Finite Group Databasebeta

← #10690916

Subgroups — #10690916

Order 768.

1010 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#74049C211yesmaximal, characteristic
2C2×#4666C211yesmaximal, characteristic
3#69955C211yesmaximal, characteristic
4C2×#4467C211yesmaximal, characteristic, complemented
5C2×#4452C211yesmaximal, characteristic, complemented
6C3×#1294C211yesmaximal, characteristic, complemented
7#67624C211yesmaximal, characteristic, complemented
8#9146—31nomaximal, Sylow
9C23×C4×D3C2211yescharacteristic
10C22×(C2×D3)⋊C4—21no—
11C2×#924C2211yescharacteristic
12C22×C4×Dic3—21no—
13C2×#897C412yescomplemented
14C2×C3×C22⋊C8C2211yescharacteristic
15C3×#380C2211yescharacteristic
16C2×C3×#95C2211yescharacteristic
17C2×C3⋊#95C2211yescharacteristic
18C2×#830C2211yescharacteristic
19C2×(C2×D3)⋊C8—22no—
20#4987C412yescomplemented
21#4955—21no—
22#4950—21no—
23#4668—22no—
24#4666—22no—
25#4666—22no—
26#4467—22no—
27#4452—22no—
28#2848—31no—
29C2×C4×C22⋊C4—31no—
30#1779—31no—
31C2×#328—31no—
32C2×#328—31no—
33#1294—31no—
34#1294D311yescharacteristic, complemented
35C24×D3D411yescharacteristic
36C23×C3×C4—21no—
37C23×C3×C4C2311yescharacteristic
38C23×Dic3—21no—
39C23×Dic3D411yescharacteristic
40C22×C4×D3—21no—
41C22×C4×D3—21no—
42C22×C4×D3—21no—
43C22×C4×D3—21no—
44C22×C4×D3—44no—
45C22×C4×D3C2×C412yes—
46C22×C4×D3D412yes—
47C22×Dic6C2×C411yescharacteristic
48C22×C3×C8—21no—
49C2×C3×C4⋊C4C2×C412yes—
50C2×C3×C22⋊C4—21no—
51C2×#217—21no—
52C2×(C2×D3)⋊C4—24no—
53C2×(C2×D3)⋊C4—28no—
54C2×(C2×D3)⋊C4—22no—
55C2×(C2×D3)⋊C4—22no—
56C2×C4⋊Dic3—22no—
57C2×C4⋊Dic3C2×C411yescharacteristic
58#924—21no—
59#924—21no—
60#924—21no—
61#924—21no—
62C2×Dic3⋊C4—21no—
63C2×C4×Dic3—24no—
64C2×C4×Dic3—44no—
65C22×C3⋊C8—21no—
66#897—44no—
67C3×C22⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×#95—21no—
70C3×#95—21no—
71C3×#95D412yes—
72C3⋊#95—21no—
73C3⋊#95—22no—
74C3⋊#95D412yes—
75#830—21no—
76#830—21no—
77(C2×D3)⋊C8—44no—
78C24×C4—31no—
79C2×C22⋊Q8—31no—
80C2×C22⋊Q8—32no—
81C22×C22⋊C4—61no—
82C22×C42—61no—
83C2×C22⋊C8—31no—
84C2×C22⋊C8—62no—
85C2×C22⋊C8C2×D311yescharacteristic
86#385—62no—
87#385—61no—
88#380—32no—
89#380C2×D311yescharacteristic
90#379—61no—
91C4×C22⋊C4—62no—
92C4×C22⋊C4—62no—
93C2×#95—31no—
94C2×#95C2×D311yescharacteristic
95#328—62no—
96#328—62no—
97C24×C3C2×D411yescharacteristic
98C23×D3—21no—
99C23×D3—21no—
100C23×D3—24no—
101C23×D3—21no—
102C23×D3—21no—
103C23×D3—21no—
104C23×D3—21no—
105C23×D3—42no—
106C23×D3C22⋊C412yes—
107C22×C3×C4—24no—
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—42no—
111C22×C3×C4—42no—
112C22×C3×C4—41no—
113C22×C3×C4—41no—
114C22×C3×C4C22×C411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C22×Dic3—21no—
121C22×Dic3—21no—
122C22×Dic3—24no—
123C22×Dic3—24no—
124C22×Dic3—22no—
125C22×Dic3—21no—
126C22×Dic3—21no—
127C22×Dic3—42no—
128C22×Dic3—41no—
129C22×Dic3—41no—
130C22×Dic3C22⋊C412yes—
131C2×C4×D3—22no—
132C2×C4×D3—22no—
133C2×C4×D3—22no—
134C2×C4×D3—22no—
135C2×C4×D3—24no—
136C2×C4×D3—24no—
137C2×C4×D3—42no—
138C2×C4×D3—42no—
139C2×C4×D3—44no—
140C2×C4×D3—44no—
141C2×C4×D3—44no—
142C2×C4×D3—44no—
143C2×Dic6—21no—
144C2×Dic6—21no—
145C2×Dic6—42no—
146C2×C3×C8—42no—
147C2×C3×C8—41no—
148C3×C4⋊C4—42no—
149C3×C22⋊C4—41no—
150C3×C22⋊C4—41no—
151#217—41no—
152#217—41no—
153(C2×D3)⋊C4—216no—
154(C2×D3)⋊C4—44no—
155(C2×D3)⋊C4—44no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—44no—
159Dic3⋊C4—41no—
160Dic3⋊C4—41no—
161C4×Dic3—48no—
162C2×C3⋊C8—42no—
163C2×C3⋊C8—41no—
164C25—31no—
165C22×Q8—31no—
166C23×C4—21no—
167C23×C4—31no—
168C23×C4—32no—
169C23×C4—32no—
170C23×C4—61no—
171C23×C4—61no—
172C23×C4—61no—
173C23×C4—61no—
174C23×C4—61no—
175C23×C4—124no—
176C23×C4C22×D311yescharacteristic
177C22×C8—21no—
178C22×C8—61no—
179C22⋊Q8—61no—
180C22⋊Q8—61no—
181C22⋊Q8—61no—
182C22⋊Q8—61no—
183C22⋊Q8—124no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—62no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4C4×D312yes—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—64no—
190C2×C22⋊C4—68no—
191C2×C22⋊C4—62no—
192C2×C22⋊C4—62no—
193C2×C22⋊C4—61no—
194C2×C42—64no—
195C2×C42—124no—
196C22⋊C8—21no—
197C22⋊C8—21no—
198C22⋊C8—61no—
199C22⋊C8—61no—
200C22⋊C8—124no—
201#95—21no—
202#95—21no—
203#95—32no—
204#95—62no—
205#95—61no—
206#95D1212yes—
207C23×C3—22no—
208C23×C3—22no—
209C23×C3C2×C22⋊C411yescharacteristic
210C23×C3#12011yescharacteristic
211C23×C3#12311yescharacteristic
212C23×C3#13511yescharacteristic
213C23×C3#13511yescharacteristic
214C23×C3#13711yescharacteristic
215C23×C3#13711yescharacteristic
216C22×D3—24no—
217C22×D3—24no—
218C22×D3—22no—
219C22×D3—22no—
220C22×D3—22no—
221C22×D3—22no—
222C22×D3—22no—
223C22×D3—22no—
224C22×D3—44no—
225C22×D3—44no—
226C22×D3—44no—
227C22×D3—44no—
228C22×D3—42no—
229C22×D3—42no—
230C22×D3—41no—
231C22×D3—41no—
232C22×D3—41no—
233C22×D3—41no—
234C22×D3#10414yes—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—24no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—42no—
243C2×C3×C4—44no—
244C2×C3×C4—44no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4C4×D411yescharacteristic
250C2×C3×C4C4×D411yescharacteristic
251C2×C3×C4#12111yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×Dic3—24no—
254C2×Dic3—24no—
255C2×Dic3—22no—
256C2×Dic3—22no—
257C2×Dic3—24no—
258C2×Dic3—42no—
259C2×Dic3—44no—
260C2×Dic3—44no—
261C2×Dic3—42no—
262C2×Dic3—41no—
263C2×Dic3—41no—
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3#9914yes—
267C4×D3—44no—
268C4×D3—44no—
269C4×D3—44no—
270C4×D3—44no—
271Dic6—81no—
272Dic6—81no—
273C3×C8—81no—
274C3⋊C8—81no—
275C24—32no—
276C24—64no—
277C24—61no—
278C24—61no—
279C24—61no—
280C24—61no—
281C24—61no—
282C24—61no—
283C24—122no—
284C24D3×D411yescharacteristic
285C2×Q8—61no—
286C2×Q8—61no—
287C2×Q8—122no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—32no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—64no—
297C22×C4—64no—
298C22×C4—64no—
299C22×C4—61no—
300C22×C4—61no—
301C22×C4—64no—
302C22×C4—62no—
303C22×C4—62no—
304C22×C4—62no—
305C22×C4—62no—
306C22×C4—62no—
307C22×C4—61no—
308C22×C4—61no—
309C22×C4—122no—
310C22×C4—122no—
311C22×C4—124no—
312C22×C4—124no—
313C22×C4—124no—
314C22×C4—124no—
315C22×C4—122no—
316C22×C4—121no—
317C22×C4—121no—
318C22×C4C2×C4×D311yescharacteristic
319C22×C4C2×D1211yescharacteristic
320C22×C4#23511yescharacteristic
321C22×C4D3×D411yescharacteristic
322C22×C4D3×D411yescharacteristic
323C22×C4#23911yescharacteristic
324C2×C8—42no—
325C2×C8—41no—
326C2×C8—122no—
327C2×C8—121no—
328C4⋊C4—42no—
329C4⋊C4—61no—
330C4⋊C4—61no—
331C4⋊C4—124no—
332C4⋊C4—121no—
333C4⋊C4—121no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—616no—
337C22⋊C4—124no—
338C22⋊C4—124no—
339C22⋊C4—121no—
340C22⋊C4—121no—
341C42—128no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—22no—
347C22×C3—22no—
348C22×C3—22no—
349C22×C3—22no—
350C22×C3#38611yescharacteristic
351C22×C3C2×#9912yes—
352C22×C3#41611yescharacteristic
353C22×C3#41911yescharacteristic
354C22×C3C2×#10412yes—
355C22×C3#45211yescharacteristic
356C22×C3#45211yescharacteristic
357C22×C3#48511yescharacteristic
358C22×C3#48511yescharacteristic
359C2×D3—24no—
360C2×D3—24no—
361C2×D3—24no—
362C2×D3—44no—
363C2×D3—44no—
364C2×D3—44no—
365C2×D3—44no—
366C2×D3—42no—
367C2×D3—42no—
368C2×D3—42no—
369C2×D3—42no—
370C3×C4—21no—
371C3×C4—21no—
372C3×C4—21no—
373C3×C4—21no—
374C3×C4—44no—
375C3×C4—82no—
376Dic3—24no—
377Dic3—44no—
378Dic3—81no—
379Dic3—81no—
380C23—22no—
381C23—22no—
382C23—34no—
383C23—64no—
384C23—64no—
385C23—62no—
386C23—62no—
387C23—62no—
388C23—62no—
389C23—62no—
390C23—62no—
391C23—124no—
392C23—124no—
393C23—124no—
394C23—124no—
395C23—122no—
396C23—122no—
397C23—121no—
398C23—121no—
399C23—121no—
400C23—121no—
401C23D3×C22⋊C411yescharacteristic
402C23#88211yescharacteristic
403C23#89411yescharacteristic
404C23#91211yescharacteristic
405C23#91511yescharacteristic
406C23#91611yescharacteristic
407C23#91811yescharacteristic
408Q8—241no—
409Q8—241no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—34no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—64no—
426C2×C4—64no—
427C2×C4—62no—
428C2×C4—62no—
429C2×C4—64no—
430C2×C4—124no—
431C2×C4—124no—
432C2×C4—124no—
433C2×C4—124no—
434C2×C4—122no—
435C2×C4—124no—
436C2×C4—124no—
437C2×C4—122no—
438C2×C4—121no—
439C2×C4—121no—
440C2×C4—122no—
441C2×C4—122no—
442C2×C4C4×D1211yescharacteristic
443C2×C4#88411yescharacteristic
444C2×C4#89311yescharacteristic
445C2×C4#89511yescharacteristic
446C8—81no—
447C8—241no—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3#150112yes—
453C2×C3#150812yes—
454C2×C3#187311yescharacteristic
455C2×C3#189611yescharacteristic
456C2×C3#189811yescharacteristic
457D3—44no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—64no—
467C22—64no—
468C22—64no—
469C22—124no—
470C22—124no—
471C22—124no—
472C22—124no—
473C22—122no—
474C22—122no—
475C22—122no—
476C22—122no—
477C22#466511yescharacteristic
478C22D3×#9912yes—
479C22#477211yescharacteristic
480C22#478311yescharacteristic
481C22#480211yescharacteristic
482C22#480811yescharacteristic
483C22D3×#10412yes—
484C22#486811yescharacteristic
485C22#488811yescharacteristic
486C4—21no—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—44no—
491C4—64no—
492C4—82no—
493C4—124no—
494C4—241no—
495C4—241no—
496C3#914611yescharacteristic, complemented, Sylow
497C2—21no—
498C2—21no—
499C2—21no—
500C2—21no—
501C2—124no—
502C2#6830112yes—
503C2#6833512yes—
504C2#6875611yescharacteristic
505C2#6887911yescharacteristic
506C2#6897011yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1294C2
socleC23×C3C2×C22⋊C4
prime radicalC24×D3D4
cosocleC23×C3×C4C23
hypercenter#1294D3
hypocenterC3#9146
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
4. C2×#4467C2semidirect
5. C2×#4452C2semidirect
6. C3×#1294C2semidirect
7. #67624C2semidirect
13. C2×#897C4semidirect
20. #4987C4semidirect
34. #1294D3semidirect
496. C3#9146semidirect

Maximal subgroups

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

subgroupindex
#740492
C2×#46662
#699552
C2×#44672
C2×#44522
C3×#12942
#676242
#91463