Finite Group Databasebeta

← #10828664

Subgroups — #10828664

Order 768.

1010 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
1C3×#3493C211yesmaximal, characteristic
2#82914C212yesmaximal, complemented
3#82234C211yesmaximal, characteristic, complemented
4#71901C212yesmaximal, complemented
5#71877C212yesmaximal, complemented
6#71712C212yesmaximal, complemented
7#71664C214yesmaximal, complemented
8#71589C211yesmaximal, characteristic, complemented
9#24081—31nomaximal, Sylow
10C3×#556—22no—
11C3×#555—22no—
12C3×#553—22no—
13C3×#552—22no—
14C3×#544C2211yescharacteristic
15C3×#538—22no—
16C3×#537C2211yescharacteristic
17C3×#529—22no—
18C3×#525C2212yes—
19C3×#520C2212yes—
20C2×C3×#117C2211yescharacteristic
21C2×#926C2211yescharacteristic
22C22×C4⋊Dic3C2211yescharacteristic
23C2×#837C2214yes—
24#5232C2214yes—
25#5207C2212yes—
26#5044—22no—
27#5043—22no—
28#5034—22no—
29#5033—22no—
30#4995C2214yes—
31#4977C2212yes—
32#4975C2212yes—
33#4946—22no—
34#4928—22no—
35#4532C2214yes—
36#4530C2212yes—
37#4530C2212yes—
38#3493D311yescharacteristic
39#3092—32no—
40#2317—31no—
41#2061—32no—
42#2040—32no—
43#1897—32no—
44#1859—34no—
45#1822—31no—
46C2×C3×#41C2312yes—
47C23×C3×C4C2311yescharacteristic
48C23×Dic3—21no—
49C3×C4⋊Q8—22no—
50C3×#126—24no—
51C3×#126—24no—
52C3×#125—22no—
53C3×#125—22no—
54C3×#124—22no—
55C3×#124—24no—
56C3×#123—24no—
57C3×#123—22no—
58C3×C22⋊Q8—22no—
59C3×C22⋊Q8—22no—
60C3×C22⋊Q8D412yes—
61C3×C22⋊Q8D412yes—
62C3×#121—24no—
63C3×#121—22no—
64C3×#121—22no—
65C3×#121D412yes—
66C3×#121D412yes—
67C3×#120—22no—
68C3×C4×Q8—22no—
69C3×C4×Q8—22no—
70C3×C4×D4—22no—
71C3×C4×D4—22no—
72C3×C4×D4—24no—
73C3×C4×D4—22no—
74C3×C4×D4—22no—
75C3×#117—24no—
76C3×#117—22no—
77C3×#117—22no—
78C3×#117—22no—
79C3×#117C2312yes—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C22⋊C4D412yes—
84C2×C3×C42D412yes—
85C2×#217—21no—
86#926—24no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3—22no—
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3C2312yes—
92C2×Dic3⋊C4—24no—
93C2×C4×Dic3C2312yes—
94C2×#208C2314yes—
95#837—28no—
96#835—24no—
97#832—24no—
98C3⋊#95—24no—
99#830—22no—
100#830—22no—
101#808—24no—
102#807—24no—
103#556—22no—
104#555—22no—
105#553—22no—
106#552—22no—
107#544C2×D311yescharacteristic
108#538—22no—
109#537C2×D311yescharacteristic
110#529—22no—
111#525C2×D312yes—
112#520C2×D312yes—
113C2×#117—31no—
114C2×#117C2×D311yescharacteristic
115C22×C4⋊C4—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#389—62no—
125#385—62no—
126#384—32no—
127#382—32no—
128#343—32no—
129#343—32no—
130#339—34no—
131C24×C3C2×D411yescharacteristic
132C3×#41—24no—
133C3×#41—24no—
134C2×C3×Q8—22no—
135C2×C3×Q8C2×D412yes—
136C2×C3×D4—22no—
137C2×C3×D4—22no—
138C2×C3×D4—22no—
139C2×C3×D4—22no—
140C2×C3×D4C2×D412yes—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—24no—
144C22×C3×C4—22no—
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D412yes—
147C22×C3×C4C2×D412yes—
148C22×C3×C4C2×D412yes—
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2411yescharacteristic
151C22×Dic3—22no—
152C22×Dic3—22no—
153C22×Dic3—42no—
154C22×Dic3—82no—
155C3×C4⋊C4—24no—
156C3×C4⋊C4—24no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4C2×D412yes—
164C3×C4⋊C4C2×D412yes—
165C3×C22⋊C4—24no—
166C3×C22⋊C4—24no—
167C3×C22⋊C4—24no—
168C3×C22⋊C4—22no—
169C3×C22⋊C4—22no—
170C3×C22⋊C4—22no—
171C3×C22⋊C4—22no—
172C3×C22⋊C4—22no—
173C3×C42—24no—
174C3×C42—22no—
175C3×C42—22no—
176C3×C42—22no—
177#217—24no—
178C4⋊Dic3—24no—
179C4⋊Dic3—24no—
180C4⋊Dic3—41no—
181C4⋊Dic3—41no—
182Dic3⋊C4—84no—
183C4×Dic3—22no—
184C4×Dic3#4114yes—
185#208—24no—
186#208—24no—
187C2×C3⋊C8—24no—
188C2×#41C22×D312yes—
189C23×C4—61no—
190C23×C4C22×D311yescharacteristic
191C2×M16—34no—
192C4⋊Q8—22no—
193#126—24no—
194#126—24no—
195#125—22no—
196#125—22no—
197#124—22no—
198#124—24no—
199#123—24no—
200#123—22no—
201C22⋊Q8—22no—
202C22⋊Q8—22no—
203C22⋊Q8#6712yes—
204C22⋊Q8#6712yes—
205#121—24no—
206#121—22no—
207#121—22no—
208#121#6712yes—
209#121#6712yes—
210#120—22no—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×D4—22no—
214C4×D4—22no—
215C4×D4—24no—
216C4×D4—22no—
217C4×D4—22no—
218#117—24no—
219#117—22no—
220#117—22no—
221#117—22no—
222#117—64no—
223#117C22×D312yes—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—32no—
226C2×C4⋊C4—61no—
227C2×C4⋊C4—62no—
228C2×C4⋊C4—62no—
229C2×C4⋊C4—61no—
230C2×C4⋊C4—64no—
231C2×C4⋊C4C22×D311yescharacteristic
232C2×C4⋊C4C22×D311yescharacteristic
233C2×C22⋊C4—61no—
234C2×C22⋊C4#6712yes—
235C2×C42—32no—
236C2×C42#6712yes—
237#107—64no—
238#106—64no—
239#104—68no—
240#103—64no—
241#102—64no—
242C22⋊C8—62no—
243C22⋊C8—62no—
244#95—64no—
245C23×C3—21no—
246C23×C3—21no—
247C23×C3—22no—
248C23×C3—41no—
249C23×C3—41no—
250C23×C3#12012yes—
251C23×C3C22×D411yescharacteristic
252C3×Q8—24no—
253C3×Q8—82no—
254C3×D4—24no—
255C3×D4—42no—
256C3×D4—42no—
257C3×D4—82no—
258C3×D4—82no—
259C3×D4—82no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—24no—
263C2×C3×C4—24no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—44no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—84no—
273C2×C3×C4#12011yescharacteristic
274C2×C3×C4#12011yescharacteristic
275C2×C3×C4C22×D412yes—
276C2×C3×C4C2×#4112yes—
277C2×C3×C42−1+412yes—
278C2×Dic3—24no—
279C2×Dic3—42no—
280C2×Dic3—42no—
281C2×Dic3—44no—
282C2×Dic3—42no—
283C2×Dic3—84no—
284C3⋊C8—44no—
285C24C2×#6711yescharacteristic
286#41—24no—
287#41—24no—
288C2×Q8—22no—
289C2×Q8C2×#6712yes—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4C2×#6712yes—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—62no—
300C22×C4—62no—
301C22×C4—122no—
302C22×C4—242no—
303C22×C4C2×#6711yescharacteristic
304C22×C4C2×#6712yes—
305C22×C4C2×#6712yes—
306C22×C4C2×#6712yes—
307C22×C4C2×#6712yes—
308C22×C4C23×D311yescharacteristic
309M16—64no—
310M16—64no—
311C2×C8—64no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—64no—
321C4⋊C4—64no—
322C4⋊C4—121no—
323C4⋊C4—121no—
324C4⋊C4—244no—
325C4⋊C4C2×#6712yes—
326C4⋊C4C2×#6712yes—
327C22⋊C4—24no—
328C22⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—22no—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—64no—
336C42—24no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—34no—
341C42—62no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—22no—
345C22×C3—22no—
346C22×C3—22no—
347C22×C3—42no—
348C22×C3—42no—
349C22×C3—42no—
350C22×C3—42no—
351C22×C3—42no—
352C22×C3C2×#12011yescharacteristic
353C22×C3#52511yescharacteristic
354C22×C3#53611yescharacteristic
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—42no—
358C3×C4—42no—
359C3×C4—42no—
360C3×C4—42no—
361C3×C4—42no—
362C3×C4#54714yes—
363Dic3—44no—
364Dic3—82no—
365C23—21no—
366C23—21no—
367C23—22no—
368C23—41no—
369C23—41no—
370C23#95312yes—
371C23C22×#6711yescharacteristic
372Q8—24no—
373Q8—82no—
374D4—24no—
375D4—42no—
376D4—42no—
377D4—82no—
378D4—82no—
379D4—82no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—64no—
393C2×C4—84no—
394C2×C4—122no—
395C2×C4—122no—
396C2×C4—124no—
397C2×C4—122no—
398C2×C4—244no—
399C2×C4#95311yescharacteristic
400C2×C4#95311yescharacteristic
401C2×C4D3×#4112yes—
402C2×C4#101012yes—
403C2×C4C22×#6712yes—
404C8—124no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—42no—
408C2×C3—82no—
409C2×C3#240611yescharacteristic
410C2×C3#302111yescharacteristic
411C2×C3#302211yescharacteristic
412C22—21no—
413C22—21no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22#582511yescharacteristic
423C22#582811yescharacteristic
424C22C2×#95311yescharacteristic
425C4—21no—
426C4—21no—
427C4—42no—
428C4—42no—
429C4—42no—
430C4—42no—
431C4—42no—
432C4—124no—
433C4—242no—
434C4#582414yes—
435C3#2408111yescharacteristic, complemented, Sylow
436C2—21no—
437C2—21no—
438C2—42no—
439C2—82no—
440C2#8409011yescharacteristic
441C2#8412211yescharacteristic
442C2#8412311yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3493C2
socleC22×C3C2×#120
hypercenter#3493D3
hypocenterC3#24081
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3493

Complements

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

NH ≅ G/Ntype
2. #82914C2semidirect
3. #82234C2semidirect
4. #71901C2semidirect
5. #71877C2semidirect
6. #71712C2semidirect
7. #71664C2semidirect
8. #71589C2semidirect
435. C3#24081semidirect

Maximal subgroups

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

subgroupindex
C3×#34932
#829142
#822342
#719012
#718772
#717122
#716642
#715892
#240813