Finite Group Databasebeta

← #10693211

Subgroups — #10693211

Order 768.

772 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 99 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#82927C211yesmaximal, characteristic
2#82919C211yesmaximal, characteristic, complemented
3C3×#1513C211yesmaximal, characteristic, complemented
4#69988C211yesmaximal, characteristic, complemented
5#69521C211yesmaximal, characteristic, complemented
6#67687C211yesmaximal, characteristic, complemented
7#67626C211yesmaximal, characteristic, complemented
8#9520—31nomaximal, Sylow
9C23×D12C2211yescharacteristic
10C3×#525C2211yescharacteristic
11C2×#949—21no—
12C3×#407C2211yescharacteristic
13C3×#380C2211yescharacteristic
14#5234—21no—
15#5207C2211yescharacteristic
16#5207C2211yescharacteristic
17C2×#830C2211yescharacteristic
18C2×#823—21no—
19C2×#823—21no—
20#5046—21no—
21#5046C412yescomplemented
22#5036—21no—
23#5036C412yescomplemented
24#5012—21no—
25#5011—21no—
26#5011—21no—
27D4×C3⋊C8—21no—
28#5001—21no—
29#5000—21no—
30C2×#809—21no—
31C2×#809—21no—
32C3×#328—21no—
33C3×#327—21no—
34#3003—31no—
35#2976—31no—
36#1786—31no—
37#1783—31no—
38#1513D311yescharacteristic, complemented
39#1305—31no—
40#1293—31no—
41C24×D3—21no—
42C2×C3×#41D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C22×D12—21no—
45C22×D12—21no—
46C22×D12—21no—
47C22×D12—21no—
48C22×D12—21no—
49C22×D12—42no—
50C22×D12—42no—
51C22×D12C2×C412yes—
52C2×C3×M16—21no—
53C3×#123—21no—
54C3×C22⋊Q8C2×C411yescharacteristic
55C3×C22⋊Q8D411yescharacteristic
56C3×#121C2×C411yescharacteristic
57C3×#121D411yescharacteristic
58C3×#120—21no—
59C3×C4×D4—21no—
60C3×C4×D4—21no—
61C3×#117D411yescharacteristic
62C2×C3×C4⋊C4—21no—
63C2×C3×C22⋊C4—21no—
64#949—42no—
65#949—42no—
66C2×#215—22no—
67C2×#213—22no—
68C2×#208—21no—
69C2×#208—21no—
70C2×#208D411yescharacteristic
71C2×#208D411yescharacteristic
72C22×C3⋊C8—21no—
73C22×C3⋊C8D411yescharacteristic
74C22×C3⋊C8D411yescharacteristic
75C3×C22⋊C8—21no—
76C3×C22⋊C8—21no—
77C3×#95—21no—
78#830—21no—
79#830—21no—
80#830—21no—
81#830—21no—
82#830—21no—
83#830—21no—
84#830—21no—
85#830C2×C412yes—
86#823—42no—
87#823—42no—
88#809—22no—
89#809—22no—
90#809—42no—
91#809—42no—
92C4⋊(C3⋊C8)—21no—
93C4⋊(C3⋊C8)—21no—
94#803—21no—
95C4×C3⋊C8—21no—
96C23×D4—31no—
97C2×AGL(1,ℤ/8)—61no—
98#525C2×D311yescharacteristic
99#450—32no—
100#450—61no—
101#447—32no—
102#447—61no—
103#436—61no—
104#435—61no—
105#435—61no—
106C8×D4—61no—
107#433—61no—
108#418—61no—
109C2×#102—61no—
110C2×#102—61no—
111C2×#100—61no—
112C2×#100—61no—
113#408—61no—
114#407—31no—
115#407—31no—
116#407C2×D311yescharacteristic
117C2×C22⋊C8—31no—
118#380C2×D311yescharacteristic
119#328—21no—
120#327—21no—
121C24×C3#4111yescharacteristic
122C23×D3—22no—
123C23×D3—44no—
124C23×D3—41no—
125C23×D3—41no—
126C23×D3—81no—
127C23×D3—81no—
128C23×D3—82no—
129C3×#41—41no—
130C3×#41—41no—
131C2×C3×Q8C22⋊C411yescharacteristic
132C2×C3×D4—21no—
133C2×C3×D4—21no—
134C2×C3×D4—21no—
135C2×C3×D4C22⋊C411yescharacteristic
136C22×C3×C4—21no—
137C22×C3×C4—41no—
138C22×C3×C4—41no—
139C22×C3×C4—41no—
140C22×C3×C4C22×C411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4#4111yescharacteristic
146C2×D12—22no—
147C2×D12—22no—
148C2×D12—22no—
149C2×D12—22no—
150C2×D12—22no—
151C2×D12—22no—
152C2×D12—22no—
153C2×D12—22no—
154C2×D12—44no—
155C2×D12—44no—
156C2×D12—44no—
157C2×D12—41no—
158C2×D12—41no—
159C2×D12—41no—
160C2×D12—41no—
161C2×D12—41no—
162C2×D12—41no—
163C2×D12—41no—
164C2×D12—41no—
165C3×M16—41no—
166C3×M16—41no—
167C2×C3×C8—41no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—41no—
171C3×C4⋊C4—41no—
172C3×C4⋊C4C22⋊C411yescharacteristic
173C3×C4⋊C4C22⋊C411yescharacteristic
174C3×C22⋊C4—21no—
175C3×C22⋊C4—21no—
176C3×C22⋊C4—21no—
177C3×C22⋊C4—21no—
178C3×C22⋊C4—21no—
179C3×C22⋊C4—81no—
180C3×C42—21no—
181#215—44no—
182#213—44no—
183#208—41no—
184#208—41no—
185#208—41no—
186#208—41no—
187#208—81no—
188C2×C3⋊C8—22no—
189C2×C3⋊C8—21no—
190C2×C3⋊C8—21no—
191C2×C3⋊C8—21no—
192C2×C3⋊C8—21no—
193C2×C3⋊C8—41no—
194C2×C3⋊C8—41no—
195C2×C3⋊C8—81no—
196C25—61no—
197C2×#41D1211yescharacteristic
198C22×D4—32no—
199C22×D4—61no—
200C22×D4—61no—
201C22×D4—61no—
202C22×D4—61no—
203C22×D4—61no—
204C22×D4—122no—
205C22×D4—122no—
206C23×C4C22×D311yescharacteristic
207AGL(1,ℤ/8)—122no—
208AGL(1,ℤ/8)—122no—
209C2×SD16—62no—
210C2×D8—62no—
211C2×M16—21no—
212C2×M16—31no—
213C2×M16—31no—
214C2×M16—61no—
215C2×M16—61no—
216C22×C8—31no—
217C22×C8—31no—
218C22×C8—61no—
219#123—21no—
220C22⋊Q8C4×D311yescharacteristic
221C22⋊Q8#6711yescharacteristic
222#121C4×D311yescharacteristic
223#121#6711yescharacteristic
224#120—21no—
225C4×D4—21no—
226C4×D4—21no—
227#117D1211yescharacteristic
228C2×C4⋊C4—21no—
229C2×C22⋊C4—21no—
230C4⋊C8—61no—
231C4⋊C8—61no—
232#102—62no—
233#102—62no—
234#102—122no—
235#102—122no—
236#100—122no—
237#100—122no—
238C22⋊C8—21no—
239C22⋊C8—21no—
240C22⋊C8—32no—
241C22⋊C8—61no—
242C22⋊C8—61no—
243C22⋊C8—61no—
244C22⋊C8—61no—
245C22⋊C8—61no—
246C22⋊C8—61no—
247C22⋊C8—61no—
248#97—61no—
249C4×C8—61no—
250#95—21no—
251C23×C3—21no—
252C23×C3—21no—
253C23×C3—21no—
254C23×C3—41no—
255C23×C3—41no—
256C23×C3C4×D411yescharacteristic
257C23×C3#12111yescharacteristic
258C23×C3#12311yescharacteristic
259C22×D3—24no—
260C22×D3—42no—
261C22×D3—42no—
262C22×D3—42no—
263C22×D3—42no—
264C22×D3—42no—
265C22×D3—42no—
266C22×D3—82no—
267C22×D3—81no—
268C22×D3—81no—
269C22×D3—82no—
270C22×D3—88no—
271C22×D3—88no—
272C22×D3—81no—
273C22×D3—81no—
274C22×D3—82no—
275C22×D3—82no—
276C3×Q8—41no—
277C3×D4—41no—
278C3×D4—41no—
279C3×D4—41no—
280C3×D4—81no—
281C3×D4—81no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—81no—
297C2×C3×C4—81no—
298C2×C3×C4—81no—
299C2×C3×C4C2×C22⋊C411yescharacteristic
300C2×C3×C4C4×D411yescharacteristic
301C2×C3×C4#12011yescharacteristic
302C2×C3×C4#12111yescharacteristic
303D12—24no—
304D12—24no—
305D12—42no—
306D12—42no—
307D12—42no—
308D12—42no—
309D12—42no—
310D12—42no—
311C3×C8—81no—
312C3⋊C8—42no—
313C3⋊C8—41no—
314C3⋊C8—41no—
315C3⋊C8—41no—
316C3⋊C8—41no—
317C24—62no—
318C24—124no—
319C24—121no—
320C24—121no—
321C24—241no—
322C24—241no—
323C24—242no—
324C24#23711yescharacteristic
325#41—41no—
326#41—41no—
327C2×Q8(C2×D3)⋊C411yescharacteristic
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—62no—
332C2×D4—62no—
333C2×D4—62no—
334C2×D4—62no—
335C2×D4—62no—
336C2×D4—62no—
337C2×D4—62no—
338C2×D4—62no—
339C2×D4—121no—
340C2×D4—121no—
341C2×D4—121no—
342C2×D4—121no—
343C2×D4—121no—
344C2×D4—121no—
345C2×D4—124no—
346C2×D4—124no—
347C2×D4—124no—
348C2×D4—121no—
349C2×D4—121no—
350C2×D4(C2×D3)⋊C411yescharacteristic
351C22×C4—21no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4C2×C4×D311yescharacteristic
356C22×C4C2×D1211yescharacteristic
357C22×C4D3×D411yescharacteristic
358C22×C4D3×D411yescharacteristic
359C22×C4#23711yescharacteristic
360C22×C4C2×#6711yescharacteristic
361SD16—124no—
362D8—124no—
363M16—41no—
364M16—41no—
365M16—121no—
366M16—121no—
367M16—121no—
368M16—121no—
369M16—241no—
370C2×C8—41no—
371C2×C8—61no—
372C2×C8—61no—
373C2×C8—61no—
374C2×C8—61no—
375C2×C8—62no—
376C2×C8—121no—
377C2×C8—121no—
378C2×C8—241no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4(C2×D3)⋊C411yescharacteristic
384C4⋊C4(C2×D3)⋊C411yescharacteristic
385C22⋊C4—21no—
386C22⋊C4—21no—
387C22⋊C4—21no—
388C22⋊C4—21no—
389C22⋊C4—21no—
390C22⋊C4—81no—
391C42—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—42no—
401C22×C3—42no—
402C22×C3—41no—
403C22×C3—41no—
404C22×C3—41no—
405C22×C3#38611yescharacteristic
406C22×C3#47111yescharacteristic
407C22×C3#47211yescharacteristic
408C2×D3—44no—
409C2×D3—82no—
410C2×D3—82no—
411C2×D3—82no—
412C2×D3—82no—
413C2×D3—84no—
414C2×D3—84no—
415C2×D3—84no—
416C2×D3—82no—
417C2×D3—82no—
418C2×D3—84no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—41no—
423C3×C4—41no—
424C3×C4—41no—
425C3×C4—161no—
426C3×C4B(4,𝔽2)11yescharacteristic
427C3×C4B(4,𝔽2)11yescharacteristic
428C23—21no—
429C23—21no—
430C23—21no—
431C23—41no—
432C23—41no—
433C23—64no—
434C23—122no—
435C23—122no—
436C23—122no—
437C23—122no—
438C23—122no—
439C23—122no—
440C23—242no—
441C23—242no—
442C23—241no—
443C23—241no—
444C23—242no—
445C23—242no—
446C23—248no—
447C23—248no—
448C23—241no—
449C23—241no—
450C23#67⋊C411yescharacteristic
451C23#88611yescharacteristic
452C23#93911yescharacteristic
453Q8—41no—
454D4—41no—
455D4—41no—
456D4—41no—
457D4—64no—
458D4—64no—
459D4—81no—
460D4—81no—
461D4—122no—
462D4—122no—
463D4—122no—
464D4—122no—
465D4—122no—
466D4—122no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4#67⋊C411yescharacteristic
485C2×C4#88211yescharacteristic
486C2×C4C2×(C2×D3)⋊C411yescharacteristic
487C2×C4#93911yescharacteristic
488C8—81no—
489C8—122no—
490C8—121no—
491C8—121no—
492C8—121no—
493C8—121no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—41no—
497C2×C3—41no—
498C2×C3—81no—
499C2×C3#188911yescharacteristic
500C2×C3#190311yescharacteristic
501C2×C3#190711yescharacteristic
502D3—84no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—42no—
512C22—42no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—124no—
517C22—242no—
518C22—242no—
519C22—242no—
520C22—242no—
521C22—244no—
522C22—244no—
523C22—244no—
524C22—244no—
525C22—242no—
526C22—242no—
527C22#495811yescharacteristic
528C22#519911yescharacteristic
529C22#520311yescharacteristic
530C4—21no—
531C4—21no—
532C4—21no—
533C4—41no—
534C4—41no—
535C4—41no—
536C4—161no—
537C4#474111yescharacteristic
538C4#474111yescharacteristic
539C3#952011yescharacteristic, complemented, Sylow
540C2—21no—
541C2—21no—
542C2—41no—
543C2—41no—
544C2—81no—
545C2—244no—
546C2#7002211yescharacteristic
547C2#7030311yescharacteristic
548C2#7040011yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1513C2
socleC22×C3#386
prime radical#82927C2
cosocleC23×C3×C4C23
hypercenter#1513D3
hypocenterC3#9520
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1513

Complements

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

NH ≅ G/Ntype
2. #82919C2semidirect
3. C3×#1513C2semidirect
4. #69988C2semidirect
5. #69521C2semidirect
6. #67687C2semidirect
7. #67626C2semidirect
21. #5046C4semidirect
23. #5036C4semidirect
38. #1513D3semidirect
539. C3#9520semidirect

Maximal subgroups

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

subgroupindex
#829272
#829192
C3×#15132
#699882
#695212
#676872
#676262
#95203