Finite Group Databasebeta

← #10692649

Subgroups — #10692649

Order 768.

720 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 93 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#73567C211yesmaximal, characteristic, complemented
3C3×#1507C211yesmaximal, characteristic, complemented
4#71673C211yesmaximal, characteristic, complemented
5#69967C211yesmaximal, characteristic, complemented
6#67683C211yesmaximal, characteristic, complemented
7#67618C211yesmaximal, characteristic, complemented
8#9413—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#937—21no—
11C2×#935—21no—
12C22×(C2×D3)⋊C4C2211yescharacteristic
13C3×#407C2211yescharacteristic
14C3×C4×C22⋊C4C2211yescharacteristic
15#5223C412yescomplemented
16C4×#217C2211yescharacteristic
17C4×#217C2211yescharacteristic
18#5207C2211yescharacteristic
19#4993C412yescomplemented
20#4957—22no—
21#4955—21no—
22#4942—21no—
23#4940—22no—
24#4669—22no—
25C3×#327—22no—
26#4537—22no—
27#4468—22no—
28#2539—31no—
29#2312—31no—
30#1507—31no—
31#1507D311yescharacteristic, complemented
32#1462—31no—
33#1297—31no—
34#1297—31no—
35C24×D3D411yescharacteristic
36C22×C3×D4—21no—
37C22×C3×D4C2×C411yescharacteristic
38C23×C3×C4C2311yescharacteristic
39C22×#67—22no—
40C23×Dic3D411yescharacteristic
41C2×C3×M16—21no—
42C3×#121—21no—
43C3×#121—21no—
44C3×#121—21no—
45C3×#121—21no—
46C2×C3×C4⋊C4C2×C411yescharacteristic
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C42—21no—
50C2×#217—21no—
51C2×#217—21no—
52C2×#217—22no—
53#937—44no—
54#935—44no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—21no—
58C2×(C2×D3)⋊C4—21no—
59C2×(C2×D3)⋊C4—22no—
60C2×(C2×D3)⋊C4—22no—
61C2×(C2×D3)⋊C4—22no—
62C2×(C2×D3)⋊C4C2×C412yes—
63C2×(C2×D3)⋊C4D412yes—
64C2×Dic3⋊C4—22no—
65C2×Dic3⋊C4—22no—
66C2×C4×Dic3—21no—
67C2×C4×Dic3—21no—
68C2×#208—21no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×#95D412yescomplemented
72C3⋊#95—22no—
73C3⋊#95C2×C412yes—
74C3⋊#95D412yescomplemented
75#830—21no—
76#830—21no—
77C2×#123—61no—
78C2×#121C2×D311yescharacteristic
79C2×#120—61no—
80C22×C22⋊C4—31no—
81#407—31no—
82#407C2×D311yescharacteristic
83#399—32no—
84#392—32no—
85#390—62no—
86#388—62no—
87#388—62no—
88#380—61no—
89#379—61no—
90C4×C22⋊C4—31no—
91C4×C22⋊C4—31no—
92C4×C22⋊C4C2×D311yescharacteristic
93#327—22no—
94#327—62no—
95#323—62no—
96C24×C3—21no—
97C24×C3C2×D411yescharacteristic
98C23×D3—21no—
99C23×D3—21no—
100C23×D3—42no—
101C23×D3—42no—
102C23×D3—81no—
103C23×D3C22⋊C412yes—
104C2×C3×D4—21no—
105C2×C3×D4—21no—
106C2×C3×D4—42no—
107C2×C3×D4—42no—
108C2×C3×D4—41no—
109C2×C3×D4—41no—
110C2×C3×D4—81no—
111C22×C3×C4—21no—
112C22×C3×C4—22no—
113C22×C3×C4—81no—
114C22×C3×C4C22×C411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C2×#67—44no—
121C2×#67—44no—
122C2×#67—82no—
123C22×Dic3—22no—
124C22×Dic3—21no—
125C22×Dic3—21no—
126C22×Dic3—22no—
127C22×Dic3—42no—
128C22×Dic3C22⋊C412yes—
129C3×M16—81no—
130C2×C3×C8—41no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C22⋊C4—42no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C42—81no—
137#217—24no—
138#217—44no—
139#217—42no—
140(C2×D3)⋊C4—22no—
141(C2×D3)⋊C4—22no—
142(C2×D3)⋊C4—22no—
143(C2×D3)⋊C4—22no—
144(C2×D3)⋊C4—44no—
145(C2×D3)⋊C4—82no—
146(C2×D3)⋊C4—82no—
147Dic3⋊C4—44no—
148Dic3⋊C4—82no—
149C4×Dic3—81no—
150C4×Dic3—81no—
151#208—81no—
152C2×C3⋊C8—41no—
153C25—31no—
154C22×D4—21no—
155C22×D4—62no—
156C22×D4C4×D311yescharacteristic
157C23×C4—31no—
158C23×C4C22×D311yescharacteristic
159C2×M16—21no—
160C2×M16—61no—
161#123—124no—
162#121—21no—
163#121—21no—
164#121—21no—
165#121—21no—
166#120—124no—
167C2×C4⋊C4—62no—
168C2×C4⋊C4—62no—
169C2×C4⋊C4C4×D311yescharacteristic
170C2×C22⋊C4—21no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—32no—
173C2×C22⋊C4—32no—
174C2×C22⋊C4—61no—
175C2×C22⋊C4—61no—
176C2×C22⋊C4—61no—
177C2×C22⋊C4—61no—
178C2×C22⋊C4—61no—
179C2×C22⋊C4—62no—
180C2×C22⋊C4—62no—
181C2×C22⋊C4—61no—
182C2×C22⋊C4—62no—
183C2×C22⋊C4—62no—
184C2×C42—21no—
185C2×C42—61no—
186C2×C42—61no—
187C22⋊C8—21no—
188C22⋊C8—21no—
189C22⋊C8—61no—
190C22⋊C8—61no—
191#95—32no—
192#95—32no—
193#95—62no—
194#95#6712yescomplemented
195C23×C3—22no—
196C23×C3—22no—
197C23×C3—41no—
198C23×C3—41no—
199C23×C3—41no—
200C23×C3—42no—
201C23×C3—41no—
202C23×C3—42no—
203C23×C3C2×C22⋊C411yescharacteristic
204C23×C3#12011yescharacteristic
205C23×C3#12311yescharacteristic
206C22×D3—22no—
207C22×D3—22no—
208C22×D3—22no—
209C22×D3—42no—
210C22×D3—42no—
211C22×D3—42no—
212C22×D3—42no—
213C22×D3—42no—
214C22×D3—42no—
215C22×D3—81no—
216C22×D3—81no—
217C22×D3—88no—
218C3×D4—81no—
219C3×D4—81no—
220C3×D4—82no—
221C3×D4—82no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—41no—
233C2×C3×C4—42no—
234C2×C3×C4C4×D411yescharacteristic
235C2×C3×C4C4×D411yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12111yescharacteristic
238#67—88no—
239C2×Dic3—24no—
240C2×Dic3—22no—
241C2×Dic3—42no—
242C2×Dic3—42no—
243C2×Dic3—42no—
244C2×Dic3—42no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—42no—
248C2×Dic3—81no—
249C3×C8—81no—
250C3⋊C8—81no—
251C24—21no—
252C24—32no—
253C24—61no—
254C24—61no—
255C24—122no—
256C24—122no—
257C24—241no—
258C24D3×D411yescharacteristic
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—81no—
266C2×D4—124no—
267C2×D4—124no—
268C2×D4—242no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—32no—
272C22×C4—62no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—62no—
276C22×C4—81no—
277C22×C4—122no—
278C22×C4C2×C4×D311yescharacteristic
279C22×C4#23511yescharacteristic
280C22×C4D3×D411yescharacteristic
281C22×C4D3×D411yescharacteristic
282C22×C4#23711yescharacteristic
283C22×C4C2×#6711yescharacteristic
284M16—81no—
285M16—241no—
286C2×C8—41no—
287C2×C8—121no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—124no—
291C4⋊C4—242no—
292C22⋊C4—42no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—64no—
296C22⋊C4—62no—
297C22⋊C4—62no—
298C22⋊C4—62no—
299C22⋊C4—62no—
300C22⋊C4—124no—
301C22⋊C4—124no—
302C22⋊C4—122no—
303C22⋊C4—242no—
304C22⋊C4—242no—
305C42—81no—
306C42—241no—
307C42—241no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—44no—
316C22×C3—82no—
317C22×C3—82no—
318C22×C3—81no—
319C22×C3—81no—
320C22×C3—81no—
321C22×C3#38611yescharacteristic
322C22×C3#42112yes—
323C22×C3#45312yes—
324C2×D3—44no—
325C2×D3—42no—
326C2×D3—42no—
327C2×D3—42no—
328C2×D3—84no—
329C2×D3—84no—
330C2×D3—82no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—82no—
336C3×C4—81no—
337Dic3—44no—
338Dic3—42no—
339Dic3—82no—
340C23—22no—
341C23—22no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—42no—
346C23—41no—
347C23—42no—
348C23—62no—
349C23—62no—
350C23—62no—
351C23—122no—
352C23—122no—
353C23—122no—
354C23—122no—
355C23—122no—
356C23—122no—
357C23—241no—
358C23—241no—
359C23—248no—
360C23D3×C22⋊C411yescharacteristic
361C23#88311yescharacteristic
362C23#93711yescharacteristic
363D4—81no—
364D4—81no—
365D4—82no—
366D4—82no—
367D4—248no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—64no—
381C2×C4—62no—
382C2×C4—122no—
383C2×C4—122no—
384C2×C4—122no—
385C2×C4—122no—
386C2×C4—122no—
387C2×C4—122no—
388C2×C4—122no—
389C2×C4—241no—
390C2×C4#67⋊C411yescharacteristic
391C2×C4#88411yescharacteristic
392C2×C4C4×#6711yescharacteristic
393C2×C4#93811yescharacteristic
394C8—81no—
395C8—241no—
396C2×C3—22no—
397C2×C3—22no—
398C2×C3—41no—
399C2×C3—161no—
400C2×C3#188711yescharacteristic
401C2×C3#189711yescharacteristic
402C2×C3#191311yescharacteristic
403D3—82no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—44no—
412C22—82no—
413C22—82no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—124no—
418C22—122no—
419C22—122no—
420C22—122no—
421C22—242no—
422C22—244no—
423C22—244no—
424C22#482112yes—
425C22#495611yescharacteristic
426C22#519812yes—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—82no—
432C4—81no—
433C4—124no—
434C4—122no—
435C4—242no—
436C3#941311yescharacteristic, complemented, Sylow
437C2—22no—
438C2—22no—
439C2—41no—
440C2—161no—
441C2—242no—
442C2#7003411yescharacteristic
443C2#7032311yescharacteristic
444C2#7038311yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1507C2
socleC22×C3#386
prime radical#82708C2
cosocleC23×C3×C4C23
hypercenter#1507D3
hypocenterC3#9413
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1507

Complements

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

NH ≅ G/Ntype
2. #73567C2semidirect
3. C3×#1507C2semidirect
4. #71673C2semidirect
5. #69967C2semidirect
6. #67683C2semidirect
7. #67618C2semidirect
15. #5223C4semidirect
19. #4993C4semidirect
31. #1507D3semidirect
71. C3×#95D4semidirect
74. C3⋊#95D4semidirect
194. #95#67semidirect
436. C3#9413semidirect

Maximal subgroups

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

subgroupindex
#827082
#735672
C3×#15072
#716732
#699672
#676832
#676182
#94133