Finite Group Databasebeta

← #10698077

Subgroups — #10698077

Order 768.

1174 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 1174 conjugacy classes of subgroups — too many to draw. Only its 89 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×#2433C211yesmaximal, characteristic, complemented
2#84049C211yesmaximal, characteristic, complemented
3#82627C211yesmaximal, characteristic, complemented
4#72129C211yesmaximal, characteristic, complemented
5#71673C211yesmaximal, characteristic, complemented
6#69444C211yesmaximal, characteristic, complemented
7#67618C211yesmaximal, characteristic, complemented
8#10326—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C2×C3×#127—21no—
11C2×C3×#121C2211yescharacteristic
12C2×C3×#120—22no—
13C2×#953—22no—
14C2×#940—21no—
15C2×#930C2211yescharacteristic
16C3×#392—22no—
17C3×#390—21no—
18C3×C4×C22⋊C4C2211yescharacteristic
19#5223—22no—
20#5221—21no—
21#5217—21no—
22C2×#833—22no—
23C4×#217C2211yescharacteristic
24#5207C2211yescharacteristic
25#5207C2211yescharacteristic
26#5039—22no—
27#5036—21no—
28#5034—21no—
29#4968—21no—
30#4537—22no—
31#4451—22no—
32#3192—31no—
33#2433—31no—
34#2433D311yescharacteristic, complemented
35#1783—31no—
36#1507—31no—
37#1507—31no—
38#1297—31no—
39C25×C3D412yes—
40C22×C3×D4—21no—
41C22×C3×D4—24no—
42C22×C3×D4—21no—
43C22×C3×D4—21no—
44C22×C3×D4—21no—
45C22×C3×D4—21no—
46C22×C3×D4—21no—
47C22×C3×D4—21no—
48C22×C3×D4—44no—
49C22×C3×D4D411yescharacteristic, complemented
50C23×C3×C4C2311yescharacteristic
51C22×#67—21no—
52C22×#67—24no—
53C22×D12D411yescharacteristic, complemented
54C3×#127—44no—
55C3×#121—21no—
56C3×#121—21no—
57C3×#121—21no—
58C3×#121—21no—
59C3×#120—48no—
60C2×C3×C4⋊C4D411yescharacteristic, complemented
61C2×C3×C22⋊C4—21no—
62C2×C3×C22⋊C4—24no—
63C2×C3×C22⋊C4—21no—
64C2×C3×C42—21no—
65#953—48no—
66C2×#217—21no—
67C2×#217—24no—
68#940—44no—
69C2×#213—22no—
70#930—21no—
71#930—21no—
72#930—21no—
73#930—21no—
74C2×(C2×D3)⋊C4—21no—
75C2×C4⋊Dic3D411yescharacteristic, complemented
76C2×C4×Dic3—21no—
77C2×#208—21no—
78C2×#208D412yescomplemented
79C3×#95D412yescomplemented
80#833—44no—
81#832—22no—
82C3⋊#95D412yescomplemented
83#830—21no—
84#830—21no—
85#830—21no—
86#830—21no—
87#809—22no—
88#808—22no—
89C23×D4C2×D311yescharacteristic
90C2×#127—21no—
91C2×#127—61no—
92C2×#121—31no—
93C2×#121C2×D311yescharacteristic
94C2×#120—22no—
95C2×#120—62no—
96#481—61no—
97#481—61no—
98#469—62no—
99#447—61no—
100#447—61no—
101C2×#100—62no—
102#407—31no—
103#407—31no—
104#392—22no—
105#392—62no—
106#390—21no—
107#390—61no—
108C4×C22⋊C4—31no—
109C4×C22⋊C4C2×D311yescharacteristic
110#327—62no—
111#327—62no—
112C24×C3—22no—
113C24×C3—22no—
114C24×C3—22no—
115C24×C3—21no—
116C24×C3—44no—
117C24×C3—44no—
118C24×C3—44no—
119C24×C3C2×D411yescharacteristic
120C23×D3—21no—
121C2×C3×D4—24no—
122C2×C3×D4—24no—
123C2×C3×D4—24no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4—42no—
127C2×C3×D4—42no—
128C2×C3×D4—42no—
129C2×C3×D4—42no—
130C2×C3×D4—48no—
131C2×C3×D4—44no—
132C2×C3×D4—44no—
133C2×C3×D4—44no—
134C2×C3×D4—44no—
135C2×C3×D4—48no—
136C2×C3×D4—41no—
137C2×C3×D4—41no—
138C2×C3×D4—41no—
139C2×C3×D4—41no—
140C2×C3×D4—41no—
141C2×C3×D4—41no—
142C2×C3×D4—84no—
143C22×C3×C4—22no—
144C22×C3×C4—21no—
145C22×C3×C4—42no—
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4#4111yescharacteristic
152C2×#67—48no—
153C2×#67—48no—
154C2×#67—44no—
155C2×#67—41no—
156C2×#67—41no—
157C2×#67—84no—
158C22×Dic3—22no—
159C22×Dic3—21no—
160C2×D12—21no—
161C2×D12—21no—
162C2×D12—42no—
163C3×C4⋊C4—21no—
164C3×C4⋊C4—21no—
165C3×C22⋊C4—24no—
166C3×C22⋊C4—48no—
167C3×C22⋊C4—41no—
168C3×C22⋊C4—41no—
169C3×C42—42no—
170#217—24no—
171#217—48no—
172#213—82no—
173(C2×D3)⋊C4—41no—
174(C2×D3)⋊C4—41no—
175C4⋊Dic3—21no—
176C4⋊Dic3—21no—
177C4×Dic3—42no—
178#208—42no—
179#208—42no—
180C2×C3⋊C8—22no—
181C2×C3⋊C8—41no—
182C25#6712yes—
183C22×D4—21no—
184C22×D4—24no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4—21no—
188C22×D4—21no—
189C22×D4—21no—
190C22×D4—21no—
191C22×D4—31no—
192C22×D4—44no—
193C22×D4—61no—
194C22×D4—64no—
195C22×D4#6711yescharacteristic, complemented
196C23×C4C22×D311yescharacteristic
197C2×D8—62no—
198C2×M16—32no—
199C2×M16—61no—
200#127—44no—
201#127—124no—
202#121—21no—
203#121—21no—
204#121—21no—
205#121—21no—
206#121—61no—
207#121—61no—
208#121—61no—
209#121—61no—
210#120—48no—
211#120—128no—
212C2×C4⋊C4—31no—
213C2×C4⋊C4#6711yescharacteristic, complemented
214C2×C22⋊C4—21no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—61no—
218C2×C22⋊C4—64no—
219C2×C22⋊C4—61no—
220C2×C42—21no—
221C2×C42—61no—
222#106—62no—
223#102—62no—
224#102—62no—
225#100—124no—
226C22⋊C8—61no—
227C22⋊C8—61no—
228C22⋊C8—61no—
229C22⋊C8—61no—
230#95—32no—
231#95#6712yescomplemented
232C23×C3—22no—
233C23×C3—22no—
234C23×C3—24no—
235C23×C3—24no—
236C23×C3—24no—
237C23×C3—41no—
238C23×C3—44no—
239C23×C3—44no—
240C23×C3—42no—
241C23×C3—42no—
242C23×C3—42no—
243C23×C3—42no—
244C23×C3—42no—
245C23×C3—42no—
246C23×C3—44no—
247C23×C3—44no—
248C23×C3—41no—
249C23×C3—42no—
250C23×C3—82no—
251C23×C3—82no—
252C23×C3—816no—
253C23×C3#12011yescharacteristic
254C23×C3#12011yescharacteristic
255C23×C3#12111yescharacteristic
256C22×D3—42no—
257C22×D3—42no—
258C22×D3—41no—
259C22×D3—42no—
260C3×D4—44no—
261C3×D4—44no—
262C3×D4—44no—
263C3×D4—44no—
264C3×D4—42no—
265C3×D4—81no—
266C3×D4—816no—
267C3×D4—82no—
268C3×D4—82no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—24no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—41no—
280C2×C3×C4#12011yescharacteristic
281C2×C3×C4#12111yescharacteristic
282C2×C3×C4#12111yescharacteristic
283C2×C3×C4#12711yescharacteristic
284#67—816no—
285#67—82no—
286#67—82no—
287C2×Dic3—24no—
288C2×Dic3—42no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—41no—
292D12—42no—
293D12—81no—
294C3⋊C8—42no—
295C3⋊C8—81no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—21no—
300C24—44no—
301C24—44no—
302C24—44no—
303C24—61no—
304C24C2×#6711yescharacteristic
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—48no—
315C2×D4—44no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—48no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—61no—
327C2×D4—61no—
328C2×D4—84no—
329C2×D4—122no—
330C2×D4—128no—
331C2×D4—124no—
332C2×D4—128no—
333C2×D4—121no—
334C2×D4—121no—
335C2×D4—244no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—42no—
339C22×C4—62no—
340C22×C4—61no—
341C22×C4D3×D411yescharacteristic
342C22×C4D3×D411yescharacteristic
343C22×C4D3×D411yescharacteristic
344C22×C4#23711yescharacteristic
345C22×C4C2×#6711yescharacteristic
346C22×C4C2×#6711yescharacteristic
347D8—242no—
348M16—122no—
349M16—122no—
350C2×C8—62no—
351C2×C8—121no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—61no—
355C4⋊C4—61no—
356C22⋊C4—24no—
357C22⋊C4—48no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—64no—
361C22⋊C4—128no—
362C22⋊C4—121no—
363C22⋊C4—121no—
364C42—42no—
365C42—122no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—44no—
374C22×C3—48no—
375C22×C3—48no—
376C22×C3—44no—
377C22×C3—84no—
378C22×C3—84no—
379C22×C3—84no—
380C22×C3—84no—
381C22×C3—84no—
382C22×C3—82no—
383C22×C3—82no—
384C22×C3—81no—
385C22×C3—81no—
386C22×C3—81no—
387C22×C3#39211yescharacteristic
388C22×C3#45312yes—
389C22×C3#45312yes—
390C2×D3—84no—
391C2×D3—81no—
392C2×D3—81no—
393C2×D3—81no—
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—44no—
398C3×C4—81no—
399C3×C4B(4,𝔽2)12yes—
400Dic3—44no—
401Dic3—81no—
402C23—22no—
403C23—22no—
404C23—24no—
405C23—24no—
406C23—24no—
407C23—41no—
408C23—44no—
409C23—44no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—44no—
417C23—44no—
418C23—41no—
419C23—42no—
420C23—82no—
421C23—82no—
422C23—816no—
423C23—122no—
424C23—122no—
425C23—121no—
426C23—122no—
427C23#93711yescharacteristic
428C23#93911yescharacteristic
429C23#95311yescharacteristic
430D4—44no—
431D4—44no—
432D4—44no—
433D4—44no—
434D4—42no—
435D4—81no—
436D4—816no—
437D4—82no—
438D4—82no—
439D4—122no—
440D4—241no—
441D4—2416no—
442D4—242no—
443D4—242no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—64no—
456C2×C4—122no—
457C2×C4—122no—
458C2×C4—122no—
459C2×C4—121no—
460C2×C4#93711yescharacteristic
461C2×C4#93811yescharacteristic
462C2×C4#93911yescharacteristic
463C2×C4#94011yescharacteristic
464C8—122no—
465C8—241no—
466C2×C3—22no—
467C2×C3—22no—
468C2×C3—41no—
469C2×C3—84no—
470C2×C3—161no—
471C2×C3#200211yescharacteristic
472C2×C3#200211yescharacteristic
473C2×C3#202111yescharacteristic
474D3—161no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—44no—
483C22—48no—
484C22—48no—
485C22—44no—
486C22—84no—
487C22—84no—
488C22—84no—
489C22—84no—
490C22—84no—
491C22—82no—
492C22—82no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22—244no—
497C22—241no—
498C22—241no—
499C22—241no—
500C22#507712yes—
501C22#522311yescharacteristic
502C22#524112yes—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—44no—
507C4—81no—
508C4—124no—
509C4—241no—
510C4#503212yes—
511C3#1032611yescharacteristic, complemented, Sylow
512C2—22no—
513C2—22no—
514C2—41no—
515C2—84no—
516C2—161no—
517C2—481no—
518C2#7167111yescharacteristic
519C2#7176911yescharacteristic
520C2#7185611yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2433C2
socleC22×C3#392
hypercenter#2433D3
hypocenterC3#10326
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2433

Complements

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

NH ≅ G/Ntype
1. C3×#2433C2semidirect
2. #84049C2semidirect
3. #82627C2semidirect
4. #72129C2semidirect
5. #71673C2semidirect
6. #69444C2semidirect
7. #67618C2semidirect
34. #2433D3semidirect
49. C22×C3×D4D4semidirect
53. C22×D12D4semidirect
60. C2×C3×C4⋊C4D4semidirect
75. C2×C4⋊Dic3D4semidirect
78. C2×#208D4semidirect
79. C3×#95D4semidirect
82. C3⋊#95D4semidirect
195. C22×D4#67semidirect
213. C2×C4⋊C4#67semidirect
231. #95#67semidirect
511. C3#10326semidirect

Maximal subgroups

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

subgroupindex
C3×#24332
#840492
#826272
#721292
#716732
#694442
#676182
#103263