Finite Group Databasebeta

← #10901357

Subgroups — #10901357

Order 768.

1068 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic
2C2×#5020C212yesmaximal, complemented
3#82076C211yesmaximal, characteristic
4#71783C212yesmaximal, complemented
5#71781C212yesmaximal, complemented
6#71694C214yesmaximal
7#71626C212yesmaximal, complemented
8#71530C211yesmaximal, characteristic
9#29688—31nomaximal, Sylow
10C3×#533—28no—
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×#125—21no—
14C2×C3×C22⋊Q8—21no—
15C2×C3×C22⋊Q8C2212yes—
16C2×C3×C4×Q8—21no—
17C2×C3×C4×Q8C2212yes—
18C2×C3×#117—21no—
19C22×C3×C42C2211yescharacteristic
20C2×#924—21no—
21C2×#869C2211yescharacteristic
22C2×#835C2214yes—
23C2×#835C2214yes—
24#5212—21no—
25#5212C2212yes—
26C2×#830C2212yes—
27#5020—28no—
28C2×#808C2214yes—
29#4935C2211yescharacteristic
30#4929—21no—
31#4928—21no—
32#4928C2212yes—
33C2×C4⋊(C3⋊C8)C2212yes—
34#4529C2214yes—
35#4524C2212yes—
36C2×#533D311yescharacteristic
37C2×#475—32no—
38#2857—31no—
39#2058—32no—
40#2043—32no—
41#2035—34no—
42#1803—32no—
43#1797—31no—
44C22×C3×Q8—21no—
45C22×C3×Q8C2312yes—
46C23×C3×C4D411yescharacteristic
47C23×C3×C4D411yescharacteristic
48C23×C3×C4C2311yescharacteristic
49C22×Dic6—21no—
50C3×C4⋊Q8—24no—
51C3×C4⋊Q8—24no—
52C3×#125—28no—
53C3×C22⋊Q8—28no—
54C3×C22⋊Q8—24no—
55C3×C22⋊Q8—24no—
56C3×C4×Q8—28no—
57C3×C4×Q8—24no—
58C3×C4×Q8D418yes—
59C3×#117—28no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C22⋊C4—22no—
67C2×C3×C42—22no—
68C2×C3×C42—24no—
69C2×C3×C42D411yescharacteristic
70C2×C3×C42D411yescharacteristic
71C2×C3×C42C2311yescharacteristic
72C2×C3×C42C2311yescharacteristic
73C2×#217—22no—
74C2×C4⋊Dic3—21no—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3—21no—
77C2×C4⋊Dic3C2312yes—
78C2×C4⋊Dic3C2312yes—
79#924—44no—
80C2×Dic3⋊C4—22no—
81C22×C3⋊C8C2314yes—
82#869—24no—
83#835—28no—
84#835—44no—
85C3⋊#95—22no—
86C3⋊#95—22no—
87#830—42no—
88#808—28no—
89C4⋊(C3⋊C8)—24no—
90#533—28no—
91C2×C4⋊Q8—31no—
92C2×C4⋊Q8C2×D311yescharacteristic
93C2×C4⋊Q8C2×D311yescharacteristic
94C2×#125—21no—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—61no—
97C2×C22⋊Q8C2×D312yes—
98C2×C4×Q8—21no—
99C2×C4×Q8C2×D312yes—
100C2×#117—21no—
101C22×C42C2×D311yescharacteristic
102#475—68no—
103C2×#106—34no—
104C2×C4⋊C8—32no—
105C2×#103—34no—
106C2×#103—34no—
107C2×C22⋊C8—32no—
108#391—61no—
109#389—32no—
110#389—61no—
111#384—31no—
112#380—32no—
113#380—61no—
114#340—34no—
115#336—32no—
116C24×C3C2×D411yescharacteristic
117C2×C3×Q8—24no—
118C2×C3×Q8—22no—
119C2×C3×Q8—22no—
120C2×C3×Q8—22no—
121C2×C3×Q8—22no—
122C2×C3×Q8—22no—
123C2×C3×Q8—42no—
124C2×C3×Q8—42no—
125C2×C3×Q8C2×D414yes—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—21no—
134C22×C3×C4—22no—
135C22×C3×C4—21no—
136C22×C3×C4—41no—
137C22×C3×C4SD1614yes—
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2411yescharacteristic
144C22×Dic3—22no—
145C22×Dic3—22no—
146C2×Dic6—42no—
147C2×Dic6—42no—
148C2×Dic6—42no—
149C3×C4⋊C4—24no—
150C3×C4⋊C4—24no—
151C3×C4⋊C4—28no—
152C3×C4⋊C4—24no—
153C3×C4⋊C4—28no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4C2×D414yes—
156C3×C22⋊C4—28no—
157C3×C42—22no—
158C3×C42—22no—
159C3×C42—28no—
160C3×C42—22no—
161C3×C42C2×D414yes—
162#217—44no—
163C4⋊Dic3—24no—
164C4⋊Dic3—22no—
165C4⋊Dic3—22no—
166C4⋊Dic3—42no—
167C4⋊Dic3—42no—
168C4⋊Dic3—42no—
169Dic3⋊C4—44no—
170C2×C3⋊C8—24no—
171C2×C3⋊C8—24no—
172C2×C3⋊C8—24no—
173C22×Q8—21no—
174C22×Q8—61no—
175C22×Q8C22×D312yes—
176C23×C4#6711yescharacteristic
177C23×C4#6711yescharacteristic
178C23×C4C22×D311yescharacteristic
179C22×C8—34no—
180C4⋊Q8—24no—
181C4⋊Q8—24no—
182C4⋊Q8—64no—
183#125—28no—
184C22⋊Q8—28no—
185C22⋊Q8—24no—
186C22⋊Q8—24no—
187C22⋊Q8—124no—
188C4×Q8—28no—
189C4×Q8—24no—
190C4×Q8#6718yes—
191#117—28no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—32no—
198C2×C4⋊C4—32no—
199C2×C4⋊C4—61no—
200C2×C4⋊C4—62no—
201C2×C4⋊C4—61no—
202C2×C4⋊C4—61no—
203C2×C4⋊C4C22×D312yes—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—62no—
206C2×C42—22no—
207C2×C42—24no—
208C2×C42#6711yescharacteristic
209C2×C42#6711yescharacteristic
210C2×C42C22×D311yescharacteristic
211C2×C42C22×D311yescharacteristic
212#106—68no—
213C4⋊C8—64no—
214#103—68no—
215#103—124no—
216C22⋊C8—122no—
217#95—62no—
218#95—62no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—21no—
222C23×C3—21no—
223C23×C3—21no—
224C23×C3—21no—
225C23×C3C2×SD1612yes—
226C23×C3C22×D411yescharacteristic
227C3×Q8—24no—
228C3×Q8—24no—
229C3×Q8—48no—
230C3×Q8—42no—
231C3×Q8—42no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—28no—
239C2×C3×C4—21no—
240C2×C3×C4—24no—
241C2×C3×C4—24no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—44no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4#12012yes—
252C2×C3×C4#12012yes—
253C2×C3×C4C2×SD1612yes—
254C2×C3×C4C2×SD1612yes—
255C2×C3×C4#13712yes—
256C2×C3×C4#13712yes—
257C2×C3×C4C22×D412yes—
258C2×C3×C4H(5,𝔽2)11yescharacteristic
259C2×C3×C42−1+411yescharacteristic
260C2×C3×C42−1+412yes—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3—42no—
267Dic6—84no—
268C3⋊C8—44no—
269C24C2×#6711yescharacteristic
270C2×Q8—24no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—122no—
279C2×Q8—122no—
280C2×Q8—122no—
281C2×Q8C2×#6714yes—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—41no—
293C22×C4—62no—
294C22×C4—62no—
295C22×C4#21514yes—
296C22×C4C2×#6711yescharacteristic
297C22×C4C2×#6711yescharacteristic
298C22×C4C2×#6711yescharacteristic
299C22×C4C2×#6711yescharacteristic
300C22×C4C2×#6711yescharacteristic
301C22×C4C23×D311yescharacteristic
302C2×C8—64no—
303C2×C8—64no—
304C2×C8—64no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—28no—
308C4⋊C4—24no—
309C4⋊C4—28no—
310C4⋊C4—22no—
311C4⋊C4—62no—
312C4⋊C4—62no—
313C4⋊C4—64no—
314C4⋊C4—122no—
315C4⋊C4—124no—
316C4⋊C4—122no—
317C4⋊C4—122no—
318C4⋊C4C2×#6714yes—
319C22⋊C4—28no—
320C22⋊C4—124no—
321C42—22no—
322C42—22no—
323C42—28no—
324C42—22no—
325C42C2×#6714yes—
326C22×C3—22no—
327C22×C3—22no—
328C22×C3—22no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3C2×#12011yescharacteristic
334C22×C3#53611yescharacteristic
335C22×C3#53711yescharacteristic
336C22×C3C22×SD1611yescharacteristic
337C22×C3C2×#13711yescharacteristic
338C22×C3C2×#13711yescharacteristic
339C22×C3#57511yescharacteristic
340C3×C4—22no—
341C3×C4—22no—
342C3×C4—22no—
343C3×C4—41no—
344C3×C4—44no—
345C3×C4—44no—
346C3×C4#44814yes—
347C3×C4#53612yes—
348C3×C4#53712yes—
349Dic3—82no—
350Dic3—82no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23C2×#21512yes—
358C23C22×#6711yescharacteristic
359Q8—24no—
360Q8—24no—
361Q8—48no—
362Q8—42no—
363Q8—42no—
364Q8—244no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—28no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—122no—
385C2×C4—122no—
386C2×C4—122no—
387C2×C4—122no—
388C2×C4—122no—
389C2×C4—122no—
390C2×C4C2×#21512yes—
391C2×C4C2×#21512yes—
392C2×C4#94212yes—
393C2×C4#95112yes—
394C2×C4#95312yes—
395C2×C4#95312yes—
396C2×C4#100911yescharacteristic
397C2×C4#101011yescharacteristic
398C2×C4#101012yes—
399C2×C4C22×#6712yes—
400C8—124no—
401C2×C3—42no—
402C2×C3#261611yescharacteristic
403C2×C3C2×#44811yescharacteristic
404C2×C3#300711yescharacteristic
405C2×C3#322911yescharacteristic
406C2×C3#323711yescharacteristic
407C2×C3#324611yescharacteristic
408C2×C3#325511yescharacteristic
409C22—22no—
410C22—22no—
411C22—22no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22C22×#21511yescharacteristic
417C22C2×#94211yescharacteristic
418C22#582211yescharacteristic
419C22C2×#95111yescharacteristic
420C22#582511yescharacteristic
421C22#583311yescharacteristic
422C22C2×#95311yescharacteristic
423C4—22no—
424C4—22no—
425C4—22no—
426C4—44no—
427C4—41no—
428C4—44no—
429C4—242no—
430C4—242no—
431C4#522714yes—
432C4#582512yes—
433C4#583312yes—
434C3#2968811yescharacteristic, complemented, Sylow
435C2—42no—
436C2#8332311yescharacteristic
437C2#8332711yescharacteristic
438C2#8336311yescharacteristic
439C2#8337411yescharacteristic
440C2C2×#522711yescharacteristic
441C2#8408411yescharacteristic
442C2#8409411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#533C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#533D3
hypocenterC3#29688
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
2. C2×#5020C2semidirect
4. #71783C2semidirect
5. #71781C2semidirect
7. #71626C2semidirect
434. C3#29688semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C2×#50202
#820762
#717832
#717812
#716942
#716262
#715302
#296883