Finite Group Databasebeta

← #10695784

Subgroups — #10695784

Order 768.

768 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C3×#3006C211yesmaximal, characteristic, complemented
2#84082C211yesmaximal, characteristic, complemented
3#82625C211yesmaximal, characteristic, complemented
4#72136C211yesmaximal, characteristic, complemented
5#71676C211yesmaximal, characteristic, complemented
6#69970C211yesmaximal, characteristic, complemented
7#67687C211yesmaximal, characteristic, complemented
8#9974—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×C3×AGL(1,ℤ/8)—21no—
11C2×C3×#121C2211yescharacteristic
12C2×#949—21no—
13C2×#930C2211yescharacteristic
14C3×#449—22no—
15C3×#448—21no—
16C3×#447—21no—
17C3×#417—21no—
18C3×#407C2211yescharacteristic
19#5238—22no—
20#5232—21no—
21#5227—21no—
22#5217—21no—
23#5217—21no—
24#5207C2211yescharacteristic
25C2×#830—22no—
26C2×#830C2211yescharacteristic
27C2×#830C2211yescharacteristic
28#5038—22no—
29#5036—21no—
30#5033—21no—
31#4967—21no—
32#4525—22no—
33#4468—22no—
34#3186—31no—
35#3006—31no—
36#3006D311yescharacteristic, complemented
37#1780—31no—
38#1471—31no—
39#1471—31no—
40#1293—31no—
41C2×C3×#41—21no—
42C2×C3×#41—21no—
43C2×C3×#41—21no—
44C2×C3×#41—21no—
45C2×C3×#41—44no—
46C22×C3×Q8D411yescharacteristic
47C22×C3×D4—21no—
48C22×C3×D4—21no—
49C22×C3×D4D411yescharacteristic
50C22×C3×D4D411yescharacteristic
51C23×C3×C4—21no—
52C23×C3×C4C2311yescharacteristic
53C22×#67—21no—
54C22×D12D411yescharacteristic
55C3×AGL(1,ℤ/8)—44no—
56C2×C3×SD16—22no—
57C2×C3×D8—22no—
58C2×C3×M16—21no—
59C3×#121—21no—
60C3×#121—21no—
61C3×#121—21no—
62C3×#121—21no—
63C2×C3×C4⋊C4D411yescharacteristic
64C2×C3×C22⋊C4—21no—
65#949—44no—
66C2×#215—22no—
67C2×#213—22no—
68C2×#213—22no—
69#930—21no—
70#930—21no—
71#930—21no—
72#930—21no—
73C2×(C2×D3)⋊C4—21no—
74C2×C4⋊Dic3D411yescharacteristic
75C2×#208—21no—
76C22×C3⋊C8—21no—
77C22×C3⋊C8D412yes—
78C3×#103—22no—
79C3×#102—22no—
80C3×C22⋊C8—21no—
81C3×C22⋊C8D412yes—
82#835—22no—
83#832—22no—
84#832—22no—
85#830—21no—
86#830—21no—
87#830—21no—
88#830—21no—
89#830—21no—
90#830—44no—
91#830D412yes—
92#809—22no—
93#807—22no—
94C22×#41C2×D311yescharacteristic
95C2×AGL(1,ℤ/8)—21no—
96C2×AGL(1,ℤ/8)—61no—
97C2×#121—31no—
98C2×#121C2×D311yescharacteristic
99#480—61no—
100#480—61no—
101#466—62no—
102#449—22no—
103#449—62no—
104#448—21no—
105#448—61no—
106#447—21no—
107#447—61no—
108#447—61no—
109#447—61no—
110#417—21no—
111#417—61no—
112#407—31no—
113#407C2×D311yescharacteristic
114C2×C22⋊C8—31no—
115C2×C22⋊C8—31no—
116C2×C22⋊C8—62no—
117#323—62no—
118#323—62no—
119C24×C3—21no—
120C24×C3—21no—
121C24×C3#4111yescharacteristic
122C23×D3—21no—
123C3×#41—24no—
124C3×#41—24no—
125C3×#41—42no—
126C3×#41—42no—
127C3×#41—42no—
128C3×#41—42no—
129C3×#41—42no—
130C3×#41—42no—
131C2×C3×Q8—21no—
132C2×C3×Q8—21no—
133C2×C3×Q8—42no—
134C2×C3×D4—21no—
135C2×C3×D4—21no—
136C2×C3×D4—21no—
137C2×C3×D4—21no—
138C2×C3×D4—42no—
139C2×C3×D4—42no—
140C2×C3×D4—42no—
141C2×C3×D4—42no—
142C2×C3×D4—41no—
143C2×C3×D4—41no—
144C2×C3×D4—41no—
145C2×C3×D4—41no—
146C2×C3×D4—82no—
147C22×C3×C4—24no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—42no—
151C22×C3×C4—42no—
152C22×C3×C4—41no—
153C22×C3×C4—41no—
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C2×#67—41no—
161C2×#67—41no—
162C2×#67—82no—
163C22×Dic3—21no—
164C2×D12—21no—
165C2×D12—21no—
166C2×D12—42no—
167C3×SD16—44no—
168C3×D8—44no—
169C3×M16—42no—
170C2×C3×C8—22no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—21no—
173C3×C22⋊C4—41no—
174C3×C22⋊C4—41no—
175#215—44no—
176#213—44no—
177#213—82no—
178(C2×D3)⋊C4—41no—
179(C2×D3)⋊C4—41no—
180C4⋊Dic3—21no—
181C4⋊Dic3—21no—
182#208—42no—
183C2×C3⋊C8—22no—
184C2×C3⋊C8—22no—
185C2×C3⋊C8—42no—
186C2×C3⋊C8—42no—
187C2×C3⋊C8—41no—
188C2×#41—21no—
189C2×#41—21no—
190C2×#41—21no—
191C2×#41—21no—
192C2×#41—44no—
193C22×Q8#6711yescharacteristic
194C22×D4—21no—
195C22×D4—21no—
196C22×D4—31no—
197C22×D4—61no—
198C22×D4#6711yescharacteristic
199C22×D4#6711yescharacteristic
200C23×C4—21no—
201C23×C4C22×D311yescharacteristic
202AGL(1,ℤ/8)—44no—
203AGL(1,ℤ/8)—124no—
204C2×SD16—22no—
205C2×SD16—62no—
206C2×D8—22no—
207C2×D8—62no—
208C2×D8—62no—
209C2×M16—21no—
210C2×M16—61no—
211C22×C8—32no—
212C22×C8—61no—
213#121—21no—
214#121—21no—
215#121—21no—
216#121—21no—
217#121—61no—
218#121—61no—
219#121—61no—
220#121—61no—
221C2×C4⋊C4—31no—
222C2×C4⋊C4#6711yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—61no—
225#107—62no—
226#103—22no—
227#103—62no—
228#102—22no—
229#102—62no—
230#102—62no—
231#102—62no—
232C22⋊C8—21no—
233C22⋊C8—32no—
234C22⋊C8—61no—
235C22⋊C8—61no—
236C22⋊C8—61no—
237C22⋊C8—61no—
238C22⋊C8—61no—
239C22⋊C8—124no—
240C22⋊C8#6712yes—
241C23×C3—22no—
242C23×C3—22no—
243C23×C3—41no—
244C23×C3—41no—
245C23×C3—41no—
246C23×C3—42no—
247C23×C3—42no—
248C23×C3—41no—
249C23×C3—42no—
250C23×C3—41no—
251C23×C3—81no—
252C23×C3—81no—
253C23×C3#12111yescharacteristic
254C23×C3#12111yescharacteristic
255C23×C3#12111yescharacteristic
256C22×D3—41no—
257C22×D3—42no—
258C22×D3—81no—
259C22×D3—81no—
260C3×Q8—22no—
261C3×Q8—22no—
262C3×Q8—41no—
263C3×Q8—41no—
264C3×D4—22no—
265C3×D4—22no—
266C3×D4—41no—
267C3×D4—41no—
268C3×D4—42no—
269C3×D4—44no—
270C3×D4—44no—
271C3×D4—42no—
272C3×D4—82no—
273C3×D4—82no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—44no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—44no—
285C2×C3×C4—41no—
286C2×C3×C4—81no—
287C2×C3×C4#12011yescharacteristic
288C2×C3×C4#12011yescharacteristic
289C2×C3×C4#12011yescharacteristic
290C2×C3×C4#12711yescharacteristic
291#67—82no—
292#67—82no—
293C2×Dic3—41no—
294C2×Dic3—81no—
295D12—42no—
296D12—42no—
297C3×C8—42no—
298C3⋊C8—42no—
299C3⋊C8—42no—
300C3⋊C8—81no—
301C24—21no—
302C24—21no—
303C24—61no—
304C24#23711yescharacteristic
305#41—24no—
306#41—24no—
307#41—42no—
308#41—42no—
309#41—42no—
310#41—42no—
311#41—42no—
312#41—42no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—42no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—61no—
329C2×D4—61no—
330C2×D4—82no—
331C2×D4—122no—
332C2×D4—121no—
333C2×D4—121no—
334C2×D4—242no—
335C22×C4—24no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—61no—
343C22×C4D3×D411yescharacteristic
344C22×C4D3×D411yescharacteristic
345C22×C4D3×D411yescharacteristic
346C22×C4C2×#6711yescharacteristic
347C22×C4C2×#6711yescharacteristic
348C22×C4C2×#6711yescharacteristic
349SD16—44no—
350SD16—124no—
351D8—44no—
352D8—124no—
353D8—242no—
354M16—42no—
355M16—122no—
356C2×C8—22no—
357C2×C8—62no—
358C2×C8—62no—
359C2×C8—122no—
360C2×C8—122no—
361C2×C8—121no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—61no—
365C4⋊C4—61no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—121no—
369C22⋊C4—121no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—44no—
379C22×C3—42no—
380C22×C3—82no—
381C22×C3—82no—
382C22×C3—81no—
383C22×C3—81no—
384C22×C3—81no—
385C22×C3—81no—
386C22×C3—81no—
387C22×C3—162no—
388C22×C3#39211yescharacteristic
389C22×C3#47212yes—
390C2×D3—81no—
391C2×D3—81no—
392C2×D3—81no—
393C2×D3—162no—
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—42no—
398C3×C4—81no—
399C3×C4B(4,𝔽2)12yes—
400Dic3—81no—
401C23—22no—
402C23—22no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—41no—
410C23—41no—
411C23—81no—
412C23—81no—
413C23—122no—
414C23—121no—
415C23—241no—
416C23—241no—
417C23#93811yescharacteristic
418C23#93911yescharacteristic
419C23#93911yescharacteristic
420Q8—22no—
421Q8—22no—
422Q8—41no—
423Q8—41no—
424D4—22no—
425D4—22no—
426D4—41no—
427D4—41no—
428D4—42no—
429D4—44no—
430D4—44no—
431D4—42no—
432D4—82no—
433D4—82no—
434D4—122no—
435D4—122no—
436D4—242no—
437D4—242no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—44no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—44no—
449C2×C4—41no—
450C2×C4—81no—
451C2×C4—121no—
452C2×C4—241no—
453C2×C4#93711yescharacteristic
454C2×C4#93711yescharacteristic
455C2×C4#94011yescharacteristic
456C2×C4#95311yescharacteristic
457C8—42no—
458C8—122no—
459C8—122no—
460C8—241no—
461C2×C3—22no—
462C2×C3—41no—
463C2×C3—41no—
464C2×C3—82no—
465C2×C3—161no—
466C2×C3#201711yescharacteristic
467C2×C3#201811yescharacteristic
468C2×C3#202111yescharacteristic
469D3—161no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—44no—
479C22—42no—
480C22—82no—
481C22—82no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22—162no—
488C22—241no—
489C22—241no—
490C22—241no—
491C22—482no—
492C22#519912yes—
493C22#522311yescharacteristic
494C4—21no—
495C4—21no—
496C4—21no—
497C4—42no—
498C4—81no—
499C4—241no—
500C4#503212yes—
501C3#997411yescharacteristic, complemented, Sylow
502C2—22no—
503C2—41no—
504C2—41no—
505C2—82no—
506C2—161no—
507C2—481no—
508C2#7167111yescharacteristic
509C2#7209511yescharacteristic
510C2#7212111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3006C2
socleC22×C3#392
hypercenter#3006D3
hypocenterC3#9974
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3006

Complements

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

NH ≅ G/Ntype
1. C3×#3006C2semidirect
2. #84082C2semidirect
3. #82625C2semidirect
4. #72136C2semidirect
5. #71676C2semidirect
6. #69970C2semidirect
7. #67687C2semidirect
36. #3006D3semidirect
501. C3#9974semidirect

Maximal subgroups

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

subgroupindex
C3×#30062
#840822
#826252
#721362
#716762
#699702
#676872
#99743