Finite Group Databasebeta

← #10695916

Subgroups — #10695916

Order 768.

686 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 97 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#83298C211yesmaximal, characteristic, complemented
2#83297C211yesmaximal, characteristic, complemented
3#74239C211yesmaximal, characteristic, complemented
4C3×#1517C211yesmaximal, characteristic, complemented
5#70762C211yesmaximal, characteristic, complemented
6#70756C211yesmaximal, characteristic, complemented
7#67631C211yesmaximal, characteristic, complemented
8#9996—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#949—21no—
11C2×#895—21no—
12C2×C4×D12C2211yescharacteristic
13C2×C4×D12C2211yescharacteristic
14C3×#431C2211yescharacteristic
15C3×#382C2211yescharacteristic
16#5232—21no—
17#5087—21no—
18#5083—21no—
19#5072—21no—
20#5072—21no—
21#5057—22no—
22#5036—21no—
23#5033—21no—
24C4×#213—22no—
25#4987—21no—
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28#4674—21no—
29#4671—21no—
30#4665—22no—
31C3×#327—22no—
32#4483—22no—
33#4459—22no—
34#3100—31no—
35#3036—31no—
36#2535—31no—
37#1517D311yescharacteristic, complemented
38#1486—31no—
39#1478—31no—
40#1286—31no—
41C2×C3×#41—21no—
42C22×D12—22no—
43C22×D12D411yescharacteristic
44C22×D12D411yescharacteristic
45C22×C4×D3—21no—
46C22×C4×D3—21no—
47C3×C4⋊Q8D411yescharacteristic, complemented
48C3×#127D411yescharacteristic, complemented
49C3×#124—21no—
50C3×#121—21no—
51C3×#121—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C42C2311yescharacteristic
56#949—42no—
57#949—42no—
58C2×#215—22no—
59C2×#213—22no—
60C2×#213—22no—
61C2×(C2×D3)⋊C4—22no—
62C2×(C2×D3)⋊C4—21no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—22no—
65C2×C4⋊Dic3D411yescharacteristic
66C2×C4⋊Dic3D411yescharacteristic
67C2×#208—21no—
68#895—42no—
69#895—42no—
70C4×D12—21no—
71C4×D12—21no—
72C4×D12—21no—
73C4×D12—21no—
74C4×D12—21no—
75C4×D12—21no—
76C4×D12—21no—
77C4×D12—21no—
78C3×C4⋊C8—21no—
79C3×C22⋊C8—21no—
80C3×C4×C8—21no—
81C3×#95—21no—
82C3×#95D412yes—
83#835—22no—
84#832—22no—
85#832—22no—
86C3⋊#95—22no—
87#830—21no—
88#809—22no—
89#807—22no—
90C4⋊(C3⋊C8)—21no—
91C4⋊(C3⋊C8)—21no—
92C4⋊(C3⋊C8)D412yescomplemented
93C4×C3⋊C8D412yescomplemented
94C2×AGL(1,ℤ/8)—61no—
95#532C2×D311yescharacteristic
96C2×#121—61no—
97C2×C4×D4—31no—
98C2×C4×D4—31no—
99#480—61no—
100#474—61no—
101#463—62no—
102#461—61no—
103#459—61no—
104#459—61no—
105#447—61no—
106C4×D8—62no—
107#431—31no—
108#431C2×D311yescharacteristic
109#423—31no—
110#417—61no—
111#397—61no—
112#392—61no—
113#386—62no—
114#382C2×D311yescharacteristic
115#380—61no—
116#327—22no—
117D4⋊C8—62no—
118D4⋊C8—62no—
119C23×D3—21no—
120C23×D3—21no—
121C3×#41—41no—
122C3×#41—41no—
123C3×#41—41no—
124C3×#41—41no—
125C2×C3×Q8—21no—
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—21no—
129C2×C3×D4—41no—
130C22×C3×C4—22no—
131C22×C3×C4—41no—
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×Dic3—21no—
136C22×Dic3—21no—
137C2×D12—21no—
138C2×D12—21no—
139C2×D12—21no—
140C2×D12—21no—
141C2×D12—21no—
142C2×D12—21no—
143C2×D12—21no—
144C2×D12—42no—
145C2×D12—44no—
146C2×D12—44no—
147C2×D12—82no—
148C2×D12D812yescomplemented
149C2×C4×D3—41no—
150C2×C4×D3—41no—
151C2×C4×D3—41no—
152C2×C4×D3—41no—
153C2×C4×D3—41no—
154C2×C4×D3—41no—
155C2×C4×D3—41no—
156C2×C4×D3—41no—
157C2×C4×D3—82no—
158C2×C3×C8—41no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—41no—
162C3×C4⋊C4—41no—
163C3×C22⋊C4—41no—
164C3×C22⋊C4—41no—
165C3×C42C2×D411yescharacteristic
166C3×C42C2×D411yescharacteristic
167C3×C42C2×D411yescharacteristic
168C3×C42#4111yescharacteristic
169#215—44no—
170#213—44no—
171#213—44no—
172(C2×D3)⋊C4—44no—
173(C2×D3)⋊C4—41no—
174(C2×D3)⋊C4—41no—
175(C2×D3)⋊C4—41no—
176(C2×D3)⋊C4—41no—
177(C2×D3)⋊C4—82no—
178C4⋊Dic3—21no—
179C4⋊Dic3—21no—
180C4⋊Dic3—21no—
181C4⋊Dic3—21no—
182#208—41no—
183#208—41no—
184C2×C3⋊C8—22no—
185C2×C3⋊C8—22no—
186C2×#41—21no—
187C22×D4—31no—
188C22×D4—31no—
189C22×D4—62no—
190C23×C4—61no—
191C23×C4—61no—
192AGL(1,ℤ/8)—122no—
193AGL(1,ℤ/8)—122no—
194C2×SD16—62no—
195C2×D8—62no—
196C2×D8—62no—
197C2×M16—61no—
198C4⋊Q8D1211yescharacteristic, complemented
199#127D1211yescharacteristic, complemented
200#124—21no—
201#121—21no—
202#121—21no—
203#121—122no—
204#121—122no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—61no—
208C4×D4—61no—
209C4×D4—61no—
210C4×D4—61no—
211C4×D4—61no—
212C4×D4—61no—
213C4×D4—61no—
214C4×D4—61no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—31no—
217C2×C4⋊C4—31no—
218C2×C22⋊C4—62no—
219C2×C22⋊C4—61no—
220C2×C22⋊C4—61no—
221C2×C22⋊C4—62no—
222C2×C42C22×D311yescharacteristic
223#107—62no—
224C4⋊C8—21no—
225C4⋊C8—32no—
226C4⋊C8—61no—
227C4⋊C8—61no—
228#103—62no—
229#102—62no—
230#102—62no—
231#102—62no—
232C22⋊C8—21no—
233C22⋊C8—61no—
234C4×C8—21no—
235C4×C8—32no—
236#95—21no—
237#95—62no—
238#95#6712yes—
239C23×C3—41no—
240C23×C3#12011yescharacteristic
241C22×D3—22no—
242C22×D3—22no—
243C22×D3—41no—
244C22×D3—42no—
245C22×D3—42no—
246C22×D3—41no—
247C22×D3—41no—
248C22×D3—41no—
249C22×D3—41no—
250C22×D3—41no—
251C22×D3—41no—
252C22×D3—41no—
253C3×Q8—41no—
254C3×Q8—41no—
255C3×D4—42no—
256C3×D4—41no—
257C3×D4—41no—
258C3×D4—81no—
259C3×D4—81no—
260C3×D4—81no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—41no—
266C2×C3×C4—42no—
267C2×C3×C4—81no—
268C2×C3×C4—81no—
269C2×C3×C4#12011yescharacteristic
270C2×C3×C4#12011yescharacteristic
271C2×C3×C4#12111yescharacteristic
272C2×C3×C4#12111yescharacteristic
273C2×C3×C4#12111yescharacteristic
274C2×C3×C4#12711yescharacteristic
275C2×C3×C4C2×D811yescharacteristic
276C2×C3×C4#13511yescharacteristic
277C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
278C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
279C2×Dic3—42no—
280C2×Dic3—41no—
281C2×Dic3—41no—
282C2×Dic3—41no—
283C2×Dic3—41no—
284D12—22no—
285D12—22no—
286D12—22no—
287D12—22no—
288D12—41no—
289D12—41no—
290D12—41no—
291D12—41no—
292D12—88no—
293C4×D3—82no—
294C4×D3—82no—
295C4×D3—81no—
296C4×D3—81no—
297C4×D3—81no—
298C4×D3—81no—
299C3×C8—81no—
300C3⋊C8—42no—
301C3⋊C8—42no—
302C24—61no—
303C24—61no—
304#41—41no—
305#41—41no—
306#41—41no—
307#41—41no—
308C2×Q8—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—32no—
313C2×D4—41no—
314C2×D4—61no—
315C2×D4—61no—
316C2×D4—61no—
317C2×D4—61no—
318C2×D4—61no—
319C2×D4—61no—
320C2×D4—61no—
321C2×D4—122no—
322C2×D4—124no—
323C2×D4—124no—
324C2×D4—242no—
325C22×C4—22no—
326C22×C4—41no—
327C22×C4—61no—
328C22×C4—61no—
329C22×C4—121no—
330C22×C4—121no—
331C22×C4—121no—
332C22×C4—121no—
333C22×C4—121no—
334C22×C4—121no—
335C22×C4—121no—
336C22×C4—121no—
337C22×C4—242no—
338C22×C4D3×D411yescharacteristic
339C22×C4D3×D411yescharacteristic
340C22×C4C2×#6711yescharacteristic
341SD16—124no—
342D8—124no—
343D8—124no—
344M16—121no—
345M16—121no—
346C2×C8—41no—
347C2×C8—62no—
348C2×C8—62no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—61no—
354C4⋊C4—61no—
355C4⋊C4—61no—
356C4⋊C4—61no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—124no—
360C22⋊C4—121no—
361C22⋊C4—121no—
362C22⋊C4—121no—
363C22⋊C4—121no—
364C22⋊C4—242no—
365C42C2×D1211yescharacteristic
366C42#23511yescharacteristic
367C42D3×D411yescharacteristic
368C42D3×D411yescharacteristic
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—81no—
372C22×C3—81no—
373C22×C3—81no—
374C22×C3#39211yescharacteristic
375C22×C3#44711yescharacteristic
376C22×C3#44911yescharacteristic
377C2×D3—42no—
378C2×D3—44no—
379C2×D3—42no—
380C2×D3—42no—
381C2×D3—42no—
382C2×D3—81no—
383C2×D3—82no—
384C2×D3—81no—
385C2×D3—81no—
386C2×D3—81no—
387C2×D3—81no—
388C2×D3—81no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—82no—
394C3×C4—81no—
395C3×C4#44711yescharacteristic
396C3×C4#44911yescharacteristic
397C3×C4#46611yescharacteristic
398C3×C4#46911yescharacteristic
399Dic3—81no—
400Dic3—81no—
401C23—41no—
402C23—62no—
403C23—62no—
404C23—121no—
405C23—122no—
406C23—122no—
407C23—121no—
408C23—121no—
409C23—121no—
410C23—121no—
411C23—121no—
412C23—121no—
413C23—121no—
414C23#93711yescharacteristic
415Q8—41no—
416Q8—41no—
417D4—42no—
418D4—41no—
419D4—41no—
420D4—62no—
421D4—62no—
422D4—62no—
423D4—62no—
424D4—81no—
425D4—81no—
426D4—81no—
427D4—121no—
428D4—121no—
429D4—121no—
430D4—121no—
431D4—248no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—122no—
441C2×C4—121no—
442C2×C4—121no—
443C2×C4—121no—
444C2×C4—121no—
445C2×C4—242no—
446C2×C4—242no—
447C2×C4—241no—
448C2×C4—241no—
449C2×C4—241no—
450C2×C4—241no—
451C2×C4#88211yescharacteristic
452C2×C4#88211yescharacteristic
453C2×C4#88411yescharacteristic
454C2×C4#88411yescharacteristic
455C2×C4D3×D811yescharacteristic
456C2×C4#91111yescharacteristic
457C2×C4#91411yescharacteristic
458C2×C4#91611yescharacteristic
459C2×C4#93011yescharacteristic
460C2×C4#94011yescharacteristic
461C8—81no—
462C8—122no—
463C8—122no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—161no—
467C2×C3#199911yescharacteristic
468C2×C3#200811yescharacteristic
469C2×C3#201011yescharacteristic
470D3—82no—
471D3—82no—
472C22—21no—
473C22—21no—
474C22—81no—
475C22—81no—
476C22—81no—
477C22—122no—
478C22—124no—
479C22—122no—
480C22—122no—
481C22—122no—
482C22—241no—
483C22—242no—
484C22—241no—
485C22—241no—
486C22—241no—
487C22—241no—
488C22—241no—
489C22#496011yescharacteristic
490C22#515611yescharacteristic
491C22#517211yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—82no—
497C4—81no—
498C4—241no—
499C4—241no—
500C4#477311yescharacteristic
501C4#477511yescharacteristic
502C4#478111yescharacteristic
503C4#478211yescharacteristic
504C3#999611yescharacteristic, complemented, Sylow
505C2—21no—
506C2—21no—
507C2—161no—
508C2—242no—
509C2—242no—
510C2#7005011yescharacteristic
511C2#7020411yescharacteristic
512C2#7040611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1517C2
socleC22×C3#392
hypercenter#1517D3
hypocenterC3#9996
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1517

Complements

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

NH ≅ G/Ntype
1. #83298C2semidirect
2. #83297C2semidirect
3. #74239C2semidirect
4. C3×#1517C2semidirect
5. #70762C2semidirect
6. #70756C2semidirect
7. #67631C2semidirect
37. #1517D3semidirect
47. C3×C4⋊Q8D4semidirect
48. C3×#127D4semidirect
92. C4⋊(C3⋊C8)D4semidirect
93. C4×C3⋊C8D4semidirect
148. C2×D12D8semidirect
198. C4⋊Q8D12semidirect
199. #127D12semidirect
504. C3#9996semidirect

Maximal subgroups

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

subgroupindex
#832982
#832972
#742392
C3×#15172
#707622
#707562
#676312
#99963