Finite Group Databasebeta

← #10876151

Subgroups — #10876151

Order 768.

1128 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4939C211yesmaximal, characteristic
2#82062C211yesmaximal, characteristic, complemented
3C3×#1817C211yesmaximal, characteristic, complemented
4#69959C212yesmaximal
5#69928C212yesmaximal, complemented
6#68682C212yesmaximal, complemented
7#68635C212yesmaximal, complemented
8#68626C214yesmaximal, complemented
9#27585—31nomaximal, Sylow
10C23×D12C2211yescharacteristic
11C22×C3×C42C2211yescharacteristic
12C22×(C2×D3)⋊C4—21no—
13C22×(C2×D3)⋊C4—21no—
14C2×#926—21no—
15C22×C4⋊Dic3C2211yescharacteristic
16C2×C3×C22⋊C8C2212yes—
17#5212—21no—
18#5212C2212yes—
19C2×#821C2214yes—
20C2×#821C2214yes—
21C2×#817C2214yescomplemented
22#4939—21no—
23#4939—21no—
24#4939C2212yes—
25#4939C2212yes—
26#4936—21no—
27#4935C2211yescharacteristic, complemented
28#4935C2211yescharacteristic, complemented
29C4×C4⋊Dic3—21no—
30C4×C4⋊Dic3C2212yescomplemented
31C3×#336C2212yescomplemented
32C3×#336C2212yescomplemented
33#4551C2214yescomplemented
34C2×#390—31no—
35#2285—31no—
36#2094—32no—
37#2071—32no—
38#1926—34no—
39#1853—32no—
40#1817D311yescharacteristic, complemented
41#1786—32no—
42C24×D3—21no—
43C23×C3×C4D411yescharacteristic
44C23×C3×C4D411yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C23×Dic3—21no—
47C22×D12—21no—
48C22×D12—24no—
49C22×D12—24no—
50C22×D12—42no—
51C22×D12C2312yes—
52C22×C3×C8C2314yes—
53C2×C3×C42—24no—
54C2×C3×C42D411yescharacteristic
55C2×C3×C42D411yescharacteristic
56C2×C3×C42C2311yescharacteristic
57C2×C3×C42C2311yescharacteristic
58C2×#217—22no—
59C2×(C2×D3)⋊C4—22no—
60C2×(C2×D3)⋊C4—22no—
61C2×(C2×D3)⋊C4—22no—
62C2×(C2×D3)⋊C4—22no—
63C2×(C2×D3)⋊C4—44no—
64C2×(C2×D3)⋊C4—44no—
65#926—44no—
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3—21no—
70C2×C4⋊Dic3—21no—
71C2×C4⋊Dic3—42no—
72C2×C4⋊Dic3C2312yes—
73C2×C4⋊Dic3C2312yes—
74C2×C4⋊Dic3C2312yes—
75C2×C4×Dic3—22no—
76C3×C22⋊C8—24no—
77C3⋊#95—22no—
78C3⋊#95—22no—
79#821—28no—
80#821—44no—
81#817—44no—
82C23×D4—31no—
83C2×#117—61no—
84C22×C4⋊C4—31no—
85C22×C22⋊C4—61no—
86C22×C22⋊C4—61no—
87C22×C42C2×D311yescharacteristic
88C2×#106—34no—
89C2×#102—34no—
90C2×#102—34no—
91C2×C22⋊C8C2×D312yes—
92#390—32no—
93#390—32no—
94#390—61no—
95#390—61no—
96#384—31no—
97#384—31no—
98#382—61no—
99#380—32no—
100#380—61no—
101C4×C4⋊C4—32no—
102C4×C4⋊C4—61no—
103#340—34no—
104#336C2×D312yescomplemented
105#336C2×D312yescomplemented
106C24×C3C2×D411yescharacteristic
107C23×D3—24no—
108C23×D3—22no—
109C23×D3—42no—
110C23×D3—42no—
111C23×D3—42no—
112C23×D3—42no—
113C23×D3—42no—
114C23×D3—42no—
115C22×C3×C4—22no—
116C22×C3×C4—22no—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—41no—
122C22×C3×C4SD1614yes—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4#4114yes—
129C22×C3×C4C2411yescharacteristic
130C22×Dic3—22no—
131C22×Dic3—22no—
132C22×Dic3—22no—
133C22×Dic3—42no—
134C22×Dic3—42no—
135C22×Dic3—42no—
136C2×D12—24no—
137C2×D12—24no—
138C2×D12—24no—
139C2×D12—22no—
140C2×D12—416no—
141C2×D12—42no—
142C2×D12—42no—
143C2×D12#4114yes—
144C2×C3×C8—24no—
145C2×C3×C8—24no—
146C2×C3×C8—24no—
147C3×C42—22no—
148C3×C42—22no—
149C3×C42—22no—
150C3×C42—22no—
151#217—44no—
152(C2×D3)⋊C4—44no—
153(C2×D3)⋊C4—44no—
154(C2×D3)⋊C4—44no—
155(C2×D3)⋊C4—44no—
156C4⋊Dic3—24no—
157C4⋊Dic3—24no—
158C4⋊Dic3—24no—
159C4⋊Dic3—42no—
160C4⋊Dic3—42no—
161C4⋊Dic3—42no—
162C4⋊Dic3—42no—
163C4⋊Dic3—42no—
164C4×Dic3—44no—
165C25—61no—
166C22×D4—32no—
167C22×D4—64no—
168C22×D4—61no—
169C22×D4—64no—
170C22×D4—122no—
171C23×C4—61no—
172C23×C4D1211yescharacteristic
173C23×C4D1211yescharacteristic
174C23×C4C22×D311yescharacteristic
175C22×C8C22×D314yes—
176#117—124no—
177C2×C4⋊C4—32no—
178C2×C4⋊C4—32no—
179C2×C4⋊C4—32no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4—122no—
186C2×C22⋊C4—62no—
187C2×C22⋊C4—62no—
188C2×C22⋊C4—62no—
189C2×C22⋊C4—62no—
190C2×C22⋊C4—62no—
191C2×C22⋊C4—124no—
192C2×C22⋊C4—124no—
193C2×C42—24no—
194C2×C42—62no—
195C2×C42D1211yescharacteristic
196C2×C42D1211yescharacteristic
197C2×C42C22×D311yescharacteristic
198C2×C42C22×D311yescharacteristic
199#106—124no—
200#102—68no—
201#102—124no—
202C22⋊C8—24no—
203#95—62no—
204#95—62no—
205C23×C3—21no—
206C23×C3—21no—
207C23×C3—21no—
208C23×C3—21no—
209C23×C3#12312yes—
210C23×C3C2×SD1612yes—
211C23×C3AGL(1,ℤ/8)12yes—
212C23×C3C22×D411yescharacteristic
213C22×D3—24no—
214C22×D3—48no—
215C22×D3—48no—
216C22×D3—48no—
217C22×D3—42no—
218C22×D3—42no—
219C22×D3—42no—
220C22×D3—42no—
221C22×D3—42no—
222C22×D3—42no—
223C22×D3—84no—
224C22×D3—84no—
225C22×D3—84no—
226C22×D3—84no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—28no—
236C2×C3×C4—28no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—44no—
241C2×C3×C4#12312yes—
242C2×C3×C4C2×SD1612yes—
243C2×C3×C4C2×SD1612yes—
244C2×C3×C4AGL(1,ℤ/8)12yes—
245C2×C3×C4C2×#4112yes—
246C2×C3×C4C2×#4112yes—
247C2×C3×C4H(5,𝔽2)12yes—
248C2×Dic3—42no—
249C2×Dic3—42no—
250C2×Dic3—42no—
251C2×Dic3—42no—
252C2×Dic3—42no—
253C2×Dic3—42no—
254C2×Dic3—42no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—84no—
258D12—28no—
259D12—48no—
260D12—44no—
261C3×C8—44no—
262C24—64no—
263C24—62no—
264C24—122no—
265C24—122no—
266C24—122no—
267C24—122no—
268C24—122no—
269C24—122no—
270C24C2×D1211yescharacteristic
271C2×D4—34no—
272C2×D4—64no—
273C2×D4—64no—
274C2×D4—62no—
275C2×D4—64no—
276C2×D4—1216no—
277C2×D4—122no—
278C2×D4—122no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—41no—
286C22×C4—62no—
287C22×C4—62no—
288C22×C4—62no—
289C22×C4—122no—
290C22×C4—122no—
291C22×C4—122no—
292C22×C4#20414yes—
293C22×C4C2×D1211yescharacteristic
294C22×C4C2×D1211yescharacteristic
295C22×C4C2×D1211yescharacteristic
296C22×C4C2×D1211yescharacteristic
297C22×C4C2×D1211yescharacteristic
298C22×C4#23714yes—
299C22×C4C23×D311yescharacteristic
300C2×C8—24no—
301C2×C8—24no—
302C2×C8—24no—
303C4⋊C4—64no—
304C4⋊C4—64no—
305C4⋊C4—64no—
306C4⋊C4—122no—
307C4⋊C4—122no—
308C4⋊C4—122no—
309C4⋊C4—122no—
310C4⋊C4—122no—
311C22⋊C4—124no—
312C22⋊C4—124no—
313C22⋊C4—124no—
314C22⋊C4—124no—
315C22⋊C4—124no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—124no—
321C22×C3—22no—
322C22×C3—22no—
323C22×C3—22no—
324C22×C3—22no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3#48114yes—
328C22×C3C2×#12311yescharacteristic
329C22×C3#52511yescharacteristic
330C22×C3#53511yescharacteristic
331C22×C3C22×SD1611yescharacteristic
332C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
333C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
334C22×C3#57511yescharacteristic
335C2×D3—44no—
336C2×D3—44no—
337C2×D3—44no—
338C2×D3—816no—
339C2×D3—82no—
340C2×D3—82no—
341C2×D3—82no—
342C2×D3—82no—
343C3×C4—22no—
344C3×C4—22no—
345C3×C4—22no—
346C3×C4—22no—
347C3×C4—41no—
348C3×C4#55114yes—
349Dic3—82no—
350Dic3—82no—
351Dic3—82no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—64no—
357C23—128no—
358C23—128no—
359C23—128no—
360C23—122no—
361C23—122no—
362C23—122no—
363C23—122no—
364C23—122no—
365C23—122no—
366C23—244no—
367C23—244no—
368C23—244no—
369C23—244no—
370C23#88612yes—
371C23C2×#20412yes—
372C23#90812yes—
373C23C22×D1211yescharacteristic
374D4—68no—
375D4—128no—
376D4—124no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—28no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—28no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—44no—
391C2×C4—122no—
392C2×C4—122no—
393C2×C4—122no—
394C2×C4—122no—
395C2×C4—122no—
396C2×C4—122no—
397C2×C4—122no—
398C2×C4—122no—
399C2×C4—122no—
400C2×C4—244no—
401C2×C4#88612yes—
402C2×C4C2×#20412yes—
403C2×C4C2×#20412yes—
404C2×C4#90812yes—
405C2×C4C2×#23712yes—
406C2×C4D3×#4112yes—
407C2×C4#100912yes—
408C8—44no—
409C2×C3—24no—
410C2×C3#248211yescharacteristic
411C2×C3C2×#48111yescharacteristic
412C2×C3#309411yescharacteristic
413C2×C3#310111yescharacteristic
414C2×C3#310911yescharacteristic
415C2×C3#320811yescharacteristic
416C2×C3#321311yescharacteristic
417D3—84no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—124no—
425C22—124no—
426C22—124no—
427C22—2416no—
428C22—242no—
429C22—242no—
430C22—242no—
431C22—242no—
432C22#473514yes—
433C22C2×#88611yescharacteristic
434C22#552411yescharacteristic
435C22#552511yescharacteristic
436C22C22×#20411yescharacteristic
437C22C2×#90811yescharacteristic
438C22C2×#90811yescharacteristic
439C22#574811yescharacteristic
440C4—22no—
441C4—22no—
442C4—22no—
443C4—22no—
444C4—41no—
445C4—242no—
446C4—242no—
447C4—242no—
448C4#565814yes—
449C3#2758511yescharacteristic, complemented, Sylow
450C2—24no—
451C2—244no—
452C2#7412911yescharacteristic
453C2C2×#473511yescharacteristic
454C2#7862411yescharacteristic
455C2#7929811yescharacteristic
456C2#7930511yescharacteristic
457C2#7934311yescharacteristic
458C2#7934911yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1817C2
socleC23×C3C22×D4
prime radicalC23×D12C22
hypercenter#1817D3
hypocenterC3#27585
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
2. #82062C2semidirect
3. C3×#1817C2semidirect
5. #69928C2semidirect
6. #68682C2semidirect
7. #68635C2semidirect
8. #68626C2semidirect
21. C2×#817C22semidirect
27. #4935C22semidirect
28. #4935C22semidirect
30. C4×C4⋊Dic3C22semidirect
31. C3×#336C22semidirect
32. C3×#336C22semidirect
33. #4551C22semidirect
40. #1817D3semidirect
104. #336C2×D3semidirect
105. #336C2×D3semidirect
449. C3#27585semidirect

Maximal subgroups

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

subgroupindex
C2×#49392
#820622
C3×#18172
#699592
#699282
#686822
#686352
#686262
#275853