Finite Group Databasebeta

← #10695612

Subgroups — #10695612

Order 768.

930 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 101 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×#3187C211yesmaximal, characteristic, complemented
2#82761C211yesmaximal, characteristic, complemented
3C2×#5036C211yesmaximal, characteristic
4#71690C211yesmaximal, characteristic, complemented
5#71686C211yesmaximal, characteristic, complemented
6#69959C211yesmaximal, characteristic, complemented
7#67677C211yesmaximal, characteristic, complemented
8#9947—31nomaximal, Sylow
9C23×D12C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C2×C3×#121C2211yescharacteristic
12C2×#947—21no—
13C22×#213—21no—
14C3×#483—21no—
15C3×#481—21no—
16C3×#465—22no—
17C3×#450—21no—
18C3×#448—21no—
19C2×C3×C22⋊C8C2211yescharacteristic
20C2×#834—22no—
21#5212C2211yescharacteristic
22C4×#217C2211yescharacteristic
23C2×#830C2211yescharacteristic
24C2×#821—22no—
25#5036—22no—
26#5036—22no—
27#4991—22no—
28#4988—21no—
29C2×#809—21no—
30#4955—22no—
31#4538—22no—
32#3187D311yescharacteristic, complemented
33C2×#447—31no—
34#2435—31no—
35#1964—31no—
36#1786—31no—
37#1520—31no—
38#1282—31no—
39C24×D3—21no—
40C22×C3×Q8D411yescharacteristic
41C22×C3×D4—21no—
42C22×C3×D4D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C22×D12—21no—
45C22×D12—21no—
46C22×D12—21no—
47C22×D12—21no—
48C22×D12—21no—
49C22×D12—44no—
50C22×D12D412yes—
51C2×C3×SD16—22no—
52C22×C3×C8D412yes—
53C3×C22⋊Q8—21no—
54C3×C22⋊Q8—21no—
55C3×C22⋊Q8—21no—
56C3×C22⋊Q8—21no—
57C3×#121—21no—
58C3×#121—21no—
59C3×#121—21no—
60C3×#121—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4D411yescharacteristic
63C2×C3×C4⋊C4D411yescharacteristic
64C2×C3×C22⋊C4—21no—
65C2×C3×C22⋊C4—21no—
66C2×#217—21no—
67C2×#217D412yes—
68#947—44no—
69C2×#213—24no—
70C2×#213—44no—
71C2×(C2×D3)⋊C4—24no—
72C2×(C2×D3)⋊C4—24no—
73C2×C4⋊Dic3—21no—
74C2×C4×Dic3—21no—
75C22×C3⋊C8—21no—
76C3×#106—22no—
77C3×#103—22no—
78C3×#102—22no—
79C3×C22⋊C8—21no—
80C3×C22⋊C8—21no—
81#834—44no—
82C3⋊#95—21no—
83C3⋊#95—21no—
84#830—21no—
85#830D412yes—
86#821—44no—
87#809—24no—
88C23×D4—31no—
89C22×D8—61no—
90C2×#124—61no—
91C2×C22⋊Q8C2×D311yescharacteristic
92C2×#121C2×D311yescharacteristic
93#483—21no—
94#481—21no—
95#465—22no—
96#450—21no—
97#448—21no—
98#447—62no—
99#447—62no—
100C2×#102—61no—
101C2×#102—62no—
102C2×#99—62no—
103C2×C22⋊C8—31no—
104C2×C22⋊C8C2×D311yescharacteristic
105#394—62no—
106#390—61no—
107#380—31no—
108#379—62no—
109C4×C22⋊C4—31no—
110#328—62no—
111C24×C3—21no—
112C24×C3C2×D411yescharacteristic
113C23×D3—22no—
114C23×D3—44no—
115C23×D3—41no—
116C23×D3—41no—
117C23×D3—42no—
118C23×D3—42no—
119C23×D3—82no—
120C2×C3×Q8—21no—
121C2×C3×Q8—21no—
122C2×C3×Q8—42no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—42no—
126C2×C3×D4—41no—
127C2×C3×D4—41no—
128C2×C3×D4—42no—
129C2×C3×D4—42no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—42no—
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C22×Dic3—22no—
141C22×Dic3—41no—
142C2×D12—24no—
143C2×D12—24no—
144C2×D12—22no—
145C2×D12—22no—
146C2×D12—24no—
147C2×D12—44no—
148C2×D12—44no—
149C2×D12—44no—
150C2×D12—42no—
151C2×D12—42no—
152C2×D12—42no—
153C2×D12—41no—
154C2×D12—41no—
155C3×SD16—82no—
156C2×C3×C8—22no—
157C2×C3×C8—42no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—21no—
162C3×C4⋊C4—41no—
163C3×C4⋊C4—41no—
164C3×C22⋊C4—41no—
165C3×C22⋊C4—41no—
166C3×C22⋊C4—41no—
167C3×C22⋊C4—41no—
168#217—22no—
169#217—22no—
170#217—42no—
171#213—48no—
172(C2×D3)⋊C4—48no—
173(C2×D3)⋊C4—84no—
174C4⋊Dic3—42no—
175C4×Dic3—42no—
176C2×C3⋊C8—22no—
177C2×C3⋊C8—42no—
178C25—61no—
179C22×Q8#6711yescharacteristic
180C22×D4—21no—
181C22×D4—32no—
182C22×D4—61no—
183C22×D4—61no—
184C22×D4—61no—
185C22×D4—61no—
186C22×D4—61no—
187C22×D4—124no—
188C22×D4#6711yescharacteristic
189C23×C4C22×D311yescharacteristic
190C2×SD16—22no—
191C2×D8—64no—
192C2×D8—124no—
193C22×C8—61no—
194C22×C8#6712yes—
195#124—124no—
196C22⋊Q8—21no—
197C22⋊Q8—21no—
198C22⋊Q8—21no—
199C22⋊Q8—21no—
200#121—21no—
201#121—21no—
202#121—21no—
203#121—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—61no—
206C2×C4⋊C4#6711yescharacteristic
207C2×C4⋊C4#6711yescharacteristic
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—32no—
211C2×C22⋊C4—64no—
212C2×C22⋊C4—61no—
213C2×C22⋊C4—64no—
214C2×C42—61no—
215#106—22no—
216#103—22no—
217#102—22no—
218#102—64no—
219#102—124no—
220#99—124no—
221C22⋊C8—21no—
222C22⋊C8—21no—
223C22⋊C8—32no—
224C22⋊C8—61no—
225#95—61no—
226#95—61no—
227C23×C3—41no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—42no—
231C23×C3—81no—
232C23×C3—81no—
233C23×C3#12011yescharacteristic
234C23×C3#12111yescharacteristic
235C23×C3#12711yescharacteristic
236C23×C3#13512yes—
237C23×C3AGL(1,ℤ/8)12yes—
238C22×D3—24no—
239C22×D3—44no—
240C22×D3—44no—
241C22×D3—44no—
242C22×D3—42no—
243C22×D3—42no—
244C22×D3—44no—
245C22×D3—88no—
246C22×D3—88no—
247C22×D3—82no—
248C22×D3—82no—
249C22×D3—82no—
250C22×D3—81no—
251C22×D3—81no—
252C3×Q8—42no—
253C3×Q8—81no—
254C3×D4—42no—
255C3×D4—42no—
256C3×D4—82no—
257C3×D4—81no—
258C3×D4—81no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—42no—
271C2×C3×C4#12011yescharacteristic
272C2×C3×C4#12011yescharacteristic
273C2×C3×C4#12111yescharacteristic
274C2×C3×C4#12111yescharacteristic
275C2×Dic3—42no—
276C2×Dic3—42no—
277C2×Dic3—42no—
278C2×Dic3—81no—
279C2×Dic3—81no—
280C2×Dic3—81no—
281D12—24no—
282D12—24no—
283D12—42no—
284D12—42no—
285D12—44no—
286D12—44no—
287C3×C8—42no—
288C3⋊C8—42no—
289C24—21no—
290C24—62no—
291C24—124no—
292C24—121no—
293C24—121no—
294C24—122no—
295C24—122no—
296C24—242no—
297C24D3×D411yescharacteristic
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—42no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—64no—
309C2×D4—64no—
310C2×D4—64no—
311C2×D4—62no—
312C2×D4—62no—
313C2×D4—122no—
314C2×D4—122no—
315C2×D4—122no—
316C2×D4—124no—
317C2×D4—124no—
318C2×D4—124no—
319C2×D4—121no—
320C2×D4—121no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—42no—
325C22×C4—62no—
326C22×C4—121no—
327C22×C4D3×D411yescharacteristic
328C22×C4D3×D411yescharacteristic
329C22×C4#23711yescharacteristic
330C22×C4C2×#6711yescharacteristic
331C22×C4C2×#6711yescharacteristic
332C22×C4C2×#6711yescharacteristic
333SD16—82no—
334D8—128no—
335C2×C8—22no—
336C2×C8—42no—
337C2×C8—62no—
338C2×C8—122no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—122no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—62no—
351C22⋊C4—62no—
352C22⋊C4—128no—
353C22⋊C4—122no—
354C22⋊C4—244no—
355C42—122no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—21no—
361C22×C3—21no—
362C22×C3—42no—
363C22×C3—42no—
364C22×C3—81no—
365C22×C3—81no—
366C22×C3—81no—
367C22×C3—162no—
368C22×C3#39211yescharacteristic
369C22×C3#44912yes—
370C22×C3#46312yes—
371C22×C3#49511yescharacteristic
372C22×C3#49611yescharacteristic
373C2×D3—48no—
374C2×D3—44no—
375C2×D3—84no—
376C2×D3—84no—
377C2×D3—88no—
378C2×D3—84no—
379C2×D3—82no—
380C2×D3—82no—
381C3×C4—21no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—81no—
385C3×C4—81no—
386C3×C4—81no—
387C3×C4B(4,𝔽2)12yes—
388Dic3—82no—
389Dic3—161no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—42no—
394C23—64no—
395C23—81no—
396C23—81no—
397C23—124no—
398C23—124no—
399C23—124no—
400C23—122no—
401C23—124no—
402C23—122no—
403C23—242no—
404C23—242no—
405C23—248no—
406C23—248no—
407C23—242no—
408C23—241no—
409C23—241no—
410C23#91412yes—
411C23#91612yes—
412C23#93711yescharacteristic
413C23#93911yescharacteristic
414C23#94011yescharacteristic
415Q8—42no—
416Q8—81no—
417D4—42no—
418D4—42no—
419D4—64no—
420D4—64no—
421D4—81no—
422D4—82no—
423D4—81no—
424D4—122no—
425D4—122no—
426D4—124no—
427D4—124no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—122no—
441C2×C4—122no—
442C2×C4—122no—
443C2×C4—241no—
444C2×C4—241no—
445C2×C4—241no—
446C2×C4#93711yescharacteristic
447C2×C4#93811yescharacteristic
448C2×C4#93911yescharacteristic
449C2×C4#95311yescharacteristic
450C8—42no—
451C8—122no—
452C2×C3—22no—
453C2×C3—21no—
454C2×C3—22no—
455C2×C3—21no—
456C2×C3—161no—
457C2×C3#200811yescharacteristic
458C2×C3#201211yescharacteristic
459C2×C3#202111yescharacteristic
460D3—84no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—42no—
468C22—42no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22—128no—
473C22—124no—
474C22—162no—
475C22—244no—
476C22—244no—
477C22—248no—
478C22—244no—
479C22—242no—
480C22—242no—
481C22#516512yes—
482C22#516811yescharacteristic
483C22#517212yes—
484C22#517611yescharacteristic
485C22#522311yescharacteristic
486C4—21no—
487C4—21no—
488C4—21no—
489C4—81no—
490C4—81no—
491C4—81no—
492C4—242no—
493C4—481no—
494C4#524412yes—
495C3#994711yescharacteristic, complemented, Sylow
496C2—22no—
497C2—21no—
498C2—22no—
499C2—21no—
500C2—161no—
501C2—244no—
502C2#7185111yescharacteristic
503C2#7205511yescharacteristic
504C2#7211911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3187C2
socleC22×C3#392
prime radicalC2×#5036C2
hypercenter#3187D3
hypocenterC3#9947
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3187

Complements

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

NH ≅ G/Ntype
1. C3×#3187C2semidirect
2. #82761C2semidirect
4. #71690C2semidirect
5. #71686C2semidirect
6. #69959C2semidirect
7. #67677C2semidirect
32. #3187D3semidirect
495. C3#9947semidirect

Maximal subgroups

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

subgroupindex
C3×#31872
#827612
C2×#50362
#716902
#716862
#699592
#676772
#99473