Finite Group Databasebeta

← #10698533

Subgroups — #10698533

Order 768.

620 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82789C211yesmaximal, characteristic
2C3×#1872C211yesmaximal, characteristic
3C2×#4538C211yesmaximal, characteristic
4#71568C211yesmaximal, characteristic
5#71559C211yesmaximal, characteristic
6C2×#4467C211yesmaximal, characteristic
7#67674C211yesmaximal, characteristic
8#10406—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×C3×C4⋊C4C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C22×Dic3⋊C4—21no—
13C2×C3×#103—21no—
14C2×C3×C22⋊C8C2211yescharacteristic
15#5230—21no—
16C2×C3⋊#95—21no—
17C2×C3⋊#95C2211yescharacteristic
18C2×C3⋊#95C2211yescharacteristic
19C2×#830C2211yescharacteristic
20#4983—21no—
21#4981—21no—
22#4979—21no—
23C2×#810—21no—
24#4942—21no—
25#4942—21no—
26C3×#340—21no—
27#4538—22no—
28#4529—21no—
29#4467—22no—
30#2659—31no—
31#1872—31no—
32#1872D311yescharacteristic
33C2×#328—31no—
34C2×#328—31no—
35#1444—31no—
36#1280—31no—
37C22×C3×Q8D411yescharacteristic
38C23×C3×C4—21no—
39C23×C3×C4C2311yescharacteristic
40C23×Dic3—21no—
41C23×Dic3D411yescharacteristic
42C23×Dic3D411yescharacteristic
43C22×Dic6D411yescharacteristic
44C22×C3×C8—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—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—42no—
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C22⋊C4—21no—
55C2×#217—21no—
56C2×C4⋊Dic3D411yescharacteristic
57#924—21no—
58#924—21no—
59#924—21no—
60#924—21no—
61C2×Dic3⋊C4—21no—
62C2×Dic3⋊C4—21no—
63C2×Dic3⋊C4—21no—
64C2×Dic3⋊C4—21no—
65C2×Dic3⋊C4—21no—
66C2×Dic3⋊C4—42no—
67C2×Dic3⋊C4—42no—
68C2×Dic3⋊C4—42no—
69C22×C3⋊C8—21no—
70C3×#103—42no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—42no—
81C3⋊#95D412yes—
82#830—21no—
83#830—21no—
84#810—42no—
85C2×C22⋊Q8—31no—
86C2×C22⋊Q8C2×D311yescharacteristic
87C22×C4⋊C4—61no—
88C22×C4⋊C4C2×D311yescharacteristic
89C2×#103—21no—
90C2×#103—61no—
91C2×C22⋊C8—31no—
92C2×C22⋊C8C2×D311yescharacteristic
93#398—61no—
94#398—61no—
95#395—61no—
96#395—61no—
97#380—61no—
98#380—61no—
99C2×#95—31no—
100C2×#95—31no—
101C2×#95—61no—
102#340—21no—
103#340—61no—
104#328—62no—
105#328—62no—
106C24×C3C2×D411yescharacteristic
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—41no—
110C2×C3×Q8—41no—
111C22×C3×C4—22no—
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—41no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4—41no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C22×C3×C4#4111yescharacteristic
127C22×C3×C4#4111yescharacteristic
128C22×Dic3—22no—
129C22×Dic3—22no—
130C22×Dic3—21no—
131C22×Dic3—21no—
132C22×Dic3—22no—
133C22×Dic3—21no—
134C22×Dic3—21no—
135C22×Dic3—21no—
136C22×Dic3—21no—
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×Dic3SD1612yes—
146C22×Dic3Q1612yes—
147C2×Dic6—21no—
148C2×Dic6—21no—
149C2×Dic6—41no—
150C2×Dic6—41no—
151C2×C3×C8—41no—
152C2×C3×C8—41no—
153C2×C3×C8—41no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—41no—
159C3×C4⋊C4—41no—
160C3×C4⋊C4—41no—
161C3×C4⋊C4—41no—
162C3×C22⋊C4—41no—
163C3×C22⋊C4—41no—
164#217—41no—
165#217—41no—
166C4⋊Dic3—21no—
167C4⋊Dic3—21no—
168Dic3⋊C4—48no—
169Dic3⋊C4—41no—
170Dic3⋊C4—41no—
171Dic3⋊C4—81no—
172Dic3⋊C4—81no—
173Dic3⋊C4—81no—
174Dic3⋊C4—81no—
175C2×C3⋊C8—41no—
176C2×C3⋊C8—41no—
177C2×C3⋊C8—41no—
178C22×Q8—31no—
179C22×Q8#6711yescharacteristic
180C23×C4—21no—
181C23×C4—31no—
182C23×C4—31no—
183C23×C4—61no—
184C23×C4C22×D311yescharacteristic
185C22×C8—21no—
186C22×C8—61no—
187C22⋊Q8—21no—
188C22⋊Q8—21no—
189C22⋊Q8—21no—
190C22⋊Q8—21no—
191C22⋊Q8—61no—
192C22⋊Q8—61no—
193C22⋊Q8—61no—
194C22⋊Q8—61no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—31no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—61no—
201C2×C4⋊C4—61no—
202C2×C4⋊C4—61no—
203C2×C4⋊C4—61no—
204C2×C4⋊C4—61no—
205C2×C4⋊C4—122no—
206C2×C4⋊C4—122no—
207C2×C4⋊C4—122no—
208C2×C4⋊C4#6711yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—61no—
211#103—42no—
212#103—122no—
213C22⋊C8—21no—
214C22⋊C8—21no—
215C22⋊C8—61no—
216C22⋊C8—61no—
217#95—32no—
218#95—61no—
219#95—61no—
220#95—61no—
221#95—61no—
222#95—61no—
223#95—61no—
224#95—61no—
225#95—122no—
226C23×C3—21no—
227C23×C3—21no—
228C23×C3—21no—
229C23×C3—21no—
230C23×C3#12011yescharacteristic
231C23×C3#12111yescharacteristic
232C23×C3#12411yescharacteristic
233C23×C3C2×SD1611yescharacteristic
234C23×C3C2×Q1611yescharacteristic
235C23×C3#13711yescharacteristic
236C23×C3#13711yescharacteristic
237C3×Q8—42no—
238C3×Q8—81no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—42no—
255C2×C3×C4—41no—
256C2×C3×C4—81no—
257C2×C3×C4—81no—
258C2×C3×C4—81no—
259C2×C3×C4—81no—
260C2×C3×C4#12111yescharacteristic
261C2×C3×C4#12111yescharacteristic
262C2×C3×C4#12311yescharacteristic
263C2×C3×C4#12311yescharacteristic
264C2×Dic3—24no—
265C2×Dic3—22no—
266C2×Dic3—22no—
267C2×Dic3—42no—
268C2×Dic3—42no—
269C2×Dic3—42no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3—41no—
273C2×Dic3—41no—
274C2×Dic3—41no—
275C2×Dic3—41no—
276C2×Dic3—42no—
277C2×Dic3—42no—
278C2×Dic3—42no—
279C2×Dic3—41no—
280C2×Dic3—41no—
281C2×Dic3—41no—
282C2×Dic3—41no—
283C2×Dic3—41no—
284C2×Dic3—41no—
285C2×Dic3—81no—
286C2×Dic3—81no—
287Dic6—42no—
288Dic6—81no—
289C3×C8—81no—
290C3⋊C8—81no—
291C24D3×D411yescharacteristic
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—61no—
297C2×Q8—61no—
298C2×Q8—121no—
299C2×Q8—121no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—32no—
304C22×C4—32no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—62no—
314C22×C4—62no—
315C22×C4—62no—
316C22×C4—61no—
317C22×C4—61no—
318C22×C4—61no—
319C22×C4—61no—
320C22×C4—61no—
321C22×C4—61no—
322C22×C4—121no—
323C22×C4—121no—
324C22×C4—121no—
325C22×C4—121no—
326C22×C4—121no—
327C22×C4—121no—
328C22×C4—121no—
329C22×C4—121no—
330C22×C4#23511yescharacteristic
331C22×C4#23511yescharacteristic
332C22×C4D3×D411yescharacteristic
333C22×C4D3×D411yescharacteristic
334C22×C4#23711yescharacteristic
335C22×C4C2×#6711yescharacteristic
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—121no—
340C2×C8—121no—
341C2×C8—121no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—61no—
351C4⋊C4—61no—
352C4⋊C4—128no—
353C4⋊C4—121no—
354C4⋊C4—121no—
355C4⋊C4—241no—
356C4⋊C4—241no—
357C4⋊C4—241no—
358C4⋊C4—241no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—121no—
362C22⋊C4—121no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3#39411yescharacteristic
376C22×C3#44811yescharacteristic
377C22×C3#45111yescharacteristic
378C22×C3#45311yescharacteristic
379C22×C3#45611yescharacteristic
380C22×C3B(4,𝔽2)11yescharacteristic
381C22×C3#45811yescharacteristic
382C22×C3#46111yescharacteristic
383C22×C3#46211yescharacteristic
384C22×C3#48711yescharacteristic
385C22×C3#49111yescharacteristic
386C3×C4—21no—
387C3×C4—21no—
388C3×C4—21no—
389C3×C4—21no—
390C3×C4—82no—
391C3×C4—81no—
392C3×C4—81no—
393Dic3—44no—
394Dic3—42no—
395Dic3—42no—
396Dic3—81no—
397Dic3—81no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23#88411yescharacteristic
403C23#88511yescharacteristic
404C23D3×SD1611yescharacteristic
405C23#91511yescharacteristic
406C23D3×Q1611yescharacteristic
407C23#91811yescharacteristic
408C23#93711yescharacteristic
409Q8—42no—
410Q8—81no—
411Q8—122no—
412Q8—241no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—64no—
431C2×C4—62no—
432C2×C4—62no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—122no—
438C2×C4—122no—
439C2×C4—122no—
440C2×C4—122no—
441C2×C4—122no—
442C2×C4—121no—
443C2×C4—121no—
444C2×C4—121no—
445C2×C4—121no—
446C2×C4—122no—
447C2×C4—122no—
448C2×C4—122no—
449C2×C4—121no—
450C2×C4—121no—
451C2×C4—121no—
452C2×C4—121no—
453C2×C4—121no—
454C2×C4—121no—
455C2×C4—241no—
456C2×C4—241no—
457C2×C4#88311yescharacteristic
458C2×C4#88411yescharacteristic
459C2×C4#92911yescharacteristic
460C2×C4#93911yescharacteristic
461C8—81no—
462C8—241no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3#159611yescharacteristic
468C2×C3#160511yescharacteristic
469C2×C3#161111yescharacteristic
470C2×C3#161411yescharacteristic
471C2×C3#203911yescharacteristic
472C2×C3#204311yescharacteristic
473C2×C3#204411yescharacteristic
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22#479211yescharacteristic
487C22#479511yescharacteristic
488C22#479711yescharacteristic
489C22#481811yescharacteristic
490C22#495911yescharacteristic
491C22#503211yescharacteristic
492C22#505311yescharacteristic
493C22#517011yescharacteristic
494C22#518611yescharacteristic
495C22#519811yescharacteristic
496C22#520511yescharacteristic
497C4—21no—
498C4—21no—
499C4—21no—
500C4—21no—
501C4—82no—
502C4—81no—
503C4—81no—
504C4—124no—
505C4—122no—
506C4—122no—
507C4—241no—
508C4—241no—
509C3#1040611yescharacteristic, complemented, Sylow
510C2—21no—
511C2—21no—
512C2—21no—
513C2—21no—
514C2#6959911yescharacteristic
515C2#7028411yescharacteristic
516C2#7031111yescharacteristic
517C2#7097611yescharacteristic
518C2#7098311yescharacteristic
519C2#7117511yescharacteristic
520C2#7117611yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1872C2
socleC23×C3#120
prime radicalC24×C3C2×D4
hypercenter#1872D3
hypocenterC3#10406
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
509. C3#10406semidirect

Maximal subgroups

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

subgroupindex
#827892
C3×#18722
C2×#45382
#715682
#715592
C2×#44672
#676742
#104063