Finite Group Databasebeta

← #10695564

Subgroups — #10695564

Order 768.

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

normal subgroups only

This group has 768 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#82708C211yesmaximal, characteristic
2#82616C211yesmaximal, characteristic, complemented
3#74129C211yesmaximal, characteristic, complemented
4C3×#1510C211yesmaximal, characteristic, complemented
5#69975C211yesmaximal, characteristic, complemented
6#69967C211yesmaximal, characteristic, complemented
7#67628C211yesmaximal, characteristic, complemented
8#9940—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#937—21no—
11C2×#935—21no—
12C22×(C2×D3)⋊C4C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C3×#407C2211yescharacteristic
15C3×#381C2211yescharacteristic
16#5223—21no—
17#5215—21no—
18C4×#217C2211yescharacteristic
19#5207C2211yescharacteristic
20#5035—22no—
21#5034—21no—
22#5033—21no—
23#4993—21no—
24#4964—21no—
25#4957—22no—
26#4955—21no—
27#4950—21no—
28#4677—21no—
29#4674—21no—
30#4670—22no—
31C3×#327—22no—
32#4555—22no—
33#4468—22no—
34#3094—31no—
35#2539—31no—
36#2482—31no—
37#1826—31no—
38#1510D311yescharacteristic, complemented
39#1462—31no—
40#1284—31no—
41C24×D3D411yescharacteristic
42C22×C3×D4—21no—
43C22×C3×D4D411yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C22×#67—22no—
46C23×Dic3D411yescharacteristic
47C2×C3×M16—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×#121—21no—
52C2×C3×C4⋊C4D411yescharacteristic
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4—21no—
55C2×#217—21no—
56C2×#217—22no—
57C2×#217—21no—
58#937—44no—
59#935—44no—
60C2×(C2×D3)⋊C4—22no—
61C2×(C2×D3)⋊C4—22no—
62C2×(C2×D3)⋊C4—21no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—21no—
65C2×(C2×D3)⋊C4—21no—
66C2×(C2×D3)⋊C4—21no—
67C2×(C2×D3)⋊C4—21no—
68C2×(C2×D3)⋊C4—22no—
69#926—21no—
70#926—21no—
71#926—21no—
72#926—21no—
73C2×C4⋊Dic3—22no—
74C2×C4⋊Dic3D411yescharacteristic, complemented
75C2×C4⋊Dic3D411yescharacteristic, complemented
76C2×Dic3⋊C4—22no—
77C2×C4×Dic3—21no—
78C2×C4×Dic3D412yes—
79C2×#208—21no—
80C3×C22⋊C8—21no—
81C3×C22⋊C8—21no—
82C3×#95—21no—
83C3×#95D412yescomplemented
84#832—22no—
85#832—22no—
86C3⋊#95—22no—
87C3⋊#95—21no—
88#830—21no—
89#830D412yescomplemented
90#808—22no—
91#807—22no—
92C2×#123—61no—
93C2×#121C2×D311yescharacteristic
94C2×#120—61no—
95C2×#117—31no—
96C22×C22⋊C4—31no—
97#482—62no—
98#481—61no—
99#480—61no—
100#428—61no—
101#418—61no—
102#407—31no—
103#407C2×D311yescharacteristic
104#399—61no—
105#399—61no—
106#397—61no—
107#392—61no—
108#390—62no—
109#388—62no—
110#385—61no—
111#381C2×D311yescharacteristic
112#379—61no—
113C4×C22⋊C4—31no—
114#339—62no—
115#327—22no—
116#323—62no—
117C24×C3—21no—
118C24×C3C2×D411yescharacteristic
119C23×D3—21no—
120C23×D3—21no—
121C23×D3—21no—
122C23×D3—42no—
123C23×D3—42no—
124C23×D3—42no—
125C2×C3×D4—21no—
126C2×C3×D4—21no—
127C2×C3×D4—42no—
128C2×C3×D4—42no—
129C2×C3×D4—42no—
130C2×C3×D4—41no—
131C2×C3×D4—41no—
132C22×C3×C4—22no—
133C22×C3×C4—21no—
134C22×C3×C4—42no—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C2×#67—44no—
142C2×#67—44no—
143C2×#67—82no—
144C22×Dic3—22no—
145C22×Dic3—21no—
146C22×Dic3—21no—
147C22×Dic3—21no—
148C22×Dic3—21no—
149C22×Dic3—21no—
150C22×Dic3—42no—
151C3×M16—42no—
152C2×C3×C8—41no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—21no—
155C3×C22⋊C4—42no—
156C3×C22⋊C4—41no—
157C3×C22⋊C4—41no—
158#217—24no—
159#217—44no—
160#217—41no—
161#217—41no—
162(C2×D3)⋊C4—44no—
163(C2×D3)⋊C4—44no—
164(C2×D3)⋊C4—41no—
165(C2×D3)⋊C4—41no—
166(C2×D3)⋊C4—41no—
167(C2×D3)⋊C4—41no—
168(C2×D3)⋊C4—82no—
169C4⋊Dic3—21no—
170C4⋊Dic3—21no—
171C4⋊Dic3—21no—
172C4⋊Dic3—21no—
173C4⋊Dic3—82no—
174Dic3⋊C4—44no—
175C4×Dic3—41no—
176C4×Dic3—41no—
177C4×Dic3—42no—
178#208—42no—
179C2×C3⋊C8—22no—
180C25—31no—
181C22×D4—21no—
182C22×D4—62no—
183C22×D4D1211yescharacteristic
184C23×C4—31no—
185C23×C4C22×D311yescharacteristic
186C2×M16—21no—
187C2×M16—61no—
188#123—124no—
189#121—21no—
190#121—21no—
191#121—21no—
192#121—21no—
193#120—124no—
194#117—61no—
195#117—61no—
196#117—61no—
197#117—61no—
198C2×C4⋊C4—31no—
199C2×C4⋊C4—31no—
200C2×C4⋊C4—62no—
201C2×C4⋊C4—62no—
202C2×C4⋊C4D1211yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—61no—
206C2×C22⋊C4—62no—
207C2×C22⋊C4—62no—
208C2×C22⋊C4—62no—
209C2×C22⋊C4—61no—
210C2×C22⋊C4—61no—
211C2×C22⋊C4—61no—
212C2×C22⋊C4—61no—
213C2×C22⋊C4—61no—
214C2×C22⋊C4—61no—
215C2×C22⋊C4—61no—
216C2×C22⋊C4—62no—
217C2×C42—32no—
218C2×C42—61no—
219#107—62no—
220#106—62no—
221#102—62no—
222#102—62no—
223C22⋊C8—21no—
224C22⋊C8—21no—
225C22⋊C8—32no—
226C22⋊C8—61no—
227#95—21no—
228#95—62no—
229#95—61no—
230#95#6712yescomplemented
231C23×C3—22no—
232C23×C3—22no—
233C23×C3—41no—
234C23×C3—41no—
235C23×C3—41no—
236C23×C3—42no—
237C23×C3—41no—
238C23×C3—42no—
239C23×C3#12011yescharacteristic
240C23×C3#12011yescharacteristic
241C23×C3#12111yescharacteristic
242C22×D3—22no—
243C22×D3—22no—
244C22×D3—22no—
245C22×D3—42no—
246C22×D3—42no—
247C22×D3—41no—
248C22×D3—41no—
249C22×D3—41no—
250C22×D3—41no—
251C22×D3—42no—
252C22×D3—42no—
253C22×D3—88no—
254C22×D3—81no—
255C22×D3—81no—
256C3×D4—42no—
257C3×D4—81no—
258C3×D4—82no—
259C3×D4—82no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—41no—
271C2×C3×C4—42no—
272C2×C3×C4#12011yescharacteristic
273C2×C3×C4#12111yescharacteristic
274C2×C3×C4#12111yescharacteristic
275C2×C3×C4#12711yescharacteristic
276#67—88no—
277C2×Dic3—24no—
278C2×Dic3—42no—
279C2×Dic3—42no—
280C2×Dic3—42no—
281C2×Dic3—42no—
282C2×Dic3—41no—
283C2×Dic3—41no—
284C2×Dic3—41no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3—81no—
288C3×C8—81no—
289C3⋊C8—42no—
290C24—21no—
291C24—61no—
292C24—61no—
293C24—61no—
294C24—122no—
295C24—122no—
296C24—122no—
297C24D3×D411yescharacteristic
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—124no—
306C2×D4—124no—
307C2×D4—242no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—42no—
311C22×C4—62no—
312C22×C4—61no—
313C22×C4—61no—
314C22×C4—61no—
315C22×C4—61no—
316C22×C4—61no—
317C22×C4—122no—
318C22×C4C2×D1211yescharacteristic
319C22×C4#23511yescharacteristic
320C22×C4D3×D411yescharacteristic
321C22×C4D3×D411yescharacteristic
322C22×C4D3×D411yescharacteristic
323C22×C4C2×#6711yescharacteristic
324M16—42no—
325M16—122no—
326C2×C8—41no—
327C2×C8—62no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—61no—
331C4⋊C4—61no—
332C4⋊C4—61no—
333C4⋊C4—61no—
334C4⋊C4—124no—
335C4⋊C4—242no—
336C22⋊C4—42no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—64no—
340C22⋊C4—124no—
341C22⋊C4—124no—
342C22⋊C4—124no—
343C22⋊C4—121no—
344C22⋊C4—121no—
345C22⋊C4—121no—
346C22⋊C4—121no—
347C22⋊C4—121no—
348C22⋊C4—121no—
349C22⋊C4—242no—
350C42—122no—
351C42—121no—
352C42—121no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—44no—
361C22×C3—82no—
362C22×C3—82no—
363C22×C3—81no—
364C22×C3—81no—
365C22×C3—81no—
366C22×C3#39211yescharacteristic
367C22×C3#45312yes—
368C22×C3#45812yes—
369C2×D3—44no—
370C2×D3—44no—
371C2×D3—42no—
372C2×D3—82no—
373C2×D3—82no—
374C2×D3—82no—
375C2×D3—81no—
376C2×D3—81no—
377C2×D3—82no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—82no—
383C3×C4—81no—
384Dic3—44no—
385Dic3—81no—
386Dic3—81no—
387Dic3—81no—
388C23—22no—
389C23—22no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—42no—
394C23—41no—
395C23—42no—
396C23—62no—
397C23—62no—
398C23—62no—
399C23—122no—
400C23—121no—
401C23—121no—
402C23—122no—
403C23—121no—
404C23—122no—
405C23—121no—
406C23—122no—
407C23—241no—
408C23—241no—
409C23—248no—
410C23#88211yescharacteristic
411C23#88411yescharacteristic
412C23#93711yescharacteristic
413D4—42no—
414D4—81no—
415D4—82no—
416D4—82no—
417D4—248no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—64no—
431C2×C4—122no—
432C2×C4—122no—
433C2×C4—122no—
434C2×C4—122no—
435C2×C4—122no—
436C2×C4—122no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4—121no—
440C2×C4—241no—
441C2×C4#88211yescharacteristic
442C2×C4#88411yescharacteristic
443C2×C4#93011yescharacteristic
444C2×C4#94011yescharacteristic
445C8—81no—
446C8—122no—
447C2×C3—22no—
448C2×C3—22no—
449C2×C3—41no—
450C2×C3—161no—
451C2×C3#200211yescharacteristic
452C2×C3#201011yescharacteristic
453C2×C3#202211yescharacteristic
454D3—82no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—44no—
463C22—82no—
464C22—82no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—124no—
469C22—124no—
470C22—122no—
471C22—242no—
472C22—242no—
473C22—242no—
474C22—242no—
475C22—241no—
476C22—241no—
477C22#474212yes—
478C22#496011yescharacteristic
479C22#519812yes—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—82no—
485C4—81no—
486C4—124no—
487C4—241no—
488C4—241no—
489C4—241no—
490C3#994011yescharacteristic, complemented, Sylow
491C2—22no—
492C2—22no—
493C2—41no—
494C2—161no—
495C2—242no—
496C2#7003111yescharacteristic
497C2#7032411yescharacteristic
498C2#7040611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1510C2
socleC22×C3#392
prime radical#82708C2
hypercenter#1510D3
hypocenterC3#9940
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1510

Complements

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

NH ≅ G/Ntype
2. #82616C2semidirect
3. #74129C2semidirect
4. C3×#1510C2semidirect
5. #69975C2semidirect
6. #69967C2semidirect
7. #67628C2semidirect
38. #1510D3semidirect
74. C2×C4⋊Dic3D4semidirect
75. C2×C4⋊Dic3D4semidirect
83. C3×#95D4semidirect
89. #830D4semidirect
230. #95#67semidirect
490. C3#9940semidirect

Maximal subgroups

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

subgroupindex
#827082
#826162
#741292
C3×#15102
#699752
#699672
#676282
#99403