Finite Group Databasebeta

← #10695468

Subgroups — #10695468

Order 768.

734 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 113 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×#4992C211yesmaximal, characteristic
2C2×#4971C211yesmaximal, characteristic, complemented
3#73952C211yesmaximal, characteristic
4C3×#1823C211yesmaximal, characteristic, complemented
5C2×#4467C211yesmaximal, characteristic, complemented
6C2×#4468C211yesmaximal, characteristic
7#67624C211yesmaximal, characteristic, complemented
8#9926—31nomaximal, Sylow
9C22×C3×C4⋊C4C2211yescharacteristic
10C22×(C2×D3)⋊C4—21no—
11C22×(C2×D3)⋊C4C2211yescharacteristic
12C2×#924C2211yescharacteristic
13C2×C3×C22⋊C8C2211yescharacteristic
14C3×#380C2211yescharacteristic
15C2×C3⋊#95—21no—
16C2×C3⋊#95C2211yescharacteristic
17C2×#830C2211yescharacteristic
18#4994—22no—
19#4992—22no—
20#4992—22no—
21#4971—22no—
22#4971—22no—
23C2×#810—21no—
24C2×#808—21no—
25#4955—21no—
26#4950—21no—
27#4677—21no—
28#4673—21no—
29C3×#340—22no—
30#4468—22no—
31#4467—22no—
32C2×#483—31no—
33#2626—31no—
34C2×#397—31no—
35#1823D311yescharacteristic, complemented
36C2×#328—31no—
37C2×#323—31no—
38#1294—31no—
39C24×D3D411yescharacteristic
40C23×C3×C4—21no—
41C23×C3×C4C2311yescharacteristic
42C23×Dic3—21no—
43C23×Dic3D411yescharacteristic
44C22×Dic6D411yescharacteristic
45C22×C3×C8—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4D412yes—
52C2×C3×C4⋊C4D412yes—
53C2×C3×C22⋊C4—21no—
54C2×#217—21no—
55C2×(C2×D3)⋊C4—24no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—21no—
58C2×(C2×D3)⋊C4—21no—
59C2×(C2×D3)⋊C4—21no—
60C2×(C2×D3)⋊C4—21no—
61C2×(C2×D3)⋊C4—21no—
62C2×(C2×D3)⋊C4—22no—
63C2×(C2×D3)⋊C4—22no—
64C2×(C2×D3)⋊C4—42no—
65C2×(C2×D3)⋊C4—42no—
66C2×C4⋊Dic3—22no—
67C2×C4⋊Dic3D411yescharacteristic
68#924—21no—
69#924—21no—
70#924—21no—
71#924—21no—
72C2×Dic3⋊C4—21no—
73C22×C3⋊C8—21no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×#95—21no—
77C3⋊#95—22no—
78C3⋊#95—24no—
79C3⋊#95—21no—
80C3⋊#95D412yes—
81#830—21no—
82#830D412yes—
83#810—24no—
84#808—24no—
85C2×C22⋊Q8—31no—
86C22×C4⋊C4C2×D311yescharacteristic
87C22×C22⋊C4—31no—
88C22×C22⋊C4—61no—
89#483—62no—
90#483—62no—
91C2×#106—61no—
92C2×#103—61no—
93C2×C22⋊C8—31no—
94C2×C22⋊C8C2×D311yescharacteristic
95#399—61no—
96#397—62no—
97#397—62no—
98#396—62no—
99#393—61no—
100#385—61no—
101#380C2×D311yescharacteristic
102#379—61no—
103C2×#95—31no—
104C2×#95—61no—
105#340—22no—
106#328—62no—
107#323—62no—
108C24×C3C2×D411yescharacteristic
109C23×D3—21no—
110C23×D3—21no—
111C23×D3—21no—
112C23×D3—21no—
113C23×D3—21no—
114C23×D3—21no—
115C23×D3—42no—
116C23×D3—42no—
117C23×D3SD1612yes—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—42no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4—41no—
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4#4111yescharacteristic
133C22×Dic3—21no—
134C22×Dic3—22no—
135C22×Dic3—21no—
136C22×Dic3—21no—
137C22×Dic3—21no—
138C22×Dic3—21no—
139C22×Dic3—42no—
140C22×Dic3—42no—
141C22×Dic3—42no—
142C22×Dic3—41no—
143C22×Dic3—41no—
144C2×Dic6—21no—
145C2×Dic6—21no—
146C2×Dic6—42no—
147C2×C3×C8—42no—
148C2×C3×C8—41no—
149C3×C4⋊C4—22no—
150C3×C4⋊C4—22no—
151C3×C4⋊C4—42no—
152C3×C4⋊C4—41no—
153C3×C4⋊C4—41no—
154C3×C22⋊C4—41no—
155C3×C22⋊C4—41no—
156#217—41no—
157#217—41no—
158(C2×D3)⋊C4—48no—
159(C2×D3)⋊C4—41no—
160(C2×D3)⋊C4—41no—
161(C2×D3)⋊C4—41no—
162(C2×D3)⋊C4—41no—
163(C2×D3)⋊C4—82no—
164(C2×D3)⋊C4—82no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167C4⋊Dic3—82no—
168Dic3⋊C4—41no—
169Dic3⋊C4—41no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—42no—
172C25—31no—
173C22×Q8—31no—
174C23×C4—21no—
175C23×C4—31no—
176C23×C4—61no—
177C23×C4C22×D311yescharacteristic
178C22×C8—21no—
179C22×C8—61no—
180C22⋊Q8—61no—
181C22⋊Q8—61no—
182C22⋊Q8—61no—
183C22⋊Q8—61no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—31no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4—62no—
192C2×C4⋊C4D1212yes—
193C2×C4⋊C4#6712yes—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—64no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4—61no—
199C2×C22⋊C4—61no—
200C2×C22⋊C4—61no—
201C2×C22⋊C4—61no—
202C2×C22⋊C4—61no—
203C2×C22⋊C4—62no—
204C2×C22⋊C4—62no—
205C2×C22⋊C4—122no—
206C2×C22⋊C4—122no—
207#106—64no—
208#103—64no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211C22⋊C8—32no—
212C22⋊C8—61no—
213#95—21no—
214#95—32no—
215#95—64no—
216#95—62no—
217#95—61no—
218C23×C3—22no—
219C23×C3—22no—
220C23×C3#12011yescharacteristic
221C23×C3#12011yescharacteristic
222C23×C3#12111yescharacteristic
223C23×C3C2×SD1611yescharacteristic
224C23×C3#13511yescharacteristic
225C23×C3AGL(1,ℤ/8)11yescharacteristic
226C23×C3#13711yescharacteristic
227C22×D3—22no—
228C22×D3—22no—
229C22×D3—22no—
230C22×D3—44no—
231C22×D3—44no—
232C22×D3—42no—
233C22×D3—42no—
234C22×D3—41no—
235C22×D3—41no—
236C22×D3—41no—
237C22×D3—42no—
238C22×D3—42no—
239C22×D3—41no—
240C22×D3—82no—
241C22×D3—82no—
242C22×D3—81no—
243C22×D3—81no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—81no—
257C2×C3×C4—81no—
258C2×C3×C4—81no—
259C2×C3×C4—81no—
260C2×C3×C4#12011yescharacteristic
261C2×C3×C4#12111yescharacteristic
262C2×C3×C4#12111yescharacteristic
263C2×C3×C4#12711yescharacteristic
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3—42no—
267C2×Dic3—41no—
268C2×Dic3—41no—
269C2×Dic3—41no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3—42no—
273C2×Dic3—42no—
274C2×Dic3—82no—
275C2×Dic3—82no—
276C2×Dic3—81no—
277Dic6—42no—
278Dic6—81no—
279C3×C8—81no—
280C3⋊C8—42no—
281C24—32no—
282C24—61no—
283C24—61no—
284C24—61no—
285C24—61no—
286C24—61no—
287C24—61no—
288C24—122no—
289C24—122no—
290C24D3×D411yescharacteristic
291C2×Q8—61no—
292C2×Q8—61no—
293C2×Q8—122no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—62no—
304C22×C4—61no—
305C22×C4—61no—
306C22×C4—61no—
307C22×C4—61no—
308C22×C4—61no—
309C22×C4—122no—
310C22×C4—122no—
311C22×C4—122no—
312C22×C4—121no—
313C22×C4—121no—
314C22×C4C2×D1211yescharacteristic
315C22×C4#23511yescharacteristic
316C22×C4D3×D411yescharacteristic
317C22×C4D3×D411yescharacteristic
318C22×C4D3×D411yescharacteristic
319C22×C4C2×#6711yescharacteristic
320C2×C8—42no—
321C2×C8—41no—
322C2×C8—62no—
323C2×C8—122no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—42no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—61no—
330C4⋊C4—61no—
331C4⋊C4—121no—
332C4⋊C4—121no—
333C4⋊C4—242no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—128no—
337C22⋊C4—121no—
338C22⋊C4—121no—
339C22⋊C4—121no—
340C22⋊C4—121no—
341C22⋊C4—121no—
342C22⋊C4—121no—
343C22⋊C4—242no—
344C22⋊C4—242no—
345C22×C3—22no—
346C22×C3—22no—
347C22×C3—22no—
348C22×C3—22no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3#39211yescharacteristic
354C22×C3#44811yescharacteristic
355C22×C3#44911yescharacteristic
356C22×C3#45011yescharacteristic
357C22×C3#45211yescharacteristic
358C22×C3#45512yes—
359C22×C3#45812yes—
360C22×C3#46511yescharacteristic
361C22×C3#46811yescharacteristic
362C2×D3—44no—
363C2×D3—44no—
364C2×D3—42no—
365C2×D3—82no—
366C2×D3—82no—
367C2×D3—82no—
368C2×D3—81no—
369C2×D3—81no—
370C2×D3—82no—
371C3×C4—21no—
372C3×C4—21no—
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—82no—
376C3×C4—82no—
377Dic3—82no—
378Dic3—81no—
379Dic3—81no—
380Dic3—81no—
381C23—22no—
382C23—22no—
383C23—62no—
384C23—62no—
385C23—62no—
386C23—124no—
387C23—122no—
388C23—122no—
389C23—124no—
390C23—121no—
391C23—122no—
392C23—121no—
393C23—121no—
394C23—121no—
395C23—122no—
396C23—242no—
397C23—241no—
398C23—241no—
399C23—242no—
400C23#88211yescharacteristic
401C23#88411yescharacteristic
402C23#91111yescharacteristic
403C23#91211yescharacteristic
404C23D3×SD1611yescharacteristic
405C23#91511yescharacteristic
406C23#93711yescharacteristic
407Q8—122no—
408Q8—241no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—122no—
426C2×C4—122no—
427C2×C4—122no—
428C2×C4—122no—
429C2×C4—122no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—122no—
434C2×C4—122no—
435C2×C4—242no—
436C2×C4—242no—
437C2×C4—241no—
438C2×C4#88211yescharacteristic
439C2×C4#88411yescharacteristic
440C2×C4#93011yescharacteristic
441C2×C4#94011yescharacteristic
442C8—81no—
443C8—122no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3#160812yes—
449C2×C3#161212yes—
450C2×C3#200011yescharacteristic
451C2×C3#200411yescharacteristic
452C2×C3#202211yescharacteristic
453D3—82no—
454C22—22no—
455C22—22no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—22no—
461C22—22no—
462C22—124no—
463C22—124no—
464C22—122no—
465C22—242no—
466C22—242no—
467C22—242no—
468C22—241no—
469C22—241no—
470C22—242no—
471C22#477611yescharacteristic
472C22#477811yescharacteristic
473C22#478011yescharacteristic
474C22#478311yescharacteristic
475C22#496011yescharacteristic
476C22#505312yes—
477C22#515811yescharacteristic
478C22#517011yescharacteristic
479C22#520212yes—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—82no—
485C4—82no—
486C4—242no—
487C4—241no—
488C4—241no—
489C4—241no—
490C3#992611yescharacteristic, complemented, Sylow
491C2—21no—
492C2—21no—
493C2—21no—
494C2—21no—
495C2—242no—
496C2#6953911yescharacteristic
497C2#7020511yescharacteristic
498C2#7034411yescharacteristic
499C2#7116512yes—
500C2#7117212yes—
subgroupG/H
centerC23#937
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1823C2
socleC23×C3#120
prime radicalC24×D3D4
hypercenter#1823D3
hypocenterC3#9926
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
2. C2×#4971C2semidirect
4. C3×#1823C2semidirect
5. C2×#4467C2semidirect
7. #67624C2semidirect
35. #1823D3semidirect
490. C3#9926semidirect

Maximal subgroups

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

subgroupindex
C2×#49922
C2×#49712
#739522
C3×#18232
C2×#44672
C2×#44682
#676242
#99263