Finite Group Databasebeta

← #11779381

Subgroups — #11779381

Order 1152.

668 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 96 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#10596048C211yesmaximal, characteristic
2C3×#5021C211yesmaximal, characteristic, complemented
3#10590373C211yesmaximal, characteristic, complemented
4#10590300C211yesmaximal, characteristic, complemented
5C3×#4535C211yesmaximal, characteristic, complemented
6#10589091C211yesmaximal, characteristic, complemented
7#10589082C211yesmaximal, characteristic, complemented
8#71160—31nomaximal
9#68261—31nomaximal
10C2×#3832—21no—
11C32×C4×Q8C2211yescharacteristic
12#64104—22no—
13#64104—21no—
14#64083—21no—
15#64083C2211yescharacteristic
16C4×C4⋊(C3⋊D3)—21no—
17C4×C4⋊(C3⋊D3)C2211yescharacteristic
18C3×#869C2211yescharacteristic
19C2×#3717—22no—
20C3×#835—21no—
21C3×#807—21no—
22C3×C4⋊(C3⋊C8)C2211yescharacteristic
23C3×C4⋊(C3⋊C8)C2211yescharacteristic
24C3×C4×C3⋊C8C2211yescharacteristic
25#63573—21no—
26#63564—21no—
27#5577—31no—
28#5577—31no—
29#5577—61no—
30C3×#477—31no—
31#5085—31no—
32#5084—31no—
33#5021D311yescharacteristic, complemented
34#4842—31no—
35#4693—31no—
36C3×Q8⋊C8—31no—
37#4535D311yescharacteristic, complemented
38#4485—31no—
39#4485—31no—
40#4483—31no—
41#4459—31no—
42C2×C32×Q8D411yescharacteristic
43#3832—42no—
44#3832—42no—
45C2×C4⋊(C3⋊D3)—22no—
46C2×C4⋊(C3⋊D3)—21no—
47C2×C4⋊(C3⋊D3)—21no—
48C2×C4⋊(C3⋊D3)—22no—
49C2×C4⋊(C3⋊D3)D411yescharacteristic
50C2×C4⋊(C3⋊D3)D412yes—
51C2×C4×C3⋊D3—22no—
52C2×C4×C3⋊D3—21no—
53C2×C3×Dic6—21no—
54C32×C4⋊C4—21no—
55C32×C4⋊C4D411yescharacteristic
56C32×C42—21no—
57C32×C42C2311yescharacteristic
58#3752—22no—
59#3752—21no—
60C4⋊(C3⋊Dic3)D411yescharacteristic
61C3×C4⋊Dic3—21no—
62C3×C4⋊Dic3—21no—
63C2×C3×C3⋊C8—21no—
64C2×C3×C3⋊C8—21no—
65C2×C3×C3⋊C8—21no—
66#3717—44no—
67#1631—91noSylow
68C2×#239—62no—
69C2×#239—61no—
70C2×#239—61no—
71C3×C4⋊Q8—31no—
72C3×C4×Q8—21no—
73C3×C4×Q8C2×D311yescharacteristic
74C3×C4×Q8C2×D311yescharacteristic
75C2×#215—62no—
76C2×#204—62no—
77#895—62no—
78#895—62no—
79#895—64no—
80#895—62no—
81#895—61no—
82#895—61no—
83#874—31no—
84#874—31no—
85#874—61no—
86#874—61no—
87#874—61no—
88#874—62no—
89C4×D12—31no—
90C4×D12—31no—
91C4×D12—61no—
92C4×D12—61no—
93C4×D12—61no—
94C4×D12—62no—
95#869C2×D311yescharacteristic
96C3×#107—61no—
97C3×C4⋊C8—31no—
98C3×C4⋊C8—31no—
99C3×#103—61no—
100C3×C4×C8—31no—
101#835—21no—
102#821—61no—
103#817—61no—
104#809—61no—
105#808—61no—
106#807—21no—
107C4⋊(C3⋊C8)C2×D311yescharacteristic
108C4⋊(C3⋊C8)C2×D311yescharacteristic
109C4×C3⋊C8C2×D311yescharacteristic
110C22×C3⋊D3—21no—
111C22×C3⋊D3—22no—
112C32×Q8—21no—
113C32×Q8SD1612yescomplemented
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4C2×D411yescharacteristic
117C2×C32×C4C2×D411yescharacteristic
118C2×C32×C4C2×D411yescharacteristic
119C2×C3⋊Dic3—21no—
120C4⋊(C3⋊D3)—21no—
121C4⋊(C3⋊D3)—21no—
122C4⋊(C3⋊D3)—44no—
123C4⋊(C3⋊D3)—41no—
124C4⋊(C3⋊D3)—41no—
125C4⋊(C3⋊D3)—41no—
126C4⋊(C3⋊D3)—41no—
127C4⋊(C3⋊D3)—42no—
128C4⋊(C3⋊D3)—82no—
129C4×C3⋊D3—44no—
130C4×C3⋊D3—41no—
131C4×C3⋊D3—41no—
132C2×C3×Dic3—41no—
133C3×Dic6—42no—
134C3×C3⋊C8—22no—
135C3×C3⋊C8—41no—
136C3×C3⋊C8—41no—
137#550—91no—
138#492—91no—
139#477—31no—
140#476—91no—
141#331—91no—
142Q8⋊C8—31no—
143D4⋊C8—91no—
144C2×C3×Q8—21no—
145C2×C3×Q8—61no—
146C2×C3×Q8D1211yescharacteristic
147C2×C3×Q8#6711yescharacteristic
148#239—124no—
149#239—122no—
150#239—122no—
151#239—124no—
152#239—122no—
153#239—122no—
154C2×D12—31no—
155C2×D12—31no—
156C2×D12—32no—
157C2×D12—32no—
158C2×D12—61no—
159C2×D12—62no—
160C2×D12—62no—
161C2×D12—64no—
162C2×D12—62no—
163C2×D12—62no—
164C2×D12—62no—
165C2×D12—61no—
166C2×D12—61no—
167C2×D12—61no—
168C2×D12—61no—
169C2×D12—64no—
170C2×D12—62no—
171C2×D12—62no—
172C2×C4×D3—64no—
173C2×C4×D3—62no—
174C2×C4×D3—62no—
175C2×C4×D3—62no—
176C2×C4×D3—61no—
177C2×C4×D3—61no—
178C2×Dic6—21no—
179C2×C3×C8—61no—
180C2×C3×C8—61no—
181C2×C3×C8—61no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—61no—
187C3×C4⋊C4—61no—
188C3×C4⋊C4D1211yescharacteristic
189C3×C4⋊C4#6711yescharacteristic
190C3×C42—22no—
191C3×C42—21no—
192C3×C42—21no—
193C3×C42—21no—
194C3×C42C22×D311yescharacteristic
195C3×C42C22×D311yescharacteristic
196#215—124no—
197(C2×D3)⋊C4—64no—
198(C2×D3)⋊C4—62no—
199(C2×D3)⋊C4—62no—
200(C2×D3)⋊C4—62no—
201(C2×D3)⋊C4—61no—
202(C2×D3)⋊C4—61no—
203C4⋊Dic3—21no—
204C4⋊Dic3—21no—
205C4⋊Dic3—31no—
206C4⋊Dic3—31no—
207C4⋊Dic3—61no—
208C2×C3⋊C8—21no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—21no—
211#204—124no—
212C22×C32#12011yescharacteristic
213C2×C3⋊D3—42no—
214C2×C3⋊D3—41no—
215C2×C3⋊D3—41no—
216C2×C3⋊D3—42no—
217C2×C3⋊D3—42no—
218C2×C3⋊D3—41no—
219C32×C4—22no—
220C32×C4—21no—
221C32×C4—41no—
222C32×C4C2×SD1611yescharacteristic
223C32×C4#13511yescharacteristic
224C32×C4AGL(1,ℤ/8)11yescharacteristic
225C32×C4#13711yescharacteristic
226C3⋊Dic3—41no—
227C3×Dic3—81no—
228C2×#41—181no—
229C2×SD16—182no—
230C4⋊Q8—31no—
231#127—91no—
232#127—181no—
233#121—182no—
234#121—181no—
235C4×Q8D3211yescharacteristic
236C4×D4—91no—
237C4×D4—181no—
238#107—61no—
239#106—181no—
240C4⋊C8—31no—
241C4⋊C8—31no—
242#103—61no—
243#102—181no—
244C4×C8—31no—
245C22×D3—62no—
246C22×D3—64no—
247C22×D3—61no—
248C22×D3—62no—
249C22×D3—61no—
250C22×D3—62no—
251C3×Q8—22no—
252C3×Q8—21no—
253C3×Q8—21no—
254C3×Q8—22no—
255C3×Q8—122no—
256C3×Q8#20412yescomplemented
257C3×Q8#21512yescomplemented
258C2×C3×C4—22no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—22no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—121no—
268C2×C3×C4C2×D1211yescharacteristic
269C2×C3×C4D3×D411yescharacteristic
270C2×C3×C4D3×D411yescharacteristic
271C2×C3×C4D3×D411yescharacteristic
272C2×C3×C4D3×D411yescharacteristic
273C2×C3×C4C2×#6711yescharacteristic
274C2×Dic3—41no—
275C2×Dic3—62no—
276C2×Dic3—61no—
277C2×Dic3—61no—
278D12—61no—
279D12—61no—
280D12—61no—
281D12—61no—
282D12—121no—
283D12—121no—
284D12—122no—
285D12—122no—
286D12—122no—
287D12—122no—
288D12—128no—
289D12—124no—
290D12—124no—
291D12—122no—
292D12—122no—
293D12—121no—
294D12—121no—
295D12—121no—
296D12—121no—
297D12—121no—
298D12—121no—
299D12—121no—
300D12—121no—
301D12—242no—
302D12—244no—
303D12—242no—
304D12—242no—
305C4×D3—128no—
306C4×D3—124no—
307C4×D3—124no—
308C4×D3—122no—
309C4×D3—122no—
310C4×D3—121no—
311C4×D3—121no—
312C4×D3—121no—
313C4×D3—121no—
314Dic6—42no—
315C3×C8—62no—
316C3×C8—121no—
317C3×C8—121no—
318C3⋊C8—22no—
319C3⋊C8—41no—
320C3⋊C8—41no—
321C2×C32#45011yescharacteristic
322C2×C32#45211yescharacteristic
323C2×C32#45311yescharacteristic
324C3⋊D3—81no—
325C3⋊D3—82no—
326#41—362no—
327#41—362no—
328C2×Q8—61no—
329C2×Q8#62611yescharacteristic
330C2×D4—91no—
331C2×D4—92no—
332C2×D4—182no—
333C2×D4—182no—
334C2×D4—181no—
335C2×D4—181no—
336C22×C4—182no—
337C22×C4—181no—
338SD16—364no—
339C2×C8—61no—
340C2×C8—61no—
341C2×C8—61no—
342C4⋊C4—21no—
343C4⋊C4—61no—
344C4⋊C4—61no—
345C4⋊C4—91no—
346C4⋊C4#62611yescharacteristic
347C22⋊C4—182no—
348C22⋊C4—181no—
349C42—21no—
350C42C2×D3211yescharacteristic
351C22×C3—21no—
352C22×C3#88211yescharacteristic
353C22×C3#93711yescharacteristic
354C2×D3—124no—
355C2×D3—122no—
356C2×D3—122no—
357C2×D3—124no—
358C2×D3—124no—
359C2×D3—122no—
360C2×D3—121no—
361C2×D3—122no—
362C2×D3—122no—
363C2×D3—121no—
364C2×D3—122no—
365C2×D3—121no—
366C2×D3—122no—
367C2×D3—121no—
368C2×D3—121no—
369C2×D3—122no—
370C2×D3—122no—
371C2×D3—121no—
372C3×C4—24no—
373C3×C4—21no—
374C3×C4—22no—
375C3×C4—22no—
376C3×C4—21no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—41no—
382C3×C4—42no—
383C3×C4—41no—
384C3×C4—41no—
385C3×C4—241no—
386C3×C4C2×#20411yescharacteristic
387C3×C4#90811yescharacteristic
388C3×C4#91611yescharacteristic
389C3×C4#91811yescharacteristic
390C3×C4#91811yescharacteristic
391C3×C4#91911yescharacteristic
392C3×C4C2×#21511yescharacteristic
393C3×C4#94911yescharacteristic
394Dic3—81no—
395Dic3—122no—
396Dic3—121no—
397Dic3—121no—
398C32#163111yescharacteristic, complemented, Sylow
399C23—181no—
400C23—182no—
401Q8—21no—
402Q8—122no—
403Q8#371712yescomplemented
404D4—181no—
405D4—181no—
406D4—362no—
407D4—364no—
408D4—361no—
409D4—361no—
410D4—361no—
411D4—361no—
412D4—722no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—121no—
416C2×C4—181no—
417C2×C4—364no—
418C2×C4—361no—
419C2×C4—361no—
420C2×C4C2×#62611yescharacteristic
421C2×C4#381111yescharacteristic
422C2×C4#381111yescharacteristic
423C8—62no—
424C8—121no—
425C8—121no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—21no—
429C2×C3#473111yescharacteristic
430C2×C3#480811yescharacteristic
431C2×C3#482211yescharacteristic
432C2×C3#504611yescharacteristic
433C2×C3#518711yescharacteristic
434C2×C3#519811yescharacteristic
435D3—242no—
436D3—244no—
437D3—241no—
438D3—242no—
439D3—241no—
440D3—242no—
441C22—361no—
442C22—362no—
443C22—362no—
444C22—361no—
445C22—362no—
446C22—361no—
447C22#6398111yescharacteristic
448C4—21no—
449C4—22no—
450C4—41no—
451C4—241no—
452C4—361no—
453C4#6382511yescharacteristic
454C4C2×#371711yescharacteristic
455C4#6394511yescharacteristic
456C4#6395011yescharacteristic
457C3—21no—
458C3#6826111yescharacteristic, complemented
459C3#7116011yescharacteristic, complemented
460C2—721no—
461C2—722no—
462C2#1059118311yescharacteristic
463C2#1059160211yescharacteristic
464C2#1059166711yescharacteristic
subgroupG/H
centerC22#63981
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4×Q8C22
socleC22×C32#120
prime radical#10596048C2
hypercenterC4×Q8D32
hypocenterC32#1631
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
2. C3×#5021C2semidirect
3. #10590373C2semidirect
4. #10590300C2semidirect
5. C3×#4535C2semidirect
6. #10589091C2semidirect
7. #10589082C2semidirect
33. #5021D3semidirect
37. #4535D3semidirect
113. C32×Q8SD16semidirect
256. C3×Q8#204semidirect
257. C3×Q8#215semidirect
398. C32#1631semidirect
403. Q8#3717semidirect
458. C3#68261semidirect
459. C3#71160semidirect

Maximal subgroups

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

subgroupindex
#105960482
C3×#50212
#105903732
#105903002
C3×#45352
#105890912
#105890822
#711603
#682613