Finite Group Databasebeta

← #10691494

Subgroups — #10691494

Order 768.

686 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2880C211yesmaximal, characteristic
2C2×#5239C211yesmaximal, characteristic
3#72138C211yesmaximal, characteristic
4#69980C211yesmaximal, characteristic, complemented
5#69923C211yesmaximal, characteristic, complemented
6#67687C211yesmaximal, characteristic, complemented
7#67678C211yesmaximal, characteristic, complemented
8#9239—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×C3×#131—21no—
11C22×#216C411yescharacteristic, complemented
12C22×#214C411yescharacteristic, complemented
13C2×#924C2211yescharacteristic
14C3×#408—21no—
15C3×#408—22no—
16C3×#407—21no—
17C3×#407C2211yescharacteristic
18C2×C3×C22⋊C8—21no—
19C2×C3×C22⋊C8C2211yescharacteristic
20#5239—21no—
21#5239—21no—
22#5239—21no—
23#5239—21no—
24C2×#837—21no—
25C2×#837—21no—
26C2×#835C411yescharacteristic, complemented
27C2×#832C411yescharacteristic, complemented
28C4×#217C2211yescharacteristic
29#5207C2211yescharacteristic
30C2×#830C2211yescharacteristic
31#4469—21no—
32#4469—21no—
33#4467—22no—
34C2×#452—31no—
35#2880D311yescharacteristic
36#1789—31no—
37#1508—31no—
38#1472—31no—
39#1293—31no—
40#1282—31no—
41C2×C3×#41—21no—
42C2×C3×#41—21no—
43C2×C3×#41—42no—
44C2×C3×#41—42no—
45C2×C3×#41D411yescharacteristic
46C2×C3×#41D411yescharacteristic
47C2×C3×#41D411yescharacteristic
48C2×C3×#41D411yescharacteristic
49C22×C3×Q8C2×C411yescharacteristic
50C22×C3×D4—21no—
51C22×C3×D4C2×C411yescharacteristic
52C23×C3×C4—21no—
53C23×C3×C4C2311yescharacteristic
54C22×Dic6C2×C411yescharacteristic
55C3×#131—42no—
56C3×#131—42no—
57C2×C3×M16—22no—
58C2×C3×M16—21no—
59C22×C3×C8—22no—
60C22×C3×C8—21no—
61C2×#217—21no—
62C2×#217—21no—
63C2×#216—21no—
64C2×#216—21no—
65C2×#216—42no—
66C2×#214—21no—
67C2×#214—21no—
68C2×#214—42no—
69C2×C4⋊Dic3C2×C411yescharacteristic
70#924—21no—
71#924—21no—
72#924—21no—
73#924—21no—
74C2×Dic3⋊C4—21no—
75C2×C4×Dic3—21no—
76C2×#208—21no—
77C22×C3⋊C8C2×C411yescharacteristic
78C22×C3⋊C8C2×C411yescharacteristic
79C3×C22⋊C8—22no—
80C3×C22⋊C8—22no—
81C3×C22⋊C8—21no—
82C3×C22⋊C8—21no—
83C3×C22⋊C8D412yes—
84C3×C22⋊C8D411yescharacteristic
85C3×C22⋊C8D411yescharacteristic
86#837—42no—
87#837—42no—
88#835—21no—
89#835—21no—
90#832—21no—
91#832—21no—
92C3⋊#95—21no—
93#830—21no—
94#830—21no—
95#830—21no—
96#830—21no—
97C22×#41C2×D311yescharacteristic
98C22×Q16—31no—
99C22×SD16—31no—
100C2×#131—21no—
101C2×C22⋊Q8—31no—
102#452—61no—
103#452—61no—
104#452—61no—
105#452—61no—
106C2×#104—61no—
107C2×#104—61no—
108C2×#103—31no—
109C2×#102—31no—
110#408—21no—
111#408—22no—
112#407—21no—
113#407—31no—
114#407C2×D311yescharacteristic
115C2×C22⋊C8—21no—
116C2×C22⋊C8—31no—
117C2×C22⋊C8C2×D311yescharacteristic
118C4×C22⋊C4—31no—
119#328—62no—
120#324—61no—
121#324—61no—
122C24×C3—21no—
123C24×C3#4111yescharacteristic
124C3×#41—22no—
125C3×#41—22no—
126C3×#41—22no—
127C3×#41—22no—
128C3×#41—41no—
129C3×#41—41no—
130C3×#41—41no—
131C3×#41—41no—
132C3×#41—44no—
133C3×#41—44no—
134C2×C3×Q8—21no—
135C2×C3×Q8—22no—
136C2×C3×Q8—41no—
137C2×C3×Q8C22⋊C411yescharacteristic
138C2×C3×Q8C22⋊C411yescharacteristic
139C2×C3×D4—21no—
140C2×C3×D4—22no—
141C2×C3×D4—21no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4—21no—
145C2×C3×D4—41no—
146C2×C3×D4—44no—
147C2×C3×D4C22⋊C411yescharacteristic
148C2×C3×D4C22⋊C411yescharacteristic
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—41no—
155C22×C3×C4—41no—
156C22×C3×C4—44no—
157C22×C3×C4C22×C411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4#4111yescharacteristic
163C22×Dic3—21no—
164C22×Dic3—21no—
165C22×Dic3—41no—
166C2×Dic6—21no—
167C2×Dic6—21no—
168C2×Dic6—41no—
169C2×Dic6—41no—
170C3×M16—44no—
171C3×M16—41no—
172C3×M16—41no—
173C2×C3×C8—21no—
174C2×C3×C8—22no—
175C2×C3×C8—21no—
176C2×C3×C8—44no—
177C2×C3×C8—41no—
178C2×C3×C8—41no—
179#217—22no—
180#217—41no—
181#217—41no—
182#217—41no—
183#216—42no—
184#216—42no—
185#214—42no—
186#214—42no—
187C4⋊Dic3—21no—
188C4⋊Dic3—21no—
189Dic3⋊C4—41no—
190Dic3⋊C4—41no—
191C4×Dic3—41no—
192C4×Dic3—41no—
193#208—41no—
194#208—41no—
195C2×C3⋊C8—21no—
196C2×C3⋊C8—21no—
197C2×C3⋊C8—41no—
198C2×C3⋊C8—41no—
199C2×C3⋊C8—41no—
200C2×#41—21no—
201C2×#41—21no—
202C2×#41—42no—
203C2×#41—42no—
204C2×#41D1211yescharacteristic
205C2×#41D1211yescharacteristic
206C2×#41#6711yescharacteristic
207C2×#41#6711yescharacteristic
208C22×Q8—31no—
209C22×Q8C4×D311yescharacteristic
210C22×D4—21no—
211C22×D4C4×D311yescharacteristic
212C23×C4—21no—
213C23×C4C22×D311yescharacteristic
214C2×Q16—61no—
215C2×Q16—61no—
216C2×Q16—122no—
217C2×SD16—61no—
218C2×SD16—61no—
219C2×SD16—122no—
220#131—42no—
221#131—42no—
222C2×M16—22no—
223C2×M16—21no—
224C2×M16—61no—
225C22×C8—22no—
226C22×C8—21no—
227C22×C8—31no—
228C22×C8—31no—
229C22⋊Q8—61no—
230C22⋊Q8—61no—
231C22⋊Q8—61no—
232C22⋊Q8—61no—
233C2×C4⋊C4—31no—
234C2×C4⋊C4—61no—
235C2×C22⋊C4—61no—
236C2×C22⋊C4—61no—
237C2×C42—61no—
238#104—122no—
239#104—122no—
240#103—61no—
241#103—61no—
242#102—61no—
243#102—61no—
244C22⋊C8—22no—
245C22⋊C8—22no—
246C22⋊C8—21no—
247C22⋊C8—21no—
248C22⋊C8—61no—
249C22⋊C8—61no—
250C22⋊C8—61no—
251C22⋊C8—61no—
252C22⋊C8#6711yescharacteristic
253C22⋊C8#6711yescharacteristic
254C22⋊C8#6712yes—
255#95—61no—
256C23×C3—21no—
257C23×C3—21no—
258C23×C3—21no—
259C23×C3—21no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3—42no—
263C23×C3—41no—
264C23×C3—41no—
265C23×C3—81no—
266C23×C3C4×D411yescharacteristic
267C23×C3#12111yescharacteristic
268C23×C3#12311yescharacteristic
269C3×Q8—21no—
270C3×Q8—21no—
271C3×Q8—21no—
272C3×Q8—42no—
273C3×Q8#9912yes—
274C3×D4—24no—
275C3×D4—21no—
276C3×D4—21no—
277C3×D4—21no—
278C3×D4—42no—
279C3×D4—42no—
280C3×D4—42no—
281C3×D4—42no—
282C3×D4#9912yes—
283C2×C3×C4—24no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4C2×C22⋊C411yescharacteristic
300C2×C3×C4C4×D411yescharacteristic
301C2×C3×C4#12011yescharacteristic
302C2×C3×C4#12111yescharacteristic
303C2×Dic3—42no—
304C2×Dic3—41no—
305C2×Dic3—41no—
306C2×Dic3—41no—
307C2×Dic3—41no—
308C2×Dic3—81no—
309C2×Dic3—81no—
310C2×Dic3—81no—
311Dic6—41no—
312Dic6—41no—
313Dic6—81no—
314C3×C8—41no—
315C3×C8—42no—
316C3×C8—41no—
317C3⋊C8—41no—
318C3⋊C8—41no—
319C3⋊C8—81no—
320C24—21no—
321C24#23511yescharacteristic
322#41—22no—
323#41—22no—
324#41—22no—
325#41—22no—
326#41—41no—
327#41—41no—
328#41—41no—
329#41—41no—
330#41—44no—
331#41—44no—
332C2×Q8—21no—
333C2×Q8—22no—
334C2×Q8—41no—
335C2×Q8—61no—
336C2×Q8—61no—
337C2×Q8—121no—
338C2×Q8—121no—
339C2×Q8(C2×D3)⋊C411yescharacteristic
340C2×Q8(C2×D3)⋊C411yescharacteristic
341C2×D4—21no—
342C2×D4—22no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—41no—
348C2×D4—44no—
349C2×D4(C2×D3)⋊C411yescharacteristic
350C2×D4(C2×D3)⋊C411yescharacteristic
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—44no—
359C22×C4—61no—
360C22×C4—61no—
361C22×C4—121no—
362C22×C4C2×C4×D311yescharacteristic
363C22×C4C2×D1211yescharacteristic
364C22×C4#23511yescharacteristic
365C22×C4C2×#6711yescharacteristic
366C22×C4C2×#6711yescharacteristic
367C22×C4C2×#6711yescharacteristic
368Q16—122no—
369Q16—122no—
370SD16—122no—
371SD16—122no—
372M16—44no—
373M16—41no—
374M16—41no—
375M16—121no—
376M16—121no—
377C2×C8—21no—
378C2×C8—22no—
379C2×C8—21no—
380C2×C8—44no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—61no—
384C2×C8—61no—
385C2×C8—121no—
386C2×C8—121no—
387C2×C8—121no—
388C4⋊C4—61no—
389C4⋊C4—61no—
390C4⋊C4—121no—
391C4⋊C4—121no—
392C22⋊C4—62no—
393C22⋊C4—121no—
394C22⋊C4—121no—
395C22⋊C4—121no—
396C42—121no—
397C42—121no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—22no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—41no—
408C22×C3—41no—
409C22×C3—41no—
410C22×C3—41no—
411C22×C3—41no—
412C22×C3—81no—
413C22×C3—81no—
414C22×C3—82no—
415C22×C3—82no—
416C22×C3#38611yescharacteristic
417C22×C3#44311yescharacteristic
418C22×C3#44411yescharacteristic
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—22no—
423C3×C4—41no—
424C3×C4C2×#9911yescharacteristic
425C3×C4C2×#9911yescharacteristic
426Dic3—82no—
427Dic3—81no—
428Dic3—81no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—41no—
434C23—41no—
435C23—42no—
436C23—41no—
437C23—41no—
438C23—81no—
439C23C4×#6711yescharacteristic
440C23#92911yescharacteristic
441C23#93011yescharacteristic
442Q8—21no—
443Q8—21no—
444Q8—21no—
445Q8—42no—
446Q8—121no—
447Q8—121no—
448Q8—241no—
449Q8#80612yes—
450D4—24no—
451D4—21no—
452D4—21no—
453D4—21no—
454D4—42no—
455D4—42no—
456D4—42no—
457D4—42no—
458D4#80612yes—
459C2×C4—24no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—122no—
476C2×C4—121no—
477C2×C4—121no—
478C2×C4—121no—
479C2×C4—121no—
480C2×C4—241no—
481C2×C4—241no—
482C2×C4—241no—
483C2×C4C2×(C2×D3)⋊C411yescharacteristic
484C2×C4C4×#6711yescharacteristic
485C2×C4#93011yescharacteristic
486C2×C4#95311yescharacteristic
487C8—41no—
488C8—42no—
489C8—41no—
490C8—121no—
491C8—121no—
492C8—241no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—41no—
496C2×C3—41no—
497C2×C3—42no—
498C2×C3—81no—
499C2×C3#187611yescharacteristic
500C2×C3#189811yescharacteristic
501C2×C3#190211yescharacteristic
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—22no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—81no—
517C22—81no—
518C22—82no—
519C22—82no—
520C22#514411yescharacteristic
521C22#514511yescharacteristic
522C22#521311yescharacteristic
523C4—21no—
524C4—21no—
525C4—22no—
526C4—21no—
527C4—41no—
528C4—242no—
529C4—241no—
530C4—241no—
531C4C2×#80611yescharacteristic
532C4C2×#80611yescharacteristic
533C3#923911yescharacteristic, complemented, Sylow
534C2—21no—
535C2—21no—
536C2—41no—
537C2—41no—
538C2—42no—
539C2—81no—
540C2#7154311yescharacteristic
541C2#7201011yescharacteristic
542C2#7202211yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2880C2
socleC22×C3#386
prime radicalC22×C3×#41C22
cosocleC23×C3×C4C23
hypercenter#2880D3
hypocenterC3#9239
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2880

Complements

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

NH ≅ G/Ntype
4. #69980C2semidirect
5. #69923C2semidirect
6. #67687C2semidirect
7. #67678C2semidirect
11. C22×#216C4semidirect
12. C22×#214C4semidirect
26. C2×#835C4semidirect
27. C2×#832C4semidirect
533. C3#9239semidirect

Maximal subgroups

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

subgroupindex
C3×#28802
C2×#52392
#721382
#699802
#699232
#676872
#676782
#92393