Finite Group Databasebeta

← #10848714

Subgroups — #10848714

Order 768.

1158 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5728C211yesmaximal, characteristic
2C2×#5093C211yesmaximal, characteristic, complemented
3C2×#4699C211yesmaximal, characteristic, complemented
4#74376C211yesmaximal, characteristic
5C3×#1990C211yesmaximal, characteristic, complemented
6C4×(C2×D3)⋊C8C211yesmaximal, characteristic
7#69579C211yesmaximal, characteristic, complemented
8#68908C214yesmaximal, complemented
9#68888C214yesmaximal
10#25600—31nomaximal, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×#946—21no—
13C2×Q8×Dic3—21no—
14C22×Dic12C2214yes—
15#5728—24no—
16#5728—24no—
17C2×#897C2212yes—
18C2×#897C2212yes—
19C2×#892—21no—
20C2×#890—21no—
21C2×#887C2212yes—
22C2×C42×D3C2211yescharacteristic
23C2×#869C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C2×C3×C4×C8C2211yescharacteristic
26C3×#384C2211yescharacteristic
27C2×(C2×D3)⋊C8C2212yes—
28#5093—24no—
29#4987—21no—
30#4987—21no—
31#4984—21no—
32#4984—21no—
33#4973C2212yes—
34C2×#810C2214yes—
35C2×#810C2214yes—
36C2×C4×C3⋊C8C2211yescharacteristic
37#4699—24no—
38C3×#340C2214yes—
39#4550C2212yes—
40C2×#533—31no—
41C2×#494—31no—
42C2×#487—31no—
43#2857—31no—
44#2001—34no—
45#1990D311yescharacteristic, complemented
46#1956—31no—
47#1878—34no—
48C4×C22⋊C8—31no—
49C22×C3×Q8—21no—
50C22×C4×D3D411yescharacteristic
51C22×C4×D3D411yescharacteristic
52C22×C4×D3C2311yescharacteristic
53C22×Dic6C2312yes—
54C22×Dic6C2312yes—
55C22×C3×C8C2312yes—
56C3×C4⋊Q8—22no—
57C3×C4⋊Q8D414yes—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4C2312yes—
62C2×C3×C4⋊C4C2312yes—
63C2×C3×C42C2311yescharacteristic
64#946—28no—
65Q8×Dic3—28no—
66C2×(C2×D3)⋊C4—22no—
67C2×(C2×D3)⋊C4—22no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3—21no—
70C2×C4⋊Dic3—22no—
71C2×C4⋊Dic3—21no—
72C2×C4⋊Dic3—21no—
73C2×C4⋊Dic3—22no—
74C2×Dic3⋊C4—22no—
75C2×C4×Dic3—22no—
76C2×C4×Dic3D411yescharacteristic
77C2×C4×Dic3D411yescharacteristic
78C2×C4×Dic3C2311yescharacteristic
79C22×C3⋊C8C2312yes—
80C2×Dic12—28no—
81C2×Dic12—28no—
82C2×Dic12—44no—
83#897—24no—
84#897—24no—
85#897—44no—
86#892—28no—
87#890—28no—
88#887—24no—
89#887—24no—
90C42×D3—22no—
91C42×D3—22no—
92#869—22no—
93#869—22no—
94#869—22no—
95#869D414yes—
96C3×C4×C8—22no—
97C3⋊#95—22no—
98C3⋊#95—22no—
99(C2×D3)⋊C8—24no—
100#810—28no—
101#810—44no—
102C4×C3⋊C8—22no—
103C22×Q16—34no—
104#533—64no—
105#533—64no—
106C2×C4⋊Q8—31no—
107C2×C4⋊Q8—31no—
108C2×C4⋊Q8C2×D311yescharacteristic
109C2×#125—61no—
110C2×C22⋊Q8—32no—
111C2×C22⋊Q8—32no—
112C2×C22⋊Q8—61no—
113C2×C4×Q8—32no—
114C2×C4×Q8—61no—
115C2×#117—61no—
116C22×C42—31no—
117#494—64no—
118#487—64no—
119C2×#103—34no—
120C2×#103—34no—
121C2×C22⋊C8—32no—
122C2×C4×C8—31no—
123C2×C4×C8C2×D311yescharacteristic
124#391—32no—
125#389—61no—
126#389—61no—
127#384C2×D311yescharacteristic
128#380—61no—
129#380—61no—
130#340C2×D314yes—
131#336—32no—
132C23×D3C2×D411yescharacteristic
133C2×C3×Q8—24no—
134C2×C3×Q8—42no—
135C2×C3×Q8—42no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2411yescharacteristic
141C22×Dic3—22no—
142C22×Dic3—22no—
143C22×Dic3C2×D411yescharacteristic
144C22×Dic3C2×D411yescharacteristic
145C22×Dic3C2×D411yescharacteristic
146C2×C4×D3—22no—
147C2×C4×D3—22no—
148C2×C4×D3—22no—
149C2×C4×D3—22no—
150C2×C4×D3—22no—
151C2×C4×D3—21no—
152C2×C4×D3—21no—
153C2×C4×D3—22no—
154C2×C4×D3—22no—
155C2×C4×D3Q1614yescomplemented
156C2×Dic6—24no—
157C2×Dic6—24no—
158C2×Dic6—22no—
159C2×Dic6—22no—
160C2×Dic6—22no—
161C2×Dic6—24no—
162C2×Dic6—24no—
163C2×Dic6C2×D414yes—
164C2×C3×C8—22no—
165C2×C3×C8—22no—
166C2×C3×C8#4114yes—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—24no—
170C3×C4⋊C4—42no—
171C3×C4⋊C4—42no—
172C3×C4⋊C4C2×D414yes—
173C3×C42C2×D412yes—
174C3×C42C2×D412yes—
175(C2×D3)⋊C4—28no—
176(C2×D3)⋊C4—44no—
177C4⋊Dic3—28no—
178C4⋊Dic3—24no—
179C4⋊Dic3—24no—
180C4⋊Dic3—44no—
181C4⋊Dic3—42no—
182C4⋊Dic3—42no—
183Dic3⋊C4—28no—
184C4×Dic3—28no—
185C4×Dic3—22no—
186C4×Dic3—22no—
187C4×Dic3—21no—
188C4×Dic3—21no—
189C2×C3⋊C8—22no—
190C2×C3⋊C8—22no—
191C2×C3⋊C8—22no—
192Dic12—416no—
193C22×Q8—21no—
194C22×Q8—32no—
195C22×Q8—32no—
196C23×C4—31no—
197C23×C4—31no—
198C23×C4—31no—
199C2×Q16—68no—
200C2×Q16—68no—
201C2×Q16—124no—
202C22×C8—32no—
203C22×C8C22×D312yes—
204C4⋊Q8—22no—
205C4⋊Q8—34no—
206C4⋊Q8—62no—
207C4⋊Q8—62no—
208C4⋊Q8—62no—
209C4⋊Q8D1214yes—
210#125—68no—
211C22⋊Q8—64no—
212C22⋊Q8—68no—
213C22⋊Q8—64no—
214C22⋊Q8—124no—
215C4×Q8—64no—
216C4×Q8—68no—
217C4×Q8—64no—
218#117—68no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—62no—
223C2×C4⋊C4—62no—
224C2×C4⋊C4—61no—
225C2×C4⋊C4—61no—
226C2×C4⋊C4—62no—
227C2×C4⋊C4—61no—
228C2×C4⋊C4—61no—
229C2×C4⋊C4C22×D312yes—
230C2×C4⋊C4C22×D312yes—
231C2×C22⋊C4—62no—
232C2×C22⋊C4—62no—
233C2×C42—31no—
234C2×C42—31no—
235C2×C42—31no—
236C2×C42—62no—
237C2×C42—62no—
238C2×C42—62no—
239C2×C42C22×D311yescharacteristic
240#103—68no—
241#103—124no—
242C22⋊C8—64no—
243C4×C8—22no—
244C4×C8—62no—
245#95—62no—
246#95—62no—
247C23×C3C22×D411yescharacteristic
248C22×D3—21no—
249C22×D3—21no—
250C22×D3—21no—
251C22×D3—21no—
252C22×D3—21no—
253C22×D3—21no—
254C22×D3C2×Q1612yes—
255C3×Q8—48no—
256C2×C3×C4—24no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4#12012yes—
263C2×C3×C4#12012yes—
264C2×C3×C4C2×Q1612yes—
265C2×C3×C4#13512yes—
266C2×C3×C4#13712yes—
267C2×C3×C4#13712yes—
268C2×C3×C4C22×D411yescharacteristic
269C2×C3×C4C22×D411yescharacteristic
270C2×C3×C4C2×#4112yes—
271C2×C3×C4H(5,𝔽2)12yes—
272C2×Dic3—21no—
273C2×Dic3—21no—
274C2×Dic3—21no—
275C2×Dic3—21no—
276C2×Dic3—24no—
277C2×Dic3—21no—
278C2×Dic3—21no—
279C2×Dic3—21no—
280C2×Dic3—21no—
281C2×Dic3—42no—
282C2×Dic3—42no—
283C2×Dic3—42no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286C2×Dic3C2×Q1612yes—
287C4×D3—28no—
288C4×D3—42no—
289C4×D3—42no—
290C4×D3—44no—
291Dic6—28no—
292Dic6—44no—
293Dic6—44no—
294Dic6—44no—
295C3×C8—24no—
296C3⋊C8—42no—
297C24—31no—
298C2×Q8—24no—
299C2×Q8—34no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—62no—
303C2×Q8—62no—
304C2×Q8—62no—
305C2×Q8—64no—
306C2×Q8—64no—
307C2×Q8—64no—
308C2×Q8—64no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—31no—
312C22×C4—34no—
313C22×C4—31no—
314C22×C4—31no—
315C22×C4—62no—
316C22×C4—62no—
317C22×C4—62no—
318C22×C4—62no—
319C22×C4—62no—
320C22×C4—62no—
321C22×C4—61no—
322C22×C4—61no—
323C22×C4—62no—
324C22×C4—62no—
325C22×C4—62no—
326C22×C4D3×D411yescharacteristic
327C22×C4D3×D411yescharacteristic
328C22×C4C23×D311yescharacteristic
329Q16—1216no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—62no—
333C2×C8—62no—
334C2×C8—62no—
335C2×C8#23914yes—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—24no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—68no—
342C4⋊C4—68no—
343C4⋊C4—64no—
344C4⋊C4—64no—
345C4⋊C4—124no—
346C4⋊C4—122no—
347C4⋊C4—122no—
348C4⋊C4C2×D1214yes—
349C22⋊C4—68no—
350C22⋊C4—124no—
351C42—68no—
352C42—62no—
353C42—62no—
354C42—61no—
355C42—61no—
356C42C2×D1212yes—
357C42D3×D412yes—
358C22×C3C2×#12011yescharacteristic
359C22×C3#53211yescharacteristic
360C22×C3#53511yescharacteristic
361C22×C3C22×Q1611yescharacteristic
362C22×C3C2×#13511yescharacteristic
363C22×C3C2×#13711yescharacteristic
364C22×C3C2×#13711yescharacteristic
365C2×D3—22no—
366C2×D3—22no—
367C2×D3—22no—
368C2×D3—41no—
369C2×D3—41no—
370C2×D3—41no—
371C2×D3—41no—
372C3×C4—44no—
373C3×C4—82no—
374C3×C4#45114yes—
375C3×C4#45214yes—
376C3×C4#54612yes—
377C3×C4#55012yes—
378Dic3—22no—
379Dic3—44no—
380Dic3—41no—
381Dic3—41no—
382Dic3—82no—
383C23—32no—
384C23—61no—
385C23—61no—
386C23—61no—
387C23—61no—
388C23—61no—
389C23—61no—
390C23C2×D3×D411yescharacteristic
391Q8—48no—
392Q8—68no—
393Q8—124no—
394Q8—124no—
395Q8—124no—
396C2×C4—24no—
397C2×C4—32no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—61no—
404C2×C4—61no—
405C2×C4—68no—
406C2×C4—61no—
407C2×C4—61no—
408C2×C4—61no—
409C2×C4—61no—
410C2×C4—64no—
411C2×C4—61no—
412C2×C4—61no—
413C2×C4—122no—
414C2×C4—122no—
415C2×C4—124no—
416C2×C4—122no—
417C2×C4—122no—
418C2×C4—122no—
419C2×C4—122no—
420C2×C4—122no—
421C2×C4#88212yes—
422C2×C4#88212yes—
423C2×C4#91212yes—
424C2×C4#91512yes—
425C2×C4#91512yes—
426C2×C4D3×Q1612yes—
427C2×C4C22×D1211yescharacteristic
428C2×C4C2×D3×D411yescharacteristic
429C2×C4C2×#23912yes—
430C2×C4#100912yes—
431C8—24no—
432C8—122no—
433C2×C3#243111yescharacteristic
434C2×C3C2×#45111yescharacteristic
435C2×C3C2×#45211yescharacteristic
436C2×C3#317011yescharacteristic
437C2×C3#318511yescharacteristic
438C2×C3#320911yescharacteristic
439C2×C3#321011yescharacteristic
440D3—42no—
441C22—62no—
442C22—62no—
443C22—62no—
444C22—121no—
445C22—121no—
446C22—121no—
447C22—121no—
448C22C2×#88211yescharacteristic
449C22#567411yescharacteristic
450C22#572511yescharacteristic
451C22C2×#91211yescharacteristic
452C22C2×#91511yescharacteristic
453C22C2×#91511yescharacteristic
454C22C2×D3×Q1611yescharacteristic
455C4—44no—
456C4—62no—
457C4—82no—
458C4—121no—
459C4—124no—
460C4—121no—
461C4—242no—
462C4#478314yes—
463C4#480914yes—
464C4#557712yes—
465C4#563012yes—
466C3#2560011yescharacteristic, complemented, Sylow
467C2—122no—
468C2#7662611yescharacteristic
469C2C2×#478311yescharacteristic
470C2C2×#480911yescharacteristic
471C2#8156911yescharacteristic
472C2#8160411yescharacteristic
473C2#8188911yescharacteristic
474C2#8193411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1990C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1990D3
hypocenterC3#25600
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
2. C2×#5093C2semidirect
3. C2×#4699C2semidirect
5. C3×#1990C2semidirect
7. #69579C2semidirect
8. #68908C2semidirect
45. #1990D3semidirect
155. C2×C4×D3Q16semidirect
466. C3#25600semidirect

Maximal subgroups

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

subgroupindex
C2×#57282
C2×#50932
C2×#46992
#743762
C3×#19902
C4×(C2×D3)⋊C82
#695792
#689082
#688882
#256003