Finite Group Databasebeta

← #10690914

Subgroups — #10690914

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#82834C211yesmaximal, characteristic
2C2×#4666C211yesmaximal, characteristic
3C2×C3×#328C211yesmaximal, characteristic, complemented
4C2×#4538C211yesmaximal, characteristic, complemented
5#69955C211yesmaximal, characteristic
6#67675C211yesmaximal, characteristic, complemented
7#67612C211yesmaximal, characteristic, complemented
8#9146—31nomaximal, Sylow
9C23×C4×D3C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C2×#946C412yescomplemented
12C22×(C2×D3)⋊C4—21no—
13C22×C4×Dic3—21no—
14C2×C3×C22⋊C8C2211yescharacteristic
15C2×C3×#95C2211yescharacteristic
16#5212C2211yescharacteristic
17C2×C3⋊#95C2211yescharacteristic
18C2×#830C2211yescharacteristic
19C2×(C2×D3)⋊C8—22no—
20#4989—22no—
21#4987C412yescomplemented
22#4955—21no—
23#4946—21no—
24#4666—22no—
25#4666—22no—
26C3×#328—22no—
27#4538—22no—
28#2848—31no—
29C2×C4×C22⋊C4—31no—
30#1779—31no—
31C2×#328—31no—
32C2×#328D311yescharacteristic, complemented
33#1294—31no—
34#1294—31no—
35C24×D3D411yescharacteristic
36C22×C3×Q8C2×C411yescharacteristic
37C23×C3×C4—21no—
38C23×C3×C4C2311yescharacteristic
39C23×Dic3—21no—
40C23×Dic3D411yescharacteristic
41C22×C4×D3—21no—
42C22×C4×D3—21no—
43C22×C4×D3—21no—
44C22×C4×D3—21no—
45C22×C4×D3—44no—
46C22×C4×D3C2×C412yes—
47C22×C4×D3D412yes—
48C22×C3×C8—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C3×C22⋊Q8—21no—
52C3×C22⋊Q8—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4C2×C411yescharacteristic
55C2×C3×C22⋊C4—21no—
56C2×#217—21no—
57#946—44no—
58C2×(C2×D3)⋊C4—24no—
59C2×(C2×D3)⋊C4—28no—
60C2×(C2×D3)⋊C4—22no—
61C2×(C2×D3)⋊C4—22no—
62C2×C4⋊Dic3C2×C412yes—
63C2×Dic3⋊C4—22no—
64C2×C4×Dic3—24no—
65C2×C4×Dic3—44no—
66C22×C3⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×#95—21no—
70C3×#95D412yes—
71C3⋊#95—21no—
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—32no—
80C2×C22⋊Q8C2×D311yescharacteristic
81C22×C22⋊C4—61no—
82C22×C42—61no—
83C2×C22⋊C8—31no—
84C2×C22⋊C8—62no—
85C2×C22⋊C8C2×D311yescharacteristic
86#385—61no—
87#385—62no—
88#380—31no—
89#380—32no—
90#379—61no—
91C4×C22⋊C4—62no—
92C4×C22⋊C4—62no—
93C2×#95—31no—
94C2×#95C2×D311yescharacteristic
95#328—22no—
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—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—42no—
110C22×C3×C4—24no—
111C22×C3×C4—22no—
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—42no—
115C22×C3×C4—42no—
116C22×C3×C4—41no—
117C22×C3×C4—41no—
118C22×C3×C4C22×C411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4#4111yescharacteristic
123C22×C3×C4#4111yescharacteristic
124C22×Dic3—21no—
125C22×Dic3—21no—
126C22×Dic3—24no—
127C22×Dic3—24no—
128C22×Dic3—22no—
129C22×Dic3—22no—
130C22×Dic3—42no—
131C22×Dic3—41no—
132C22×Dic3—41no—
133C22×Dic3C22⋊C412yes—
134C2×C4×D3—22no—
135C2×C4×D3—22no—
136C2×C4×D3—22no—
137C2×C4×D3—22no—
138C2×C4×D3—24no—
139C2×C4×D3—24no—
140C2×C4×D3—42no—
141C2×C4×D3—42no—
142C2×C4×D3—44no—
143C2×C4×D3—44no—
144C2×C4×D3—44no—
145C2×C4×D3—44no—
146C2×C3×C8—42no—
147C2×C3×C8—41no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—41no—
151C3×C4⋊C4—41no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—41no—
154#217—41no—
155#217—41no—
156(C2×D3)⋊C4—216no—
157(C2×D3)⋊C4—44no—
158(C2×D3)⋊C4—44no—
159C4⋊Dic3—42no—
160Dic3⋊C4—44no—
161C4×Dic3—48no—
162C2×C3⋊C8—42no—
163C2×C3⋊C8—41no—
164C25—31no—
165C22×Q8C4×D311yescharacteristic
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—21no—
180C22⋊Q8—21no—
181C22⋊Q8—21no—
182C22⋊Q8—21no—
183C22⋊Q8—124no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—32no—
186C2×C4⋊C4—62no—
187C2×C4⋊C4C4×D311yescharacteristic
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—32no—
203#95—62no—
204#95—61no—
205#95—61no—
206#95#6712yes—
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—
235C3×Q8—81no—
236C3×Q8—81no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—24no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—42no—
245C2×C3×C4—44no—
246C2×C3×C4—44no—
247C2×C3×C4—42no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4C4×D411yescharacteristic
253C2×C3×C4C4×D411yescharacteristic
254C2×C3×C4#12111yescharacteristic
255C2×C3×C4#12111yescharacteristic
256C2×Dic3—24no—
257C2×Dic3—24no—
258C2×Dic3—22no—
259C2×Dic3—22no—
260C2×Dic3—24no—
261C2×Dic3—42no—
262C2×Dic3—44no—
263C2×Dic3—44no—
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3—42no—
267C2×Dic3—42no—
268C2×Dic3#9914yes—
269C4×D3—44no—
270C4×D3—44no—
271C4×D3—44no—
272C4×D3—44no—
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—21no—
286C2×Q8—21no—
287C2×Q8—42no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—32no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—64no—
298C22×C4—64no—
299C22×C4—64no—
300C22×C4—61no—
301C22×C4—61no—
302C22×C4—64no—
303C22×C4—62no—
304C22×C4—62no—
305C22×C4—62no—
306C22×C4—62no—
307C22×C4—62no—
308C22×C4—62no—
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×C4#23511yescharacteristic
320C22×C4D3×D411yescharacteristic
321C22×C4D3×D411yescharacteristic
322C22×C4#23711yescharacteristic
323C22×C4C2×#6711yescharacteristic
324C2×C8—42no—
325C2×C8—41no—
326C2×C8—122no—
327C2×C8—121no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—122no—
333C4⋊C4—124no—
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—81no—
376C3×C4—81no—
377Dic3—24no—
378Dic3—44no—
379Dic3—82no—
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#88311yescharacteristic
403C23#91511yescharacteristic
404C23#91611yescharacteristic
405C23#91811yescharacteristic
406C23#91911yescharacteristic
407C23#93711yescharacteristic
408Q8—81no—
409Q8—81no—
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—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—64no—
427C2×C4—64no—
428C2×C4—62no—
429C2×C4—62no—
430C2×C4—64no—
431C2×C4—124no—
432C2×C4—124no—
433C2×C4—124no—
434C2×C4—124no—
435C2×C4—122no—
436C2×C4—124no—
437C2×C4—124no—
438C2×C4—122no—
439C2×C4—122no—
440C2×C4—122no—
441C2×C4—122no—
442C2×C4#67⋊C411yescharacteristic
443C2×C4#88411yescharacteristic
444C2×C4C4×#6711yescharacteristic
445C2×C4#93811yescharacteristic
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—22no—
461C22—22no—
462C22—21no—
463C22—21no—
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—
477C22D3×#9912yes—
478C22#480211yescharacteristic
479C22#480411yescharacteristic
480C22#481411yescharacteristic
481C22#481511yescharacteristic
482C22D3×#10412yes—
483C22#495611yescharacteristic
484C22#517311yescharacteristic
485C22#518711yescharacteristic
486C4—21no—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—44no—
491C4—64no—
492C4—81no—
493C4—81no—
494C4—124no—
495C4—242no—
496C3#914611yescharacteristic, complemented, Sylow
497C2—21no—
498C2—21no—
499C2—21no—
500C2—21no—
501C2—124no—
502C2#6833312yes—
503C2#6833512yes—
504C2#7003611yescharacteristic
505C2#7026911yescharacteristic
506C2#7038211yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#328C2
socleC23×C3C2×C22⋊C4
prime radicalC24×D3D4
cosocleC23×C3×C4C23
hypercenterC2×#328D3
hypocenterC3#9146
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
3. C2×C3×#328C2semidirect
4. C2×#4538C2semidirect
6. #67675C2semidirect
7. #67612C2semidirect
11. C2×#946C4semidirect
21. #4987C4semidirect
32. C2×#328D3semidirect
496. C3#9146semidirect

Maximal subgroups

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

subgroupindex
#828342
C2×#46662
C2×C3×#3282
C2×#45382
#699552
#676752
#676122
#91463