Finite Group Databasebeta

← #10690175

Subgroups — #10690175

Order 768.

908 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#74005C211yesmaximal, characteristic
2C2×#4673C211yesmaximal, characteristic
3#69963C211yesmaximal, characteristic, complemented
4C2×#4468C211yesmaximal, characteristic
5#69525C211yesmaximal, characteristic, complemented
6C3×#1294C211yesmaximal, characteristic, complemented
7#67612C211yesmaximal, characteristic, complemented
8#9013—31nomaximal, Sylow
9C23×Dic6C2211yescharacteristic
10C22×(C2×D3)⋊C4—21no—
11C22×(C2×D3)⋊C4C2211yescharacteristic
12C2×D3×C22⋊C4—21no—
13C2×C3×C22⋊C8C2211yescharacteristic
14C3×#380C2211yescharacteristic
15C2×C3×#95C2211yescharacteristic
16#5212C2211yescharacteristic
17C2×C3⋊#95—21no—
18C2×#830C2211yescharacteristic
19C2×#816—22no—
20#4989—22no—
21C2×#810—22no—
22#4942—21no—
23#4673—22no—
24#4673C414yescomplemented
25#4668—22no—
26#4468—22no—
27C2×#393—31no—
28#2315—31no—
29#1787—31no—
30#1787—31no—
31C2×#323—31no—
32#1294—31no—
33#1294D311yescharacteristic, complemented
34C24×D3D411yescharacteristic
35C23×C3×C4—21no—
36C23×C3×C4C2311yescharacteristic
37C23×Dic3—21no—
38C23×Dic3D411yescharacteristic
39C22×C4×D3—22no—
40C22×Dic6—21no—
41C22×Dic6—21no—
42C22×Dic6—21no—
43C22×Dic6—21no—
44C22×Dic6—21no—
45C22×Dic6—44no—
46C22×Dic6C2×C412yescomplemented
47C22×C3×C8—21no—
48C2×C3×C4⋊C4D412yes—
49C2×C3×C22⋊C4—21no—
50C2×#217—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—21no—
53C2×(C2×D3)⋊C4—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—22no—
56C2×(C2×D3)⋊C4—24no—
57C2×(C2×D3)⋊C4—44no—
58C2×(C2×D3)⋊C4C2×C412yes—
59C2×(C2×D3)⋊C4D412yes—
60C2×C4⋊Dic3D412yes—
61C2×Dic3⋊C4—22no—
62C22×C3⋊C8—21no—
63D3×C22⋊C4—42no—
64D3×C22⋊C4—42no—
65C3×C22⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×#95—21no—
68C3×#95—21no—
69C3×#95C2×C412yescomplemented
70C3⋊#95—22no—
71C3⋊#95—21no—
72C3⋊#95—24no—
73#830—21no—
74#830—21no—
75#816—44no—
76#810—44no—
77C23×Q8—31no—
78C22×C22⋊C4—31no—
79C22×C22⋊C4—61no—
80C22×C22⋊C4—61no—
81C2×#103—62no—
82C2×#103—62no—
83C2×C22⋊C8—31no—
84C2×C22⋊C8C2×D311yescharacteristic
85#393—34no—
86#393—62no—
87#385—62no—
88#385—62no—
89#380—31no—
90#380—61no—
91#380C2×D311yescharacteristic
92C2×#95—61no—
93C2×#95C2×D311yescharacteristic
94#323—62no—
95C24×C3C2×D411yescharacteristic
96C23×D3—21no—
97C23×D3—21no—
98C23×D3—21no—
99C23×D3—21no—
100C23×D3—42no—
101C23×D3—42no—
102C23×D3C22⋊C412yes—
103C23×D3SD1612yes—
104C23×D3Q1612yes—
105C22×C3×C4—22no—
106C22×C3×C4—22no—
107C22×C3×C4—42no—
108C22×C3×C4—42no—
109C22×C3×C4—42no—
110C22×C3×C4—41no—
111C22×C3×C4—41no—
112C22×C3×C4C22×C411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4#4111yescharacteristic
117C22×C3×C4#4111yescharacteristic
118C22×Dic3—24no—
119C22×Dic3—22no—
120C22×Dic3—21no—
121C22×Dic3—21no—
122C22×Dic3—22no—
123C22×Dic3—42no—
124C22×Dic3—41no—
125C22×Dic3—41no—
126C22×Dic3—42no—
127C22×Dic3C22⋊C412yes—
128C2×C4×D3—44no—
129C2×C4×D3—82no—
130C2×C4×D3—82no—
131C2×Dic6—24no—
132C2×Dic6—24no—
133C2×Dic6—22no—
134C2×Dic6—22no—
135C2×Dic6—44no—
136C2×Dic6—44no—
137C2×Dic6—44no—
138C2×Dic6—44no—
139C2×Dic6—42no—
140C2×Dic6—42no—
141C2×Dic6—41no—
142C2×Dic6—41no—
143C2×C3×C8—42no—
144C2×C3×C8—41no—
145C3×C4⋊C4—42no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148#217—41no—
149#217—41no—
150(C2×D3)⋊C4—22no—
151(C2×D3)⋊C4—22no—
152(C2×D3)⋊C4—22no—
153(C2×D3)⋊C4—22no—
154(C2×D3)⋊C4—48no—
155(C2×D3)⋊C4—44no—
156C4⋊Dic3—42no—
157Dic3⋊C4—82no—
158C2×C3⋊C8—42no—
159C2×C3⋊C8—41no—
160C25—31no—
161C22×Q8—32no—
162C22×Q8—61no—
163C22×Q8—61no—
164C22×Q8—61no—
165C22×Q8—61no—
166C22×Q8—61no—
167C22×Q8—124no—
168C23×C4—21no—
169C23×C4—31no—
170C23×C4—62no—
171C23×C4—61no—
172C23×C4C22×D311yescharacteristic
173C22×C8—21no—
174C22×C8—61no—
175C2×C4⋊C4—32no—
176C2×C4⋊C4—62no—
177C2×C4⋊C4D1212yes—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—32no—
180C2×C22⋊C4—32no—
181C2×C22⋊C4—61no—
182C2×C22⋊C4—61no—
183C2×C22⋊C4—61no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—64no—
186C2×C22⋊C4—62no—
187C2×C22⋊C4—61no—
188C2×C22⋊C4—122no—
189C2×C22⋊C4—122no—
190C2×C22⋊C4—124no—
191#103—124no—
192#103—124no—
193C22⋊C8—21no—
194C22⋊C8—21no—
195C22⋊C8—61no—
196C22⋊C8—61no—
197#95—21no—
198#95—21no—
199#95—64no—
200#95—62no—
201#95—61no—
202#95C4×D312yescomplemented
203C23×C3—22no—
204C23×C3—22no—
205C23×C3C2×C22⋊C411yescharacteristic
206C23×C3#12011yescharacteristic
207C23×C3#12311yescharacteristic
208C23×C3C2×SD1611yescharacteristic
209C23×C3C2×Q1611yescharacteristic
210C23×C3AGL(1,ℤ/8)11yescharacteristic
211C23×C3#13711yescharacteristic
212C22×D3—22no—
213C22×D3—22no—
214C22×D3—22no—
215C22×D3—22no—
216C22×D3—22no—
217C22×D3—22no—
218C22×D3—42no—
219C22×D3—42no—
220C22×D3—42no—
221C22×D3—42no—
222C22×D3—44no—
223C22×D3—44no—
224C22×D3—82no—
225C22×D3—82no—
226C22×D3—81no—
227C22×D3—81no—
228C22×D3#10314yescomplemented
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—82no—
242C2×C3×C4—82no—
243C2×C3×C4C4×D411yescharacteristic
244C2×C3×C4C4×D411yescharacteristic
245C2×C3×C4#12111yescharacteristic
246C2×C3×C4#12111yescharacteristic
247C2×Dic3—24no—
248C2×Dic3—22no—
249C2×Dic3—42no—
250C2×Dic3—44no—
251C2×Dic3—42no—
252C2×Dic3—42no—
253C2×Dic3—44no—
254C2×Dic3—42no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—42no—
258C2×Dic3—81no—
259C4×D3—88no—
260Dic6—24no—
261Dic6—24no—
262Dic6—44no—
263Dic6—44no—
264Dic6—42no—
265Dic6—42no—
266C3×C8—81no—
267C3⋊C8—81no—
268C24—32no—
269C24—32no—
270C24—32no—
271C24—61no—
272C24—61no—
273C24—61no—
274C24—61no—
275C24—122no—
276C24—122no—
277C24D3×D411yescharacteristic
278C2×Q8—64no—
279C2×Q8—64no—
280C2×Q8—62no—
281C2×Q8—62no—
282C2×Q8—122no—
283C2×Q8—122no—
284C2×Q8—124no—
285C2×Q8—124no—
286C2×Q8—124no—
287C2×Q8—124no—
288C2×Q8—121no—
289C2×Q8—121no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—32no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—64no—
299C22×C4—62no—
300C22×C4—61no—
301C22×C4—61no—
302C22×C4—62no—
303C22×C4—124no—
304C22×C4—122no—
305C22×C4—121no—
306C22×C4—121no—
307C22×C4—122no—
308C22×C4—242no—
309C22×C4—242no—
310C22×C4C2×C4×D311yescharacteristic
311C22×C4C2×D1211yescharacteristic
312C22×C4#23511yescharacteristic
313C22×C4D3×D411yescharacteristic
314C22×C4D3×D411yescharacteristic
315C22×C4#23911yescharacteristic
316C2×C8—42no—
317C2×C8—41no—
318C2×C8—122no—
319C2×C8—121no—
320C4⋊C4—42no—
321C4⋊C4—122no—
322C4⋊C4—242no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—62no—
326C22⋊C4—62no—
327C22⋊C4—62no—
328C22⋊C4—62no—
329C22⋊C4—128no—
330C22⋊C4—124no—
331C22⋊C4—121no—
332C22⋊C4—121no—
333C22×C3—22no—
334C22×C3—22no—
335C22×C3—22no—
336C22×C3—22no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3#38611yescharacteristic
342C22×C3C2×#10311yescharacteristic
343C22×C3#41711yescharacteristic
344C22×C3#45011yescharacteristic
345C22×C3#45111yescharacteristic
346C22×C3#45512yes—
347C22×C3B(4,𝔽2)12yes—
348C22×C3#48311yescharacteristic
349C22×C3#48411yescharacteristic
350C2×D3—24no—
351C2×D3—44no—
352C2×D3—44no—
353C2×D3—42no—
354C2×D3—42no—
355C2×D3—44no—
356C2×D3—82no—
357C2×D3—82no—
358C2×D3—82no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—82no—
364C3×C4—82no—
365Dic3—42no—
366Dic3—44no—
367Dic3—82no—
368C23—22no—
369C23—22no—
370C23—34no—
371C23—62no—
372C23—62no—
373C23—62no—
374C23—62no—
375C23—62no—
376C23—62no—
377C23—122no—
378C23—122no—
379C23—124no—
380C23—124no—
381C23—122no—
382C23—122no—
383C23—241no—
384C23—241no—
385C23—242no—
386C23—242no—
387C23D3×C22⋊C411yescharacteristic
388C23#88211yescharacteristic
389C23#89411yescharacteristic
390C23#91111yescharacteristic
391C23D3×SD1611yescharacteristic
392C23#91511yescharacteristic
393C23D3×Q1611yescharacteristic
394Q8—64no—
395Q8—64no—
396Q8—124no—
397Q8—124no—
398Q8—122no—
399Q8—122no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—64no—
413C2×C4—62no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—122no—
417C2×C4—124no—
418C2×C4—122no—
419C2×C4—124no—
420C2×C4—122no—
421C2×C4—122no—
422C2×C4—122no—
423C2×C4—122no—
424C2×C4—122no—
425C2×C4—248no—
426C2×C4—241no—
427C2×C4C4×D1211yescharacteristic
428C2×C4#88411yescharacteristic
429C2×C4#89311yescharacteristic
430C2×C4#89511yescharacteristic
431C8—81no—
432C8—241no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3#160112yes—
438C2×C3#160212yes—
439C2×C3#187211yescharacteristic
440C2×C3#188511yescharacteristic
441C2×C3#188911yescharacteristic
442D3—44no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—64no—
452C22—124no—
453C22—122no—
454C22—122no—
455C22—124no—
456C22—124no—
457C22—242no—
458C22—242no—
459C22—242no—
460C22#466511yescharacteristic
461C22#474112yes—
462C22#477111yescharacteristic
463C22#477611yescharacteristic
464C22D3×#10311yescharacteristic
465C22#480911yescharacteristic
466C22#482512yes—
467C22#486211yescharacteristic
468C22#488411yescharacteristic
469C4—21no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—82no—
474C4—82no—
475C4—124no—
476C4—122no—
477C4—242no—
478C3#901311yescharacteristic, complemented, Sylow
479C2—21no—
480C2—21no—
481C2—21no—
482C2—21no—
483C2—124no—
484C2#6828612yes—
485C2#6831812yes—
486C2#6875911yescharacteristic
487C2#6887811yescharacteristic
488C2#6896211yescharacteristic
subgroupG/H
centerC23#882
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1294C2
socleC23×C3#120
prime radicalC24×D3D4
cosocleC23×C3×C4C23
hypercenter#1294D3
hypocenterC3#9013
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
3. #69963C2semidirect
5. #69525C2semidirect
6. C3×#1294C2semidirect
7. #67612C2semidirect
24. #4673C4semidirect
33. #1294D3semidirect
46. C22×Dic6C2×C4semidirect
69. C3×#95C2×C4semidirect
202. #95C4×D3semidirect
228. C22×D3#103semidirect
478. C3#9013semidirect

Maximal subgroups

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

subgroupindex
#740052
C2×#46732
#699632
C2×#44682
#695252
C3×#12942
#676122
#90133