Finite Group Databasebeta

← #10853310

Subgroups — #10853310

Order 768.

786 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2551C211yesmaximal, characteristic, complemented
2#73638C211yesmaximal, characteristic
3#71792C212yesmaximal
4#71779C212yesmaximal, complemented
5#69629C212yesmaximal
6#69602C212yesmaximal, complemented
7#69584C211yesmaximal, characteristic
8#69574C211yesmaximal, characteristic, complemented
9#69484C211yesmaximal, characteristic, complemented
10C4×#809C211yesmaximal, characteristic
11#69467C211yesmaximal, characteristic, complemented
12#26024—31nomaximal, Sylow
13C2×C3×C4⋊Q8—21no—
14C2×C3×C4⋊Q8C2212yes—
15C2×#892—21no—
16C2×C4×D12C2211yescharacteristic
17C3×#395—22no—
18C3×#389—21no—
19C3×#389C2212yes—
20C3×#384C2211yescharacteristic
21C3×C4×C4⋊C4C2211yescharacteristic
22C3×C4×C4⋊C4C2211yescharacteristic
23#4987—21no—
24C2×#809C2211yescharacteristic
25C2×#809C2211yescharacteristic
26C2×#809C2212yes—
27C2×#808C2212yes—
28#4939C2212yes—
29#4935C2211yescharacteristic
30C4×C4⋊Dic3C2212yes—
31C2×C4⋊(C3⋊C8)C2211yescharacteristic
32C2×C4⋊(C3⋊C8)C2212yes—
33C2×C4×C3⋊C8C2211yescharacteristic
34#4670—22no—
35#4529C2212yes—
36#4529C2211yescharacteristic
37#4529C2211yescharacteristic
38#4529C2212yes—
39#4529C2212yes—
40#4529C2212yes—
41#4524C2212yes—
42#2551D311yescharacteristic, complemented
43#2354—31no—
44#2065—32no—
45#2023—32no—
46#1987—32no—
47#1948—31no—
48#1939—31no—
49#1920—32no—
50#1834—31no—
51#1802—31no—
52C4×#102—31no—
53C22×C3×Q8—22no—
54C22×D12C2311yescharacteristic
55C22×C4×D3—21no—
56C3×C4⋊Q8—24no—
57C3×C4⋊Q8—42no—
58C3×C4⋊Q8—44no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4D411yescharacteristic
64C2×C3×C4⋊C4D411yescharacteristic
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C42—21no—
69C2×C3×C42C2312yes—
70C2×C3×C42C2311yescharacteristic
71C2×(C2×D3)⋊C4—21no—
72C2×(C2×D3)⋊C4—22no—
73C2×(C2×D3)⋊C4—22no—
74C2×C4⋊Dic3—21no—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3C2312yes—
77C2×C4⋊Dic3C2311yescharacteristic
78C2×C4×Dic3—22no—
79C22×C3⋊C8C2311yescharacteristic
80C22×C3⋊C8C2311yescharacteristic
81C22×C3⋊C8C2312yes—
82#892—44no—
83C4×D12—22no—
84C4×D12Q814yescomplemented
85C3×#95—21no—
86C3×#95—22no—
87C3×#95D412yes—
88C3⋊#95—22no—
89#809—22no—
90#809—22no—
91#809—42no—
92#808—24no—
93C4⋊(C3⋊C8)—22no—
94C4⋊(C3⋊C8)—42no—
95C4×C3⋊C8—22no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8C2×D312yes—
98C2×C4×D4—31no—
99C2×#117—61no—
100C2×#106—32no—
101C2×C4⋊C8—31no—
102C2×C4⋊C8—32no—
103C2×#102—31no—
104C2×#102—31no—
105C2×#102—32no—
106C2×C4×C8—31no—
107#395—22no—
108#390—32no—
109#389—21no—
110#389C2×D312yes—
111#388—62no—
112#384—31no—
113#384C2×D311yescharacteristic
114#380—61no—
115C4×C4⋊C4—32no—
116C4×C4⋊C4C2×D311yescharacteristic
117C4×C4⋊C4C2×D311yescharacteristic
118#340—32no—
119#340—32no—
120#340—32no—
121#340—31no—
122#340—32no—
123#340—31no—
124#336—32no—
125C23×D3—21no—
126C2×C3×Q8—44no—
127C2×C3×Q8—44no—
128C2×C3×Q8—82no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—22no—
132C22×C3×C4C2×D412yes—
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D412yes—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2411yescharacteristic
137C22×Dic3—21no—
138C22×Dic3—22no—
139C2×D12—21no—
140C2×D12—21no—
141C2×D12—21no—
142C2×D12—21no—
143C2×D12—21no—
144C2×D12C2×Q812yes—
145C2×C4×D3—24no—
146C2×C4×D3—42no—
147C2×C4×D3—42no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—22no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—44no—
154C3×C4⋊C4—44no—
155C3×C4⋊C4—42no—
156C3×C4⋊C4—81no—
157C3×C4⋊C4SD1614yes—
158C3×C4⋊C4#4112yes—
159C3×C4⋊C4#4112yes—
160C3×C42—22no—
161C3×C42—22no—
162C3×C42—42no—
163C3×C42C2×Q812yes—
164C3×C42#4112yes—
165(C2×D3)⋊C4—24no—
166(C2×D3)⋊C4—44no—
167(C2×D3)⋊C4—82no—
168C4⋊Dic3—21no—
169C4⋊Dic3—22no—
170C4⋊Dic3—22no—
171C4⋊Dic3—42no—
172C4⋊Dic3—81no—
173C4⋊Dic3C2×Q812yes—
174C4×Dic3—44no—
175C2×C3⋊C8—21no—
176C2×C3⋊C8—21no—
177C2×C3⋊C8—22no—
178C2×C3⋊C8—22no—
179C2×C3⋊C8—21no—
180C2×C3⋊C8—21no—
181C2×C3⋊C8—21no—
182C2×C3⋊C8—22no—
183C2×C3⋊C8—21no—
184C22×Q8—22no—
185C22×D4—31no—
186C23×C4—61no—
187C22×C8—31no—
188C22×C8—32no—
189C22×C8—31no—
190C4⋊Q8—24no—
191C4⋊Q8—42no—
192C4⋊Q8—44no—
193C4×D4—34no—
194C4×D4—62no—
195#117—124no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—31no—
201C2×C4⋊C4—32no—
202C2×C4⋊C4—61no—
203C2×C4⋊C4—61no—
204C2×C4⋊C4#6711yescharacteristic
205C2×C4⋊C4#6711yescharacteristic
206C2×C4⋊C4C22×D312yes—
207C2×C4⋊C4C22×D311yescharacteristic
208C2×C4⋊C4C22×D311yescharacteristic
209C2×C22⋊C4—61no—
210C2×C22⋊C4—62no—
211C2×C22⋊C4—62no—
212C2×C42—21no—
213C2×C42—62no—
214C2×C42C22×D312yes—
215C2×C42C22×D311yescharacteristic
216#106—64no—
217C4⋊C8—62no—
218C4⋊C8—122no—
219#102—62no—
220#102—62no—
221#102—122no—
222C4×C8—62no—
223#95—21no—
224#95—22no—
225#95—62no—
226#95#6712yes—
227C23×C3C22×D411yescharacteristic
228C22×D3—22no—
229C22×D3—41no—
230C22×D3—41no—
231C22×D3—41no—
232C22×D3—41no—
233C22×D3—41no—
234C22×D3—41no—
235C3×Q8—88no—
236C2×C3×C4—22no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4C22⋊Q812yes—
253C2×C3×C4C22⋊Q812yes—
254C2×C3×C4C2×SD1614yes—
255C2×C3×C4C2×SD1612yes—
256C2×C3×C4AGL(1,ℤ/8)12yes—
257C2×C3×C4C22×Q811yescharacteristic
258C2×C3×C4C2×#4111yescharacteristic
259C2×C3×C4C2×#4111yescharacteristic
260C2×C3×C4C2×#4111yescharacteristic
261C2×C3×C4H(5,𝔽2)11yescharacteristic
262C2×C3×C42−1+411yescharacteristic
263C2×Dic3—22no—
264C2×Dic3—41no—
265C2×Dic3—41no—
266C2×Dic3—42no—
267C2×Dic3—42no—
268C2×Dic3—42no—
269D12—22no—
270D12—22no—
271D12—41no—
272D12—41no—
273C4×D3—48no—
274C3⋊C8—41no—
275C3⋊C8—42no—
276C3⋊C8—41no—
277C24—61no—
278C2×Q8—44no—
279C2×Q8—44no—
280C2×Q8—82no—
281C2×D4—32no—
282C2×D4—61no—
283C2×D4—61no—
284C2×D4—61no—
285C2×D4—61no—
286C2×D4—61no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—61no—
291C22×C4—64no—
292C22×C4—62no—
293C22×C4—122no—
294C22×C4—122no—
295C22×C4C2×#6712yes—
296C22×C4C2×#6712yes—
297C22×C4C2×#6711yescharacteristic
298C22×C4C2×#6711yescharacteristic
299C22×C4C23×D311yescharacteristic
300C2×C8—61no—
301C2×C8—62no—
302C2×C8—61no—
303C2×C8—61no—
304C2×C8—61no—
305C2×C8—61no—
306C2×C8—62no—
307C2×C8—62no—
308C2×C8—61no—
309C4⋊C4—22no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—32no—
315C4⋊C4—44no—
316C4⋊C4—44no—
317C4⋊C4—42no—
318C4⋊C4—61no—
319C4⋊C4—62no—
320C4⋊C4—62no—
321C4⋊C4—81no—
322C4⋊C4—122no—
323C4⋊C4—241no—
324C4⋊C4#21514yes—
325C4⋊C4#23712yes—
326C4⋊C4#23912yes—
327C22⋊C4—64no—
328C22⋊C4—124no—
329C22⋊C4—242no—
330C42—22no—
331C42—22no—
332C42—42no—
333C42—124no—
334C42D3×Q812yes—
335C42#23912yes—
336C22×C3C2×C22⋊Q811yescharacteristic
337C22×C3#54611yescharacteristic
338C22×C3#54811yescharacteristic
339C22×C3C22×SD1611yescharacteristic
340C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
341C22×C3#57611yescharacteristic
342C22×C3#57811yescharacteristic
343C2×D3—42no—
344C2×D3—42no—
345C2×D3—42no—
346C2×D3—81no—
347C2×D3—81no—
348C2×D3—81no—
349C2×D3—81no—
350C3×C4—22no—
351C3×C4—24no—
352C3×C4—42no—
353C3×C4—42no—
354C3×C4—42no—
355C3×C4—82no—
356C3×C4#47614yes—
357C3×C4#53311yescharacteristic
358C3×C4#54411yescharacteristic
359C3×C4#55511yescharacteristic
360C3×C4#55611yescharacteristic
361Dic3—42no—
362Dic3—82no—
363C23—62no—
364C23—121no—
365C23—121no—
366C23—121no—
367C23—121no—
368C23—121no—
369C23—121no—
370C23C22×#6711yescharacteristic
371Q8—88no—
372D4—62no—
373D4—62no—
374D4—121no—
375D4—121no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—62no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—128no—
396C2×C4—122no—
397C2×C4—122no—
398C2×C4—122no—
399C2×C4#93212yes—
400C2×C4C2×#21514yes—
401C2×C4C2×#21512yes—
402C2×C4#94612yes—
403C2×C4#94612yes—
404C2×C4C2×#23711yescharacteristic
405C2×C4#100411yescharacteristic
406C2×C4C2×D3×Q811yescharacteristic
407C2×C4C2×#23911yescharacteristic
408C2×C4C2×#23911yescharacteristic
409C2×C4#100711yescharacteristic
410C8—121no—
411C8—122no—
412C8—121no—
413C2×C3#243911yescharacteristic
414C2×C3C2×#47611yescharacteristic
415C2×C3#308311yescharacteristic
416C2×C3#329811yescharacteristic
417C2×C3#332511yescharacteristic
418C2×C3#336211yescharacteristic
419C2×C3#336711yescharacteristic
420D3—82no—
421C22—122no—
422C22—122no—
423C22—122no—
424C22—241no—
425C22—241no—
426C22—241no—
427C22—241no—
428C22C2×#93211yescharacteristic
429C22#580611yescharacteristic
430C22C22×#21511yescharacteristic
431C22C2×#94611yescharacteristic
432C22#581711yescharacteristic
433C22#583511yescharacteristic
434C22#583811yescharacteristic
435C4—22no—
436C4—24no—
437C4—42no—
438C4—42no—
439C4—42no—
440C4—82no—
441C4—122no—
442C4—242no—
443C4#508414yes—
444C4#572811yescharacteristic
445C4#572911yescharacteristic
446C4#573211yescharacteristic
447C4#573411yescharacteristic
448C3#2602411yescharacteristic, complemented, Sylow
449C2—242no—
450C2C2×#508411yescharacteristic
451C2#8325611yescharacteristic
452C2#8347511yescharacteristic
453C2#8348711yescharacteristic
454C2#8355011yescharacteristic
455C2#8356311yescharacteristic
456C2#8370311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2551C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2551D3
hypocenterC3#26024
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2551

Complements

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

NH ≅ G/Ntype
1. C3×#2551C2semidirect
4. #71779C2semidirect
6. #69602C2semidirect
8. #69574C2semidirect
9. #69484C2semidirect
11. #69467C2semidirect
42. #2551D3semidirect
84. C4×D12Q8semidirect
448. C3#26024semidirect

Maximal subgroups

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

subgroupindex
C3×#25512
#736382
#717922
#717792
#696292
#696022
#695842
#695742
#694842
C4×#8092
#694672
#260243