Finite Group Databasebeta

← #10695465

Subgroups — #10695465

Order 768.

734 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2626C211yesmaximal, characteristic
2C2×#5043C211yesmaximal, characteristic, complemented
3C2×#4953C211yesmaximal, characteristic
4C2×#4538C211yesmaximal, characteristic, complemented
5C2×#4525C211yesmaximal, characteristic
6#69501C211yesmaximal, characteristic, complemented
7#67624C211yesmaximal, characteristic, complemented
8#9926—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×C3×C22⋊C4C2211yescharacteristic
11C22×#217—21no—
12C22×C4⋊Dic3C2211yescharacteristic
13C3×#399—21no—
14C3×#396—22no—
15C3×#393—21no—
16C3×#385—21no—
17C3×#380C2211yescharacteristic
18C3×#379—21no—
19C2×#835—21no—
20C2×C3⋊#95—21no—
21C2×C3⋊#95C2211yescharacteristic
22C2×#830C2211yescharacteristic
23C2×#830C2211yescharacteristic
24#5043—22no—
25#5043—22no—
26C2×#808—21no—
27#4953—22no—
28#4953—22no—
29#4538—22no—
30#4529—22no—
31#4525—22no—
32C2×#483—31no—
33#2626D311yescharacteristic
34C2×#397—31no—
35#1823—31no—
36C2×#328—31no—
37C2×#323—31no—
38#1294—31no—
39C25×C3D411yescharacteristic
40C22×C3×Q8D411yescharacteristic
41C23×C3×C4D411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C23×Dic3—21no—
44C23×Dic3—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—22no—
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C4⋊C4D412yes—
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4—21no—
55C2×C3×C22⋊C4—21no—
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—21no—
61C2×C3×C22⋊C4—22no—
62C2×C3×C22⋊C4—22no—
63C2×#217—24no—
64C2×#217—42no—
65C2×#217—42no—
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3—21no—
70C2×C4⋊Dic3—21no—
71C2×C4⋊Dic3D412yes—
72C22×C3⋊C8—21no—
73C22×C3⋊C8—21no—
74C3×#95—22no—
75C3×#95—21no—
76#835—24no—
77C3⋊#95—24no—
78C3⋊#95—21no—
79C3⋊#95D412yes—
80#830—21no—
81#830—21no—
82#830—21no—
83#830D412yes—
84#808—24no—
85C2×C22⋊Q8C2×D311yescharacteristic
86C22×C4⋊C4—31no—
87C22×C22⋊C4—61no—
88C22×C22⋊C4C2×D311yescharacteristic
89#483—62no—
90#483—62no—
91C2×#106—61no—
92C2×#103—61no—
93C2×C22⋊C8—31no—
94C2×C22⋊C8—31no—
95#399—21no—
96#397—62no—
97#397—62no—
98#396—22no—
99#393—21no—
100#385—21no—
101#380C2×D311yescharacteristic
102#379—21no—
103C2×#95—31no—
104C2×#95—61no—
105#340—62no—
106#328—62no—
107#323—62no—
108C24×C3—21no—
109C24×C3—21no—
110C24×C3—21no—
111C24×C3—21no—
112C24×C3—21no—
113C24×C3—21no—
114C24×C3—42no—
115C24×C3—42no—
116C24×C3SD1612yes—
117C24×C3C2×D411yescharacteristic
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—42no—
121C22×C3×C4—21no—
122C22×C3×C4—22no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—42no—
128C22×C3×C4—42no—
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—22no—
136C22×Dic3—22no—
137C22×Dic3—42no—
138C22×Dic3—42no—
139C22×Dic3—41no—
140C22×Dic3—41no—
141C22×Dic3—41no—
142C22×Dic3—41no—
143C22×Dic3—41no—
144C22×Dic3—41no—
145C22×Dic3—41no—
146C22×Dic3—41no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—42no—
150C3×C4⋊C4—41no—
151C3×C4⋊C4—41no—
152C3×C4⋊C4—82no—
153C3×C22⋊C4—41no—
154C3×C22⋊C4—41no—
155C3×C22⋊C4—41no—
156C3×C22⋊C4—41no—
157C3×C22⋊C4—41no—
158C3×C22⋊C4—41no—
159C3×C22⋊C4—41no—
160C3×C22⋊C4—41no—
161C3×C22⋊C4—82no—
162C3×C22⋊C4—82no—
163#217—48no—
164C4⋊Dic3—22no—
165C4⋊Dic3—22no—
166C4⋊Dic3—41no—
167C4⋊Dic3—41no—
168C2×C3⋊C8—22no—
169C2×C3⋊C8—42no—
170C2×C3⋊C8—42no—
171C2×C3⋊C8—41no—
172C25#6711yescharacteristic
173C22×Q8#6711yescharacteristic
174C23×C4—61no—
175C23×C4—61no—
176C23×C4#6711yescharacteristic
177C23×C4C22×D311yescharacteristic
178C22×C8—61no—
179C22×C8—61no—
180C22⋊Q8—21no—
181C22⋊Q8—21no—
182C22⋊Q8—21no—
183C22⋊Q8—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
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#6711yescharacteristic
193C2×C4⋊C4#6712yes—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—64no—
205C2×C22⋊C4—122no—
206C2×C22⋊C4—122no—
207#106—64no—
208#103—64no—
209C22⋊C8—32no—
210C22⋊C8—61no—
211C22⋊C8—61no—
212C22⋊C8—61no—
213#95—22no—
214#95—21no—
215#95—32no—
216#95—64no—
217#95—61no—
218C23×C3—22no—
219C23×C3—22no—
220C23×C3—22no—
221C23×C3—22no—
222C23×C3—22no—
223C23×C3—44no—
224C23×C3—44no—
225C23×C3—42no—
226C23×C3—42no—
227C23×C3—41no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—42no—
231C23×C3—42no—
232C23×C3—41no—
233C23×C3—82no—
234C23×C3—82no—
235C23×C3—81no—
236C23×C3—81no—
237C23×C3#12011yescharacteristic
238C23×C3#12011yescharacteristic
239C23×C3#12111yescharacteristic
240C23×C3C2×SD1611yescharacteristic
241C23×C3#13511yescharacteristic
242C23×C3AGL(1,ℤ/8)11yescharacteristic
243C23×C3#13711yescharacteristic
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—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—81no—
263C2×C3×C4#12011yescharacteristic
264C2×C3×C4#12111yescharacteristic
265C2×C3×C4#12111yescharacteristic
266C2×C3×C4#12711yescharacteristic
267C2×Dic3—42no—
268C2×Dic3—42no—
269C2×Dic3—42no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3—42no—
273C2×Dic3—82no—
274C2×Dic3—82no—
275C2×Dic3—81no—
276C2×Dic3—81no—
277C2×Dic3—81no—
278C2×Dic3—81no—
279C3⋊C8—42no—
280C3⋊C8—81no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24—21no—
287C24—42no—
288C24—42no—
289C24#21512yes—
290C24C2×#6711yescharacteristic
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—42no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—62no—
303C22×C4—62no—
304C22×C4—122no—
305C22×C4—122no—
306C22×C4—121no—
307C22×C4—121no—
308C22×C4—121no—
309C22×C4—121no—
310C22×C4—121no—
311C22×C4—121no—
312C22×C4—121no—
313C22×C4—121no—
314C22×C4D3×D411yescharacteristic
315C22×C4D3×D411yescharacteristic
316C22×C4D3×D411yescharacteristic
317C22×C4#23711yescharacteristic
318C22×C4C2×#6711yescharacteristic
319C22×C4C2×#6711yescharacteristic
320C2×C8—62no—
321C2×C8—122no—
322C2×C8—122no—
323C2×C8—121no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—42no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—62no—
330C4⋊C4—62no—
331C4⋊C4—82no—
332C4⋊C4—121no—
333C4⋊C4—121no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C22⋊C4—128no—
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—44no—
354C22×C3—44no—
355C22×C3—42no—
356C22×C3—82no—
357C22×C3—82no—
358C22×C3—82no—
359C22×C3—82no—
360C22×C3—81no—
361C22×C3—81no—
362C22×C3#39211yescharacteristic
363C22×C3#44811yescharacteristic
364C22×C3#44911yescharacteristic
365C22×C3#45011yescharacteristic
366C22×C3#45211yescharacteristic
367C22×C3#45512yes—
368C22×C3#45812yes—
369C22×C3#46511yescharacteristic
370C22×C3#46811yescharacteristic
371C3×C4—21no—
372C3×C4—21no—
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—82no—
376C3×C4—81no—
377C3×C4—81no—
378C3×C4—81no—
379Dic3—82no—
380Dic3—82no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—44no—
387C23—42no—
388C23—42no—
389C23—44no—
390C23—41no—
391C23—41no—
392C23—42no—
393C23—42no—
394C23—41no—
395C23—41no—
396C23—82no—
397C23—81no—
398C23—81no—
399C23—82no—
400C23#93711yescharacteristic
401C23#93911yescharacteristic
402C23C2×#21511yescharacteristic
403C23#94211yescharacteristic
404C23#94911yescharacteristic
405C23#95011yescharacteristic
406C23#95311yescharacteristic
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—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—81no—
426C2×C4—122no—
427C2×C4—122no—
428C2×C4—122no—
429C2×C4—122no—
430C2×C4—122no—
431C2×C4—122no—
432C2×C4—242no—
433C2×C4—242no—
434C2×C4—241no—
435C2×C4—241no—
436C2×C4—241no—
437C2×C4—241no—
438C2×C4#93711yescharacteristic
439C2×C4#93811yescharacteristic
440C2×C4#93911yescharacteristic
441C2×C4#94011yescharacteristic
442C8—122no—
443C8—241no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—82no—
449C2×C3#160812yes—
450C2×C3#161212yes—
451C2×C3#200011yescharacteristic
452C2×C3#200411yescharacteristic
453C2×C3#202211yescharacteristic
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—44no—
463C22—42no—
464C22—44no—
465C22—82no—
466C22—82no—
467C22—82no—
468C22—81no—
469C22—81no—
470C22—82no—
471C22#504611yescharacteristic
472C22#504711yescharacteristic
473C22#504811yescharacteristic
474C22#504911yescharacteristic
475C22#522311yescharacteristic
476C22#522711yescharacteristic
477C22#523811yescharacteristic
478C22#524212yes—
479C22#524312yes—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—81no—
485C4—81no—
486C4—81no—
487C4—82no—
488C4—242no—
489C4—242no—
490C3#992611yescharacteristic, complemented, Sylow
491C2—21no—
492C2—21no—
493C2—21no—
494C2—21no—
495C2—82no—
496C2#7169911yescharacteristic
497C2#7184911yescharacteristic
498C2#7185711yescharacteristic
499C2#7188212yes—
500C2#7189012yes—
subgroupG/H
centerC23#953
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2626C2
socleC23×C3#120
prime radicalC25×C3D4
hypercenter#2626D3
hypocenterC3#9926
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2626

Complements

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

NH ≅ G/Ntype
2. C2×#5043C2semidirect
4. C2×#4538C2semidirect
6. #69501C2semidirect
7. #67624C2semidirect
490. C3#9926semidirect

Maximal subgroups

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

subgroupindex
C3×#26262
C2×#50432
C2×#49532
C2×#45382
C2×#45252
#695012
#676242
#99263