Finite Group Databasebeta

← #10695559

Subgroups — #10695559

Order 768.

824 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 83 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#84090C211yesmaximal, characteristic, complemented
2#84082C211yesmaximal, characteristic
3#73550C211yesmaximal, characteristic, complemented
4C3×#1789C211yesmaximal, characteristic, complemented
5#69978C211yesmaximal, characteristic, complemented
6#69971C211yesmaximal, characteristic, complemented
7#67616C211yesmaximal, characteristic, complemented
8#9939—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×#949—21no—
11C2×#935—21no—
12C2×#930C2211yescharacteristic
13C22×Dic3⋊C4C2211yescharacteristic
14C2×C3×#104—22no—
15C3×#407C2211yescharacteristic
16C3×C4×C22⋊C4C2211yescharacteristic
17#5238—22no—
18#5232—21no—
19#5230—21no—
20#5227—21no—
21#5222—21no—
22#5220—21no—
23#5217—21no—
24#5211C2211yescharacteristic
25#5210—22no—
26#5207C2211yescharacteristic
27#4958—21no—
28#4958—21no—
29#4940—22no—
30#4672—21no—
31#4659—21no—
32#4470—22no—
33#4469—22no—
34#3006—31no—
35#2591—31no—
36#2406—31no—
37#1789D311yescharacteristic, complemented
38#1510—31no—
39#1463—31no—
40#1284—31no—
41C2×C3×#41—21no—
42C2×C3×#41—21no—
43C2×C3×#41—21no—
44C2×C3×#41—21no—
45C2×C3×#41—44no—
46C22×C3×Q8D411yescharacteristic
47C22×C3×D4—21no—
48C22×C3×D4D411yescharacteristic
49C23×C3×C4—21no—
50C23×C3×C4C2311yescharacteristic
51C22×#67—21no—
52C23×Dic3D411yescharacteristic
53C23×Dic3D411yescharacteristic
54C22×D12D411yescharacteristic
55C2×C3×M16—21no—
56C2×C3×C22⋊C4—21no—
57C2×C3×C42D412yes—
58C2×#217—22no—
59C2×#217—22no—
60C2×#217—21no—
61#949—44no—
62C2×#215—22no—
63#935—44no—
64C2×#213—22no—
65#930—21no—
66#930—21no—
67#930—21no—
68#930—21no—
69C2×(C2×D3)⋊C4—22no—
70C2×(C2×D3)⋊C4—21no—
71C2×(C2×D3)⋊C4—22no—
72C2×C4⋊Dic3D411yescharacteristic
73C2×Dic3⋊C4—22no—
74C2×Dic3⋊C4—21no—
75C2×Dic3⋊C4—21no—
76C2×Dic3⋊C4—21no—
77C2×Dic3⋊C4—21no—
78C2×Dic3⋊C4—21no—
79C2×Dic3⋊C4—21no—
80C2×#208—21no—
81C3×#104—44no—
82C3×C22⋊C8—21no—
83C3×C22⋊C8—21no—
84C3×#95—21no—
85#835—22no—
86#832—22no—
87C3⋊#95—22no—
88C3⋊#95—22no—
89C3⋊#95—22no—
90C3⋊#95—21no—
91C3⋊#95D412yes—
92#830—21no—
93#830D412yes—
94C22×#41C2×D311yescharacteristic
95C2×AGL(1,ℤ/8)—61no—
96C2×#123—61no—
97C2×#121—31no—
98C22×C4⋊C4—31no—
99#449—62no—
100#448—61no—
101#447—61no—
102C2×#104—22no—
103#417—61no—
104#407—31no—
105#407C2×D311yescharacteristic
106#400—61no—
107#395—61no—
108#394—61no—
109#394—61no—
110#388—62no—
111#386—61no—
112#386—61no—
113#386—61no—
114#385—62no—
115#381—31no—
116C4×C22⋊C4C2×D311yescharacteristic
117#327—62no—
118#324—62no—
119C24×C3—21no—
120C24×C3C2×D411yescharacteristic
121C23×D3—21no—
122C3×#41—24no—
123C3×#41—24no—
124C3×#41—42no—
125C3×#41—42no—
126C3×#41—42no—
127C3×#41—42no—
128C3×#41—42no—
129C3×#41—42no—
130C2×C3×Q8—21no—
131C2×C3×Q8—21no—
132C2×C3×Q8—42no—
133C2×C3×D4—24no—
134C2×C3×D4—21no—
135C2×C3×D4—21no—
136C2×C3×D4—42no—
137C2×C3×D4—42no—
138C2×C3×D4—41no—
139C2×C3×D4—41no—
140C22×C3×C4—21no—
141C22×C3×C4—22no—
142C22×C3×C4—42no—
143C22×C3×C4—42no—
144C22×C3×C4—42no—
145C22×C3×C4—41no—
146C22×C3×C4—41no—
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4#4111yescharacteristic
153C2×#67—44no—
154C2×#67—41no—
155C2×#67—41no—
156C22×Dic3—21no—
157C22×Dic3—21no—
158C22×Dic3—21no—
159C22×Dic3—21no—
160C22×Dic3—22no—
161C22×Dic3—21no—
162C22×Dic3—21no—
163C22×Dic3—21no—
164C22×Dic3—42no—
165C22×Dic3—42no—
166C2×D12—21no—
167C2×D12—21no—
168C2×D12—42no—
169C3×M16—42no—
170C2×C3×C8—41no—
171C3×C22⋊C4—42no—
172C3×C42—42no—
173#217—44no—
174#217—42no—
175#217—44no—
176#215—44no—
177#213—44no—
178(C2×D3)⋊C4—44no—
179(C2×D3)⋊C4—41no—
180(C2×D3)⋊C4—41no—
181(C2×D3)⋊C4—82no—
182C4⋊Dic3—21no—
183C4⋊Dic3—21no—
184Dic3⋊C4—44no—
185Dic3⋊C4—41no—
186Dic3⋊C4—41no—
187Dic3⋊C4—41no—
188Dic3⋊C4—41no—
189#208—42no—
190C2×C3⋊C8—22no—
191C2×#41—21no—
192C2×#41—21no—
193C2×#41—21no—
194C2×#41—21no—
195C2×#41—44no—
196C22×Q8D1211yescharacteristic
197C22×D4—21no—
198C22×D4—31no—
199C22×D4—61no—
200C22×D4D1211yescharacteristic
201C23×C4—21no—
202C23×C4—31no—
203C23×C4—31no—
204C23×C4C22×D311yescharacteristic
205AGL(1,ℤ/8)—124no—
206C2×SD16—62no—
207C2×D8—62no—
208C2×M16—21no—
209C2×M16—61no—
210#123—124no—
211#121—61no—
212#121—61no—
213#121—61no—
214#121—61no—
215C2×C4⋊C4—31no—
216C2×C4⋊C4—62no—
217C2×C4⋊C4—61no—
218C2×C4⋊C4—61no—
219C2×C4⋊C4—61no—
220C2×C4⋊C4—61no—
221C2×C4⋊C4—61no—
222C2×C4⋊C4—61no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—62no—
225C2×C22⋊C4—62no—
226C2×C22⋊C4—62no—
227C2×C22⋊C4—61no—
228C2×C22⋊C4—61no—
229C2×C22⋊C4—62no—
230C2×C42#6712yes—
231#104—44no—
232#103—62no—
233#102—62no—
234C22⋊C8—21no—
235C22⋊C8—21no—
236C22⋊C8—32no—
237C22⋊C8—61no—
238#95—21no—
239#95—32no—
240#95—62no—
241#95—62no—
242#95—62no—
243#95—61no—
244C23×C3—22no—
245C23×C3—22no—
246C23×C3—22no—
247C23×C3—41no—
248C23×C3—42no—
249C23×C3—41no—
250C23×C3—41no—
251C23×C3—42no—
252C23×C3#12011yescharacteristic
253C23×C3#12011yescharacteristic
254C23×C3#12111yescharacteristic
255C22×D3—42no—
256C22×D3—42no—
257C22×D3—41no—
258C22×D3—42no—
259C3×Q8—22no—
260C3×Q8—22no—
261C3×Q8—41no—
262C3×Q8—41no—
263C3×D4—22no—
264C3×D4—22no—
265C3×D4—41no—
266C3×D4—41no—
267C3×D4—44no—
268C3×D4—44no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—44no—
277C2×C3×C4—44no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4#12011yescharacteristic
284C2×C3×C4#12111yescharacteristic
285C2×C3×C4#12111yescharacteristic
286C2×C3×C4#12711yescharacteristic
287#67—82no—
288#67—82no—
289C2×Dic3—22no—
290C2×Dic3—22no—
291C2×Dic3—22no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—42no—
297C2×Dic3—42no—
298C2×Dic3—42no—
299C2×Dic3—42no—
300C2×Dic3—41no—
301C2×Dic3—41no—
302C2×Dic3—42no—
303C2×Dic3—42no—
304C2×Dic3—81no—
305D12—42no—
306D12—81no—
307C3×C8—81no—
308C3⋊C8—42no—
309C24—21no—
310C24—61no—
311C24D3×D411yescharacteristic
312#41—24no—
313#41—24no—
314#41—42no—
315#41—42no—
316#41—42no—
317#41—42no—
318#41—42no—
319#41—42no—
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8—42no—
323C2×D4—24no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—61no—
331C2×D4—61no—
332C2×D4—122no—
333C2×D4—124no—
334C2×D4—121no—
335C2×D4—121no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—61no—
344C22×C4—61no—
345C22×C4—61no—
346C22×C4—62no—
347C22×C4—61no—
348C22×C4—61no—
349C22×C4—61no—
350C22×C4—61no—
351C22×C4—122no—
352C22×C4—122no—
353C22×C4C2×D1211yescharacteristic
354C22×C4#23511yescharacteristic
355C22×C4D3×D411yescharacteristic
356C22×C4D3×D411yescharacteristic
357C22×C4D3×D411yescharacteristic
358C22×C4C2×#6711yescharacteristic
359SD16—124no—
360D8—124no—
361M16—42no—
362M16—122no—
363C2×C8—41no—
364C2×C8—62no—
365C4⋊C4—61no—
366C4⋊C4—61no—
367C4⋊C4—124no—
368C4⋊C4—121no—
369C4⋊C4—121no—
370C4⋊C4—121no—
371C4⋊C4—121no—
372C22⋊C4—42no—
373C22⋊C4—124no—
374C22⋊C4—124no—
375C22⋊C4—122no—
376C22⋊C4—124no—
377C22⋊C4—121no—
378C22⋊C4—121no—
379C22⋊C4—242no—
380C42—42no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—44no—
389C22×C3—44no—
390C22×C3—42no—
391C22×C3—82no—
392C22×C3—81no—
393C22×C3—81no—
394C22×C3#39211yescharacteristic
395C22×C3#45412yes—
396C22×C3B(4,𝔽2)12yes—
397C2×D3—84no—
398C2×D3—81no—
399C2×D3—81no—
400C2×D3—81no—
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—21no—
404C3×C4—21no—
405C3×C4—42no—
406C3×C4—82no—
407Dic3—42no—
408Dic3—82no—
409Dic3—81no—
410Dic3—81no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—41no—
415C23—42no—
416C23—41no—
417C23—41no—
418C23—42no—
419C23—122no—
420C23—122no—
421C23—121no—
422C23—122no—
423C23#88211yescharacteristic
424C23#88411yescharacteristic
425C23#93711yescharacteristic
426Q8—22no—
427Q8—22no—
428Q8—41no—
429Q8—41no—
430D4—22no—
431D4—22no—
432D4—41no—
433D4—41no—
434D4—44no—
435D4—44no—
436D4—122no—
437D4—241no—
438D4—242no—
439D4—242no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—62no—
455C2×C4—62no—
456C2×C4—62no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4—121no—
460C2×C4—121no—
461C2×C4—122no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4—121no—
466C2×C4—121no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—241no—
470C2×C4#88211yescharacteristic
471C2×C4#88411yescharacteristic
472C2×C4#93011yescharacteristic
473C2×C4#94011yescharacteristic
474C8—81no—
475C8—122no—
476C2×C3—22no—
477C2×C3—22no—
478C2×C3—41no—
479C2×C3—82no—
480C2×C3#200311yescharacteristic
481C2×C3#201011yescharacteristic
482C2×C3#202111yescharacteristic
483D3—161no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—44no—
492C22—44no—
493C22—42no—
494C22—82no—
495C22—81no—
496C22—81no—
497C22—244no—
498C22—241no—
499C22—241no—
500C22—241no—
501C22#482712yes—
502C22#496011yescharacteristic
503C22#503212yes—
504C4—21no—
505C4—21no—
506C4—21no—
507C4—21no—
508C4—42no—
509C4—82no—
510C4—122no—
511C4—242no—
512C4—241no—
513C4—241no—
514C3#993911yescharacteristic, complemented, Sylow
515C2—22no—
516C2—22no—
517C2—41no—
518C2—82no—
519C2—481no—
520C2#6950011yescharacteristic
521C2#7038811yescharacteristic
522C2#7040611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1789C2
socleC22×C3#392
prime radical#84082C2
hypercenter#1789D3
hypocenterC3#9939
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1789

Complements

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

NH ≅ G/Ntype
1. #84090C2semidirect
3. #73550C2semidirect
4. C3×#1789C2semidirect
5. #69978C2semidirect
6. #69971C2semidirect
7. #67616C2semidirect
37. #1789D3semidirect
514. C3#9939semidirect

Maximal subgroups

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

subgroupindex
#840902
#840822
#735502
C3×#17892
#699782
#699712
#676162
#99393