Finite Group Databasebeta

← #10848713

Subgroups — #10848713

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×#5470C211yesmaximal, characteristic
2C3×#2857C211yesmaximal, characteristic
3C2×#5092C211yesmaximal, characteristic, complemented
4C2×#5079C211yesmaximal, characteristic, complemented
5#71700C214yesmaximal
6C4×#830C211yesmaximal, characteristic
7#69690C214yesmaximal, complemented
8#69582C211yesmaximal, characteristic, complemented
9#69579C211yesmaximal, characteristic, complemented
10#25600—31nomaximal, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×C22⋊Q8C2212yes—
13C22×C3×C42C2211yescharacteristic
14C22×#216C2214yes—
15C2×#926—21no—
16C2×#924—21no—
17C2×#924C2212yes—
18#5470—24no—
19#5470—24no—
20C2×#870—21no—
21C2×#869C2211yescharacteristic
22C2×#869C2211yescharacteristic
23C2×C4×Dic6—21no—
24C2×C4×Dic6C2212yes—
25C3×#391C2212yes—
26C3×#389—21no—
27C3×#389—21no—
28C3×#384C2211yescharacteristic
29C3×#380—21no—
30C3×#380—21no—
31C2×#835C2214yes—
32C2×#830C2212yes—
33#5092—24no—
34#5079—24no—
35C2×#810C2214yes—
36C2×C4×C3⋊C8C2211yescharacteristic
37C2×C4×C3⋊C8C2211yescharacteristic
38#4529C2214yes—
39#4524C2212yes—
40C2×#533—31no—
41C2×#494—31no—
42C2×#487—31no—
43#2857D311yescharacteristic
44#2001—34no—
45#1990—31no—
46#1956—31no—
47#1878—34no—
48C4×C22⋊C8—31no—
49C22×C3×Q8C2312yes—
50C23×C3×C4D411yescharacteristic
51C23×C3×C4D411yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C22×Dic6—21no—
54C22×Dic6C2312yes—
55C3×C4⋊Q8—22no—
56C3×C4⋊Q8—22no—
57C3×C22⋊Q8—44no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4C2312yes—
64C2×C3×C22⋊C4—22no—
65C2×C3×C42—22no—
66C2×C3×C42—22no—
67C2×C3×C42D411yescharacteristic
68C2×C3×C42D411yescharacteristic
69C2×C3×C42C2311yescharacteristic
70C2×C3×C42C2311yescharacteristic
71C2×#217—22no—
72C2×#216—28no—
73C2×#216—28no—
74C2×#216—44no—
75#926—28no—
76C2×C4⋊Dic3—21no—
77C2×C4⋊Dic3—21no—
78C2×C4⋊Dic3—21no—
79C2×C4⋊Dic3C2312yes—
80#924—28no—
81#924—24no—
82#924—24no—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—22no—
85C2×C4×Dic3—22no—
86C22×C3⋊C8C2312yes—
87C22×C3⋊C8C2312yes—
88#870—28no—
89#869—22no—
90#869—22no—
91#869D414yes—
92#869D414yes—
93C4×Dic6—28no—
94C4×Dic6—24no—
95C4×Dic6—24no—
96C3×#95—22no—
97C3×#95—22no—
98#835—28no—
99#830—24no—
100#810—44no—
101C4×C3⋊C8—22no—
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—61no—
112C2×C22⋊Q8C2×D312yes—
113C2×C4×Q8—32no—
114C2×C4×Q8—61no—
115C2×#117—61no—
116C22×C42C2×D311yescharacteristic
117#494—64no—
118#487—64no—
119C2×#103—34no—
120C2×#103—34no—
121C2×C22⋊C8—32no—
122C2×C4×C8—31no—
123C2×C4×C8—31no—
124#391C2×D312yes—
125#389—21no—
126#389—21no—
127#384C2×D311yescharacteristic
128#380—21no—
129#380—21no—
130#340—34no—
131#336—32no—
132C24×C3C2×D411yescharacteristic
133C2×C3×Q8—22no—
134C2×C3×Q8—22no—
135C2×C3×Q8—24no—
136C2×C3×Q8—24no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4Q1614yescomplemented
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2411yescharacteristic
155C22×Dic3—22no—
156C22×Dic3—22no—
157C2×Dic6—24no—
158C2×Dic6—24no—
159C2×Dic6—24no—
160C2×Dic6—22no—
161C2×Dic6—42no—
162C2×Dic6—42no—
163C2×Dic6C2×D414yes—
164C3×C4⋊C4—24no—
165C3×C4⋊C4—44no—
166C3×C4⋊C4—42no—
167C3×C4⋊C4—42no—
168C3×C4⋊C4—42no—
169C3×C4⋊C4—42no—
170C3×C22⋊C4—44no—
171C3×C42—22no—
172C3×C42—22no—
173C3×C42—21no—
174C3×C42—21no—
175C3×C42C2×D412yes—
176C3×C42C2×D412yes—
177#217—28no—
178#216—416no—
179C4⋊Dic3—24no—
180C4⋊Dic3—24no—
181C4⋊Dic3—24no—
182C4⋊Dic3—22no—
183C4⋊Dic3C2×D414yes—
184Dic3⋊C4—28no—
185Dic3⋊C4—28no—
186C4×Dic3—28no—
187C2×C3⋊C8—22no—
188C2×C3⋊C8—22no—
189C2×C3⋊C8—22no—
190C2×C3⋊C8—22no—
191C2×C3⋊C8—22no—
192C2×C3⋊C8#4114yes—
193C22×Q8—32no—
194C22×Q8—61no—
195C22×Q8C22×D312yes—
196C23×C4#6711yescharacteristic
197C23×C4#6711yescharacteristic
198C23×C4C22×D311yescharacteristic
199C2×Q16—68no—
200C2×Q16—68no—
201C2×Q16—124no—
202C22×C8—32no—
203C22×C8—32no—
204C4⋊Q8—22no—
205C4⋊Q8—22no—
206C4⋊Q8—34no—
207C4⋊Q8—34no—
208C4⋊Q8—62no—
209C4⋊Q8—62no—
210#125—68no—
211C22⋊Q8—44no—
212C22⋊Q8—64no—
213C22⋊Q8—64no—
214C22⋊Q8—68no—
215C4×Q8—64no—
216C4×Q8—64no—
217C4×Q8—68no—
218#117—68no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—32no—
225C2×C4⋊C4—61no—
226C2×C4⋊C4—61no—
227C2×C4⋊C4—62no—
228C2×C4⋊C4—62no—
229C2×C4⋊C4—61no—
230C2×C4⋊C4C22×D312yes—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—62no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—62no—
236C2×C42#6711yescharacteristic
237C2×C42#6711yescharacteristic
238C2×C42C22×D311yescharacteristic
239C2×C42C22×D311yescharacteristic
240#103—68no—
241#103—124no—
242C22⋊C8—64no—
243C4×C8—62no—
244C4×C8—62no—
245#95—22no—
246#95—22no—
247C23×C3—21no—
248C23×C3—21no—
249C23×C3—21no—
250C23×C3—21no—
251C23×C3—21no—
252C23×C3—21no—
253C23×C3C2×Q1612yes—
254C23×C3C22×D411yescharacteristic
255C3×Q8—44no—
256C3×Q8—44no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—28no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—44no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4#12012yes—
276C2×C3×C4#12012yes—
277C2×C3×C4C2×Q1612yes—
278C2×C3×C4C2×Q1612yes—
279C2×C3×C4#13512yes—
280C2×C3×C4#13712yes—
281C2×C3×C4#13712yes—
282C2×C3×C4C22×D411yescharacteristic
283C2×C3×C4C22×D411yescharacteristic
284C2×C3×C4C2×#4112yes—
285C2×C3×C4H(5,𝔽2)12yes—
286C2×Dic3—24no—
287C2×Dic3—24no—
288C2×Dic3—42no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—42no—
292Dic6—28no—
293Dic6—44no—
294Dic6—48no—
295C3⋊C8—24no—
296C3⋊C8—42no—
297C24C2×#6711yescharacteristic
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—34no—
303C2×Q8—64no—
304C2×Q8—62no—
305C2×Q8—64no—
306C2×Q8—64no—
307C2×Q8—122no—
308C2×Q8—122no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—62no—
321C22×C4—62no—
322C22×C4#21614yescomplemented
323C22×C4C2×#6711yescharacteristic
324C22×C4C2×#6711yescharacteristic
325C22×C4C2×#6711yescharacteristic
326C22×C4C2×#6711yescharacteristic
327C22×C4C2×#6711yescharacteristic
328C22×C4C23×D311yescharacteristic
329Q16—1216no—
330C2×C8—34no—
331C2×C8—62no—
332C2×C8—62no—
333C2×C8—62no—
334C2×C8—62no—
335C2×C8—62no—
336C4⋊C4—24no—
337C4⋊C4—34no—
338C4⋊C4—44no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—68no—
344C4⋊C4—68no—
345C4⋊C4—64no—
346C4⋊C4—64no—
347C4⋊C4—64no—
348C4⋊C4—62no—
349C22⋊C4—44no—
350C22⋊C4—68no—
351C42—22no—
352C42—22no—
353C42—21no—
354C42—21no—
355C42—68no—
356C42D3×D412yes—
357C42D3×D412yes—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—41no—
362C22×C3—41no—
363C22×C3—41no—
364C22×C3—41no—
365C22×C3C2×#12011yescharacteristic
366C22×C3#53211yescharacteristic
367C22×C3#53511yescharacteristic
368C22×C3C22×Q1611yescharacteristic
369C22×C3C2×#13511yescharacteristic
370C22×C3C2×#13711yescharacteristic
371C22×C3C2×#13711yescharacteristic
372C3×C4—22no—
373C3×C4—41no—
374C3×C4—41no—
375C3×C4—82no—
376C3×C4—82no—
377C3×C4#45114yes—
378C3×C4#45214yes—
379C3×C4#54612yes—
380C3×C4#55012yes—
381Dic3—44no—
382Dic3—44no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23C2×#21612yes—
390C23C22×#6711yescharacteristic
391Q8—44no—
392Q8—44no—
393Q8—68no—
394Q8—128no—
395Q8—124no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—28no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—64no—
415C2×C4—64no—
416C2×C4—122no—
417C2×C4—122no—
418C2×C4—122no—
419C2×C4—122no—
420C2×C4#93712yes—
421C2×C4#93712yes—
422C2×C4#94212yes—
423C2×C4C2×#21612yes—
424C2×C4C2×#21612yes—
425C2×C4#95012yes—
426C2×C4#95112yes—
427C2×C4C2×D3×D411yescharacteristic
428C2×C4C2×D3×D411yescharacteristic
429C2×C4#100412yes—
430C2×C4D3×#4112yes—
431C8—64no—
432C8—122no—
433C2×C3—42no—
434C2×C3#243111yescharacteristic
435C2×C3C2×#45111yescharacteristic
436C2×C3C2×#45211yescharacteristic
437C2×C3#317011yescharacteristic
438C2×C3#318511yescharacteristic
439C2×C3#320911yescharacteristic
440C2×C3#321011yescharacteristic
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22C2×#93711yescharacteristic
449C22#580511yescharacteristic
450C22C2×#94211yescharacteristic
451C22C22×#21611yescharacteristic
452C22C2×#95011yescharacteristic
453C22C2×#95111yescharacteristic
454C22#583211yescharacteristic
455C4—22no—
456C4—41no—
457C4—41no—
458C4—82no—
459C4—82no—
460C4—124no—
461C4—124no—
462C4#504714yes—
463C4#505014yes—
464C4#559912yes—
465C4#561212yes—
466C3#2560011yescharacteristic, complemented, Sylow
467C2—42no—
468C2#8265211yescharacteristic
469C2C2×#504711yescharacteristic
470C2C2×#505011yescharacteristic
471C2#8322211yescharacteristic
472C2#8331811yescharacteristic
473C2#8334411yescharacteristic
474C2#8334811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2857C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2857D3
hypocenterC3#25600
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×#5092C2semidirect
4. C2×#5079C2semidirect
7. #69690C2semidirect
8. #69582C2semidirect
9. #69579C2semidirect
148. C22×C3×C4Q16semidirect
322. C22×C4#216semidirect
466. C3#25600semidirect

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×#50922
C2×#50792
#717002
C4×#8302
#696902
#695822
#695792
#256003