Finite Group Databasebeta

← #10695750

Subgroups — #10695750

Order 768.

772 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 117 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
1C2×C3×#481C211yesmaximal, characteristic, complemented
2#82680C211yesmaximal, characteristic
3C2×#4968C211yesmaximal, characteristic, complemented
4C2×#4537C211yesmaximal, characteristic, complemented
5#71569C211yesmaximal, characteristic, complemented
6C2×#4470C211yesmaximal, characteristic, complemented
7#67674C211yesmaximal, characteristic, complemented
8#9968—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C22×C3×C4⋊C4C2211yescharacteristic
11C2×#939—22no—
12C2×#930C2211yescharacteristic
13C3×#481—22no—
14C3×#481—22no—
15C2×C3×#106—21no—
16C2×C3×#102—21no—
17C2×C3×C22⋊C8C2211yescharacteristic
18#5223—21no—
19C2×C3⋊#95—22no—
20C2×C3⋊#95C2211yescharacteristic
21C2×C3⋊#95C2211yescharacteristic
22C2×#830C2211yescharacteristic
23#4993—21no—
24#4968—22no—
25#4968—22no—
26C2×#809—21no—
27C2×#808—21no—
28#4960—21no—
29#4949—21no—
30#4537—22no—
31#4470—22no—
32C2×#481—31no—
33C2×#481D311yescharacteristic, complemented
34#2656—31no—
35C2×#327—31no—
36C2×#327—31no—
37#1445—31no—
38#1280—31no—
39C22×C3×D4—21no—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4—21no—
42C23×C3×C4C2311yescharacteristic
43C22×#67—21no—
44C22×#67—22no—
45C23×Dic3—21no—
46C23×Dic3D412yes—
47C22×D12D411yescharacteristic
48C22×C3×C8—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×#121—21no—
52C3×#121—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C4⋊C4D412yescomplemented
60C2×C3×C22⋊C4—21no—
61C2×#217—22no—
62#939—44no—
63#939—44no—
64#930—21no—
65#930—21no—
66#930—21no—
67#930—21no—
68C2×(C2×D3)⋊C4—22no—
69C2×(C2×D3)⋊C4—21no—
70C2×C4⋊Dic3D411yescharacteristic
71C2×Dic3⋊C4—22no—
72C22×C3⋊C8—21no—
73C3×#106—24no—
74C3×#102—24no—
75C3×C22⋊C8—21no—
76C3×C22⋊C8D412yescomplemented
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81C3⋊#95—44no—
82#830—21no—
83#830D412yescomplemented
84#809—24no—
85#808—24no—
86C2×#121—31no—
87C2×#121—62no—
88C2×#121C2×D311yescharacteristic
89C22×C4⋊C4C2×D311yescharacteristic
90#481—22no—
91#481—22no—
92#481—62no—
93#481—62no—
94C2×#106—21no—
95C2×#106—61no—
96C2×#102—21no—
97C2×#102—61no—
98C2×C22⋊C8—31no—
99C2×C22⋊C8C2×D311yescharacteristic
100#399—61no—
101#399—61no—
102#392—61no—
103#392—61no—
104C2×#95—31no—
105C2×#95—31no—
106C2×#95—62no—
107#327—62no—
108#327—62no—
109C24×C3—21no—
110C24×C3C2×D411yescharacteristic
111C23×D3—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—42no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—42no—
118C2×C3×D4—42no—
119C22×C3×C4—21no—
120C22×C3×C4—22no—
121C22×C3×C4—42no—
122C22×C3×C4—42no—
123C22×C3×C4—42no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4#4111yescharacteristic
132C2×#67—44no—
133C2×#67—41no—
134C2×#67—41no—
135C2×#67—42no—
136C2×#67—42no—
137C2×#67—82no—
138C2×#67—82no—
139C22×Dic3—22no—
140C22×Dic3—22no—
141C22×Dic3—22no—
142C22×Dic3—21no—
143C22×Dic3—42no—
144C22×Dic3—42no—
145C22×Dic3—41no—
146C22×Dic3—42no—
147C22×Dic3—41no—
148C22×Dic3—42no—
149C22×Dic3SD1614yes—
150C2×D12—21no—
151C2×D12—21no—
152C2×D12—42no—
153C2×C3×C8—22no—
154C2×C3×C8—42no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—41no—
160C3×C4⋊C4—41no—
161C3×C22⋊C4—41no—
162C3×C22⋊C4—41no—
163#217—44no—
164(C2×D3)⋊C4—44no—
165(C2×D3)⋊C4—41no—
166(C2×D3)⋊C4—41no—
167C4⋊Dic3—21no—
168C4⋊Dic3—21no—
169Dic3⋊C4—44no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—42no—
172C22×D4—21no—
173C22×D4—31no—
174C22×D4—61no—
175C22×D4—62no—
176C22×D4#6711yescharacteristic
177C23×C4—21no—
178C23×C4—32no—
179C23×C4—61no—
180C23×C4C22×D311yescharacteristic
181C22×C8—21no—
182C22×C8—61no—
183#121—21no—
184#121—21no—
185#121—21no—
186#121—21no—
187#121—61no—
188#121—61no—
189#121—61no—
190#121—61no—
191#121—124no—
192#121—124no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—31no—
199C2×C4⋊C4—62no—
200C2×C4⋊C4#6711yescharacteristic
201C2×C4⋊C4#6712yescomplemented
202C2×C22⋊C4—21no—
203C2×C22⋊C4—62no—
204C2×C22⋊C4—62no—
205C2×C22⋊C4—61no—
206#106—24no—
207#106—64no—
208#102—24no—
209#102—64no—
210C22⋊C8—21no—
211C22⋊C8—32no—
212C22⋊C8—61no—
213C22⋊C8#6712yescomplemented
214#95—61no—
215#95—61no—
216#95—61no—
217#95—61no—
218#95—124no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—22no—
222C23×C3—41no—
223C23×C3—42no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3—42no—
227C23×C3#12011yescharacteristic
228C23×C3#12111yescharacteristic
229C23×C3#12711yescharacteristic
230C23×C3C2×SD1612yes—
231C23×C3AGL(1,ℤ/8)12yes—
232C22×D3—41no—
233C22×D3—42no—
234C22×D3—41no—
235C22×D3—41no—
236C22×D3—42no—
237C3×D4—42no—
238C3×D4—81no—
239C3×D4—82no—
240C3×D4—81no—
241C3×D4—81no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—44no—
249C2×C3×C4—41no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—82no—
253C2×C3×C4—81no—
254C2×C3×C4—81no—
255C2×C3×C4#12011yescharacteristic
256C2×C3×C4#12011yescharacteristic
257C2×C3×C4#12111yescharacteristic
258C2×C3×C4#12111yescharacteristic
259#67—88no—
260#67—82no—
261#67—81no—
262#67—81no—
263C2×Dic3—24no—
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3—44no—
267C2×Dic3—44no—
268C2×Dic3—42no—
269C2×Dic3—41no—
270C2×Dic3—42no—
271C2×Dic3—82no—
272C2×Dic3—82no—
273C2×Dic3—81no—
274C2×Dic3—81no—
275D12—42no—
276D12—81no—
277C3×C8—42no—
278C3⋊C8—42no—
279C24—21no—
280C24—61no—
281C24D3×D411yescharacteristic
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—61no—
290C2×D4—61no—
291C2×D4—122no—
292C2×D4—124no—
293C2×D4—122no—
294C2×D4—122no—
295C2×D4—121no—
296C2×D4—121no—
297C2×D4—242no—
298C2×D4—242no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—34no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—62no—
308C22×C4—62no—
309C22×C4—62no—
310C22×C4—61no—
311C22×C4—122no—
312C22×C4—122no—
313C22×C4—121no—
314C22×C4—122no—
315C22×C4—122no—
316C22×C4—121no—
317C22×C4D3×D411yescharacteristic
318C22×C4D3×D411yescharacteristic
319C22×C4#23711yescharacteristic
320C22×C4C2×#6711yescharacteristic
321C22×C4C2×#6711yescharacteristic
322C22×C4C2×#6711yescharacteristic
323C2×C8—22no—
324C2×C8—42no—
325C2×C8—62no—
326C2×C8—122no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—22no—
330C4⋊C4—21no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—61no—
334C4⋊C4—61no—
335C4⋊C4—124no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—124no—
339C22⋊C4—124no—
340C22⋊C4—121no—
341C22⋊C4—121no—
342C22×C3—22no—
343C22×C3—22no—
344C22×C3—22no—
345C22×C3—22no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—82no—
351C22×C3—82no—
352C22×C3—81no—
353C22×C3—81no—
354C22×C3—81no—
355C22×C3#39211yescharacteristic
356C22×C3#45012yes—
357C22×C3#45311yescharacteristic
358C22×C3#45411yescharacteristic
359C22×C3#45812yes—
360C22×C3#46012yes—
361C22×C3#49211yescharacteristic
362C22×C3#49611yescharacteristic
363C2×D3—82no—
364C2×D3—82no—
365C2×D3—81no—
366C2×D3—81no—
367C2×D3—81no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—21no—
372C3×C4—82no—
373C3×C4—81no—
374Dic3—44no—
375Dic3—82no—
376Dic3—81no—
377C23—21no—
378C23—21no—
379C23—22no—
380C23—42no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—42no—
385C23—122no—
386C23—121no—
387C23—121no—
388C23—121no—
389C23—122no—
390C23D3×SD1612yes—
391C23#91412yes—
392C23#93711yescharacteristic
393C23#93911yescharacteristic
394C23#94011yescharacteristic
395D4—42no—
396D4—81no—
397D4—82no—
398D4—81no—
399D4—81no—
400D4—122no—
401D4—241no—
402D4—248no—
403D4—242no—
404D4—241no—
405D4—241no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—64no—
417C2×C4—82no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—122no—
421C2×C4—122no—
422C2×C4—124no—
423C2×C4—124no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4—121no—
427C2×C4—242no—
428C2×C4—242no—
429C2×C4—241no—
430C2×C4—241no—
431C2×C4#93711yescharacteristic
432C2×C4#93811yescharacteristic
433C2×C4#93911yescharacteristic
434C2×C4#95311yescharacteristic
435C8—42no—
436C8—122no—
437C2×C3—21no—
438C2×C3—22no—
439C2×C3—21no—
440C2×C3—161no—
441C2×C3#161312yes—
442C2×C3#161612yes—
443C2×C3#200611yescharacteristic
444C2×C3#201411yescharacteristic
445C2×C3#202211yescharacteristic
446D3—161no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—82no—
456C22—82no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22—242no—
461C22—242no—
462C22—241no—
463C22—241no—
464C22—241no—
465C22#505312yes—
466C22#516312yes—
467C22#516912yes—
468C22#517511yescharacteristic
469C22#517611yescharacteristic
470C22#519811yescharacteristic
471C22#520111yescharacteristic
472C22#522311yescharacteristic
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—81no—
478C4—82no—
479C4—124no—
480C4—242no—
481C4—241no—
482C3#996811yescharacteristic, complemented, Sylow
483C2—21no—
484C2—22no—
485C2—21no—
486C2—161no—
487C2—481no—
488C2#7116812yes—
489C2#7117112yes—
490C2#7170611yescharacteristic
491C2#7205611yescharacteristic
492C2#7209811yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#481C2
socleC23×C3#120
prime radical#82680C2
hypercenterC2×#481D3
hypocenterC3#9968
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#481

Complements

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

NH ≅ G/Ntype
1. C2×C3×#481C2semidirect
3. C2×#4968C2semidirect
4. C2×#4537C2semidirect
5. #71569C2semidirect
6. C2×#4470C2semidirect
7. #67674C2semidirect
33. C2×#481D3semidirect
59. C2×C3×C4⋊C4D4semidirect
76. C3×C22⋊C8D4semidirect
83. #830D4semidirect
201. C2×C4⋊C4#67semidirect
213. C22⋊C8#67semidirect
482. C3#9968semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4812
#826802
C2×#49682
C2×#45372
#715692
C2×#44702
#676742
#99683