Finite Group Databasebeta

← #10901361

Subgroups — #10901361

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×#5470C211yesmaximal, characteristic
2C3×#2857C211yesmaximal, characteristic
3C2×#5094C212yesmaximal, complemented
4#71530C211yesmaximal, characteristic
5#69633C212yesmaximal, complemented
6#69605C212yesmaximal, complemented
7#69588C212yesmaximal, complemented
8#69526C214yesmaximal
9#29688—31nomaximal, Sylow
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×C22⋊Q8—21no—
12C22×C3×C42C2211yescharacteristic
13C2×#926—21no—
14C2×#924—21no—
15C2×#924C2212yes—
16#5470—28no—
17C2×#870—21no—
18C2×#869C2211yescharacteristic
19C2×#869C2211yescharacteristic
20C2×C4×Dic6—21no—
21C2×C4×Dic6C2212yes—
22C3×#391—21no—
23C3×#389—21no—
24C3×#389C2212yes—
25C3×#384C2211yescharacteristic
26C3×#380—21no—
27C3×#380C2212yes—
28C2×#830C2212yes—
29#5094—28no—
30C2×#810C2214yes—
31C2×#810C2214yes—
32C2×#808C2214yes—
33C2×C4⋊(C3⋊C8)C2212yes—
34#4529C2214yes—
35#4524C2212yes—
36C2×#533—31no—
37C2×#475—32no—
38#2857D311yescharacteristic
39#2058—32no—
40#2043—32no—
41#2035—34no—
42#1803—32no—
43#1797—31no—
44C22×C3×Q8—21no—
45C23×C3×C4D411yescharacteristic
46C23×C3×C4D411yescharacteristic
47C23×C3×C4C2311yescharacteristic
48C22×Dic6—21no—
49C22×Dic6C2312yes—
50C3×C4⋊Q8—24no—
51C3×C22⋊Q8—44no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—22no—
56C2×C3×C4⋊C4C2312yes—
57C2×C3×C4⋊C4C2312yes—
58C2×C3×C22⋊C4—22no—
59C2×C3×C42—24no—
60C2×C3×C42D411yescharacteristic
61C2×C3×C42D411yescharacteristic
62C2×C3×C42C2311yescharacteristic
63C2×C3×C42C2311yescharacteristic
64C2×#217—22no—
65#926—28no—
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3C2312yes—
70#924—28no—
71#924—24no—
72#924—24no—
73C2×Dic3⋊C4—22no—
74C2×Dic3⋊C4—22no—
75C2×C4×Dic3—22no—
76C22×C3⋊C8C2314yes—
77#870—28no—
78#869—24no—
79#869—24no—
80C4×Dic6—28no—
81C4×Dic6—24no—
82C4×Dic6D418yes—
83C3×#95—22no—
84C3×#95—22no—
85#830—42no—
86#810—28no—
87#810—44no—
88#808—28no—
89C4⋊(C3⋊C8)—24no—
90#533—68no—
91C2×C4⋊Q8—31no—
92C2×C4⋊Q8—31no—
93C2×C4⋊Q8C2×D311yescharacteristic
94C2×#125—61no—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—32no—
97C2×C22⋊Q8—61no—
98C2×C4×Q8—32no—
99C2×C4×Q8—61no—
100C2×#117—61no—
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—21no—
109#389—21no—
110#389C2×D312yes—
111#384C2×D311yescharacteristic
112#380—21no—
113#380C2×D312yes—
114#340—34no—
115#336—32no—
116C24×C3C2×D411yescharacteristic
117C2×C3×Q8—42no—
118C2×C3×Q8—42no—
119C2×C3×Q8—42no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—21no—
126C22×C3×C4—22no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—41no—
131C22×C3×C4SD1614yes—
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2411yescharacteristic
138C22×Dic3—22no—
139C22×Dic3—22no—
140C2×Dic6—22no—
141C2×Dic6—22no—
142C2×Dic6—22no—
143C2×Dic6—22no—
144C2×Dic6—24no—
145C2×Dic6—22no—
146C2×Dic6—42no—
147C2×Dic6—42no—
148C2×Dic6C2×D414yes—
149C3×C4⋊C4—24no—
150C3×C4⋊C4—22no—
151C3×C4⋊C4—22no—
152C3×C4⋊C4—42no—
153C3×C4⋊C4—42no—
154C3×C4⋊C4—42no—
155C3×C4⋊C4—44no—
156C3×C22⋊C4—44no—
157C3×C42—22no—
158C3×C42—22no—
159C3×C42—22no—
160C3×C42C2×D414yes—
161#217—28no—
162C4⋊Dic3—24no—
163C4⋊Dic3—24no—
164C4⋊Dic3—24no—
165C4⋊Dic3—22no—
166C4⋊Dic3C2×D414yes—
167Dic3⋊C4—28no—
168Dic3⋊C4—28no—
169C4×Dic3—28no—
170C2×C3⋊C8—24no—
171C2×C3⋊C8—24no—
172C2×C3⋊C8—24no—
173C22×Q8—21no—
174C22×Q8—32no—
175C22×Q8—61no—
176C23×C4#6711yescharacteristic
177C23×C4#6711yescharacteristic
178C23×C4C22×D311yescharacteristic
179C22×C8—34no—
180C4⋊Q8—24no—
181C4⋊Q8—64no—
182C4⋊Q8—64no—
183#125—68no—
184C22⋊Q8—44no—
185C22⋊Q8—64no—
186C22⋊Q8—64no—
187C22⋊Q8—68no—
188C4×Q8—38no—
189C4×Q8—68no—
190C4×Q8—64no—
191#117—68no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—32no—
197C2×C4⋊C4—61no—
198C2×C4⋊C4—61no—
199C2×C4⋊C4—62no—
200C2×C4⋊C4—62no—
201C2×C4⋊C4—61no—
202C2×C4⋊C4C22×D312yes—
203C2×C4⋊C4C22×D312yes—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—62no—
206C2×C42—24no—
207C2×C42—62no—
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—22no—
218#95—22no—
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—84no—
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—28no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—44no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4#12012yes—
248C2×C3×C4#12012yes—
249C2×C3×C4C2×SD1612yes—
250C2×C3×C4C2×SD1612yes—
251C2×C3×C4#13712yes—
252C2×C3×C4#13712yes—
253C2×C3×C4C22×D412yes—
254C2×C3×C4H(5,𝔽2)11yescharacteristic
255C2×C3×C42−1+412yes—
256C2×C3×C42−1+411yescharacteristic
257C2×Dic3—24no—
258C2×Dic3—24no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263Dic6—24no—
264Dic6—24no—
265Dic6—48no—
266Dic6—42no—
267Dic6—42no—
268C3⋊C8—44no—
269C24C2×#6711yescharacteristic
270C2×Q8—34no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—64no—
275C2×Q8—62no—
276C2×Q8—62no—
277C2×Q8—62no—
278C2×Q8—62no—
279C2×Q8—62no—
280C2×Q8—122no—
281C2×Q8—122no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—41no—
293C22×C4—62no—
294C22×C4—62no—
295C22×C4#21414yes—
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—22no—
307C4⋊C4—22no—
308C4⋊C4—34no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—44no—
312C4⋊C4—42no—
313C4⋊C4—64no—
314C4⋊C4—64no—
315C4⋊C4—68no—
316C4⋊C4—64no—
317C4⋊C4—68no—
318C4⋊C4—62no—
319C22⋊C4—44no—
320C22⋊C4—68no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—68no—
325C42D3×D414yes—
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—82no—
345C3×C4—82no—
346C3×C4#44814yes—
347C3×C4#53612yes—
348C3×C4#53712yes—
349Dic3—44no—
350Dic3—44no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23C2×#21412yes—
358C23C22×#6711yescharacteristic
359Q8—64no—
360Q8—64no—
361Q8—84no—
362Q8—128no—
363Q8—122no—
364Q8—122no—
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—44no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—64no—
385C2×C4—64no—
386C2×C4—122no—
387C2×C4—122no—
388C2×C4—122no—
389C2×C4—122no—
390C2×C4C2×#21412yes—
391C2×C4C2×#21412yes—
392C2×C4#93712yes—
393C2×C4#93712yes—
394C2×C4#94212yes—
395C2×C4#95112yes—
396C2×C4C2×D3×D412yes—
397C2×C4#100712yes—
398C2×C4#100911yescharacteristic
399C2×C4#101011yescharacteristic
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×#21411yescharacteristic
417C22C2×#93711yescharacteristic
418C22C2×#94211yescharacteristic
419C22#581611yescharacteristic
420C22#582211yescharacteristic
421C22C2×#95111yescharacteristic
422C22#583311yescharacteristic
423C4—22no—
424C4—22no—
425C4—22no—
426C4—41no—
427C4—82no—
428C4—82no—
429C4—124no—
430C4—124no—
431C4#504014yes—
432C4#562812yes—
433C4#562912yes—
434C3#2968811yescharacteristic, complemented, Sylow
435C2—42no—
436C2C2×#504011yescharacteristic
437C2#8273511yescharacteristic
438C2#8282011yescharacteristic
439C2#8322311yescharacteristic
440C2#8332011yescharacteristic
441C2#8337211yescharacteristic
442C2#8337411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2857C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2857D3
hypocenterC3#29688
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2857

Complements

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

NH ≅ G/Ntype
3. C2×#5094C2semidirect
5. #69633C2semidirect
6. #69605C2semidirect
7. #69588C2semidirect
434. C3#29688semidirect

Maximal subgroups

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

subgroupindex
C2×#54702
C3×#28572
C2×#50942
#715302
#696332
#696052
#695882
#695262
#296883