Finite Group Databasebeta

← #10695555

Subgroups — #10695555

Order 768.

824 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C3×#3006C211yesmaximal, characteristic
2#84090C211yesmaximal, characteristic, complemented
3#82674C211yesmaximal, characteristic, complemented
4#72138C211yesmaximal, characteristic, complemented
5#71674C211yesmaximal, characteristic, complemented
6#69971C211yesmaximal, characteristic, complemented
7#67682C211yesmaximal, characteristic, complemented
8#9939—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×C3×AGL(1,ℤ/8)—21no—
11C2×C3×#121C2211yescharacteristic
12C2×#935—21no—
13C22×Dic3⋊C4C2211yescharacteristic
14C3×#449—22no—
15C3×#448—21no—
16C3×#447—21no—
17C3×#417—21no—
18C3×#407C2211yescharacteristic
19C2×#837—22no—
20#5230—21no—
21#5222—21no—
22#5222—21no—
23#5220—21no—
24#5220—21no—
25#5220—21no—
26#5211C2211yescharacteristic
27#5210—22no—
28C4×#217C2211yescharacteristic
29#5207C2211yescharacteristic
30#4985—21no—
31#4945—22no—
32#4537—22no—
33#4469—22no—
34#3006D311yescharacteristic
35#2591—31no—
36#2406—31no—
37#1789—31no—
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×D4—21no—
49C22×C3×D4D411yescharacteristic
50C22×C3×D4D411yescharacteristic
51C23×C3×C4—21no—
52C23×C3×C4C2311yescharacteristic
53C23×Dic3D411yescharacteristic
54C23×Dic3D411yescharacteristic
55C3×AGL(1,ℤ/8)—44no—
56C2×C3×SD16—22no—
57C2×C3×D8—22no—
58C2×C3×M16—21no—
59C3×#121—21no—
60C3×#121—21no—
61C3×#121—21no—
62C3×#121—21no—
63C2×C3×C4⋊C4D411yescharacteristic
64C2×C3×C22⋊C4—21no—
65C2×#217—22no—
66C2×#217—22no—
67C2×#217—22no—
68C2×#217—21no—
69C2×#217—21no—
70C2×#217—22no—
71#935—44no—
72C2×Dic3⋊C4—22no—
73C2×Dic3⋊C4—21no—
74C2×Dic3⋊C4—21no—
75C2×Dic3⋊C4—21no—
76C2×Dic3⋊C4—21no—
77C2×Dic3⋊C4—21no—
78C2×Dic3⋊C4—21no—
79C2×C4×Dic3D412yes—
80C2×#208—21no—
81C3×#103—22no—
82C3×#102—22no—
83C3×C22⋊C8—21no—
84C3×C22⋊C8D412yes—
85#837—44no—
86C3⋊#95—22no—
87C3⋊#95—22no—
88C3⋊#95—22no—
89C3⋊#95—21no—
90C3⋊#95—21no—
91C3⋊#95D412yes—
92#830—21no—
93#830—21no—
94C22×#41C2×D311yescharacteristic
95C2×AGL(1,ℤ/8)—21no—
96C2×#123—61no—
97C2×#121C2×D311yescharacteristic
98C22×C4⋊C4—31no—
99#449—22no—
100#448—21no—
101#447—21no—
102C2×#104—62no—
103#417—21no—
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⋊C4—31no—
117#327—62no—
118#324—62no—
119C24×C3—21no—
120C24×C3—21no—
121C24×C3C2×D411yescharacteristic
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—21no—
137C2×C3×D4—21no—
138C2×C3×D4—42no—
139C2×C3×D4—42no—
140C2×C3×D4—42no—
141C2×C3×D4—41no—
142C2×C3×D4—41no—
143C2×C3×D4—44no—
144C2×C3×D4—41no—
145C2×C3×D4—41no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—42no—
149C22×C3×C4—42no—
150C22×C3×C4—42no—
151C22×C3×C4—41no—
152C22×C3×C4—41no—
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4#4111yescharacteristic
159C22×Dic3—21no—
160C22×Dic3—21no—
161C22×Dic3—21no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—21no—
165C22×Dic3—21no—
166C22×Dic3—21no—
167C22×Dic3—42no—
168C22×Dic3—42no—
169C3×SD16—44no—
170C3×D8—44no—
171C3×M16—42no—
172C2×C3×C8—22no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—21no—
175C3×C22⋊C4—41no—
176C3×C22⋊C4—41no—
177#217—44no—
178#217—44no—
179#217—42no—
180#217—44no—
181#217—42no—
182#217—82no—
183Dic3⋊C4—44no—
184Dic3⋊C4—41no—
185Dic3⋊C4—41no—
186Dic3⋊C4—41no—
187Dic3⋊C4—41no—
188C4×Dic3—42no—
189#208—42no—
190C2×C3⋊C8—41no—
191C2×#41—21no—
192C2×#41—21no—
193C2×#41—21no—
194C2×#41—21no—
195C2×#41—44no—
196C22×Q8#6711yescharacteristic
197C22×D4—21no—
198C22×D4—21no—
199C22×D4#6711yescharacteristic
200C22×D4#6711yescharacteristic
201C23×C4—21no—
202C23×C4—31no—
203C23×C4—31no—
204C23×C4C22×D311yescharacteristic
205AGL(1,ℤ/8)—44no—
206C2×SD16—22no—
207C2×D8—22no—
208C2×M16—21no—
209C2×M16—61no—
210#123—124no—
211#121—21no—
212#121—21no—
213#121—21no—
214#121—21no—
215C2×C4⋊C4—62no—
216C2×C4⋊C4—61no—
217C2×C4⋊C4—61no—
218C2×C4⋊C4—61no—
219C2×C4⋊C4—61no—
220C2×C4⋊C4—61no—
221C2×C4⋊C4—61no—
222C2×C4⋊C4#6711yescharacteristic
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—32no—
231#104—124no—
232#103—22no—
233#102—22no—
234C22⋊C8—21no—
235C22⋊C8—61no—
236C22⋊C8—61no—
237C22⋊C8#6712yes—
238#95—32no—
239#95—62no—
240#95—62no—
241#95—62no—
242#95—61no—
243#95—61no—
244C23×C3—22no—
245C23×C3—22no—
246C23×C3—22no—
247C23×C3—41no—
248C23×C3—42no—
249C23×C3—41no—
250C23×C3—42no—
251C23×C3—42no—
252C23×C3—41no—
253C23×C3—42no—
254C23×C3—41no—
255C23×C3—42no—
256C23×C3#12011yescharacteristic
257C23×C3#12011yescharacteristic
258C23×C3#12111yescharacteristic
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—42no—
268C3×D4—44no—
269C3×D4—44no—
270C3×D4—81no—
271C3×D4—82no—
272C3×D4—82no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—44no—
281C2×C3×C4—44no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—42no—
285C2×C3×C4—41no—
286C2×C3×C4#12011yescharacteristic
287C2×C3×C4#12111yescharacteristic
288C2×C3×C4#12111yescharacteristic
289C2×C3×C4#12711yescharacteristic
290C2×Dic3—22no—
291C2×Dic3—22no—
292C2×Dic3—22no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—41no—
297C2×Dic3—42no—
298C2×Dic3—42no—
299C2×Dic3—42no—
300C2×Dic3—42no—
301C2×Dic3—42no—
302C2×Dic3—41no—
303C2×Dic3—42no—
304C2×Dic3—42no—
305C2×Dic3—42no—
306C2×Dic3—81no—
307C3×C8—42no—
308C3⋊C8—81no—
309C24—21no—
310C24—21no—
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—21no—
327C2×D4—21no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—44no—
334C2×D4—41no—
335C2×D4—41no—
336C22×C4—21no—
337C22×C4—21no—
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—62no—
348C22×C4—61no—
349C22×C4—61no—
350C22×C4—61no—
351C22×C4—122no—
352C22×C4—122no—
353C22×C4D3×D411yescharacteristic
354C22×C4D3×D411yescharacteristic
355C22×C4#23711yescharacteristic
356C22×C4C2×#6711yescharacteristic
357C22×C4C2×#6711yescharacteristic
358C22×C4C2×#6711yescharacteristic
359SD16—44no—
360D8—44no—
361M16—42no—
362M16—122no—
363C2×C8—22no—
364C2×C8—121no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—124no—
368C4⋊C4—121no—
369C4⋊C4—121no—
370C4⋊C4—121no—
371C4⋊C4—121no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—124no—
375C22⋊C4—124no—
376C22⋊C4—122no—
377C22⋊C4—124no—
378C22⋊C4—122no—
379C22⋊C4—242no—
380C42—122no—
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—84no—
393C22×C3—81no—
394C22×C3—81no—
395C22×C3—81no—
396C22×C3—81no—
397C22×C3—81no—
398C22×C3#39211yescharacteristic
399C22×C3#45412yes—
400C22×C3B(4,𝔽2)12yes—
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—21no—
404C3×C4—21no—
405C3×C4—42no—
406C3×C4—81no—
407Dic3—42no—
408Dic3—82no—
409Dic3—82no—
410Dic3—81no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—41no—
415C23—42no—
416C23—41no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—41no—
421C23—41no—
422C23—42no—
423C23#93711yescharacteristic
424C23#93711yescharacteristic
425C23#93811yescharacteristic
426Q8—22no—
427Q8—22no—
428Q8—41no—
429Q8—41no—
430D4—22no—
431D4—22no—
432D4—41no—
433D4—41no—
434D4—42no—
435D4—44no—
436D4—44no—
437D4—81no—
438D4—82no—
439D4—82no—
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—41no—
453C2×C4—62no—
454C2×C4—62no—
455C2×C4—62no—
456C2×C4—121no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4—121no—
460C2×C4—122no—
461C2×C4—122no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4—122no—
466C2×C4—121no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—241no—
470C2×C4#93911yescharacteristic
471C2×C4#93911yescharacteristic
472C2×C4#94011yescharacteristic
473C2×C4#95311yescharacteristic
474C8—42no—
475C8—241no—
476C2×C3—22no—
477C2×C3—22no—
478C2×C3—41no—
479C2×C3—82no—
480C2×C3—161no—
481C2×C3#200311yescharacteristic
482C2×C3#201011yescharacteristic
483C2×C3#202111yescharacteristic
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—44no—
492C22—44no—
493C22—42no—
494C22—82no—
495C22—84no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22#503212yes—
502C22#520112yes—
503C22#522311yescharacteristic
504C4—21no—
505C4—21no—
506C4—21no—
507C4—21no—
508C4—42no—
509C4—81no—
510C4—122no—
511C4—242no—
512C4—242no—
513C4—241no—
514C3#993911yescharacteristic, complemented, Sylow
515C2—22no—
516C2—22no—
517C2—41no—
518C2—82no—
519C2—161no—
520C2#7167111yescharacteristic
521C2#7209611yescharacteristic
522C2#7211811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#3006C2
socleC22×C3#392
hypercenter#3006D3
hypocenterC3#9939
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3006

Complements

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

NH ≅ G/Ntype
2. #84090C2semidirect
3. #82674C2semidirect
4. #72138C2semidirect
5. #71674C2semidirect
6. #69971C2semidirect
7. #67682C2semidirect
514. C3#9939semidirect

Maximal subgroups

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

subgroupindex
C3×#30062
#840902
#826742
#721382
#716742
#699712
#676822
#99393