Finite Group Databasebeta

← #10692233

Subgroups — #10692233

Order 768.

678 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 678 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82735C211yesmaximal, characteristic
2#78600C211yesmaximal, characteristic
3C3×#1520C211yesmaximal, characteristic, complemented
4#71691C211yesmaximal, characteristic, complemented
5#69522C211yesmaximal, characteristic
6#67680C211yesmaximal, characteristic, complemented
7#67626C211yesmaximal, characteristic, complemented
8#9345—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C2×#942C411yescharacteristic, complemented
12C2×#942C411yescharacteristic, complemented
13C2×#901—21no—
14C2×C3×C22⋊C8C2211yescharacteristic
15C3×#380C2211yescharacteristic
16#5207C2211yescharacteristic
17#5207C2211yescharacteristic
18C2×#830C2211yescharacteristic
19#5110—21no—
20#5050—21no—
21#5047—21no—
22#5047—21no—
23#5046—21no—
24C2×#810—21no—
25#4966C411yescharacteristic, complemented
26#4966C411yescharacteristic, complemented
27C2×#809—21no—
28#4727—22no—
29#4726—21no—
30D3×C22⋊C8—21no—
31C3×#328—21no—
32C3×#328—21no—
33#4528—22no—
34#3007—31no—
35#2880—31no—
36#1788—31no—
37#1520D311yescharacteristic, complemented
38#1472—31no—
39#1305—31no—
40#1298—31no—
41C22×C3×Q8C2×C411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C22×#67—21no—
44C2×#235—21no—
45C2×#235—21no—
46C2×#235—42no—
47C2×#235—42no—
48C2×#235C2×C411yescharacteristic
49C2×#235C2×C411yescharacteristic
50C2×#235D411yescharacteristic
51C2×#235D411yescharacteristic
52C22×D12D411yescharacteristic
53C22×C4×D3—21no—
54C22×Dic6D411yescharacteristic
55C22×C3×C8—21no—
56C3×C22⋊Q8—21no—
57C3×C22⋊Q8—21no—
58C3×C22⋊Q8—21no—
59C3×C22⋊Q8—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4C2×C411yescharacteristic
63C2×C3×C22⋊C4—21no—
64C2×C3×C22⋊C4—21no—
65C2×#216—21no—
66C2×#216—21no—
67#942—42no—
68#942—42no—
69C2×#215—21no—
70C2×#215—21no—
71C2×#208—21no—
72C2×#208C2×C411yescharacteristic
73C2×#208C2×C411yescharacteristic
74C22×C3⋊C8—21no—
75#901—42no—
76#901—42no—
77C2×ΓL(1,𝔽25)—22no—
78C2×D3×C8—22no—
79C3×C22⋊C8—21no—
80C3×C22⋊C8D411yescharacteristic
81C3×C22⋊C8D411yescharacteristic
82C3×#95—21no—
83#830—21no—
84#830—21no—
85#830—21no—
86#830—21no—
87#830—21no—
88#830D412yes—
89(C2×D3)⋊C8—22no—
90(C2×D3)⋊C8—22no—
91#810—21no—
92#810—21no—
93#810—42no—
94#809—21no—
95#809—21no—
96#809—42no—
97C22×#41—31no—
98C2×#137—31no—
99C2×#137—31no—
100C2×#131—61no—
101C2×C22⋊Q8C2×D311yescharacteristic
102#452—61no—
103#452—61no—
104#451—61no—
105#450—61no—
106#417—31no—
107#417—31no—
108C2×#103—61no—
109C2×#102—61no—
110#408—62no—
111#408—61no—
112#407—31no—
113#407—31no—
114#407—61no—
115C2×C22⋊C8—31no—
116C2×C22⋊C8—61no—
117C2×C22⋊C8C2×D311yescharacteristic
118#380C2×D311yescharacteristic
119#328—21no—
120#328—21no—
121#324—62no—
122C24×C3#4111yescharacteristic
123C23×D3—21no—
124C2×C3×Q8—21no—
125C2×C3×Q8—21no—
126C2×C3×Q8—41no—
127C2×C3×Q8—41no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—41no—
131C22×C3×C4—41no—
132C22×C3×C4—41no—
133C22×C3×C4C22×C411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4#4111yescharacteristic
139C2×#67—21no—
140C2×#67—21no—
141C2×#67—21no—
142C2×#67—21no—
143C2×#67—44no—
144C22×Dic3—21no—
145#235—22no—
146#235—22no—
147#235—22no—
148#235—22no—
149#235—41no—
150#235—41no—
151#235—41no—
152#235—41no—
153#235—44no—
154#235—44no—
155C2×D12—22no—
156C2×D12—21no—
157C2×D12—41no—
158C2×D12C22⋊C411yescharacteristic
159C2×D12C22⋊C411yescharacteristic
160C2×C4×D3—21no—
161C2×C4×D3—21no—
162C2×C4×D3—21no—
163C2×C4×D3—21no—
164C2×C4×D3—44no—
165C2×Dic6—22no—
166C2×Dic6—21no—
167C2×Dic6—41no—
168C2×Dic6C22⋊C411yescharacteristic
169C2×Dic6C22⋊C411yescharacteristic
170C2×C3×C8—21no—
171C2×C3×C8—21no—
172C2×C3×C8—41no—
173C2×C3×C8—41no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—41no—
177C3×C4⋊C4—41no—
178C3×C4⋊C4—41no—
179C3×C4⋊C4—41no—
180C3×C22⋊C4—41no—
181C3×C22⋊C4—41no—
182C3×C22⋊C4—81no—
183#216—42no—
184#216—42no—
185#215—42no—
186#215—42no—
187#208—41no—
188#208—41no—
189#208—41no—
190#208—41no—
191C2×C3⋊C8—21no—
192C2×C3⋊C8—21no—
193C2×C3⋊C8—22no—
194C2×C3⋊C8—41no—
195C2×C3⋊C8—41no—
196C2×C3⋊C8—41no—
197ΓL(1,𝔽25)—44no—
198D3×C8—44no—
199C2×#41—31no—
200C2×#41—31no—
201C2×#41—31no—
202C2×#41—31no—
203C2×#41—61no—
204C2×#41—61no—
205C2×#41—122no—
206C2×#41—122no—
207C22×Q8—31no—
208C22×Q8C4×D311yescharacteristic
209C22×D4—31no—
210C22×D4—61no—
211C23×C4—61no—
212C23×C4C22×D311yescharacteristic
213#137—122no—
214#137—122no—
215C2×Q16—61no—
216C2×Q16—61no—
217C2×SD16—61no—
218C2×SD16—61no—
219#131—122no—
220#131—122no—
221C2×M16—31no—
222C2×M16—31no—
223C2×M16—62no—
224C2×M16—61no—
225C22×C8—21no—
226C22×C8—62no—
227C22×C8—61no—
228C22⋊Q8—21no—
229C22⋊Q8—21no—
230C22⋊Q8—21no—
231C22⋊Q8—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4C4×D311yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237#103—61no—
238#103—61no—
239#103—122no—
240#102—61no—
241#102—61no—
242#102—122no—
243C22⋊C8—21no—
244C22⋊C8—32no—
245C22⋊C8—62no—
246C22⋊C8—62no—
247C22⋊C8—61no—
248C22⋊C8—61no—
249C22⋊C8—61no—
250C22⋊C8—61no—
251C22⋊C8—61no—
252C22⋊C8#6711yescharacteristic
253C22⋊C8#6711yescharacteristic
254#95—21no—
255C23×C3—21no—
256C23×C3—21no—
257C23×C3—41no—
258C23×C3—41no—
259C23×C3C4×D411yescharacteristic
260C23×C3#12111yescharacteristic
261C23×C3#12311yescharacteristic
262C22×D3—21no—
263C22×D3—21no—
264C22×D3—42no—
265C22×D3—41no—
266C22×D3—41no—
267C22×D3—81no—
268C3×Q8—41no—
269C3×Q8—41no—
270C3×Q8—81no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—42no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—81no—
284C2×C3×C4—81no—
285C2×C3×C4—81no—
286C2×C3×C4—81no—
287C2×C3×C4C2×C22⋊C411yescharacteristic
288C2×C3×C4C4×D411yescharacteristic
289C2×C3×C4#12011yescharacteristic
290C2×C3×C4#12111yescharacteristic
291#67—24no—
292#67—42no—
293#67—42no—
294#67—42no—
295C2×Dic3—21no—
296C2×Dic3—21no—
297C2×Dic3—42no—
298D12—21no—
299D12—21no—
300D12—21no—
301D12—42no—
302D12#9912yes—
303C4×D3—24no—
304C4×D3—42no—
305C4×D3—42no—
306C4×D3—42no—
307Dic6—21no—
308Dic6—21no—
309Dic6—21no—
310Dic6—42no—
311Dic6#9912yes—
312C3×C8—41no—
313C3×C8—41no—
314C3⋊C8—42no—
315C3⋊C8—41no—
316C3⋊C8—41no—
317C3⋊C8—81no—
318C24—61no—
319C24#23711yescharacteristic
320#41—62no—
321#41—62no—
322#41—62no—
323#41—62no—
324#41—121no—
325#41—121no—
326#41—121no—
327#41—121no—
328#41—124no—
329#41—124no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—31no—
333C2×Q8—31no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—62no—
337C2×Q8—61no—
338C2×Q8—121no—
339C2×D4—31no—
340C2×D4—31no—
341C2×D4—62no—
342C2×D4—61no—
343C2×D4—61no—
344C2×D4—61no—
345C2×D4—61no—
346C2×D4—61no—
347C2×D4—121no—
348C2×D4—124no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—61no—
355C22×C4—61no—
356C22×C4—61no—
357C22×C4—61no—
358C22×C4—61no—
359C22×C4—124no—
360C22×C4C2×C4×D311yescharacteristic
361C22×C4#23511yescharacteristic
362C22×C4D3×D411yescharacteristic
363C22×C4D3×D411yescharacteristic
364C22×C4D3×D411yescharacteristic
365C22×C4C2×#6711yescharacteristic
366Q16—122no—
367Q16—122no—
368SD16—122no—
369SD16—122no—
370M16—124no—
371M16—121no—
372M16—121no—
373M16—121no—
374M16—121no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—61no—
380C2×C8—61no—
381C2×C8—62no—
382C2×C8—124no—
383C2×C8—121no—
384C2×C8—121no—
385C2×C8—121no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—81no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—81no—
404C22×C3—81no—
405C22×C3#38611yescharacteristic
406C22×C3#47111yescharacteristic
407C22×C3#47311yescharacteristic
408C2×D3—22no—
409C2×D3—41no—
410C2×D3—41no—
411C2×D3—41no—
412C2×D3—41no—
413C2×D3—41no—
414C2×D3—82no—
415C2×D3—82no—
416C3×C4—21no—
417C3×C4—21no—
418C3×C4—21no—
419C3×C4—81no—
420C3×C4—81no—
421C3×C4—161no—
422C3×C4C2×#9911yescharacteristic
423C3×C4C2×#9911yescharacteristic
424Dic3—22no—
425Dic3—41no—
426C23—21no—
427C23—21no—
428C23—41no—
429C23—41no—
430C23—61no—
431C23—61no—
432C23—122no—
433C23—121no—
434C23—121no—
435C23—241no—
436C23#67⋊C411yescharacteristic
437C23#88311yescharacteristic
438C23#93811yescharacteristic
439Q8—32no—
440Q8—41no—
441Q8—41no—
442Q8—61no—
443Q8—61no—
444Q8—61no—
445Q8—81no—
446Q8—122no—
447D4—32no—
448D4—64no—
449D4—61no—
450D4—61no—
451D4—61no—
452D4—122no—
453D4—122no—
454D4—122no—
455D4—122no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—64no—
469C2×C4—61no—
470C2×C4—61no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—122no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4—122no—
479C2×C4D3×C22⋊C411yescharacteristic
480C2×C4#88411yescharacteristic
481C2×C4C4×#6711yescharacteristic
482C2×C4#93711yescharacteristic
483C8—41no—
484C8—41no—
485C8—122no—
486C8—121no—
487C8—121no—
488C8—241no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—41no—
492C2×C3—41no—
493C2×C3#188111yescharacteristic
494C2×C3#189811yescharacteristic
495C2×C3#190811yescharacteristic
496D3—42no—
497D3—81no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—62no—
507C22—81no—
508C22—81no—
509C22—121no—
510C22—121no—
511C22—121no—
512C22—121no—
513C22—121no—
514C22—242no—
515C22—242no—
516C22#495611yescharacteristic
517C22#517711yescharacteristic
518C22#519211yescharacteristic
519C4—21no—
520C4—21no—
521C4—21no—
522C4—62no—
523C4—81no—
524C4—81no—
525C4—121no—
526C4—161no—
527C4D3×#9911yescharacteristic
528C4D3×#9911yescharacteristic
529C3#934511yescharacteristic, complemented, Sylow
530C2—21no—
531C2—21no—
532C2—41no—
533C2—41no—
534C2—122no—
535C2—241no—
536C2#7003611yescharacteristic
537C2#7028611yescharacteristic
538C2#7041611yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1520C2
socleC22×C3#386
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenter#1520D3
hypocenterC3#9345
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1520

Complements

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

NH ≅ G/Ntype
3. C3×#1520C2semidirect
4. #71691C2semidirect
6. #67680C2semidirect
7. #67626C2semidirect
11. C2×#942C4semidirect
12. C2×#942C4semidirect
25. #4966C4semidirect
26. #4966C4semidirect
37. #1520D3semidirect
529. C3#9345semidirect

Maximal subgroups

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

subgroupindex
#827352
#786002
C3×#15202
#716912
#695222
#676802
#676262
#93453