Finite Group Databasebeta

← #10691747

Subgroups — #10691747

Order 768.

606 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 107 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#78612C211yesmaximal, characteristic
2#73338C211yesmaximal, characteristic
3#69529C211yesmaximal, characteristic, complemented
4#69509C211yesmaximal, characteristic
5C2×#4452C211yesmaximal, characteristic, complemented
6C3×#1294C211yesmaximal, characteristic, complemented
7#67678C211yesmaximal, characteristic, complemented
8#9275—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C2×C4×#67—21no—
11C2×#924C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13C3×#380C2211yescharacteristic
14C2×C3×#95C2211yescharacteristic
15#5213—21no—
16C4×#217C2211yescharacteristic
17C2×#830C2211yescharacteristic
18#5114—22no—
19#4738C412yescomplemented
20#4735C412yescomplemented
21#4731—21no—
22#4721—21no—
23#4669—21no—
24#4666—21no—
25#4665—21no—
26#4654—21no—
27#4527—21no—
28#4527—21no—
29#4452—21no—
30#4452—21no—
31#4451—22no—
32#3187—31no—
33#2364—31no—
34#1996—31no—
35#1519—31no—
36C2×#328—31no—
37#1294D311yescharacteristic, complemented
38#1282—31no—
39C23×C3×C4—21no—
40C23×C3×C4C2311yescharacteristic
41C22×#67—21no—
42C22×D12D411yescharacteristic
43C22×C4×D3—21no—
44C22×Dic6C2×C411yescharacteristic, complemented
45C22×C3×C8—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C22⋊C4—21no—
48C2×#217—21no—
49C2×#217D411yescharacteristic
50C2×#217D411yescharacteristic
51#930—21no—
52#930—21no—
53#930C2×C412yes—
54#930D412yes—
55C4×#67—42no—
56C4×#67—42no—
57C2×(C2×D3)⋊C4—21no—
58C2×(C2×D3)⋊C4—21no—
59C2×(C2×D3)⋊C4—21no—
60C2×(C2×D3)⋊C4—21no—
61C2×C4⋊Dic3C2×C411yescharacteristic, complemented
62C2×C4⋊Dic3D411yescharacteristic
63#924—21no—
64#924—21no—
65#924—21no—
66#924—21no—
67C2×Dic3⋊C4—21no—
68C2×Dic3⋊C4—21no—
69C2×C4×Dic3—21no—
70C2×C4×Dic3—21no—
71C22×C3⋊C8—21no—
72C2×#204—22no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8C2×C412yescomplemented
75C3×#95—21no—
76C3×#95—21no—
77C3×#95D411yescharacteristic
78C3×#95D411yescharacteristic
79C3⋊#95—21no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82#830—21no—
83#830—21no—
84#821—22no—
85#817—22no—
86#816—22no—
87C2×C22⋊Q8—31no—
88C2×#121—31no—
89C2×C4×D4—61no—
90#483—61no—
91#481—32no—
92#465—62no—
93#450—61no—
94#448—32no—
95C2×C22⋊C8—31no—
96C2×C22⋊C8C2×D311yescharacteristic
97#388—61no—
98#387—61no—
99#386—61no—
100#386—61no—
101#380C2×D311yescharacteristic
102C4×C22⋊C4—31no—
103C4×C22⋊C4—61no—
104C2×#95C2×D311yescharacteristic
105#342—61no—
106#342—61no—
107#328—61no—
108#328—61no—
109#327—62no—
110C24×C3C2×D411yescharacteristic
111C23×D3—21no—
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—42no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4—41no—
118C22×C3×C4—81no—
119C22×C3×C4—81no—
120C22×C3×C4C22×C411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C2×#67—22no—
127C2×#67—22no—
128C2×#67—41no—
129C2×#67—41no—
130C2×#67—41no—
131C2×#67—41no—
132C22×Dic3—21no—
133C22×Dic3—21no—
134C22×Dic3—21no—
135C22×Dic3—21no—
136C22×Dic3—21no—
137C2×D12—21no—
138C2×D12—81no—
139C2×D12C22⋊C412yes—
140C2×C4×D3—42no—
141C2×C4×D3—42no—
142C2×C4×D3—81no—
143C2×Dic6—21no—
144C2×Dic6—21no—
145C2×Dic6—81no—
146C2×C3×C8—22no—
147C2×C3×C8—81no—
148C3×C4⋊C4—81no—
149C3×C22⋊C4—41no—
150C3×C22⋊C4—41no—
151#217—21no—
152#217—21no—
153#217—21no—
154#217—41no—
155#217—41no—
156#217SD1612yescomplemented
157(C2×D3)⋊C4—22no—
158(C2×D3)⋊C4—22no—
159(C2×D3)⋊C4—42no—
160(C2×D3)⋊C4—42no—
161(C2×D3)⋊C4—81no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165C4⋊Dic3C22⋊C412yes—
166Dic3⋊C4—42no—
167Dic3⋊C4—41no—
168Dic3⋊C4—41no—
169C4×Dic3—42no—
170C4×Dic3—81no—
171C2×C3⋊C8—41no—
172C2×C3⋊C8—81no—
173#204—82no—
174C22×Q8—31no—
175C22×D4—31no—
176C22×D4—61no—
177C23×C4—21no—
178C23×C4—61no—
179C23×C4C22×D311yescharacteristic
180C2×SD16—62no—
181C22×C8—21no—
182C22×C8—61no—
183C22⋊Q8—61no—
184C22⋊Q8—61no—
185C22⋊Q8—61no—
186C22⋊Q8—61no—
187#121—32no—
188#121—32no—
189#121—61no—
190#121—61no—
191C4×D4—122no—
192C4×D4—122no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4—31no—
196C2×C4⋊C4—61no—
197C2×C4⋊C4—61no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—31no—
200C2×C22⋊C4—31no—
201C2×C22⋊C4—61no—
202C2×C22⋊C4—61no—
203C2×C22⋊C4—61no—
204C2×C22⋊C4—61no—
205C2×C22⋊C4—61no—
206C2×C42—61no—
207C2×C42—61no—
208#106—62no—
209#103—62no—
210#102—62no—
211C22⋊C8—21no—
212C22⋊C8—61no—
213C22⋊C8—61no—
214C22⋊C8C4×D312yescomplemented
215#95—21no—
216#95—21no—
217#95—61no—
218#95—61no—
219#95—61no—
220#95D1211yescharacteristic
221#95D1211yescharacteristic
222C23×C3—41no—
223C23×C3—41no—
224C23×C3C4×D411yescharacteristic
225C23×C3#12011yescharacteristic
226C23×C3#12111yescharacteristic
227C23×C3C2×SD1611yescharacteristic
228C23×C3#13511yescharacteristic
229C23×C3AGL(1,ℤ/8)11yescharacteristic
230C23×C3#13711yescharacteristic
231C22×D3—22no—
232C22×D3—41no—
233C22×D3—41no—
234C22×D3—41no—
235C22×D3—41no—
236C22×D3—41no—
237C22×D3—41no—
238C2×C3×C4—22no—
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—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—82no—
251C2×C3×C4—82no—
252C2×C3×C4—81no—
253C2×C3×C4—81no—
254C2×C3×C4—81no—
255C2×C3×C4C2×C22⋊C411yescharacteristic
256C2×C3×C4C4×D411yescharacteristic
257C2×C3×C4#12111yescharacteristic
258C2×C3×C4#12311yescharacteristic
259#67—42no—
260#67—42no—
261#67—42no—
262#67—42no—
263C2×Dic3—22no—
264C2×Dic3—22no—
265C2×Dic3—41no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268C2×Dic3—41no—
269C2×Dic3—41no—
270C2×Dic3—41no—
271C2×Dic3—41no—
272C2×Dic3—41no—
273C2×Dic3—41no—
274C2×Dic3—81no—
275C2×Dic3—81no—
276D12—42no—
277D12—81no—
278C4×D3—84no—
279Dic6—42no—
280Dic6—81no—
281C3×C8—42no—
282C3⋊C8—81no—
283C24—61no—
284C24D3×D411yescharacteristic
285C2×Q8—61no—
286C2×Q8—61no—
287C2×Q8—241no—
288C2×D4—32no—
289C2×D4—62no—
290C2×D4—62no—
291C2×D4—61no—
292C2×D4—121no—
293C2×D4—121no—
294C2×D4—121no—
295C2×D4—121no—
296C2×D4—241no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—61no—
304C22×C4—61no—
305C22×C4—61no—
306C22×C4—61no—
307C22×C4—61no—
308C22×C4—81no—
309C22×C4—81no—
310C22×C4—122no—
311C22×C4—122no—
312C22×C4—241no—
313C22×C4C2×C4×D311yescharacteristic
314C22×C4C2×D1211yescharacteristic
315C22×C4#23511yescharacteristic
316C22×C4D3×D411yescharacteristic
317C22×C4D3×D411yescharacteristic
318C22×C4#23911yescharacteristic
319SD16—242no—
320C2×C8—22no—
321C2×C8—81no—
322C2×C8—121no—
323C2×C8—241no—
324C4⋊C4—32no—
325C4⋊C4—61no—
326C4⋊C4—61no—
327C4⋊C4—61no—
328C4⋊C4—81no—
329C4⋊C4—122no—
330C4⋊C4—121no—
331C4⋊C4—121no—
332C22⋊C4—32no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—61no—
336C22⋊C4—62no—
337C22⋊C4—62no—
338C22⋊C4—61no—
339C22⋊C4—61no—
340C22⋊C4—122no—
341C22⋊C4—122no—
342C22⋊C4—121no—
343C22⋊C4—121no—
344C22⋊C4—241no—
345C42—122no—
346C42—241no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3—41no—
357C22×C3#38611yescharacteristic
358C22×C3C4×SD1611yescharacteristic
359C22×C3#44011yescharacteristic
360C22×C3#45011yescharacteristic
361C22×C3#45211yescharacteristic
362C22×C3#46111yescharacteristic
363C22×C3#46311yescharacteristic
364C2×D3—42no—
365C2×D3—42no—
366C2×D3—42no—
367C2×D3—81no—
368C2×D3—81no—
369C2×D3—81no—
370C2×D3—81no—
371C3×C4—21no—
372C3×C4—21no—
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—42no—
376C3×C4—81no—
377C3×C4—161no—
378Dic3—42no—
379Dic3—42no—
380Dic3—81no—
381Dic3—81no—
382Dic3—81no—
383C23—41no—
384C23—41no—
385C23—62no—
386C23—121no—
387C23—121no—
388C23—121no—
389C23—121no—
390C23—121no—
391C23—121no—
392C23#88211yescharacteristic
393C23#88411yescharacteristic
394C23#89311yescharacteristic
395C23#91111yescharacteristic
396C23D3×SD1611yescharacteristic
397C23#91611yescharacteristic
398C23#91811yescharacteristic
399Q8—122no—
400Q8—241no—
401D4—122no—
402D4—122no—
403D4—122no—
404D4—122no—
405D4—122no—
406D4—241no—
407C2×C4—22no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—62no—
420C2×C4—62no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—121no—
427C2×C4—121no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—121no—
434C2×C4—121no—
435C2×C4—244no—
436C2×C4—241no—
437C2×C4—241no—
438C2×C4C4×D1211yescharacteristic
439C2×C4D3×C22⋊C411yescharacteristic
440C2×C4#89411yescharacteristic
441C2×C4#89511yescharacteristic
442C8—42no—
443C8—241no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3#187711yescharacteristic
451C2×C3#190611yescharacteristic
452C2×C3#191311yescharacteristic
453D3—82no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—122no—
465C22—122no—
466C22—122no—
467C22—241no—
468C22—241no—
469C22—241no—
470C22—241no—
471C22#466511yescharacteristic
472C22#477611yescharacteristic
473C22#478711yescharacteristic
474C22#480811yescharacteristic
475C22#481811yescharacteristic
476C22#485211yescharacteristic
477C22#488911yescharacteristic
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—42no—
483C4—81no—
484C4—122no—
485C4—122no—
486C4—161no—
487C4—241no—
488C4—241no—
489C4—241no—
490C3#927511yescharacteristic, complemented, Sylow
491C2—21no—
492C2—21no—
493C2—21no—
494C2—21no—
495C2—21no—
496C2—21no—
497C2—242no—
498C2#6877811yescharacteristic
499C2#6889611yescharacteristic
500C2#6896411yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1294C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1294D3
hypocenterC3#9275
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
3. #69529C2semidirect
5. C2×#4452C2semidirect
6. C3×#1294C2semidirect
7. #67678C2semidirect
19. #4738C4semidirect
20. #4735C4semidirect
37. #1294D3semidirect
44. C22×Dic6C2×C4semidirect
61. C2×C4⋊Dic3C2×C4semidirect
74. C3×C22⋊C8C2×C4semidirect
156. #217SD16semidirect
214. C22⋊C8C4×D3semidirect
490. C3#9275semidirect

Maximal subgroups

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

subgroupindex
#786122
#733382
#695292
#695092
C2×#44522
C3×#12942
#676782
#92753