Finite Group Databasebeta

← #10900928

Subgroups — #10900928

Order 768.

1292 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#888C211yesmaximal, characteristic, complemented
2C2×#4968C211yesmaximal, characteristic
3C2×#4735C211yesmaximal, characteristic
4#73999C211yesmaximal, characteristic
5C3×#1824C211yesmaximal, characteristic, complemented
6#69910C211yesmaximal, characteristic, complemented
7#69502C211yesmaximal, characteristic, complemented
8#68898C214yesmaximal, complemented
9#68827C214yesmaximal, complemented
10#29639—31nomaximal, Sylow
11C23×Dic6—21no—
12C22×C3×C4⋊C4C2211yescharacteristic
13C2×#930C2211yescharacteristic
14C22×C4⋊Dic3C2211yescharacteristic
15C22×Dic3⋊C4—21no—
16C22×C4×Dic3C2211yescharacteristic
17C2×#888—24no—
18C2×#888—24no—
19C2×#888—22no—
20C2×#888C2214yes—
21C2×#886—21no—
22C2×C3×C22⋊C8C2211yescharacteristic
23C3×#380C2211yescharacteristic
24C2×#830C2211yescharacteristic
25C2×#821C2212yes—
26C2×#817C2212yes—
27#4988C2212yes—
28#4984C2212yes—
29#4968—24no—
30C2×#809C2212yes—
31C2×#808C2212yes—
32#4958—21no—
33#4735—24no—
34#4670—22no—
35C3×#340C2214yes—
36#4550C2214yes—
37#4529C2214yes—
38C22×C4⋊Q8—31no—
39C2×#481—31no—
40C2×#481—31no—
41#2854—31no—
42#2058—34no—
43#2023—34no—
44#1824—31no—
45#1824D311yescharacteristic, complemented
46#1817—31no—
47C23×C3×C4—21no—
48C23×C3×C4C2311yescharacteristic
49C22×#67—21no—
50C23×Dic3—21no—
51C23×Dic3D411yescharacteristic
52C23×Dic3D411yescharacteristic
53C22×D12C2311yescharacteristic
54C22×Dic6—28no—
55C22×Dic6—24no—
56C22×Dic6—48no—
57C22×C3×C8C2312yes—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4D414yes—
61C2×C3×C4⋊C4C2312yes—
62C2×C3×C4⋊C4C2312yes—
63C2×C3×C22⋊C4—21no—
64#930—22no—
65#930—22no—
66C2×(C2×D3)⋊C4—21no—
67C2×(C2×D3)⋊C4—22no—
68C2×(C2×D3)⋊C4—22no—
69C2×C4⋊Dic3—22no—
70C2×C4⋊Dic3—21no—
71C2×C4⋊Dic3—22no—
72C2×C4⋊Dic3D414yes—
73C2×C4⋊Dic3C2311yescharacteristic
74C2×C4⋊Dic3C2312yes—
75C2×Dic3⋊C4—28no—
76C2×Dic3⋊C4—24no—
77C2×C4×Dic3—22no—
78C2×C4×Dic3—22no—
79C2×C4×Dic3D412yes—
80C2×C4×Dic3C2312yes—
81C22×C3⋊C8C2312yes—
82#888—216no—
83#888—28no—
84#888—28no—
85#886—44no—
86C3×C22⋊C8—22no—
87C3×#95—21no—
88C3⋊#95—22no—
89#830—22no—
90#821—24no—
91#817—24no—
92#809—24no—
93#808—24no—
94C23×Q8—61no—
95C2×C4⋊Q8—34no—
96C2×C4⋊Q8—62no—
97C2×C4⋊Q8—64no—
98C2×C4⋊Q8—64no—
99C2×#123—61no—
100C2×#121—31no—
101C22×C4⋊C4—31no—
102C22×C4⋊C4—61no—
103C22×C4⋊C4C2×D311yescharacteristic
104C22×C42—31no—
105#481—64no—
106#481—64no—
107C2×#106—32no—
108C2×#106—32no—
109C2×#102—32no—
110C2×#102—32no—
111C2×C22⋊C8—31no—
112C2×C22⋊C8C2×D311yescharacteristic
113#390—32no—
114#389—32no—
115#388—62no—
116#386—61no—
117#380C2×D311yescharacteristic
118#340—34no—
119#340C2×D314yes—
120#336—34no—
121C24×C3C2×D411yescharacteristic
122C23×D3—21no—
123C22×C3×C4—24no—
124C22×C3×C4—24no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—44no—
128C22×C3×C4C2×D412yes—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D412yes—
131C22×C3×C4C2411yescharacteristic
132C2×#67—42no—
133C2×#67—42no—
134C2×#67—42no—
135C22×Dic3—22no—
136C22×Dic3—22no—
137C22×Dic3—22no—
138C22×Dic3—24no—
139C22×Dic3—24no—
140C22×Dic3—22no—
141C22×Dic3—21no—
142C22×Dic3—44no—
143C22×Dic3SD1614yes—
144C22×Dic3C2×D412yes—
145C22×Dic3C2×D412yes—
146C2×D12—21no—
147C2×D12—21no—
148C2×D12—24no—
149C2×Dic6—216no—
150C2×Dic6—432no—
151C2×Dic6—48no—
152C2×Dic6—48no—
153C2×C3×C8—22no—
154C2×C3×C8—24no—
155C3×C4⋊C4—24no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—24no—
158C3×C4⋊C4C2×D414yes—
159C3×C22⋊C4—42no—
160(C2×D3)⋊C4—42no—
161(C2×D3)⋊C4—44no—
162(C2×D3)⋊C4—44no—
163C4⋊Dic3—24no—
164C4⋊Dic3—22no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167C4⋊Dic3—44no—
168C4⋊Dic3C2×D414yes—
169Dic3⋊C4—216no—
170C4×Dic3—24no—
171C4×Dic3—22no—
172C4×Dic3—22no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—24no—
175C22×Q8—64no—
176C22×Q8—68no—
177C22×Q8—128no—
178C22×D4—31no—
179C22×D4—61no—
180C23×C4—21no—
181C23×C4—31no—
182C23×C4—31no—
183C23×C4—61no—
184C23×C4C22×D311yescharacteristic
185C22×C8—32no—
186C22×C8C22×D312yes—
187C4⋊Q8—68no—
188C4⋊Q8—616no—
189C4⋊Q8—68no—
190#123—124no—
191#121—62no—
192#121—62no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—34no—
196C2×C4⋊C4—32no—
197C2×C4⋊C4—31no—
198C2×C4⋊C4—64no—
199C2×C4⋊C4—68no—
200C2×C4⋊C4—62no—
201C2×C4⋊C4—61no—
202C2×C4⋊C4—62no—
203C2×C4⋊C4D1214yes—
204C2×C4⋊C4C22×D312yes—
205C2×C4⋊C4C22×D312yes—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—62no—
208C2×C22⋊C4—62no—
209C2×C22⋊C4—61no—
210C2×C42—32no—
211C2×C42—32no—
212C2×C42—62no—
213C2×C42—62no—
214#106—64no—
215#106—64no—
216#102—64no—
217#102—64no—
218C22⋊C8—22no—
219C22⋊C8—62no—
220#95—21no—
221#95—62no—
222C23×C3—22no—
223C23×C3—22no—
224C23×C3#12012yes—
225C23×C3C2×SD1612yes—
226C23×C3#13712yes—
227C23×C3C22×D411yescharacteristic
228C22×D3—41no—
229C22×D3—42no—
230C22×D3—41no—
231C22×D3—42no—
232C22×D3—41no—
233C2×C3×C4—22no—
234C2×C3×C4—24no—
235C2×C3×C4—24no—
236C2×C3×C4—24no—
237C2×C3×C4—48no—
238C2×C3×C4—44no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4#12012yes—
243C2×C3×C4C22×D411yescharacteristic
244C2×C3×C4C22×D411yescharacteristic
245C2×C3×C42−1+414yes—
246#67—84no—
247C2×Dic3—28no—
248C2×Dic3—22no—
249C2×Dic3—22no—
250C2×Dic3—22no—
251C2×Dic3—22no—
252C2×Dic3—28no—
253C2×Dic3—24no—
254C2×Dic3—48no—
255C2×Dic3—44no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258C2×Dic3—41no—
259C2×Dic3C2×SD1614yes—
260C2×Dic3AGL(1,ℤ/8)14yes—
261D12—42no—
262D12—42no—
263Dic6—432no—
264C3×C8—42no—
265C3⋊C8—42no—
266C24—61no—
267C24D3×D411yescharacteristic
268C2×Q8—616no—
269C2×Q8—128no—
270C2×Q8—1232no—
271C2×Q8—128no—
272C2×D4—61no—
273C2×D4—61no—
274C2×D4—64no—
275C2×D4—122no—
276C2×D4—122no—
277C2×D4—122no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—34no—
283C22×C4—32no—
284C22×C4—32no—
285C22×C4—44no—
286C22×C4—64no—
287C22×C4—62no—
288C22×C4—62no—
289C22×C4—62no—
290C22×C4—64no—
291C22×C4—62no—
292C22×C4—61no—
293C22×C4—124no—
294C22×C4C2×D1212yes—
295C22×C4D3×D411yescharacteristic
296C22×C4D3×D412yes—
297C22×C4C23×D311yescharacteristic
298C2×C8—22no—
299C2×C8—24no—
300C2×C8—62no—
301C2×C8—64no—
302C4⋊C4—24no—
303C4⋊C4—22no—
304C4⋊C4—24no—
305C4⋊C4—34no—
306C4⋊C4—616no—
307C4⋊C4—64no—
308C4⋊C4—62no—
309C4⋊C4—61no—
310C4⋊C4—61no—
311C4⋊C4—124no—
312C4⋊C4C2×D1214yes—
313C22⋊C4—42no—
314C22⋊C4—124no—
315C22⋊C4—124no—
316C22⋊C4—122no—
317C42—64no—
318C42—62no—
319C42—62no—
320C22×C3—22no—
321C22×C3—24no—
322C22×C3—24no—
323C22×C3—22no—
324C22×C3#44814yes—
325C22×C3C2×#12011yescharacteristic
326C22×C3#53612yes—
327C22×C3C22×SD1611yescharacteristic
328C22×C3C2×AGL(1,ℤ/8)12yes—
329C22×C3C2×#13711yescharacteristic
330C2×D3—81no—
331C2×D3—82no—
332C2×D3—82no—
333C2×D3—81no—
334C2×D3—81no—
335C3×C4—22no—
336C3×C4—44no—
337C3×C4—82no—
338C3×C4#53612yes—
339C3×C4#53612yes—
340Dic3—24no—
341Dic3—24no—
342Dic3—44no—
343Dic3—81no—
344C23—22no—
345C23—22no—
346C23—122no—
347C23—121no—
348C23—121no—
349C23—121no—
350C23—122no—
351C23#88212yes—
352C23D3×SD1612yes—
353C23#91812yes—
354C23C2×D3×D411yescharacteristic
355Q8—1232no—
356D4—122no—
357D4—122no—
358D4—244no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—34no—
364C2×C4—34no—
365C2×C4—48no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—68no—
371C2×C4—62no—
372C2×C4—62no—
373C2×C4—62no—
374C2×C4—62no—
375C2×C4—68no—
376C2×C4—64no—
377C2×C4—128no—
378C2×C4—124no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4#88212yes—
383C2×C4C22×D1211yescharacteristic
384C2×C4C2×D3×D411yescharacteristic
385C2×C4#101014yes—
386C8—42no—
387C8—122no—
388C2×C3—24no—
389C2×C3#261411yescharacteristic
390C2×C3C2×#44811yescharacteristic
391C2×C3#300311yescharacteristic
392C2×C3#322712yes—
393C2×C3#323912yes—
394D3—161no—
395C22—24no—
396C22—22no—
397C22—22no—
398C22—24no—
399C22—242no—
400C22—241no—
401C22—241no—
402C22—242no—
403C22—241no—
404C22#480614yes—
405C22C2×#88211yescharacteristic
406C22#564812yes—
407C22C2×D3×SD1611yescharacteristic
408C22C2×#91811yescharacteristic
409C22D3×AGL(1,ℤ/8)12yes—
410C4—22no—
411C4—44no—
412C4—64no—
413C4—64no—
414C4—82no—
415C4—124no—
416C4—241no—
417C4#552612yes—
418C4#564812yes—
419C3#2963911yescharacteristic, complemented, Sylow
420C2—24no—
421C2—481no—
422C2#7413511yescharacteristic
423C2C2×#480611yescharacteristic
424C2#7892611yescharacteristic
425C2#8131412yes—
426C2#8145912yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1824C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1824D3
hypocenterC3#29639
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
1. C22×#888C2semidirect
5. C3×#1824C2semidirect
6. #69910C2semidirect
7. #69502C2semidirect
8. #68898C2semidirect
9. #68827C2semidirect
45. #1824D3semidirect
419. C3#29639semidirect

Maximal subgroups

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

subgroupindex
C22×#8882
C2×#49682
C2×#47352
#739992
C3×#18242
#699102
#695022
#688982
#688272
#296393