Finite Group Databasebeta

← #10698501

Subgroups — #10698501

Order 768.

570 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2627C211yesmaximal, characteristic
2#71692C211yesmaximal, characteristic
3C2×#4538C211yesmaximal, characteristic
4#71565C211yesmaximal, characteristic
5C2×#4528C211yesmaximal, characteristic
6#69505C211yesmaximal, characteristic
7#67624C211yesmaximal, characteristic
8#10401—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×C3×C4⋊C4C2211yescharacteristic
11C22×C4⋊Dic3C2211yescharacteristic
12C3×#400—21no—
13C3×#398—21no—
14C3×#398—21no—
15C3×#395—21no—
16C3×#385—21no—
17C3×#381—21no—
18C3×#380C2211yescharacteristic
19C2×#835—21no—
20C2×C3⋊#95—21no—
21C2×C3⋊#95—21no—
22C2×C3⋊#95C2211yescharacteristic
23C2×#830C2211yescharacteristic
24C2×#830C2211yescharacteristic
25C2×#808—21no—
26#4538—22no—
27#4529—21no—
28#4529—21no—
29#4528—22no—
30#2627D311yescharacteristic
31#1872—31no—
32#1853—31no—
33C2×#328—31no—
34C2×#324—31no—
35#1441—31no—
36#1294—31no—
37C22×C3×Q8D411yescharacteristic
38C23×C3×C4D411yescharacteristic
39C23×C3×C4D411yescharacteristic
40C23×C3×C4C2311yescharacteristic
41C23×Dic3—21no—
42C23×Dic3—21no—
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—21no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3—42no—
64C22×C3⋊C8—21no—
65C22×C3⋊C8—21no—
66C3×#95—21no—
67C3×#95—21no—
68C3×#95—21no—
69C3×#95—21no—
70C3×#95—21no—
71C3×#95—21no—
72C3×#95—21no—
73#835—42no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—42no—
77C3⋊#95—42no—
78#830—21no—
79#830—21no—
80#830—21no—
81#830D412yes—
82#808—42no—
83C2×C22⋊Q8C2×D311yescharacteristic
84C22×C4⋊C4—31no—
85C22×C4⋊C4C2×D311yescharacteristic
86C2×#106—61no—
87C2×#103—61no—
88C2×C22⋊C8—31no—
89C2×C22⋊C8—31no—
90#400—21no—
91#398—21no—
92#398—21no—
93#395—21no—
94#385—21no—
95#381—21no—
96#380C2×D311yescharacteristic
97C2×#95—31no—
98C2×#95—61no—
99C2×#95—61no—
100#340—61no—
101#340—61no—
102#328—62no—
103#324—62no—
104C24×C3C2×D411yescharacteristic
105C2×C3×Q8—21no—
106C2×C3×Q8—21no—
107C2×C3×Q8—41no—
108C2×C3×Q8—41no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4SD1612yes—
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4#4111yescharacteristic
131C22×C3×C4#4111yescharacteristic
132C22×C3×C4#4111yescharacteristic
133C22×Dic3—22no—
134C22×Dic3—22no—
135C22×Dic3—41no—
136C22×Dic3—41no—
137C22×Dic3—41no—
138C22×Dic3—41no—
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—41no—
150C3×C4⋊C4—41no—
151C3×C4⋊C4—41no—
152C3×C4⋊C4—41no—
153C3×C4⋊C4—41no—
154C3×C4⋊C4—41no—
155C3×C4⋊C4—41no—
156C3×C4⋊C4—41no—
157C3×C4⋊C4—81no—
158C3×C4⋊C4—81no—
159C3×C22⋊C4—41no—
160C3×C22⋊C4—41no—
161C3×C22⋊C4—41no—
162C3×C22⋊C4—41no—
163C4⋊Dic3—22no—
164C4⋊Dic3—22no—
165C4⋊Dic3—41no—
166C4⋊Dic3—41no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—41no—
169C2×C3⋊C8—41no—
170C2×C3⋊C8—41no—
171C2×C3⋊C8—41no—
172C2×C3⋊C8—41no—
173C22×Q8#6711yescharacteristic
174C23×C4—61no—
175C23×C4—61no—
176C23×C4#6711yescharacteristic
177C23×C4#6711yescharacteristic
178C23×C4C22×D311yescharacteristic
179C22×C8—61no—
180C22×C8—61no—
181C22⋊Q8—21no—
182C22⋊Q8—21no—
183C22⋊Q8—21no—
184C22⋊Q8—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—61no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4—122no—
197C2×C4⋊C4#6711yescharacteristic
198C2×C4⋊C4#6711yescharacteristic
199C2×C4⋊C4#6711yescharacteristic
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202#106—122no—
203#103—122no—
204C22⋊C8—32no—
205C22⋊C8—61no—
206C22⋊C8—61no—
207C22⋊C8—61no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—61no—
216#95—61no—
217#95—122no—
218#95—122no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—21no—
222C23×C3—21no—
223C23×C3#12011yescharacteristic
224C23×C3#12111yescharacteristic
225C23×C3#12311yescharacteristic
226C23×C3C2×SD1611yescharacteristic
227C23×C3#13511yescharacteristic
228C23×C3#13711yescharacteristic
229C23×C3#13711yescharacteristic
230C3×Q8—42no—
231C3×Q8—81no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—42no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—81no—
264C2×C3×C4#12111yescharacteristic
265C2×C3×C4#12111yescharacteristic
266C2×C3×C4#12311yescharacteristic
267C2×C3×C4#12411yescharacteristic
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—
279C2×Dic3—81no—
280C2×Dic3—81no—
281C2×Dic3—81no—
282C3⋊C8—42no—
283C3⋊C8—81no—
284C24C2×#6711yescharacteristic
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—41no—
288C2×Q8—41no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—62no—
307C22×C4—62no—
308C22×C4—121no—
309C22×C4—121no—
310C22×C4—121no—
311C22×C4—121no—
312C22×C4—121no—
313C22×C4—121no—
314C22×C4—121no—
315C22×C4—121no—
316C22×C4—121no—
317C22×C4—121no—
318C22×C4—121no—
319C22×C4—121no—
320C22×C4#21512yes—
321C22×C4D3×D411yescharacteristic
322C22×C4#23711yescharacteristic
323C22×C4#23711yescharacteristic
324C22×C4#23711yescharacteristic
325C22×C4C2×#6711yescharacteristic
326C22×C4C2×#6711yescharacteristic
327C2×C8—62no—
328C2×C8—121no—
329C2×C8—121no—
330C2×C8—121no—
331C2×C8—121no—
332C2×C8—121no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—62no—
344C4⋊C4—62no—
345C4⋊C4—81no—
346C4⋊C4—81no—
347C4⋊C4—121no—
348C4⋊C4—121no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—21no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3#39411yescharacteristic
366C22×C3#44811yescharacteristic
367C22×C3#45211yescharacteristic
368C22×C3#45411yescharacteristic
369C22×C3#45611yescharacteristic
370C22×C3B(4,𝔽2)11yescharacteristic
371C22×C3#45811yescharacteristic
372C22×C3#46511yescharacteristic
373C22×C3#47011yescharacteristic
374C22×C3#48311yescharacteristic
375C22×C3#48511yescharacteristic
376C3×C4—21no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—42no—
381C3×C4—81no—
382C3×C4—81no—
383C3×C4—81no—
384C3×C4—81no—
385C3×C4—81no—
386Dic3—82no—
387Dic3—82no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23#93511yescharacteristic
393C23#93911yescharacteristic
394C23C2×#21511yescharacteristic
395C23#94211yescharacteristic
396C23#95011yescharacteristic
397C23#95111yescharacteristic
398C23#95311yescharacteristic
399Q8—42no—
400Q8—81no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—81no—
433C2×C4—122no—
434C2×C4—122no—
435C2×C4—122no—
436C2×C4—122no—
437C2×C4—122no—
438C2×C4—122no—
439C2×C4—241no—
440C2×C4—241no—
441C2×C4—241no—
442C2×C4—241no—
443C2×C4—241no—
444C2×C4—241no—
445C2×C4—241no—
446C2×C4—241no—
447C2×C4#93511yescharacteristic
448C2×C4#93611yescharacteristic
449C2×C4#93811yescharacteristic
450C2×C4#93911yescharacteristic
451C8—122no—
452C8—241no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3#159911yescharacteristic
458C2×C3#160311yescharacteristic
459C2×C3#161111yescharacteristic
460C2×C3#161511yescharacteristic
461C2×C3#203511yescharacteristic
462C2×C3#203811yescharacteristic
463C2×C3#204411yescharacteristic
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22#504311yescharacteristic
477C22#504511yescharacteristic
478C22#504811yescharacteristic
479C22#505211yescharacteristic
480C22#522211yescharacteristic
481C22#522711yescharacteristic
482C22#523911yescharacteristic
483C22#524311yescharacteristic
484C22#524411yescharacteristic
485C22#524511yescharacteristic
486C22#524611yescharacteristic
487C4—21no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—42no—
492C4—81no—
493C4—81no—
494C4—81no—
495C4—81no—
496C4—81no—
497C4—242no—
498C4—242no—
499C3#1040111yescharacteristic, complemented, Sylow
500C2—21no—
501C2—21no—
502C2—21no—
503C2—21no—
504C2#7169411yescharacteristic
505C2#7185011yescharacteristic
506C2#7186311yescharacteristic
507C2#7188711yescharacteristic
508C2#7188911yescharacteristic
509C2#7189811yescharacteristic
510C2#7190211yescharacteristic
subgroupG/H
centerC23#953
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2627C2
socleC23×C3#120
prime radicalC24×C3C2×D4
hypercenter#2627D3
hypocenterC3#10401
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2627

Complements

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

NH ≅ G/Ntype
499. C3#10401semidirect

Maximal subgroups

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

subgroupindex
C3×#26272
#716922
C2×#45382
#715652
C2×#45282
#695052
#676242
#104013