Finite Group Databasebeta

← #10849220

Subgroups — #10849220

Order 768.

1346 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5718C211yesmaximal, characteristic
2C2×C3×#486C211yesmaximal, characteristic, complemented
3#83243C211yesmaximal, characteristic, complemented
4C2×#5084C211yesmaximal, characteristic, complemented
5#71761C211yesmaximal, characteristic, complemented
6#70211C214yesmaximal, complemented
7#70169C214yesmaximal, complemented
8#69788C211yesmaximal, characteristic, complemented
9C4×#821C211yesmaximal, characteristic, complemented
10#25649—31nomaximal, Sylow
11C22×#237—21no—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×#127C2211yescharacteristic
14C2×#940C2212yes—
15C2×#939—22no—
16C2×#938—21no—
17C2×D4×Dic3C2212yes—
18#5718—24no—
19#5718—24no—
20C2×C4×D12C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C3×#486—24no—
23C2×C3×#102C2214yes—
24C2×C3×C4×C8C2211yescharacteristic
25#5230—22no—
26C2×#832C2214yes—
27C2×#821C2212yes—
28#5084—24no—
29#4975—21no—
30#4973—21no—
31#4972C2214yes—
32C2×#809C2212yes—
33C2×#808C2212yes—
34C4×C4⋊Dic3C2211yescharacteristic
35C2×C4⋊(C3⋊C8)C2211yescharacteristic
36#4551C2212yes—
37#4550C2214yes—
38C2×#550—31no—
39C2×#486D311yescharacteristic, complemented
40C2×#476—31no—
41#2567—31no—
42#2041—34no—
43#1984—31no—
44#1982—31no—
45#1864—34no—
46C4×#102—31no—
47C22×C3×Q8—21no—
48C22×C3×D4—21no—
49C22×C3×D4—21no—
50C22×C3×D4C2312yes—
51C22×#67—22no—
52C22×#67—22no—
53C23×Dic3—22no—
54C2×#237—28no—
55C2×#237—44no—
56C2×#237—44no—
57C22×D12C2311yescharacteristic
58C22×C4×D3—21no—
59C22×Dic6D411yescharacteristic
60C22×C3×C8C2312yes—
61C3×C4⋊Q8—22no—
62C3×C4⋊Q8—22no—
63C3×#127—22no—
64C3×#127D414yes—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C42C2311yescharacteristic
68C2×#217—22no—
69#940—24no—
70#940—24no—
71#939—216no—
72#938—28no—
73D4×Dic3—24no—
74D4×Dic3—24no—
75C2×(C2×D3)⋊C4—21no—
76C2×C4⋊Dic3D411yescharacteristic
77C2×C4⋊Dic3C2311yescharacteristic
78C2×C4⋊Dic3C2312yes—
79C2×Dic3⋊C4—24no—
80C2×Dic3⋊C4D412yes—
81C2×C4×Dic3—21no—
82C2×C4×Dic3C2312yes—
83C22×C3⋊C8C2312yes—
84C4×D12—22no—
85C4×D12D414yes—
86C4×Dic6—22no—
87C4×Dic6—22no—
88C3×#102—28no—
89C3×C4×C8—22no—
90#832—28no—
91C3⋊#95—21no—
92C3⋊#95—24no—
93#821—24no—
94#809—24no—
95#808—24no—
96C4⋊(C3⋊C8)—22no—
97C22×#41—61no—
98#550—64no—
99#550—64no—
100C2×C4⋊Q8C2×D311yescharacteristic
101C2×#127—32no—
102C2×#127C2×D311yescharacteristic
103C2×#121—61no—
104C2×#121—62no—
105C2×C4×Q8—31no—
106C2×C4×D4—32no—
107C2×C4×D4—31no—
108#486—24no—
109#476—64no—
110C2×#106—32no—
111C2×C4⋊C8—31no—
112C2×#102—34no—
113C2×#102—32no—
114C2×#102—32no—
115C2×#102C2×D314yes—
116C2×C4×C8C2×D311yescharacteristic
117#395—62no—
118#391—61no—
119#389—34no—
120#384—61no—
121C4×C4⋊C4—31no—
122#340—32no—
123#336—34no—
124C24×C3—22no—
125C23×D3—21no—
126C2×C3×Q8—42no—
127C2×C3×Q8—82no—
128C2×C3×D4—24no—
129C2×C3×D4—24no—
130C2×C3×D4—22no—
131C2×C3×D4—24no—
132C2×C3×D4—24no—
133C2×C3×D4—42no—
134C2×C3×D4—42no—
135C2×C3×D4—82no—
136C2×C3×D4C2×D414yes—
137C22×C3×C4—22no—
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2411yescharacteristic
141C2×#67—28no—
142C2×#67—28no—
143C2×#67—44no—
144C2×#67—44no—
145C2×#67—84no—
146C22×Dic3—28no—
147C22×Dic3—22no—
148C22×Dic3—21no—
149C22×Dic3—44no—
150C22×Dic3—44no—
151C22×Dic3C2×D412yes—
152C22×Dic3C2×D412yes—
153#237—432no—
154C2×D12—22no—
155C2×D12—22no—
156C2×D12—21no—
157C2×D12C2×D412yes—
158C2×C4×D3—24no—
159C2×C4×D3—42no—
160C2×C4×D3—42no—
161C2×Dic6—22no—
162C2×Dic6—22no—
163C2×Dic6SD1614yes—
164C2×C3×C8—22no—
165C2×C3×C8—22no—
166C2×C3×C8—22no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—42no—
170C3×C42C2×D412yes—
171C3×C42C2×D412yes—
172#217—28no—
173(C2×D3)⋊C4—24no—
174C4⋊Dic3—22no—
175C4⋊Dic3—21no—
176C4⋊Dic3—22no—
177C4⋊Dic3C2×D412yes—
178C4⋊Dic3#4114yescomplemented
179Dic3⋊C4—24no—
180Dic3⋊C4—84no—
181C4×Dic3—22no—
182C4×Dic3—22no—
183C4×Dic3—42no—
184C2×C3⋊C8—22no—
185C2×C3⋊C8—24no—
186C2×#41—68no—
187C2×#41—124no—
188C2×#41—124no—
189C22×Q8—21no—
190C22×Q8—31no—
191C22×D4—21no—
192C22×D4—21no—
193C22×D4—31no—
194C22×D4—62no—
195C22×D4—62no—
196C22×D4C22×D312yes—
197C23×C4—62no—
198C23×C4—61no—
199C22×C8—32no—
200C22×C8C22×D312yes—
201C4⋊Q8—22no—
202C4⋊Q8—22no—
203#127—22no—
204#127—64no—
205#127—64no—
206#127#6714yes—
207#121—68no—
208#121—616no—
209C4×Q8—62no—
210C4×Q8—62no—
211C4×D4—34no—
212C4×D4—64no—
213C4×D4—64no—
214C4×D4—62no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—31no—
217C2×C4⋊C4—32no—
218C2×C4⋊C4—31no—
219C2×C4⋊C4—32no—
220C2×C4⋊C4—64no—
221C2×C4⋊C4C22×D312yes—
222C2×C22⋊C4—62no—
223C2×C22⋊C4—61no—
224C2×C42—32no—
225C2×C42—61no—
226C2×C42C22×D311yescharacteristic
227#106—64no—
228C4⋊C8—62no—
229#102—28no—
230#102—64no—
231#102—64no—
232#102—68no—
233C4×C8—22no—
234#95—64no—
235#95—61no—
236C23×C3—24no—
237C23×C3—44no—
238C23×C3—42no—
239C23×C3—42no—
240C23×C3—44no—
241C23×C3C22×D411yescharacteristic
242C22×D3—22no—
243C22×D3—42no—
244C22×D3—41no—
245C22×D3—41no—
246C22×D3—42no—
247C3×Q8—84no—
248C3×D4—28no—
249C3×D4—42no—
250C3×D4—42no—
251C3×D4—48no—
252C3×D4—84no—
253C2×C3×C4—22no—
254C2×C3×C4—42no—
255C2×C3×C4—44no—
256C2×C3×C4#12012yes—
257C2×C3×C4#12012yes—
258C2×C3×C4C2×SD1612yes—
259C2×C3×C4AGL(1,ℤ/8)12yes—
260C2×C3×C4C22×D411yescharacteristic
261C2×C3×C4C22×D411yescharacteristic
262C2×C3×C4C2×#4112yes—
263C2×C3×C42−1+412yes—
264#67—416no—
265#67—88no—
266C2×Dic3—22no—
267C2×Dic3—22no—
268C2×Dic3—22no—
269C2×Dic3—22no—
270C2×Dic3—24no—
271C2×Dic3—24no—
272C2×Dic3—416no—
273C2×Dic3—41no—
274C2×Dic3—42no—
275C2×Dic3—41no—
276C2×Dic3—42no—
277C2×Dic3C2×SD1614yes—
278D12—22no—
279D12—22no—
280D12—42no—
281C4×D3—48no—
282Dic6—28no—
283C3×C8—42no—
284C3⋊C8—42no—
285C24—22no—
286C24—61no—
287#41—1232no—
288C2×Q8—34no—
289C2×Q8—42no—
290C2×Q8—62no—
291C2×Q8—62no—
292C2×Q8—82no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—22no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—32no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—68no—
302C2×D4—68no—
303C2×D4—62no—
304C2×D4—62no—
305C2×D4—61no—
306C2×D4—82no—
307C2×D4—124no—
308C2×D4—124no—
309C2×D4—244no—
310C2×D4C2×#6714yes—
311C22×C4—22no—
312C22×C4—32no—
313C22×C4—32no—
314C22×C4—68no—
315C22×C4—62no—
316C22×C4—61no—
317C22×C4—64no—
318C22×C4—124no—
319C22×C4—124no—
320C22×C4—122no—
321C22×C4—122no—
322C22×C4D3×D411yescharacteristic
323C22×C4D3×D411yescharacteristic
324C22×C4C23×D311yescharacteristic
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—62no—
329C2×C8—64no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—32no—
333C4⋊C4—34no—
334C4⋊C4—42no—
335C4⋊C4—64no—
336C4⋊C4—62no—
337C4⋊C4—62no—
338C4⋊C4—61no—
339C4⋊C4—244no—
340C22⋊C4—68no—
341C22⋊C4—64no—
342C42—62no—
343C42—62no—
344C42—122no—
345C42D3×D412yes—
346C42C2×#6712yes—
347C22×C3—44no—
348C22×C3—44no—
349C22×C3—44no—
350C22×C3—84no—
351C22×C3—84no—
352C22×C3C2×#12011yescharacteristic
353C22×C3#54812yes—
354C22×C3C22×SD1611yescharacteristic
355C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
356C22×C3#57612yes—
357C2×D3—42no—
358C2×D3—42no—
359C2×D3—42no—
360C2×D3—82no—
361C2×D3—82no—
362C3×C4—22no—
363C3×C4—82no—
364C3×C4#45014yes—
365C3×C4#53211yescharacteristic
366C3×C4#53611yescharacteristic
367C3×C4#54712yes—
368Dic3—24no—
369Dic3—44no—
370Dic3—42no—
371Dic3—42no—
372C23—24no—
373C23—44no—
374C23—44no—
375C23—42no—
376C23—42no—
377C23—62no—
378C23—122no—
379C23—122no—
380C23—121no—
381C23—121no—
382C23C2×D3×D411yescharacteristic
383Q8—68no—
384Q8—84no—
385D4—28no—
386D4—42no—
387D4—42no—
388D4—48no—
389D4—62no—
390D4—62no—
391D4—84no—
392D4—1216no—
393D4—122no—
394D4—248no—
395C2×C4—22no—
396C2×C4—34no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—62no—
400C2×C4—62no—
401C2×C4—64no—
402C2×C4—64no—
403C2×C4—62no—
404C2×C4—62no—
405C2×C4—1216no—
406C2×C4—122no—
407C2×C4—121no—
408C2×C4—122no—
409C2×C4—121no—
410C2×C4—128no—
411C2×C4D3×SD1612yes—
412C2×C4#91412yes—
413C2×C4#93712yes—
414C2×C4#93712yes—
415C2×C4C2×D3×D411yescharacteristic
416C2×C4D3×#4112yes—
417C2×C4#101012yes—
418C2×C4C22×#6711yescharacteristic
419C8—42no—
420C8—122no—
421C2×C3—84no—
422C2×C3#243211yescharacteristic
423C2×C3C2×#45011yescharacteristic
424C2×C3#300811yescharacteristic
425C2×C3#336212yes—
426C2×C3#336612yes—
427D3—82no—
428C22—44no—
429C22—44no—
430C22—44no—
431C22—84no—
432C22—84no—
433C22—122no—
434C22—122no—
435C22—122no—
436C22—242no—
437C22—242no—
438C22#567712yes—
439C22#575712yes—
440C22C2×D3×SD1611yescharacteristic
441C22C2×#91411yescharacteristic
442C22C2×#93711yescharacteristic
443C4—22no—
444C4—64no—
445C4—82no—
446C4—122no—
447C4—122no—
448C4—124no—
449C4#516914yes—
450C4#566811yescharacteristic
451C4#567511yescharacteristic
452C4#582412yes—
453C3#2564911yescharacteristic, complemented, Sylow
454C2—84no—
455C2—242no—
456C2#8158612yes—
457C2#8158812yes—
458C2#8295911yescharacteristic
459C2#8383311yescharacteristic
460C2C2×#516911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#486C2
socleC23×C3C22×D4
prime radicalC2×#5718C2
hypercenterC2×#486D3
hypocenterC3#25649
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#486

Complements

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

NH ≅ G/Ntype
2. C2×C3×#486C2semidirect
3. #83243C2semidirect
4. C2×#5084C2semidirect
5. #71761C2semidirect
6. #70211C2semidirect
7. #70169C2semidirect
8. #69788C2semidirect
9. C4×#821C2semidirect
39. C2×#486D3semidirect
178. C4⋊Dic3#41semidirect
453. C3#25649semidirect

Maximal subgroups

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

subgroupindex
C2×#57182
C2×C3×#4862
#832432
C2×#50842
#717612
#702112
#701692
#697882
C4×#8212
#256493