Finite Group Databasebeta

← #10695466

Subgroups — #10695466

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
1#82822C211yesmaximal, characteristic
2C2×#5043C211yesmaximal, characteristic, complemented
3C2×#4674C211yesmaximal, characteristic
4C2×C3×#328C211yesmaximal, characteristic, complemented
5#69924C211yesmaximal, characteristic, complemented
6C2×#4468C211yesmaximal, characteristic
7#67612C211yesmaximal, characteristic, complemented
8#9926—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×(C2×D3)⋊C4—21no—
11C22×(C2×D3)⋊C4C2211yescharacteristic
12C22×C4⋊Dic3C2211yescharacteristic
13C2×C3×C22⋊C8C2211yescharacteristic
14C2×C3×#95C2211yescharacteristic
15#5231—21no—
16C2×#835—21no—
17#5212C2211yescharacteristic
18C2×C3⋊#95—21no—
19C2×#830C2211yescharacteristic
20#5043—22no—
21#5043—22no—
22#4994—22no—
23#4993—21no—
24C2×#808—21no—
25#4955—21no—
26#4946—21no—
27#4674—22no—
28#4674—22no—
29C3×#328—22no—
30#4551—22no—
31#4468—22no—
32C2×#483—31no—
33#2626—31no—
34C2×#397—31no—
35#1823—31no—
36C2×#328D311yescharacteristic, complemented
37C2×#323—31no—
38#1294—31no—
39C24×D3D411yescharacteristic
40C22×C3×Q8D411yescharacteristic
41C23×C3×C4—21no—
42C23×C3×C4C2311yescharacteristic
43C23×Dic3—21no—
44C23×Dic3D411yescharacteristic
45C22×C3×C8—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C22⋊C4—21no—
53C2×#217—21no—
54C2×(C2×D3)⋊C4—24no—
55C2×(C2×D3)⋊C4—21no—
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—22no—
62C2×(C2×D3)⋊C4—22no—
63C2×(C2×D3)⋊C4—42no—
64C2×(C2×D3)⋊C4—42no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3—21no—
70C2×C4⋊Dic3D412yes—
71C2×C4⋊Dic3D412yes—
72C2×Dic3⋊C4—22no—
73C22×C3⋊C8—21no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×#95—21no—
77C3×#95D412yes—
78#835—24no—
79C3⋊#95—24no—
80C3⋊#95—22no—
81C3⋊#95—21no—
82#830—21no—
83#830D412yes—
84#808—24no—
85C2×C22⋊Q8C2×D311yescharacteristic
86C22×C4⋊C4—31no—
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#380—31no—
102#379—61no—
103C2×#95—61no—
104C2×#95C2×D311yescharacteristic
105#340—62no—
106#328—22no—
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—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—42no—
121C22×C3×C4—22no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—42no—
125C22×C3×C4—42no—
126C22×C3×C4—42no—
127C22×C3×C4—41no—
128C22×C3×C4—41no—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4#4111yescharacteristic
135C22×Dic3—21no—
136C22×Dic3—21no—
137C22×Dic3—22no—
138C22×Dic3—22no—
139C22×Dic3—21no—
140C22×Dic3—42no—
141C22×Dic3—41no—
142C22×Dic3—41no—
143C22×Dic3—41no—
144C22×Dic3—41no—
145C22×Dic3—41no—
146C22×Dic3—41no—
147C2×C3×C8—42no—
148C2×C3×C8—41no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—41no—
152C3×C4⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C22⋊C4—41no—
155#217—41no—
156#217—41no—
157(C2×D3)⋊C4—48no—
158(C2×D3)⋊C4—41no—
159(C2×D3)⋊C4—41no—
160(C2×D3)⋊C4—41no—
161(C2×D3)⋊C4—41no—
162(C2×D3)⋊C4—82no—
163(C2×D3)⋊C4—82no—
164C4⋊Dic3—22no—
165C4⋊Dic3—22no—
166C4⋊Dic3—41no—
167C4⋊Dic3—41no—
168C4⋊Dic3—42no—
169Dic3⋊C4—82no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—42no—
172C25—31no—
173C22×Q8D1211yescharacteristic
174C23×C4—21no—
175C23×C4—31no—
176C23×C4—61no—
177C23×C4C22×D311yescharacteristic
178C22×C8—21no—
179C22×C8—61no—
180C22⋊Q8—21no—
181C22⋊Q8—21no—
182C22⋊Q8—21no—
183C22⋊Q8—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—32no—
186C2×C4⋊C4—32no—
187C2×C4⋊C4—61no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4—61no—
192C2×C4⋊C4—62no—
193C2×C4⋊C4D1211yescharacteristic
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—64no—
215#95—62no—
216#95—61no—
217#95#6712yes—
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—
244C3×Q8—42no—
245C3×Q8—81no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—82no—
260C2×C3×C4—82no—
261C2×C3×C4#12011yescharacteristic
262C2×C3×C4#12111yescharacteristic
263C2×C3×C4#12111yescharacteristic
264C2×C3×C4#12711yescharacteristic
265C2×Dic3—42no—
266C2×Dic3—42no—
267C2×Dic3—41no—
268C2×Dic3—42no—
269C2×Dic3—42no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3—42no—
273C2×Dic3—42no—
274C2×Dic3—81no—
275C2×Dic3—81no—
276C2×Dic3—81no—
277C2×Dic3—81no—
278C2×Dic3—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—21no—
292C2×Q8—21no—
293C2×Q8—42no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—62no—
303C22×C4—61no—
304C22×C4—61no—
305C22×C4—62no—
306C22×C4—61no—
307C22×C4—122no—
308C22×C4—121no—
309C22×C4—121no—
310C22×C4—121no—
311C22×C4—121no—
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—21no—
325C4⋊C4—21no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—62no—
329C4⋊C4—62no—
330C4⋊C4—122no—
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—81no—
377C3×C4—81no—
378Dic3—82no—
379Dic3—82no—
380Dic3—81no—
381C23—22no—
382C23—22no—
383C23—62no—
384C23—62no—
385C23—62no—
386C23—124no—
387C23—122no—
388C23—124no—
389C23—122no—
390C23—121no—
391C23—121no—
392C23—122no—
393C23—121no—
394C23—121no—
395C23—122no—
396C23—242no—
397C23—241no—
398C23—242no—
399C23—241no—
400C23#88211yescharacteristic
401C23#88411yescharacteristic
402C23D3×SD1611yescharacteristic
403C23#91411yescharacteristic
404C23#91811yescharacteristic
405C23#91911yescharacteristic
406C23#93711yescharacteristic
407Q8—42no—
408Q8—81no—
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—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4—122no—
427C2×C4—121no—
428C2×C4—122no—
429C2×C4—122no—
430C2×C4—122no—
431C2×C4—122no—
432C2×C4—122no—
433C2×C4—241no—
434C2×C4—241no—
435C2×C4—241no—
436C2×C4—241no—
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—21no—
455C22—21no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—21no—
461C22—21no—
462C22—124no—
463C22—122no—
464C22—124no—
465C22—242no—
466C22—242no—
467C22—242no—
468C22—241no—
469C22—242no—
470C22—241no—
471C22#474212yes—
472C22#480611yescharacteristic
473C22#480711yescharacteristic
474C22#481011yescharacteristic
475C22#481211yescharacteristic
476C22#482512yes—
477C22#496011yescharacteristic
478C22#516911yescharacteristic
479C22#518711yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—82no—
485C4—81no—
486C4—81no—
487C4—242no—
488C4—242no—
489C4—241no—
490C3#992611yescharacteristic, complemented, Sylow
491C2—21no—
492C2—21no—
493C2—21no—
494C2—21no—
495C2—242no—
496C2#6832012yes—
497C2#6832112yes—
498C2#7003111yescharacteristic
499C2#7027211yescharacteristic
500C2#7038611yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#328C2
socleC23×C3#120
prime radicalC24×D3D4
hypercenterC2×#328D3
hypocenterC3#9926
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
2. C2×#5043C2semidirect
4. C2×C3×#328C2semidirect
5. #69924C2semidirect
7. #67612C2semidirect
36. C2×#328D3semidirect
490. C3#9926semidirect

Maximal subgroups

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

subgroupindex
#828222
C2×#50432
C2×#46742
C2×C3×#3282
#699242
C2×#44682
#676122
#99263