Finite Group Databasebeta

← #10695567

Subgroups — #10695567

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#82115C211yesmaximal, characteristic
2#82113C211yesmaximal, characteristic, complemented
3#78624C211yesmaximal, characteristic, complemented
4C3×#1462C211yesmaximal, characteristic, complemented
5#69978C211yesmaximal, characteristic, complemented
6#69975C211yesmaximal, characteristic, complemented
7#67682C211yesmaximal, characteristic, complemented
8#9940—31nomaximal, Sylow
9C22×C3×C22⋊C4C2211yescharacteristic
10C2×#953—21no—
11C2×#930C2211yescharacteristic
12C2×#929—21no—
13C2×#926C2211yescharacteristic
14C3×#407C2211yescharacteristic
15C3×#407C2211yescharacteristic
16#5247—21no—
17#5211C2211yescharacteristic
18#5210—21no—
19C4×#217C2211yescharacteristic
20#5131—21no—
21#5121—21no—
22#4960—21no—
23#4957—22no—
24#4953—21no—
25#4949—21no—
26#4949—21no—
27#4944—22no—
28#4737—22no—
29#4736—21no—
30#4735—21no—
31C3×#323—22no—
32#4555—22no—
33#4470—22no—
34#3094—31no—
35#2539—31no—
36#2482—31no—
37#1826—31no—
38#1510—31no—
39#1462D311yescharacteristic, complemented
40#1284—31no—
41C25×C3D411yescharacteristic
42C23×C3×C4D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C22×#67—21no—
45C22×#67—22no—
46C22×D12D411yescharacteristic
47C2×C3×M16—21no—
48C2×C3×M16—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4—21no—
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4—21no—
55#953—44no—
56C2×#217—21no—
57C2×#217—22no—
58C2×#217—22no—
59C2×#217—21no—
60C2×#217—21no—
61C2×#217—22no—
62#930—21no—
63#930—21no—
64#930—21no—
65#930—21no—
66#929—44no—
67C2×(C2×D3)⋊C4—22no—
68C2×(C2×D3)⋊C4—21no—
69#926—21no—
70#926—21no—
71#926—21no—
72#926—21no—
73C2×C4⋊Dic3D411yescharacteristic
74C2×C4⋊Dic3D411yescharacteristic, complemented
75C2×C4⋊Dic3D411yescharacteristic, complemented
76C2×Dic3⋊C4—22no—
77C2×Dic3⋊C4—22no—
78C2×C4×Dic3—21no—
79C2×C4×Dic3D412yes—
80C3×C22⋊C8—21no—
81C3×C22⋊C8—21no—
82C3×C22⋊C8—21no—
83C3×C22⋊C8D412yescomplemented
84C3⋊#95—22no—
85C3⋊#95—21no—
86C3⋊#95—21no—
87C3⋊#95D412yescomplemented
88#821—22no—
89#821—22no—
90#818—22no—
91#817—22no—
92C2×#123—61no—
93C2×#121—31no—
94C2×#120—61no—
95C2×#117—31no—
96C22×C22⋊C4C2×D311yescharacteristic
97#482—62no—
98#481—61no—
99#480—61no—
100#428—61no—
101#418—61no—
102#407C2×D311yescharacteristic
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#381—31no—
112#379—61no—
113C4×C22⋊C4—31no—
114#339—62no—
115#327—62no—
116#323—22no—
117C24×C3—21no—
118C24×C3—21no—
119C24×C3—21no—
120C24×C3—42no—
121C24×C3—42no—
122C24×C3—42no—
123C24×C3C2×D411yescharacteristic
124C23×D3—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—42no—
129C22×C3×C4—42no—
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4#4111yescharacteristic
136C2×#67—44no—
137C2×#67—42no—
138C2×#67—42no—
139C2×#67—44no—
140C2×#67—41no—
141C2×#67—41no—
142C2×#67—82no—
143C22×Dic3—22no—
144C22×Dic3—22no—
145C22×Dic3—21no—
146C22×Dic3—21no—
147C22×Dic3—21no—
148C2×D12—21no—
149C2×D12—21no—
150C2×D12—42no—
151C3×M16—42no—
152C3×M16—42no—
153C2×C3×C8—22no—
154C2×C3×C8—41no—
155C3×C22⋊C4—41no—
156C3×C22⋊C4—41no—
157C3×C22⋊C4—41no—
158C3×C22⋊C4—41no—
159#217—24no—
160#217—44no—
161#217—42no—
162#217—44no—
163#217—41no—
164#217—41no—
165#217—82no—
166(C2×D3)⋊C4—44no—
167(C2×D3)⋊C4—41no—
168(C2×D3)⋊C4—41no—
169C4⋊Dic3—21no—
170C4⋊Dic3—21no—
171C4⋊Dic3—21no—
172C4⋊Dic3—21no—
173C4⋊Dic3—21no—
174C4⋊Dic3—21no—
175Dic3⋊C4—44no—
176Dic3⋊C4—82no—
177C4×Dic3—41no—
178C4×Dic3—41no—
179C4×Dic3—42no—
180C25D1211yescharacteristic
181C22×D4—31no—
182C22×D4—61no—
183C22×D4—62no—
184C23×C4D1211yescharacteristic
185C23×C4C22×D311yescharacteristic
186C2×M16—21no—
187C2×M16—21no—
188#123—124no—
189#121—61no—
190#121—61no—
191#121—61no—
192#121—61no—
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—31no—
201C2×C4⋊C4—62no—
202C2×C4⋊C4—62no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—61no—
210C2×C22⋊C4—62no—
211C2×C22⋊C4—62no—
212C2×C22⋊C4—62no—
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—21no—
226C22⋊C8#6712yescomplemented
227#95—32no—
228#95—62no—
229#95—61no—
230#95—61no—
231C23×C3—22no—
232C23×C3—22no—
233C23×C3—22no—
234C23×C3—22no—
235C23×C3—22no—
236C23×C3—41no—
237C23×C3—42no—
238C23×C3—41no—
239C23×C3—41no—
240C23×C3—42no—
241C23×C3—41no—
242C23×C3—41no—
243C23×C3—42no—
244C23×C3—42no—
245C23×C3—81no—
246C23×C3—81no—
247C23×C3—88no—
248C23×C3#12011yescharacteristic
249C23×C3#12011yescharacteristic
250C23×C3#12111yescharacteristic
251C22×D3—41no—
252C22×D3—41no—
253C22×D3—42no—
254C22×D3—41no—
255C22×D3—42no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—42no—
263C2×C3×C4—41no—
264C2×C3×C4—42no—
265C2×C3×C4—81no—
266C2×C3×C4#12011yescharacteristic
267C2×C3×C4#12111yescharacteristic
268C2×C3×C4#12111yescharacteristic
269C2×C3×C4#12711yescharacteristic
270#67—88no—
271#67—82no—
272#67—82no—
273C2×Dic3—24no—
274C2×Dic3—42no—
275C2×Dic3—42no—
276C2×Dic3—41no—
277C2×Dic3—41no—
278C2×Dic3—42no—
279C2×Dic3—42no—
280C2×Dic3—42no—
281C2×Dic3—41no—
282C2×Dic3—41no—
283C2×Dic3—41no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286D12—42no—
287D12—81no—
288C3×C8—42no—
289C3×C8—81no—
290C24—21no—
291C24—21no—
292C24—21no—
293C24—42no—
294C24—42no—
295C24—42no—
296C24—61no—
297C24C2×D1211yescharacteristic
298C2×D4—61no—
299C2×D4—61no—
300C2×D4—122no—
301C2×D4—124no—
302C2×D4—122no—
303C2×D4—122no—
304C2×D4—124no—
305C2×D4—121no—
306C2×D4—121no—
307C2×D4—242no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—62no—
314C22×C4—62no—
315C22×C4—61no—
316C22×C4—61no—
317C22×C4—61no—
318C22×C4#23511yescharacteristic
319C22×C4D3×D411yescharacteristic
320C22×C4D3×D411yescharacteristic
321C22×C4D3×D411yescharacteristic
322C22×C4D3×D411yescharacteristic
323C22×C4C2×#6711yescharacteristic
324M16—42no—
325M16—42no—
326C2×C8—22no—
327C2×C8—41no—
328C4⋊C4—61no—
329C4⋊C4—61no—
330C4⋊C4—61no—
331C4⋊C4—61no—
332C4⋊C4—61no—
333C4⋊C4—61no—
334C4⋊C4—124no—
335C4⋊C4—242no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—64no—
341C22⋊C4—124no—
342C22⋊C4—124no—
343C22⋊C4—122no—
344C22⋊C4—124no—
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—44no—
362C22×C3—44no—
363C22×C3—42no—
364C22×C3—82no—
365C22×C3—82no—
366C22×C3—82no—
367C22×C3—82no—
368C22×C3—81no—
369C22×C3—81no—
370C22×C3#39211yescharacteristic
371C22×C3#45312yes—
372C22×C3#45812yes—
373C2×D3—82no—
374C2×D3—82no—
375C2×D3—81no—
376C2×D3—81no—
377C2×D3—81no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—81no—
383Dic3—44no—
384Dic3—82no—
385Dic3—81no—
386Dic3—81no—
387Dic3—81no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—41no—
394C23—42no—
395C23—41no—
396C23—41no—
397C23—42no—
398C23—41no—
399C23—41no—
400C23—42no—
401C23—42no—
402C23—81no—
403C23—81no—
404C23—88no—
405C23—121no—
406C23—121no—
407C23—122no—
408C23—121no—
409C23—122no—
410C23#88211yescharacteristic
411C23#88211yescharacteristic
412C23#93011yescharacteristic
413D4—122no—
414D4—241no—
415D4—248no—
416D4—242no—
417D4—242no—
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—64no—
428C2×C4—81no—
429C2×C4—122no—
430C2×C4—122no—
431C2×C4—121no—
432C2×C4—122no—
433C2×C4—122no—
434C2×C4—122no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4—122no—
440C2×C4—122no—
441C2×C4#88411yescharacteristic
442C2×C4#88411yescharacteristic
443C2×C4#93711yescharacteristic
444C2×C4#94011yescharacteristic
445C8—42no—
446C8—81no—
447C2×C3—22no—
448C2×C3—22no—
449C2×C3—41no—
450C2×C3—82no—
451C2×C3#200211yescharacteristic
452C2×C3#201011yescharacteristic
453C2×C3#202211yescharacteristic
454D3—161no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—44no—
463C22—44no—
464C22—44no—
465C22—42no—
466C22—82no—
467C22—82no—
468C22—82no—
469C22—82no—
470C22—81no—
471C22—81no—
472C22—242no—
473C22—242no—
474C22—241no—
475C22—241no—
476C22—241no—
477C22#474212yes—
478C22#474712yes—
479C22#496011yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—81no—
485C4—124no—
486C4—242no—
487C4—241no—
488C4—241no—
489C4—241no—
490C3#994011yescharacteristic, complemented, Sylow
491C2—22no—
492C2—22no—
493C2—41no—
494C2—82no—
495C2—481no—
496C2#6999311yescharacteristic
497C2#7003111yescharacteristic
498C2#7008211yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1462C2
socleC22×C3#392
prime radical#82115C2
hypercenter#1462D3
hypocenterC3#9940
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
2. #82113C2semidirect
3. #78624C2semidirect
4. C3×#1462C2semidirect
5. #69978C2semidirect
6. #69975C2semidirect
7. #67682C2semidirect
39. #1462D3semidirect
74. C2×C4⋊Dic3D4semidirect
75. C2×C4⋊Dic3D4semidirect
83. C3×C22⋊C8D4semidirect
87. C3⋊#95D4semidirect
226. C22⋊C8#67semidirect
490. C3#9940semidirect

Maximal subgroups

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

subgroupindex
#821152
#821132
#786242
C3×#14622
#699782
#699752
#676822
#99403