Finite Group Databasebeta

← #10828626

Subgroups — #10828626

Order 768.

1086 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3492C211yesmaximal, characteristic
2#82914C212yesmaximal, complemented
3C2×#4977C211yesmaximal, characteristic, complemented
4#71901C212yesmaximal, complemented
5#71877C212yesmaximal, complemented
6#71711C212yesmaximal, complemented
7#71665C214yesmaximal, complemented
8#71562C211yesmaximal, characteristic, complemented
9#24077—31nomaximal, Sylow
10C3×#554—22no—
11C3×#553—22no—
12C3×#552—22no—
13C3×#547—22no—
14C3×#544C2211yescharacteristic
15C3×#539—22no—
16C3×#537C2211yescharacteristic
17C3×#525—22no—
18C3×#525C2212yes—
19C3×#520C2212yes—
20C22×C3×C4⋊C4C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C22×C4×Dic3C2211yescharacteristic
23C2×#837C2214yes—
24#5232C2214yes—
25C2×C3⋊#95—22no—
26#5207C2212yes—
27#5044—22no—
28#5043—22no—
29#5034—22no—
30#5033—22no—
31#4995C2214yes—
32#4977—22no—
33#4977C2212yes—
34#4977C2212yes—
35#4532C2214yes—
36#4530C2212yes—
37#4530C2212yes—
38#3492D311yescharacteristic
39#3092—32no—
40C2×#382—31no—
41#2061—32no—
42#2040—32no—
43#1896—32no—
44#1860—34no—
45#1821—31no—
46C2×C3×#41—22no—
47C2×C3×#41C2312yes—
48C23×C3×C4D412yes—
49C23×C3×C4C2311yescharacteristic
50C23×Dic3—21no—
51C23×Dic3—21no—
52C3×C4⋊Q8—22no—
53C3×#126—24no—
54C3×#125—24no—
55C3×#125—22no—
56C3×#123—24no—
57C3×#123—24no—
58C3×#123—22no—
59C3×C22⋊Q8—22no—
60C3×C22⋊Q8—24no—
61C3×C22⋊Q8—22no—
62C3×C22⋊Q8—22no—
63C3×C22⋊Q8D412yes—
64C3×C22⋊Q8D412yes—
65C3×#121—22no—
66C3×#121—22no—
67C3×#121—22no—
68C3×#121D412yes—
69C3×#121D412yes—
70C3×#120—22no—
71C3×C4×Q8—22no—
72C3×C4×D4—24no—
73C3×C4×D4—22no—
74C3×C4×D4—24no—
75C3×C4×D4—22no—
76C3×C4×D4—22no—
77C3×#117—22no—
78C3×#117—22no—
79C3×#117C2312yes—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4D412yes—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3—22no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3C2312yes—
90C2×C4×Dic3—21no—
91C2×C4×Dic3—24no—
92C2×C4×Dic3C2312yes—
93C2×#208C2314yes—
94#837—28no—
95#835—24no—
96#832—24no—
97C3⋊#95—24no—
98C3⋊#95—24no—
99#830—22no—
100#830—22no—
101#808—24no—
102#807—24no—
103#554—22no—
104#553—22no—
105#552—22no—
106#547—22no—
107#544C2×D311yescharacteristic
108#539—22no—
109#537C2×D311yescharacteristic
110#525—22no—
111#525C2×D312yes—
112#520C2×D312yes—
113C22×C4⋊C4—31no—
114C22×C4⋊C4C2×D311yescharacteristic
115C22×C42—31no—
116#484—62no—
117#483—62no—
118#481—62no—
119#480—62no—
120#428—34no—
121C2×#104—34no—
122#417—34no—
123#407—32no—
124#382—32no—
125#382—32no—
126#382—62no—
127C2×#95—62no—
128#343—32no—
129#343—32no—
130#339—34no—
131C24×C3C2×D411yescharacteristic
132C3×#41—24no—
133C3×#41—24no—
134C3×#41—84no—
135C2×C3×Q8—22no—
136C2×C3×Q8C2×D412yes—
137C2×C3×D4—22no—
138C2×C3×D4—22no—
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C2×C3×D4C2×D412yes—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—24no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—44no—
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2411yescharacteristic
155C22×Dic3—24no—
156C22×Dic3—22no—
157C22×Dic3—22no—
158C22×Dic3—42no—
159C22×Dic3—42no—
160C22×Dic3—44no—
161C22×Dic3—41no—
162C22×Dic3—41no—
163C3×C4⋊C4—24no—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—24no—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4C2×D412yes—
173C3×C4⋊C4C2×D412yes—
174C3×C22⋊C4—24no—
175C3×C22⋊C4—24no—
176C3×C22⋊C4—22no—
177C3×C22⋊C4—22no—
178C3×C22⋊C4—22no—
179C3×C22⋊C4—22no—
180C3×C22⋊C4—22no—
181C3×C42—24no—
182C3×C42—22no—
183C4⋊Dic3—24no—
184C4⋊Dic3—42no—
185C4×Dic3—24no—
186C4×Dic3—22no—
187C4×Dic3#4114yescomplemented
188#208—24no—
189#208—24no—
190C2×C3⋊C8—24no—
191C2×#41—22no—
192C2×#41C22×D312yes—
193C23×C4—61no—
194C23×C4—61no—
195C23×C4#6712yes—
196C23×C4C22×D311yescharacteristic
197C2×M16—34no—
198C4⋊Q8—22no—
199#126—24no—
200#125—24no—
201#125—22no—
202#123—24no—
203#123—24no—
204#123—22no—
205C22⋊Q8—22no—
206C22⋊Q8—24no—
207C22⋊Q8—22no—
208C22⋊Q8—22no—
209C22⋊Q8#6712yes—
210C22⋊Q8#6712yes—
211#121—22no—
212#121—22no—
213#121—22no—
214#121#6712yes—
215#121#6712yes—
216#120—22no—
217C4×Q8—22no—
218C4×D4—24no—
219C4×D4—22no—
220C4×D4—24no—
221C4×D4—22no—
222C4×D4—22no—
223#117—22no—
224#117—22no—
225#117C22×D312yes—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—32no—
230C2×C4⋊C4—62no—
231C2×C4⋊C4—62no—
232C2×C4⋊C4—61no—
233C2×C4⋊C4#6712yes—
234C2×C4⋊C4C22×D311yescharacteristic
235C2×C4⋊C4C22×D311yescharacteristic
236C2×C42—32no—
237C2×C42—64no—
238C2×C42—61no—
239#107—64no—
240#106—64no—
241#104—68no—
242#103—64no—
243#102—64no—
244C22⋊C8—62no—
245C22⋊C8—62no—
246#95—64no—
247#95—64no—
248C23×C3—21no—
249C23×C3—21no—
250C23×C3—22no—
251C23×C3—22no—
252C23×C3—22no—
253C23×C3#12011yescharacteristic
254C23×C3#12011yescharacteristic
255C23×C3C22×D411yescharacteristic
256C3×Q8—24no—
257C3×Q8—82no—
258C3×D4—24no—
259C3×D4—42no—
260C3×D4—42no—
261C3×D4—44no—
262C3×D4—44no—
263C3×D4—82no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—24no—
267C2×C3×C4—24no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—44no—
274C2×C3×C4—44no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—82no—
278C2×C3×C4—84no—
279C2×C3×C4#12012yes—
280C2×C3×C4C22×D412yes—
281C2×C3×C4C2×#4112yes—
282C2×C3×C42−1+412yes—
283C2×Dic3—24no—
284C2×Dic3—48no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3—44no—
288C2×Dic3—42no—
289C2×Dic3—82no—
290C2×Dic3—82no—
291C3⋊C8—44no—
292C24C2×#6711yescharacteristic
293#41—24no—
294#41—24no—
295#41—84no—
296C2×Q8—22no—
297C2×Q8C2×#6712yes—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4C2×#6712yes—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—44no—
310C22×C4—64no—
311C22×C4—62no—
312C22×C4—62no—
313C22×C4—122no—
314C22×C4—122no—
315C22×C4—124no—
316C22×C4—121no—
317C22×C4—121no—
318C22×C4C2×#6711yescharacteristic
319C22×C4C2×#6712yes—
320C22×C4C2×#6712yes—
321C22×C4C2×#6712yes—
322C22×C4C2×#6712yes—
323C22×C4C23×D311yescharacteristic
324M16—64no—
325M16—64no—
326C2×C8—64no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—64no—
337C4⋊C4—122no—
338C4⋊C4C2×#6712yes—
339C4⋊C4C2×#6712yes—
340C22⋊C4—24no—
341C22⋊C4—24no—
342C22⋊C4—22no—
343C22⋊C4—22no—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—22no—
347C42—24no—
348C42—22no—
349C42—34no—
350C42—64no—
351C42—62no—
352C22×C3—24no—
353C22×C3—24no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—42no—
361C22×C3—42no—
362C22×C3—42no—
363C22×C3#45412yes—
364C22×C3#45512yes—
365C22×C3C2×#12011yescharacteristic
366C22×C3#52511yescharacteristic
367C22×C3#53611yescharacteristic
368C3×C4—22no—
369C3×C4—42no—
370C3×C4—42no—
371C3×C4—42no—
372C3×C4—42no—
373C3×C4—42no—
374C3×C4#54714yes—
375Dic3—44no—
376Dic3—82no—
377C23—22no—
378C23—22no—
379C23—21no—
380C23—21no—
381C23—22no—
382C23#95311yescharacteristic
383C23#95311yescharacteristic
384C23C22×#6711yescharacteristic
385Q8—24no—
386Q8—82no—
387D4—24no—
388D4—44no—
389D4—44no—
390D4—42no—
391D4—42no—
392D4—82no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—64no—
407C2×C4—82no—
408C2×C4—84no—
409C2×C4—128no—
410C2×C4—122no—
411C2×C4—122no—
412C2×C4—124no—
413C2×C4—122no—
414C2×C4—242no—
415C2×C4—242no—
416C2×C4#95312yes—
417C2×C4D3×#4112yes—
418C2×C4#101012yes—
419C2×C4C22×#6712yes—
420C8—124no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—22no—
424C2×C3—22no—
425C2×C3—82no—
426C2×C3#240611yescharacteristic
427C2×C3C2×#45511yescharacteristic
428C2×C3C2×#45411yescharacteristic
429C22—24no—
430C22—24no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22#524212yes—
441C22#524512yes—
442C22#582511yescharacteristic
443C22#582811yescharacteristic
444C22C2×#95311yescharacteristic
445C4—22no—
446C4—42no—
447C4—42no—
448C4—42no—
449C4—42no—
450C4—42no—
451C4—124no—
452C4—242no—
453C4#582414yes—
454C3#2407711yescharacteristic, complemented, Sylow
455C2—21no—
456C2—22no—
457C2—22no—
458C2—21no—
459C2—82no—
460C2#8409011yescharacteristic
461C2C2×#524211yescharacteristic
462C2C2×#524511yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3492C2
socleC22×C3C2×#120
hypercenter#3492D3
hypocenterC3#24077
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3492

Complements

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

NH ≅ G/Ntype
2. #82914C2semidirect
3. C2×#4977C2semidirect
4. #71901C2semidirect
5. #71877C2semidirect
6. #71711C2semidirect
7. #71665C2semidirect
8. #71562C2semidirect
187. C4×Dic3#41semidirect
454. C3#24077semidirect

Maximal subgroups

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

subgroupindex
C3×#34922
#829142
C2×#49772
#719012
#718772
#717112
#716652
#715622
#240773