264C23×C3H(5,𝔽2)14yes—
265C22×D3—22no—
266C22×D3—24no—
267C22×D3—44no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—24no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—24no—
274C2×C3×C4—24no—
275C2×C3×C4—24no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—48no—
285C2×C3×C4—44no—
286C2×C3×C4C2×C22⋊C412yes—
287C2×C3×C4C2×C22⋊C412yes—
288C2×C3×C4C2×C22⋊C412yes—
289C2×C3×C4C2×C22⋊C412yes—
290#67—48no—
291#67—44no—
292#67—44no—
293#67—48no—
294C2×Dic3—24no—
295C2×Dic3—22no—
296C2×Dic3—24no—
297C2×Dic3—44no—
298C2×Dic3—48no—
299C2×Dic3—44no—
300C2×Dic3—88no—
301D12—44no—
302C4×D3—48no—
303Dic6—44no—
304C24—24no—
305C24—32no—
306C24—44no—
307C24C2×C4×D312yes—
308C24C2×D1211yescharacteristic
309C24C2×D1211yescharacteristic
310C24C2×#6712yes—
311C24C23×D311yescharacteristic
312#41—128no—
313#41—128no—
314C2×Q8—62no—
315C2×Q8—122no—
316C2×D4—64no—
317C2×D4—64no—
318C2×D4—62no—
319C2×D4—68no—
320C2×D4—62no—
321C2×D4—62no—
322C2×D4—122no—
323C2×D4—128no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—32no—
328C22×C4—34no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—64no—
336C22×C4—64no—
337C22×C4—124no—
338C22×C4—124no—
339C22×C4—128no—
340C22×C4(C2×D3)⋊C412yes—
341C22×C4(C2×D3)⋊C412yes—
342C22×C4(C2×D3)⋊C412yes—
343C22×C4(C2×D3)⋊C412yes—
344C22×C4(C2×D3)⋊C412yes—
345C22×C4(C2×D3)⋊C412yes—
346C22×C4(C2×D3)⋊C412yes—
347C22×C4(C2×D3)⋊C412yes—
348C22×C4C2×C4×D311yescharacteristic
349C22×C4C2×C4×D312yes—
350C22×C4C2×C4×D311yescharacteristic
351C22×C4C2×D1212yes—
352C22×C4C2×D1212yes—
353C22×C4C2×#6712yes—
354C22×C4C2×#6711yescharacteristic
355C22×C4C2×#6711yescharacteristic
356C4⋊C4—44no—
357C4⋊C4—62no—
358C4⋊C4—62no—
359C4⋊C4—64no—
360C4⋊C4—64no—
361C22⋊C4—28no—
362C22⋊C4—48no—
363C22⋊C4—64no—
364C22⋊C4—64no—
365C22⋊C4—62no—
366C22⋊C4—62no—
367C22⋊C4—64no—
368C22⋊C4—64no—
369C22⋊C4—68no—
370C22⋊C4—124no—
371C42—44no—
372C42—64no—
373C22×C3—28no—
374C22×C3—22no—
375C22×C3—22no—
376C22×C3—22no—
377C22×C3—22no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—44no—
383C22×C3—48no—
384C22×C3—48no—
385C22×C3—44no—
386C22×C3#41014yes—
387C22×C3C22×C22⋊C411yescharacteristic
388C22×C3#51812yes—
389C22×C3#53412yes—
390C22×C3#53412yes—
391C2×D3—42no—
392C2×D3—44no—
393C2×D3—84no—
394C3×C4—22no—
395C3×C4—22no—
396C3×C4—22no—
397C3×C4—22no—
398C3×C4—84no—
399Dic3—42no—
400Dic3—44no—
401Dic3—84no—
402C23—22no—
403C23—24no—
404C23—24no—
405C23—22no—
406C23—24no—
407C23—24no—
408C23—24no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—416no—
414C23—62no—
415C23—64no—
416C23—124no—
417C23C2×(C2×D3)⋊C412yes—
418C23C2×(C2×D3)⋊C412yes—
419C23C22×C4×D311yescharacteristic
420C23C22×D1211yescharacteristic
421C23#100414yes—
422C23C22×#6711yescharacteristic
423Q8—124no—
424D4—124no—
425D4—128no—
426D4—124no—
427D4—128no—
428D4—124no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—48no—
446C2×C4—44no—
447C2×C4—64no—
448C2×C4—64no—
449C2×C4—62no—
450C2×C4—128no—
451C2×C4—124no—
452C2×C4—128no—
453C2×C4—124no—
454C2×C4—248no—
455C2×C4C2×(C2×D3)⋊C412yes—
456C2×C4C2×(C2×D3)⋊C412yes—
457C2×C4C2×(C2×D3)⋊C412yes—
458C2×C4C2×(C2×D3)⋊C412yes—
459C2×C3—22no—
460C2×C3—22no—
461C2×C3—22no—
462C2×C3—24no—
463C2×C3—42no—
464C2×C3#230811yescharacteristic
465C2×C3C2×#41012yes—
466D3—82no—
467C22—28no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—44no—
477C22—48no—
478C22—48no—
479C22—44no—
480C22—124no—
481C22—122no—
482C22—244no—
483C22#500614yes—
484C22#548612yes—
485C22#549312yes—
486C22C22×(C2×D3)⋊C411yescharacteristic
487C22#580412yes—
488C4—22no—
489C4—22no—
490C4—22no—
491C4—22no—
492C4—84no—
493C4—124no—
494C4—122no—
495C4—244no—
496C3#2174811yescharacteristic, complemented, Sylow
497C2—22no—
498C2—22no—
499C2—24no—
500C2—22no—
501C2—42no—
502C2—242no—
503C2#8212111yescharacteristic
504C2C2×#500612yes—
subgroupG/H
centerC22C22×(C2×D3)⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#2280C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#5791C2
cosocleC24×C3C24
hypercenter#2280D3
hypocenterC3#21748
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24×C4 ◁ #2280

Complements

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

NH ≅ G/Ntype
2. C3×#2280C2semidirect
3. C2×#5211C2semidirect
4. #71595C2semidirect
5. #69537C2semidirect
6. #69534C2semidirect
7. #69496C2semidirect
18. C2×#930C4semidirect
20. C2×C4×#67C4semidirect
22. C2×#924C4semidirect
34. #2280D3semidirect
56. C22×D12C2×C4semidirect
57. C22×C4×D3C2×C4semidirect
58. C22×Dic6C2×C4semidirect
83. C2×(C2×D3)⋊C4C2×C4semidirect
86. C2×C4⋊Dic3C2×C4semidirect
91. C2×Dic3⋊C4C2×C4semidirect
92. C2×C4×Dic3C2×C4semidirect
496. C3#21748semidirect

Maximal subgroups

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

subgroupindex
C2×#57912
C3×#22802
C2×#52112
#715952
#695372
#695342
#694962
#217483