Finite Group Databasebeta

← #10804169

Subgroups — #10804169

Order 768.

1446 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5524C211yesmaximal, characteristic
2D3×#407C212yesmaximal, complemented
3C3×#1822C211yesmaximal, characteristic, complemented
4#71589C211yesmaximal, characteristic, complemented
5#69989C212yesmaximal, complemented
6#68993C214yesmaximal, complemented
7#68992C214yesmaximal, complemented
8#21754—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C23×C4×D3C2211yescharacteristic
11C2×C3×#117C2211yescharacteristic
12C2×D3×M16C2214yes—
13#5524—22no—
14#5524—22no—
15#5524—22no—
16#5524—22no—
17C2×#897C2212yes—
18C2×#895C2212yes—
19C2×#886C412yes—
20C2×#882C412yes—
21C2×C4×D12C414yes—
22C3×#407C2212yes—
23#5207C2212yes—
24C2×#824C2214yescomplemented
25C2×#823C2214yescomplemented
26#5126C2214yes—
27#4726—22no—
28#4726—22no—
29D3×C22⋊C8—22no—
30D3×C22⋊C8—22no—
31C3×#343C2214yescomplemented
32#4530C2214yescomplemented
33C2×#525—31no—
34C2×#407—32no—
35#1901—34no—
36#1900—34no—
37#1822—31no—
38#1822D311yescharacteristic, complemented
39#1785—32no—
40C24×D3C2×C411yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C22×#67C2×C412yes—
43C23×Dic3C2×C411yescharacteristic
44C2×#235—22no—
45C2×#235—22no—
46C2×#235—44no—
47C22×D12C2×C414yes—
48C22×D12C2311yescharacteristic
49C22×C4×D3—44no—
50C22×C4×D3C2×C412yes—
51C22×C4×D3C2×C412yes—
52C22×C4×D3D412yes—
53C22×C4×D3D412yes—
54C22×C4×D3D412yes—
55C22×C4×D3D412yes—
56C22×C4×D3C2312yes—
57C22×Dic6C2311yescharacteristic
58C2×C3×M16C2314yes—
59C3×#117—22no—
60C3×#117—22no—
61C2×C3×C4⋊C4C2312yes—
62C2×C3×C22⋊C4C2×C412yes—
63C2×C3×C42C2×C412yes—
64C2×(C2×D3)⋊C4C2×C414yes—
65C2×(C2×D3)⋊C4C2×C414yes—
66C2×C4⋊Dic3C2×C414yes—
67C2×#208C2314yes—
68D3×M16—44no—
69D3×M16—44no—
70C2×ΓL(1,𝔽25)—24no—
71C2×D3×C8—24no—
72#897—22no—
73#897—22no—
74#897—22no—
75#897—22no—
76#895—22no—
77#895—22no—
78#895—22no—
79#895—22no—
80#886—22no—
81#886—24no—
82#886—22no—
83#882—22no—
84#882—24no—
85#882—22no—
86C4×D12—24no—
87C4×D12—24no—
88C4×D12—24no—
89C4×D12—24no—
90C3×C22⋊C8—22no—
91C3×C22⋊C8—22no—
92#830—22no—
93#830—22no—
94#824—44no—
95#823—44no—
96(C2×D3)⋊C8—24no—
97(C2×D3)⋊C8—24no—
98C22×#41—31no—
99C24×C4—31no—
100C22×M16—34no—
101#525—62no—
102#525—62no—
103#525—62no—
104#525—62no—
105C2×#123—32no—
106C2×C22⋊Q8—32no—
107C2×#121—32no—
108C2×#120—32no—
109C2×C4×D4—34no—
110C2×#117C2×D311yescharacteristic
111C2×#101—34no—
112C2×#100—34no—
113#407—32no—
114#407—34no—
115#407—62no—
116#407—62no—
117#407C2×D312yes—
118C2×C22⋊C8—62no—
119C2×C22⋊C8—62no—
120#343—34no—
121#343C2×D314yescomplemented
122C24×C3C22×C411yescharacteristic
123C23×D3—44no—
124C23×D3—42no—
125C23×D3C22⋊C411yescharacteristic
126C23×D3C22⋊C412yes—
127C23×D3C22⋊C411yescharacteristic
128C23×D3C22×C412yes—
129C23×D3C22×C412yes—
130C22×C3×C4—24no—
131C22×C3×C4C22×C411yescharacteristic
132C22×C3×C4C22×C414yes—
133C22×C3×C4C2×D412yes—
134C22×C3×C4C2×D412yes—
135C22×C3×C4C2411yescharacteristic
136C2×#67—22no—
137C2×#67—22no—
138C2×#67—44no—
139C22×Dic3—42no—
140C22×Dic3C22⋊C411yescharacteristic
141C22×Dic3C22⋊C412yes—
142C22×Dic3C22⋊C411yescharacteristic
143C22×Dic3C22×C412yes—
144C22×Dic3C22×C412yes—
145#235—48no—
146#235—48no—
147C2×D12—24no—
148C2×D12—24no—
149C2×D12—22no—
150C2×D12—24no—
151C2×D12—48no—
152C2×C4×D3—24no—
153C2×C4×D3—24no—
154C2×C4×D3—24no—
155C2×C4×D3—44no—
156C2×C4×D3—44no—
157C2×C4×D3—44no—
158C2×C4×D3—44no—
159C2×C4×D3—42no—
160C2×C4×D3—42no—
161C2×C4×D3—44no—
162C2×C4×D3C22⋊C412yes—
163C2×C4×D3C22⋊C412yes—
164C2×C4×D3C22⋊C412yes—
165C2×C4×D3C22⋊C412yes—
166C2×C4×D3C2×D412yes—
167C2×C4×D3C2×D412yes—
168C2×C4×D3C2×D412yes—
169C2×C4×D3C2×D412yes—
170C2×Dic6—22no—
171C2×Dic6—24no—
172C3×M16—24no—
173C3×M16—24no—
174C2×C3×C8—24no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—22no—
177C3×C22⋊C4—22no—
178C3×C22⋊C4—22no—
179C3×C42—24no—
180(C2×D3)⋊C4—24no—
181(C2×D3)⋊C4—24no—
182(C2×D3)⋊C4—24no—
183(C2×D3)⋊C4—24no—
184C4⋊Dic3—24no—
185C4⋊Dic3—24no—
186#208—24no—
187#208—24no—
188C2×C3⋊C8—24no—
189ΓL(1,𝔽25)—48no—
190D3×C8—48no—
191C25—31no—
192C2×#41—62no—
193C2×#41—62no—
194C2×#41—124no—
195C22×Q8—31no—
196C22×D4—32no—
197C22×D4—34no—
198C22×D4—31no—
199C23×C4—31no—
200C23×C4—32no—
201C23×C4—32no—
202C23×C4—32no—
203C23×C4—32no—
204C23×C4—32no—
205C23×C4—32no—
206C23×C4—32no—
207C23×C4—124no—
208C23×C4C22×D311yescharacteristic
209C2×M16—34no—
210C2×M16—64no—
211C2×M16—124no—
212C2×M16—124no—
213C2×M16C22×D314yes—
214C22×C8—64no—
215#123—62no—
216#123—64no—
217#123—62no—
218C22⋊Q8—62no—
219C22⋊Q8—62no—
220C22⋊Q8—62no—
221C22⋊Q8—62no—
222#121—62no—
223#121—62no—
224#121—62no—
225#121—62no—
226#120—62no—
227#120—64no—
228#120—62no—
229C4×D4—64no—
230C4×D4—64no—
231C4×D4—64no—
232C4×D4—64no—
233#117—22no—
234#117—22no—
235C2×C4⋊C4—34no—
236C2×C4⋊C4C22×D312yes—
237C2×C22⋊C4—34no—
238C2×C22⋊C4—34no—
239C2×C22⋊C4C4×D312yes—
240C2×C42C4×D312yes—
241#101—124no—
242#100—124no—
243C22⋊C8—22no—
244C22⋊C8—22no—
245C22⋊C8—64no—
246C22⋊C8—64no—
247C22⋊C8—62no—
248C22⋊C8—62no—
249C23×C3—22no—
250C23×C3—42no—
251C23×C3C2×C22⋊C411yescharacteristic
252C23×C3C2×C22⋊C411yescharacteristic
253C23×C3C23×C411yescharacteristic
254C22×D3—22no—
255C22×D3—22no—
256C22×D3—24no—
257C22×D3—24no—
258C22×D3—24no—
259C22×D3—22no—
260C22×D3—24no—
261C22×D3—41no—
262C22×D3—42no—
263C22×D3—41no—
264C22×D3—42no—
265C22×D3—42no—
266C22×D3—42no—
267C22×D3—416no—
268C22×D3—44no—
269C22×D3C2×C22⋊C412yes—
270C22×D3C2×C22⋊C412yes—
271C2×C3×C4—22no—
272C2×C3×C4—24no—
273C2×C3×C4—24no—
274C2×C3×C4—24no—
275C2×C3×C4—44no—
276C2×C3×C4C2×C22⋊C412yes—
277C2×C3×C4C22×D412yes—
278C2×C3×C4H(5,𝔽2)14yes—
279#67—44no—
280#67—44no—
281C2×Dic3—22no—
282C2×Dic3—22no—
283C2×Dic3—22no—
284C2×Dic3—24no—
285C2×Dic3—41no—
286C2×Dic3—42no—
287C2×Dic3—41no—
288C2×Dic3C2×C22⋊C412yes—
289C2×Dic3C2×C22⋊C412yes—
290D12—48no—
291D12—48no—
292D12—44no—
293C4×D3—24no—
294C4×D3—24no—
295C4×D3—24no—
296C4×D3—24no—
297C4×D3—42no—
298C4×D3—42no—
299C4×D3—42no—
300C4×D3—42no—
301C4×D3—48no—
302Dic6—44no—
303C3×C8—44no—
304C3⋊C8—44no—
305C24—31no—
306C24—32no—
307C24—32no—
308C24—31no—
309C24—32no—
310C24—122no—
311C24—124no—
312C24C2×C4×D311yescharacteristic
313#41—128no—
314#41—128no—
315C2×Q8—62no—
316C2×Q8—64no—
317C2×D4—64no—
318C2×D4—64no—
319C2×D4—62no—
320C2×D4—62no—
321C2×D4—62no—
322C2×D4—64no—
323C2×D4—128no—
324C2×D4—124no—
325C22×C4—24no—
326C22×C4—32no—
327C22×C4—31no—
328C22×C4—32no—
329C22×C4—31no—
330C22×C4—32no—
331C22×C4—32no—
332C22×C4—32no—
333C22×C4—32no—
334C22×C4—32no—
335C22×C4—32no—
336C22×C4—32no—
337C22×C4—32no—
338C22×C4—32no—
339C22×C4—64no—
340C22×C4—64no—
341C22×C4—64no—
342C22×C4—124no—
343C22×C4—124no—
344C22×C4—124no—
345C22×C4—124no—
346C22×C4—122no—
347C22×C4—122no—
348C22×C4—124no—
349C22×C4—122no—
350C22×C4C2×C4×D311yescharacteristic
351C22×C4C2×C4×D314yes—
352C22×C4D3×D412yes—
353C22×C4D3×D412yes—
354C22×C4C23×D311yescharacteristic
355M16—24no—
356M16—24no—
357M16—64no—
358M16—64no—
359M16—128no—
360C2×C8—24no—
361C2×C8—64no—
362C2×C8—128no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—64no—
366C4⋊C4—64no—
367C22⋊C4—22no—
368C22⋊C4—22no—
369C22⋊C4—64no—
370C22⋊C4—64no—
371C22⋊C4—64no—
372C22⋊C4—64no—
373C42—24no—
374C22×C3—22no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—22no—
378C22×C3—22no—
379C22×C3—44no—
380C22×C3C22×C22⋊C411yescharacteristic
381C22×C3#51812yes—
382C2×D3—22no—
383C2×D3—22no—
384C2×D3—22no—
385C2×D3—22no—
386C2×D3—22no—
387C2×D3—44no—
388C2×D3—44no—
389C2×D3—44no—
390C2×D3—48no—
391C2×D3—44no—
392C2×D3—42no—
393C2×D3—44no—
394C2×D3—84no—
395C2×D3#41314yes—
396C3×C4—22no—
397C3×C4—44no—
398C3×C4#53414yes—
399Dic3—22no—
400Dic3—22no—
401Dic3—42no—
402C23—22no—
403C23—32no—
404C23—32no—
405C23—42no—
406C23—64no—
407C23—64no—
408C23—64no—
409C23—62no—
410C23—62no—
411C23—62no—
412C23—64no—
413C23—121no—
414C23—122no—
415C23—121no—
416C23—122no—
417C23—122no—
418C23—122no—
419C23—124no—
420C23—1216no—
421C23D3×C22⋊C411yescharacteristic
422C23D3×C22⋊C411yescharacteristic
423C23C22×C4×D311yescharacteristic
424Q8—124no—
425D4—128no—
426D4—128no—
427D4—124no—
428D4—124no—
429D4—124no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—32no—
435C2×C4—32no—
436C2×C4—44no—
437C2×C4—62no—
438C2×C4—62no—
439C2×C4—64no—
440C2×C4—64no—
441C2×C4—64no—
442C2×C4—64no—
443C2×C4—62no—
444C2×C4—64no—
445C2×C4—122no—
446C2×C4—122no—
447C2×C4—122no—
448C2×C4—122no—
449C2×C4—128no—
450C2×C4—121no—
451C2×C4—122no—
452C2×C4—121no—
453C2×C4D3×C22⋊C412yes—
454C2×C4C2×D3×D412yes—
455C2×C4#100914yes—
456C8—44no—
457C8—124no—
458C2×C3—22no—
459C2×C3—41no—
460C2×C3—41no—
461C2×C3#230811yescharacteristic
462C2×C3C2×#41312yes—
463D3—24no—
464D3—42no—
465D3—82no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—21no—
470C22—21no—
471C22—34no—
472C22—44no—
473C22—62no—
474C22—62no—
475C22—62no—
476C22—62no—
477C22—62no—
478C22—124no—
479C22—124no—
480C22—124no—
481C22—128no—
482C22—124no—
483C22—124no—
484C22—122no—
485C22—244no—
486C22C2×D3×C22⋊C411yescharacteristic
487C22#552112yes—
488C4—22no—
489C4—44no—
490C4—62no—
491C4—62no—
492C4—122no—
493C4#565314yes—
494C3#2175411yescharacteristic, complemented, Sylow
495C2—22no—
496C2—41no—
497C2—41no—
498C2—64no—
499C2—122no—
500C2—242no—
501C2#7411911yescharacteristic
502C2D3×#41312yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1822C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#5524C2
cosocleC22×C3×C4C24
hypercenter#1822D3
hypocenterC3#21754
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1822

Complements

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

NH ≅ G/Ntype
2. D3×#407C2semidirect
3. C3×#1822C2semidirect
4. #71589C2semidirect
5. #69989C2semidirect
6. #68993C2semidirect
7. #68992C2semidirect
24. C2×#824C22semidirect
25. C2×#823C22semidirect
31. C3×#343C22semidirect
32. #4530C22semidirect
38. #1822D3semidirect
121. #343C2×D3semidirect
494. C3#21754semidirect

Maximal subgroups

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

subgroupindex
C2×#55242
D3×#4072
C3×#18222
#715892
#699892
#689932
#689922
#217543