Finite Group Databasebeta

← #10695556

Subgroups — #10695556

Order 768.

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

normal subgroups only

This group has 824 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×#2591C211yesmaximal, characteristic, complemented
2#82624C211yesmaximal, characteristic
3#82192C211yesmaximal, characteristic, complemented
4#71674C211yesmaximal, characteristic, complemented
5#71552C211yesmaximal, characteristic, complemented
6#69454C211yesmaximal, characteristic, complemented
7#67616C211yesmaximal, characteristic, complemented
8#9939—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C22×C3×C4⋊C4C2211yescharacteristic
12C2×#932—21no—
13C2×#929—21no—
14C3×#400—21no—
15C3×#394—21no—
16C3×#388—22no—
17C3×#386—21no—
18C3×#386—21no—
19C3×C4×C22⋊C4C2211yescharacteristic
20#5213—21no—
21#5211C2211yescharacteristic
22#5207C2211yescharacteristic
23#5207C2211yescharacteristic
24#5040—21no—
25#5037—22no—
26#5036—21no—
27#4991—21no—
28#4989—22no—
29#4979—21no—
30#4966—21no—
31C2×#805—22no—
32#4537—22no—
33#4528—22no—
34#3006—31no—
35#2591D311yescharacteristic, complemented
36#2406—31no—
37#1789—31no—
38#1510—31no—
39#1463—31no—
40#1284—31no—
41C22×C3×D4—21no—
42C22×C3×D4D411yescharacteristic
43C23×C3×C4D411yescharacteristic
44C23×C3×C4D411yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C22×#67—21no—
47C2×#235—21no—
48C2×#235—21no—
49C2×#235—21no—
50C2×#235—21no—
51C2×#235—44no—
52C22×D12D411yescharacteristic
53C22×C4×D3—21no—
54C22×Dic6D411yescharacteristic
55C3×#121—21no—
56C3×#121—21no—
57C3×#121—21no—
58C3×#121—21no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4D411yescharacteristic
66C2×C3×C22⋊C4—22no—
67C2×C3×C22⋊C4—21no—
68C2×C3×C22⋊C4—21no—
69C2×C3×C22⋊C4—22no—
70C2×C3×C42D412yes—
71C2×#217—21no—
72C2×#214—22no—
73#932—44no—
74C2×#213—22no—
75#929—44no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×Dic3⋊C4—22no—
79C2×#208—21no—
80C2×#208—21no—
81C3×#95—22no—
82C3×#95—22no—
83C3×#95—21no—
84C3⋊#95—22no—
85C3⋊#95—21no—
86C3⋊#95D412yes—
87#830—21no—
88#830—21no—
89#830—21no—
90#830D412yes—
91#810—22no—
92#809—22no—
93#805—44no—
94C22×#41—31no—
95C2×AGL(1,ℤ/8)—61no—
96C2×#123—61no—
97C2×#121C2×D311yescharacteristic
98C22×C4⋊C4C2×D311yescharacteristic
99#449—62no—
100#448—61no—
101#447—61no—
102C2×#104—62no—
103#417—61no—
104#407—31no—
105#407—31no—
106#400—21no—
107#395—61no—
108#394—21no—
109#394—61no—
110#388—22no—
111#386—21no—
112#386—21no—
113#386—61no—
114#385—62no—
115#381—31no—
116C4×C22⋊C4C2×D311yescharacteristic
117#327—62no—
118#324—62no—
119C24×C3—21no—
120C24×C3C2×D411yescharacteristic
121C23×D3—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—42no—
125C2×C3×D4—44no—
126C2×C3×D4—41no—
127C2×C3×D4—41no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—22no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—42no—
137C22×C3×C4—42no—
138C22×C3×C4—42no—
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4#4111yescharacteristic
145C2×#67—24no—
146C2×#67—42no—
147C2×#67—41no—
148C2×#67—41no—
149C22×Dic3—21no—
150C22×Dic3—22no—
151#235—24no—
152#235—24no—
153#235—42no—
154#235—42no—
155#235—42no—
156#235—42no—
157#235—42no—
158#235—42no—
159C2×D12—21no—
160C2×D12—21no—
161C2×D12—42no—
162C2×C4×D3—42no—
163C2×C4×D3—42no—
164C2×C4×D3—41no—
165C2×C4×D3—41no—
166C2×Dic6—21no—
167C2×Dic6—21no—
168C2×Dic6—42no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—41no—
172C3×C4⋊C4—41no—
173C3×C4⋊C4—41no—
174C3×C4⋊C4—41no—
175C3×C22⋊C4—44no—
176C3×C22⋊C4—42no—
177C3×C22⋊C4—41no—
178C3×C22⋊C4—41no—
179C3×C22⋊C4—82no—
180C3×C42—42no—
181#217—42no—
182#214—44no—
183#213—44no—
184(C2×D3)⋊C4—44no—
185(C2×D3)⋊C4—44no—
186Dic3⋊C4—44no—
187#208—42no—
188#208—42no—
189C2×C3⋊C8—22no—
190C2×C3⋊C8—41no—
191C2×#41—61no—
192C2×#41—61no—
193C2×#41—61no—
194C2×#41—61no—
195C2×#41—124no—
196C22×Q8—31no—
197C22×D4—21no—
198C22×D4—31no—
199C22×D4—61no—
200C22×D4#6711yescharacteristic
201C23×C4—61no—
202C23×C4#6711yescharacteristic
203C23×C4#6711yescharacteristic
204C23×C4C22×D311yescharacteristic
205AGL(1,ℤ/8)—124no—
206C2×SD16—62no—
207C2×D8—62no—
208C2×M16—61no—
209C2×M16—61no—
210#123—124no—
211#121—21no—
212#121—21no—
213#121—21no—
214#121—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—62no—
222C2×C4⋊C4#6711yescharacteristic
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—62no—
228C2×C22⋊C4—62no—
229C2×C22⋊C4—61no—
230C2×C42#6712yes—
231#104—124no—
232#103—62no—
233#102—62no—
234C22⋊C8—32no—
235C22⋊C8—61no—
236C22⋊C8—61no—
237C22⋊C8—61no—
238#95—22no—
239#95—22no—
240#95—21no—
241#95—32no—
242#95—62no—
243#95—61no—
244C23×C3—22no—
245C23×C3—22no—
246C23×C3—41no—
247C23×C3—42no—
248C23×C3—42no—
249C23×C3—41no—
250C23×C3—42no—
251C23×C3#12011yescharacteristic
252C23×C3#12011yescharacteristic
253C23×C3#12111yescharacteristic
254C22×D3—22no—
255C22×D3—42no—
256C22×D3—41no—
257C22×D3—41no—
258C22×D3—42no—
259C3×D4—42no—
260C3×D4—81no—
261C3×D4—82no—
262C3×D4—82no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
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—42no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—81no—
285C2×C3×C4#12011yescharacteristic
286C2×C3×C4#12111yescharacteristic
287C2×C3×C4#12111yescharacteristic
288C2×C3×C4#12711yescharacteristic
289#67—44no—
290#67—44no—
291C2×Dic3—22no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3—42no—
295C2×Dic3—42no—
296C2×Dic3—42no—
297D12—22no—
298D12—22no—
299D12—41no—
300D12—41no—
301C4×D3—44no—
302C4×D3—44no—
303Dic6—22no—
304Dic6—22no—
305Dic6—41no—
306Dic6—41no—
307C3⋊C8—42no—
308C3⋊C8—81no—
309C24—21no—
310C24—61no—
311C24C2×#6711yescharacteristic
312#41—64no—
313#41—64no—
314#41—122no—
315#41—122no—
316#41—122no—
317#41—122no—
318#41—122no—
319#41—122no—
320C2×Q8—61no—
321C2×Q8—61no—
322C2×Q8—122no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—42no—
326C2×D4—44no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—64no—
330C2×D4—61no—
331C2×D4—61no—
332C2×D4—122no—
333C2×D4—122no—
334C2×D4—121no—
335C2×D4—121no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—61no—
348C22×C4—62no—
349C22×C4—122no—
350C22×C4—122no—
351C22×C4—121no—
352C22×C4—121no—
353C22×C4D3×D411yescharacteristic
354C22×C4D3×D411yescharacteristic
355C22×C4D3×D411yescharacteristic
356C22×C4#23711yescharacteristic
357C22×C4C2×#6711yescharacteristic
358C22×C4C2×#6711yescharacteristic
359SD16—124no—
360D8—124no—
361M16—122no—
362M16—122no—
363C2×C8—62no—
364C2×C8—121no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—124no—
372C22⋊C4—44no—
373C22⋊C4—42no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—82no—
377C22⋊C4—124no—
378C22⋊C4—124no—
379C22⋊C4—122no—
380C42—42no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—44no—
389C22×C3—84no—
390C22×C3—81no—
391C22×C3—81no—
392C22×C3—81no—
393C22×C3#39211yescharacteristic
394C22×C3#45412yes—
395C22×C3B(4,𝔽2)12yes—
396C2×D3—44no—
397C2×D3—42no—
398C2×D3—82no—
399C2×D3—81no—
400C2×D3—81no—
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—21no—
404C3×C4—21no—
405C3×C4—42no—
406C3×C4—82no—
407C3×C4—81no—
408C3×C4—81no—
409Dic3—42no—
410Dic3—82no—
411C23—22no—
412C23—22no—
413C23—41no—
414C23—42no—
415C23—42no—
416C23—41no—
417C23—42no—
418C23—62no—
419C23—122no—
420C23—121no—
421C23—121no—
422C23—122no—
423C23#93711yescharacteristic
424C23#93911yescharacteristic
425C23#95311yescharacteristic
426Q8—62no—
427Q8—62no—
428Q8—121no—
429Q8—121no—
430D4—42no—
431D4—62no—
432D4—62no—
433D4—81no—
434D4—82no—
435D4—82no—
436D4—121no—
437D4—121no—
438D4—124no—
439D4—124no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—62no—
462C2×C4—81no—
463C2×C4—124no—
464C2×C4—124no—
465C2×C4—121no—
466C2×C4—121no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—122no—
470C2×C4#93711yescharacteristic
471C2×C4#93811yescharacteristic
472C2×C4#93911yescharacteristic
473C2×C4#94011yescharacteristic
474C8—122no—
475C8—241no—
476C2×C3—22no—
477C2×C3—22no—
478C2×C3—41no—
479C2×C3—161no—
480C2×C3#200311yescharacteristic
481C2×C3#201011yescharacteristic
482C2×C3#202111yescharacteristic
483D3—82no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—44no—
492C22—84no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22—124no—
497C22—122no—
498C22—242no—
499C22—241no—
500C22—241no—
501C22#506212yes—
502C22#522311yescharacteristic
503C22#524412yes—
504C4—21no—
505C4—21no—
506C4—21no—
507C4—21no—
508C4—42no—
509C4—82no—
510C4—81no—
511C4—81no—
512C4—122no—
513C4—242no—
514C3#993911yescharacteristic, complemented, Sylow
515C2—22no—
516C2—22no—
517C2—41no—
518C2—161no—
519C2—242no—
520C2#7171511yescharacteristic
521C2#7173011yescharacteristic
522C2#7185111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2591C2
socleC22×C3#392
prime radical#82624C2
hypercenter#2591D3
hypocenterC3#9939
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2591

Complements

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

NH ≅ G/Ntype
1. C3×#2591C2semidirect
3. #82192C2semidirect
4. #71674C2semidirect
5. #71552C2semidirect
6. #69454C2semidirect
7. #67616C2semidirect
35. #2591D3semidirect
514. C3#9939semidirect

Maximal subgroups

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

subgroupindex
C3×#25912
#826242
#821922
#716742
#715522
#694542
#676162
#99393