Finite Group Databasebeta

← #10691257

Subgroups — #10691257

Order 768.

690 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 105 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#82710C211yesmaximal, characteristic
2#74004C211yesmaximal, characteristic, complemented
3C3×#1519C211yesmaximal, characteristic, complemented
4#71675C211yesmaximal, characteristic, complemented
5C2×#4468C211yesmaximal, characteristic, complemented
6#67676C211yesmaximal, characteristic, complemented
7#67625C211yesmaximal, characteristic, complemented
8#9205—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#937—21no—
11C22×(C2×D3)⋊C4C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13C3×#380C2211yescharacteristic
14#5223C412yescomplemented
15#5220—21no—
16#5212C2211yescharacteristic
17#5212C2211yescharacteristic
18C2×#830C2211yescharacteristic
19#4992C412yescomplemented
20#4990—22no—
21#4961—22no—
22#4955—21no—
23Dic3×C22⋊C4—21no—
24#4670—22no—
25C3×#327—22no—
26#4537—22no—
27#4468—22no—
28#2617—31no—
29#2312—31no—
30#1519—31no—
31#1519D311yescharacteristic, complemented
32C2×#323—31no—
33#1289—31no—
34#1289—31no—
35C24×D3D411yescharacteristic
36C22×C3×D4—21no—
37C22×C3×D4C2×C411yescharacteristic
38C23×C3×C4C2311yescharacteristic
39C22×#67—22no—
40C23×Dic3D411yescharacteristic
41C22×C3×C8—21no—
42C3×#121—21no—
43C3×#121—21no—
44C3×#121—21no—
45C3×#121—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4C2×C411yescharacteristic
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×#217—21no—
51C2×#217—21no—
52C2×#217—22no—
53#937—42no—
54#937—42no—
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×C4⋊Dic3—21no—
65C2×C4⋊Dic3—21no—
66C2×C4×Dic3—22no—
67C22×C3⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×#95D412yescomplemented
71C3⋊#95—22no—
72C3⋊#95—22no—
73C3⋊#95C2×C412yes—
74C3⋊#95D412yescomplemented
75#830—21no—
76#830—21no—
77C2×#121C2×D311yescharacteristic
78C2×#120—61no—
79C22×C22⋊C4—31no—
80C2×C22⋊C8—31no—
81C2×C22⋊C8C2×D311yescharacteristic
82#397—32no—
83#394—62no—
84#392—32no—
85#388—62no—
86#388—62no—
87#386—61no—
88#380—31no—
89#380—31no—
90#380C2×D311yescharacteristic
91#379—61no—
92C4×C22⋊C4—61no—
93#327—22no—
94#327—62no—
95#323—62no—
96C24×C3—21no—
97C24×C3C2×D411yescharacteristic
98C23×D3—21no—
99C23×D3—21no—
100C23×D3—21no—
101C23×D3—21no—
102C23×D3—21no—
103C23×D3—21no—
104C23×D3—81no—
105C23×D3—81no—
106C23×D3C22⋊C412yes—
107C2×C3×D4—21no—
108C2×C3×D4—21no—
109C2×C3×D4—41no—
110C2×C3×D4—41no—
111C2×C3×D4—41no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—81no—
116C22×C3×C4—21no—
117C22×C3×C4—22no—
118C22×C3×C4—81no—
119C22×C3×C4C22×C411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4#4111yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C2×#67—44no—
126C2×#67—82no—
127C2×#67—82no—
128C22×Dic3—21no—
129C22×Dic3—21no—
130C22×Dic3—22no—
131C22×Dic3—22no—
132C22×Dic3—81no—
133C22×Dic3C22⋊C412yes—
134C2×C3×C8—41no—
135C2×C3×C8—81no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—81no—
139C3×C22⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143#217—41no—
144#217—41no—
145#217—41no—
146#217—41no—
147#217—82no—
148(C2×D3)⋊C4—22no—
149(C2×D3)⋊C4—22no—
150(C2×D3)⋊C4—22no—
151(C2×D3)⋊C4—22no—
152(C2×D3)⋊C4—44no—
153(C2×D3)⋊C4—82no—
154(C2×D3)⋊C4—82no—
155C4⋊Dic3—81no—
156C4⋊Dic3—81no—
157C4×Dic3—82no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—81no—
160C25—31no—
161C22×D4—21no—
162C22×D4—62no—
163C22×D4C4×D311yescharacteristic
164C23×C4—31no—
165C23×C4C22×D311yescharacteristic
166C22×C8—21no—
167C22×C8—61no—
168#121—21no—
169#121—21no—
170#121—21no—
171#121—21no—
172#120—122no—
173#120—122no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4C4×D311yescharacteristic
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—32no—
181C2×C22⋊C4—32no—
182C2×C22⋊C4—61no—
183C2×C22⋊C4—61no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—61no—
186C2×C22⋊C4—62no—
187C2×C22⋊C4—61no—
188C2×C22⋊C4—61no—
189C2×C22⋊C4—62no—
190C2×C22⋊C4—62no—
191C2×C22⋊C4—62no—
192C2×C42—62no—
193C22⋊C8—21no—
194C22⋊C8—21no—
195C22⋊C8—61no—
196C22⋊C8—61no—
197#95—32no—
198#95—32no—
199#95—62no—
200#95—62no—
201#95#6712yescomplemented
202C23×C3—41no—
203C23×C3—41no—
204C23×C3—42no—
205C23×C3—42no—
206C23×C3—41no—
207C23×C3—41no—
208C23×C3—41no—
209C23×C3C2×C22⋊C411yescharacteristic
210C23×C3#12011yescharacteristic
211C23×C3#12311yescharacteristic
212C23×C3#13511yescharacteristic
213C23×C3#13511yescharacteristic
214C23×C3AGL(1,ℤ/8)11yescharacteristic
215C23×C3AGL(1,ℤ/8)11yescharacteristic
216C22×D3—22no—
217C22×D3—22no—
218C22×D3—22no—
219C22×D3—42no—
220C22×D3—42no—
221C22×D3—42no—
222C22×D3—42no—
223C22×D3—41no—
224C22×D3—41no—
225C22×D3—41no—
226C22×D3—41no—
227C22×D3—81no—
228C22×D3—81no—
229C22×D3—82no—
230C22×D3—82no—
231C22×D3—82no—
232C22×D3—82no—
233C3×D4—81no—
234C3×D4—81no—
235C3×D4—81no—
236C3×D4—81no—
237C3×D4—81no—
238C3×D4—81no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—42no—
251C2×C3×C4C4×D411yescharacteristic
252C2×C3×C4C4×D411yescharacteristic
253C2×C3×C4#12111yescharacteristic
254C2×C3×C4#12111yescharacteristic
255#67—88no—
256C2×Dic3—22no—
257C2×Dic3—42no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265C2×Dic3—81no—
266C3×C8—81no—
267C3⋊C8—81no—
268C24—21no—
269C24—32no—
270C24—61no—
271C24—61no—
272C24—61no—
273C24—61no—
274C24—61no—
275C24—61no—
276C24—241no—
277C24—241no—
278C24D3×D411yescharacteristic
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—81no—
288C2×D4—124no—
289C2×D4—242no—
290C2×D4—242no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—32no—
294C22×C4—61no—
295C22×C4—61no—
296C22×C4—62no—
297C22×C4—62no—
298C22×C4—81no—
299C22×C4—241no—
300C22×C4C2×C4×D311yescharacteristic
301C22×C4#23511yescharacteristic
302C22×C4D3×D411yescharacteristic
303C22×C4D3×D411yescharacteristic
304C22×C4#23711yescharacteristic
305C22×C4C2×#6711yescharacteristic
306C2×C8—41no—
307C2×C8—81no—
308C2×C8—121no—
309C2×C8—241no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—81no—
313C4⋊C4—241no—
314C4⋊C4—241no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—62no—
320C22⋊C4—62no—
321C22⋊C4—62no—
322C22⋊C4—62no—
323C22⋊C4—124no—
324C22⋊C4—121no—
325C22⋊C4—121no—
326C22⋊C4—121no—
327C22⋊C4—121no—
328C22⋊C4—242no—
329C22⋊C4—242no—
330C22⋊C4—242no—
331C42—242no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—41no—
341C22×C3—41no—
342C22×C3—82no—
343C22×C3—82no—
344C22×C3—81no—
345C22×C3—81no—
346C22×C3—81no—
347C22×C3#38611yescharacteristic
348C22×C3#41611yescharacteristic
349C22×C3#41811yescharacteristic
350C22×C3#44911yescharacteristic
351C22×C3#44911yescharacteristic
352C22×C3#48211yescharacteristic
353C22×C3#48211yescharacteristic
354C2×D3—44no—
355C2×D3—42no—
356C2×D3—42no—
357C2×D3—42no—
358C2×D3—82no—
359C2×D3—82no—
360C2×D3—82no—
361C2×D3—82no—
362C2×D3—82no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—82no—
368C3×C4—81no—
369Dic3—42no—
370Dic3—82no—
371Dic3—82no—
372C23—41no—
373C23—41no—
374C23—42no—
375C23—42no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—62no—
380C23—62no—
381C23—62no—
382C23—122no—
383C23—122no—
384C23—122no—
385C23—122no—
386C23—121no—
387C23—121no—
388C23—121no—
389C23—121no—
390C23—241no—
391C23—241no—
392C23—242no—
393C23—242no—
394C23—242no—
395C23—242no—
396C23D3×C22⋊C411yescharacteristic
397C23#88311yescharacteristic
398C23#91111yescharacteristic
399C23#91211yescharacteristic
400C23#91411yescharacteristic
401C23#91611yescharacteristic
402C23#93711yescharacteristic
403D4—81no—
404D4—81no—
405D4—81no—
406D4—81no—
407D4—81no—
408D4—81no—
409D4—248no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—62no—
423C2×C4—122no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4—122no—
427C2×C4—122no—
428C2×C4—122no—
429C2×C4—122no—
430C2×C4—122no—
431C2×C4—241no—
432C2×C4#67⋊C411yescharacteristic
433C2×C4#88411yescharacteristic
434C2×C4C4×#6711yescharacteristic
435C2×C4#93811yescharacteristic
436C8—81no—
437C8—241no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—161no—
445C2×C3#187511yescharacteristic
446C2×C3#191111yescharacteristic
447C2×C3#191311yescharacteristic
448D3—82no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—82no—
460C22—82no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22—124no—
465C22—122no—
466C22—122no—
467C22—122no—
468C22—242no—
469C22—242no—
470C22—242no—
471C22—242no—
472C22—242no—
473C22#477011yescharacteristic
474C22#477211yescharacteristic
475C22#477911yescharacteristic
476C22#478411yescharacteristic
477C22#495611yescharacteristic
478C22#515811yescharacteristic
479C22#517211yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—82no—
485C4—81no—
486C4—122no—
487C4—242no—
488C4—242no—
489C3#920511yescharacteristic, complemented, Sylow
490C2—21no—
491C2—21no—
492C2—21no—
493C2—21no—
494C2—21no—
495C2—21no—
496C2—161no—
497C2—242no—
498C2#7003411yescharacteristic
499C2#7020111yescharacteristic
500C2#7040811yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1519C2
socleC22×C3#386
prime radical#82710C2
cosocleC23×C3×C4C23
hypercenter#1519D3
hypocenterC3#9205
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1519

Complements

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

NH ≅ G/Ntype
2. #74004C2semidirect
3. C3×#1519C2semidirect
4. #71675C2semidirect
5. C2×#4468C2semidirect
6. #67676C2semidirect
7. #67625C2semidirect
14. #5223C4semidirect
19. #4992C4semidirect
31. #1519D3semidirect
70. C3×#95D4semidirect
74. C3⋊#95D4semidirect
201. #95#67semidirect
489. C3#9205semidirect

Maximal subgroups

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

subgroupindex
#827102
#740042
C3×#15192
#716752
C2×#44682
#676762
#676252
#92053