Finite Group Databasebeta

← #10827638

Subgroups — #10827638

Order 768.

1474 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 1474 conjugacy classes of subgroups — too many to draw. Only its 199 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3490C211yesmaximal, characteristic
2#82602C212yesmaximal, complemented
3C2×#4943C211yesmaximal, characteristic, complemented
4#71894C212yesmaximal, complemented
5#71849C212yesmaximal, complemented
6#71670C212yesmaximal, complemented
7#71655C214yesmaximal, complemented
8#71542C211yesmaximal, characteristic, complemented
9#23985—31nomaximal, Sylow
10C24×Dic3C2211yescharacteristic
11C3×#551—24no—
12C3×#546—24no—
13C3×#543C2211yescharacteristic
14C3×#538—22no—
15C3×#534C2211yescharacteristic
16C3×#525—22no—
17C2×C3×#120C2212yes—
18C3×#518C2212yes—
19C22×C3×C22⋊C4C2211yescharacteristic
20C22×#217C2211yescharacteristic
21C2×#935C2214yes—
22#5247C2212yes—
23#5220—24no—
24C2×#834C2214yes—
25C2×C3⋊#95—22no—
26#4946—24no—
27#4943—22no—
28#4943C2212yes—
29#4943C2212yes—
30#4943C2214yes—
31#4526C2212yes—
32#4526C2212yes—
33#4526C2214yes—
34#3490D311yescharacteristic
35#2849—32no—
36C2×#381—31no—
37#2030—32no—
38#2022—32no—
39#1898—32no—
40#1890—34no—
41#1819—31no—
42C25×C3C2311yescharacteristic
43C2×C3×#41—22no—
44C22×C3×D4D414yes—
45C22×C3×D4C2312yes—
46C23×C3×C4D412yes—
47C23×Dic3—21no—
48C23×Dic3—21no—
49C23×Dic3—24no—
50C23×Dic3—24no—
51C23×Dic3—21no—
52C23×Dic3—42no—
53C23×Dic3C2312yes—
54C3×#126—24no—
55C3×#124—24no—
56C3×#123—24no—
57C3×#123—24no—
58C3×#123—22no—
59C3×C22⋊Q8—22no—
60C3×C22⋊Q8—22no—
61C3×C22⋊Q8—24no—
62C3×#121—22no—
63C3×#121—24no—
64C3×#121—22no—
65C3×#120—22no—
66C3×#120—22no—
67C3×#120—24no—
68C3×#120—22no—
69C3×C4×D4—24no—
70C3×C4×D4—24no—
71C3×C4×D4—24no—
72C3×#117—22no—
73C3×#117—22no—
74C2×C3×C22⋊C4—24no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4D412yes—
77C2×C3×C22⋊C4D414yes—
78C2×C3×C22⋊C4C2311yescharacteristic
79C2×C3×C22⋊C4C2311yescharacteristic
80C2×C3×C22⋊C4C2312yes—
81C2×#217—21no—
82C2×#217—24no—
83C2×#217—24no—
84C2×#217—42no—
85C2×#217C2312yes—
86C2×#217C2314yes—
87#935—28no—
88#935—44no—
89C2×Dic3⋊C4—24no—
90#834—44no—
91C3⋊#95—24no—
92C3⋊#95—24no—
93C3⋊#95—28no—
94C24×C4—31no—
95#551—24no—
96#546—24no—
97#543C2×D311yescharacteristic
98#538—22no—
99#534C2×D311yescharacteristic
100#525—22no—
101C2×#123—34no—
102C2×#120C2×D312yes—
103#518C2×D312yes—
104C22×C22⋊C4—31no—
105C22×C22⋊C4C2×D311yescharacteristic
106C2×#99—34no—
107#386—64no—
108#385—64no—
109#381—34no—
110#381—32no—
111#381—32no—
112#381—62no—
113#379—32no—
114C2×#95—62no—
115#342—32no—
116#342—32no—
117#342—34no—
118C24×C3—22no—
119C24×C3—22no—
120C24×C3—44no—
121C24×C3C2×D411yescharacteristic
122C24×C3C2×D414yes—
123C24×C3C2×D412yes—
124C24×C3C2×D411yescharacteristic
125C24×C3C2411yescharacteristic
126C3×#41—84no—
127C2×C3×Q8—22no—
128C2×C3×D4—24no—
129C2×C3×D4—22no—
130C2×C3×D4—22no—
131C2×C3×D4—24no—
132C2×C3×D4—24no—
133C2×C3×D4—22no—
134C2×C3×D4—24no—
135C2×C3×D4—48no—
136C22×C3×C4—22no—
137C22×C3×C4—24no—
138C22×C3×C4—22no—
139C22×C3×C4—44no—
140C22×C3×C4C2×D412yes—
141C22×C3×C4C2×D412yes—
142C22×C3×C4C2×D414yes—
143C22×C3×C4C2×D412yes—
144C22×Dic3—24no—
145C22×Dic3—24no—
146C22×Dic3—22no—
147C22×Dic3—24no—
148C22×Dic3—22no—
149C22×Dic3—24no—
150C22×Dic3—22no—
151C22×Dic3—24no—
152C22×Dic3—416no—
153C22×Dic3—42no—
154C22×Dic3—42no—
155C22×Dic3—42no—
156C22×Dic3—42no—
157C22×Dic3—42no—
158C22×Dic3—44no—
159C22×Dic3#4114yescomplemented
160C3×C4⋊C4—24no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—24no—
164C3×C22⋊C4—24no—
165C3×C22⋊C4—24no—
166C3×C22⋊C4—24no—
167C3×C22⋊C4—24no—
168C3×C22⋊C4—24no—
169C3×C22⋊C4—22no—
170C3×C22⋊C4—22no—
171C3×C22⋊C4—22no—
172C3×C22⋊C4—22no—
173C3×C42—24no—
174#217—24no—
175#217—24no—
176#217—24no—
177#217—48no—
178#217—48no—
179#217—42no—
180Dic3⋊C4—48no—
181C25C22×D311yescharacteristic
182C2×#41—22no—
183C22×D4#6714yes—
184C22×D4C22×D312yes—
185C23×C4—32no—
186C23×C4—64no—
187C23×C4—61no—
188C23×C4—61no—
189C23×C4—61no—
190C23×C4—64no—
191C23×C4—122no—
192C23×C4#6712yes—
193#126—24no—
194#124—24no—
195#123—24no—
196#123—24no—
197#123—22no—
198#123—68no—
199#123—124no—
200C22⋊Q8—22no—
201C22⋊Q8—22no—
202C22⋊Q8—24no—
203#121—22no—
204#121—24no—
205#121—22no—
206#120—22no—
207#120—22no—
208#120—24no—
209#120—22no—
210C4×D4—24no—
211C4×D4—24no—
212C4×D4—24no—
213#117—22no—
214#117—22no—
215C2×C4⋊C4—64no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—32no—
219C2×C22⋊C4—34no—
220C2×C22⋊C4—61no—
221C2×C22⋊C4—64no—
222C2×C22⋊C4—64no—
223C2×C22⋊C4—122no—
224C2×C22⋊C4#6712yes—
225C2×C22⋊C4#6714yes—
226C2×C22⋊C4C22×D311yescharacteristic
227C2×C22⋊C4C22×D311yescharacteristic
228C2×C22⋊C4C22×D312yes—
229#99—124no—
230#95—64no—
231#95—64no—
232#95—68no—
233C23×C3—21no—
234C23×C3—21no—
235C23×C3—24no—
236C23×C3—21no—
237C23×C3—21no—
238C23×C3—28no—
239C23×C3—28no—
240C23×C3—22no—
241C23×C3—24no—
242C23×C3—42no—
243C23×C3—44no—
244C23×C3—42no—
245C23×C3—48no—
246C23×C3—48no—
247C23×C3—44no—
248C23×C3#12012yes—
249C23×C3#12012yes—
250C23×C3C22×D411yescharacteristic
251C23×C3C22×D412yes—
252C23×C3C2×#4112yes—
253C23×C3H(5,𝔽2)12yes—
254C3×Q8—82no—
255C3×D4—42no—
256C3×D4—42no—
257C3×D4—44no—
258C3×D4—48no—
259C3×D4—84no—
260C3×D4—82no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—24no—
264C2×C3×C4—22no—
265C2×C3×C4—24no—
266C2×C3×C4—44no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—44no—
270C2×C3×C4—82no—
271C2×C3×C4—84no—
272C2×Dic3—24no—
273C2×Dic3—28no—
274C2×Dic3—48no—
275C2×Dic3—48no—
276C2×Dic3—42no—
277C2×Dic3—42no—
278C2×Dic3—44no—
279C2×Dic3—48no—
280C2×Dic3—44no—
281C2×Dic3—42no—
282C2×Dic3—44no—
283C2×Dic3—84no—
284C24—22no—
285C24—22no—
286C24—44no—
287C24C2×#6711yescharacteristic
288C24C2×#6714yes—
289C24C2×#6712yes—
290C24C2×#6711yescharacteristic
291C24C23×D311yescharacteristic
292#41—84no—
293C2×Q8—22no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—24no—
301C2×D4—48no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—34no—
306C22×C4—44no—
307C22×C4—64no—
308C22×C4—64no—
309C22×C4—62no—
310C22×C4—64no—
311C22×C4—62no—
312C22×C4—64no—
313C22×C4—62no—
314C22×C4—64no—
315C22×C4—1216no—
316C22×C4—122no—
317C22×C4—122no—
318C22×C4—122no—
319C22×C4—122no—
320C22×C4—122no—
321C22×C4—124no—
322C22×C4C2×#6712yes—
323C22×C4C2×#6712yes—
324C22×C4C2×#6714yes—
325C22×C4C2×#6712yes—
326C4⋊C4—24no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—128no—
331C22⋊C4—24no—
332C22⋊C4—24no—
333C22⋊C4—24no—
334C22⋊C4—24no—
335C22⋊C4—24no—
336C22⋊C4—22no—
337C22⋊C4—22no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—64no—
341C22⋊C4—64no—
342C22⋊C4—64no—
343C22⋊C4—128no—
344C22⋊C4—122no—
345C22⋊C4—128no—
346C42—24no—
347C22×C3—24no—
348C22×C3—24no—
349C22×C3—22no—
350C22×C3—22no—
351C22×C3—24no—
352C22×C3—22no—
353C22×C3—22no—
354C22×C3—24no—
355C22×C3—42no—
356C22×C3—42no—
357C22×C3—416no—
358C22×C3—42no—
359C22×C3—42no—
360C22×C3—42no—
361C22×C3—44no—
362C22×C3—84no—
363C22×C3#45814yes—
364C22×C3C2×#12011yescharacteristic
365C22×C3#52511yescharacteristic
366C22×C3#53511yescharacteristic
367C22×C3#54614yes—
368C3×C4—42no—
369C3×C4—42no—
370C3×C4—42no—
371C3×C4—44no—
372Dic3—44no—
373Dic3—84no—
374Dic3—82no—
375C23—24no—
376C23—21no—
377C23—21no—
378C23—28no—
379C23—21no—
380C23—28no—
381C23—21no—
382C23—22no—
383C23—24no—
384C23—42no—
385C23—44no—
386C23—42no—
387C23—48no—
388C23—48no—
389C23—44no—
390C23#95312yes—
391C23#95312yes—
392C23D3×#4112yes—
393C23#100912yes—
394C23C22×#6711yescharacteristic
395C23C22×#6712yes—
396Q8—82no—
397D4—48no—
398D4—42no—
399D4—44no—
400D4—42no—
401D4—84no—
402D4—82no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—64no—
413C2×C4—68no—
414C2×C4—82no—
415C2×C4—84no—
416C2×C4—128no—
417C2×C4—128no—
418C2×C4—124no—
419C2×C4—122no—
420C2×C4—122no—
421C2×C4—128no—
422C2×C4—124no—
423C2×C4—124no—
424C2×C4—122no—
425C2×C4—244no—
426C2×C3—21no—
427C2×C3—24no—
428C2×C3—21no—
429C2×C3—24no—
430C2×C3—42no—
431C2×C3—82no—
432C2×C3#240411yescharacteristic
433C2×C3C2×#45812yes—
434C22—24no—
435C22—24no—
436C22—22no—
437C22—22no—
438C22—24no—
439C22—22no—
440C22—22no—
441C22—24no—
442C22—42no—
443C22—42no—
444C22—416no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—44no—
449C22—84no—
450C22#524314yes—
451C22#582811yescharacteristic
452C22#582914yes—
453C22#583011yescharacteristic
454C22C2×#95311yescharacteristic
455C4—42no—
456C4—42no—
457C4—42no—
458C4—44no—
459C4—124no—
460C4—242no—
461C4—244no—
462C3#2398511yescharacteristic, complemented, Sylow
463C2—21no—
464C2—24no—
465C2—24no—
466C2—21no—
467C2—42no—
468C2—82no—
469C2#8409611yescharacteristic
470C2C2×#524312yes—
subgroupG/H
centerC22C2×#953
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
Fitting, prime radicalC3×#3490C2
socleC22×C3C2×#120
hypercenter#3490D3
hypocenterC3#23985
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ #3490

Complements

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

NH ≅ G/Ntype
2. #82602C2semidirect
3. C2×#4943C2semidirect
4. #71894C2semidirect
5. #71849C2semidirect
6. #71670C2semidirect
7. #71655C2semidirect
8. #71542C2semidirect
159. C22×Dic3#41semidirect
462. C3#23985semidirect

Maximal subgroups

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

subgroupindex
C3×#34902
#826022
C2×#49432
#718942
#718492
#716702
#716552
#715422
#239853