Finite Group Databasebeta

← #11787225

Subgroups — #11787225

Order 1152.

917 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 917 conjugacy classes of subgroups — too many to draw. Only its 134 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#10595995C211yesmaximal, characteristic
2C2×#63818C211yesmaximal, characteristic, complemented
3C4×#3722C211yesmaximal, characteristic, complemented
4C2×#63570C212yesmaximal, complemented
5C3×#4532C212yesmaximal, complemented
6#68755—32nomaximal
7C2×#3828C2211yescharacteristic
8C22×C4×C3⋊D3C2211yescharacteristic
9C32×#117C2211yescharacteristic
10#64106C411yescharacteristic, complemented
11#64106C411yescharacteristic, complemented
12#64104C411yescharacteristic, complemented
13#64104C411yescharacteristic, complemented
14#64095—21no—
15#64091—21no—
16C4×C4⋊(C3⋊D3)—21no—
17C4×C4⋊(C3⋊D3)—21no—
18C2×C3×C4×Dic3C2212yes—
19C2×C3×#208C2212yes—
20C2×#3722—21no—
21C2×#3712—21no—
22#63818—24no—
23C2×#3710—21no—
24#63579—21no—
25#63570—24no—
26C2×D3×M16—32no—
27#5524—32no—
28#5524—61no—
29C2×#825—32no—
30C4×(C2×D3)⋊C4—32no—
31C2×#805—32no—
32C3×#339—32no—
33#4532D312yescomplemented
34C23×C3⋊D3D411yescharacteristic
35C22×C32×C4C2311yescharacteristic
36C2×#638—21no—
37C22×C3⋊Dic3D411yescharacteristic
38#3828—22no—
39#3828—22no—
40C2×C4⋊(C3⋊D3)—21no—
41C2×C4⋊(C3⋊D3)—21no—
42C2×C4⋊(C3⋊D3)C2×C411yescharacteristic
43C2×C4×C3⋊D3—22no—
44C2×C4×C3⋊D3—22no—
45C2×C4×C3⋊D3—21no—
46C2×C4×C3⋊D3C2×C411yescharacteristic
47C2×C4×C3⋊D3C2×C411yescharacteristic
48C2×C4×C3⋊D3D411yescharacteristic
49C2×C4×C3⋊D3D411yescharacteristic
50C2×#634C2×C411yescharacteristic
51C22×C3×Dic3—22no—
52C32×C4⋊C4C2×C411yescharacteristic
53C32×C4⋊C4C2×C411yescharacteristic
54C32×C22⋊C4—21no—
55C32×C42—21no—
56#3752—21no—
57#3752—21no—
58#3752—21no—
59#3752—21no—
60C4⋊(C3⋊Dic3)—21no—
61C4⋊(C3⋊Dic3)—21no—
62C3×C4×Dic3—22no—
63C3×C4×Dic3—22no—
64C3×#208—22no—
65C3×#208D414yescomplemented
66C2×C3×C3⋊C8—22no—
67#3722—24no—
68#3712—24no—
69#3710—24no—
70#1871—91noSylow
71C2×#235—32no—
72C2×#235—61no—
73C22×C4×D3—32no—
74C22×C4×D3—61no—
75C2×C3×M16—32no—
76C3×#117—21no—
77C3×#117C2×D312yes—
78C2×C3×C42—32no—
79C2×(C2×D3)⋊C4—62no—
80C2×C4×Dic3C2×D312yes—
81C2×#208C2×D312yes—
82D3×M16—64no—
83D3×M16—64no—
84C2×ΓL(1,𝔽25)—62no—
85C2×D3×C8—62no—
86#897—32no—
87#897—32no—
88#897—61no—
89#897—61no—
90#895—32no—
91#895—32no—
92#895—61no—
93#895—61no—
94#886—62no—
95#886—62no—
96#882—62no—
97#882—62no—
98C4×D12—62no—
99C4×D12—62no—
100C4×D12—62no—
101C4×D12—62no—
102C3⋊#95—62no—
103#825—64no—
104#805—64no—
105C23×C32C2×D411yescharacteristic
106C22×C3⋊D3—21no—
107C22×C3⋊D3—22no—
108C22×C3⋊D3—22no—
109C22×C3⋊D3—41no—
110C22×C3⋊D3C22⋊C411yescharacteristic
111C22×C3⋊D3C22⋊C411yescharacteristic
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4C22×C411yescharacteristic
115C2×C32×C4C2×D412yes—
116C2×C32×C4C2×D411yescharacteristic
117C2×C32×C4#4112yes—
118#638—42no—
119C2×C3⋊Dic3—21no—
120C2×C3⋊Dic3—22no—
121C2×C3⋊Dic3C22⋊C411yescharacteristic
122C2×C3⋊Dic3C22⋊C411yescharacteristic
123C4⋊(C3⋊D3)—22no—
124C4⋊(C3⋊D3)—42no—
125C4⋊(C3⋊D3)—42no—
126C4×C3⋊D3—22no—
127C4×C3⋊D3—22no—
128C4×C3⋊D3—21no—
129C4×C3⋊D3—21no—
130C4×C3⋊D3—21no—
131C4×C3⋊D3—21no—
132C4×C3⋊D3—41no—
133C4×C3⋊D3—41no—
134#634—22no—
135C2×C3×Dic3—24no—
136C2×C3×Dic3—42no—
137C2×C3×Dic3—42no—
138C3×C3⋊C8—24no—
139C22×M16—91no—
140#525—91no—
141C2×#104—92no—
142C4×C22⋊C4—91no—
143#339—32no—
144C23×D3—32no—
145C23×D3—61no—
146C22×C3×C4—21no—
147C22×C3×C4—62no—
148C22×C3×C4C22×D312yes—
149C2×#67—62no—
150C2×#67—62no—
151C22×Dic3—22no—
152C22×Dic3—32no—
153C22×Dic3—61no—
154#235—62no—
155#235—62no—
156#235—62no—
157#235—62no—
158#235—122no—
159#235—122no—
160C2×D12—32no—
161C2×D12—61no—
162C2×D12—62no—
163C2×D12—62no—
164C2×D12—62no—
165C2×D12—62no—
166C2×C4×D3—32no—
167C2×C4×D3—32no—
168C2×C4×D3—32no—
169C2×C4×D3—32no—
170C2×C4×D3—61no—
171C2×C4×D3—61no—
172C2×C4×D3—61no—
173C2×C4×D3—61no—
174C2×C4×D3—62no—
175C2×C4×D3—62no—
176C2×C4×D3—62no—
177C2×C4×D3—62no—
178C2×C4×D3—62no—
179C2×C4×D3—62no—
180C2×C4×D3—122no—
181C2×C4×D3—122no—
182C2×Dic6—32no—
183C2×Dic6—61no—
184C3×M16—34no—
185C3×M16—62no—
186C2×C3×C8—62no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4C4×D312yes—
190C3×C4⋊C4C4×D312yes—
191C3×C22⋊C4—22no—
192C3×C22⋊C4—22no—
193C3×C42—22no—
194C3×C42—22no—
195C3×C42—62no—
196C3×C42—62no—
197(C2×D3)⋊C4—68no—
198(C2×D3)⋊C4—62no—
199(C2×D3)⋊C4—62no—
200(C2×D3)⋊C4—62no—
201(C2×D3)⋊C4—62no—
202(C2×D3)⋊C4—62no—
203(C2×D3)⋊C4—62no—
204(C2×D3)⋊C4—62no—
205(C2×D3)⋊C4—62no—
206C4⋊Dic3—62no—
207C4⋊Dic3—62no—
208C4⋊Dic3—62no—
209C4⋊Dic3—62no—
210C4×Dic3—22no—
211C4×Dic3—22no—
212#208—22no—
213#208D1214yescomplemented
214C2×C3⋊C8—22no—
215ΓL(1,𝔽25)—68no—
216D3×C8—68no—
217C22×C32—21no—
218C22×C32—21no—
219C22×C32C2×C22⋊C411yescharacteristic
220C22×C32#12112yes—
221C2×C3⋊D3—21no—
222C2×C3⋊D3—21no—
223C2×C3⋊D3—21no—
224C2×C3⋊D3—21no—
225C2×C3⋊D3—21no—
226C2×C3⋊D3—44no—
227C2×C3⋊D3—41no—
228C2×C3⋊D3—41no—
229C2×C3⋊D3—41no—
230C2×C3⋊D3#10412yescomplemented
231C32×C4—41no—
232C32×C4—41no—
233C32×C4C4×D412yes—
234C32×C4#12011yescharacteristic
235C32×C4#12311yescharacteristic
236C3⋊Dic3—21no—
237C3⋊Dic3—21no—
238C3⋊Dic3—41no—
239C3×Dic3—44no—
240C2×#41—91no—
241C23×C4—91no—
242C2×M16—32no—
243C2×M16—181no—
244C2×M16—184no—
245C22×C8—181no—
246#123—181no—
247C22⋊Q8—91no—
248C22⋊Q8—91no—
249#121—91no—
250#121—91no—
251#120—181no—
252C4×D4—181no—
253C4×D4—181no—
254#117D3211yescharacteristic
255C2×C22⋊C4—181no—
256C2×C42—32no—
257#104—184no—
258#95—181no—
259C23×C3—21no—
260C23×C3D3×D412yes—
261C22×D3—32no—
262C22×D3—32no—
263C22×D3—61no—
264C22×D3—61no—
265C22×D3—64no—
266C22×D3—62no—
267C22×D3—62no—
268C22×D3—62no—
269C22×D3—62no—
270C22×D3—122no—
271C22×D3—122no—
272C22×D3—122no—
273C22×D3—122no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—64no—
283C2×C3×C4—122no—
284C2×C3×C4—122no—
285C2×C3×C4C2×C4×D312yes—
286C2×C3×C4C2×D1212yes—
287C2×C3×C4#23512yes—
288C2×C3×C4D3×D412yes—
289C2×C3×C4D3×D412yes—
290C2×C3×C4#23912yes—
291#67—124no—
292#67—122no—
293#67—122no—
294C2×Dic3—24no—
295C2×Dic3—32no—
296C2×Dic3—32no—
297C2×Dic3—42no—
298C2×Dic3—42no—
299C2×Dic3—61no—
300C2×Dic3—61no—
301C2×Dic3—62no—
302C2×Dic3—62no—
303C2×Dic3—62no—
304C2×Dic3—62no—
305C2×Dic3—122no—
306D12—62no—
307D12—62no—
308D12—122no—
309D12—124no—
310D12—124no—
311D12—124no—
312D12—124no—
313C4×D3—62no—
314C4×D3—62no—
315C4×D3—62no—
316C4×D3—62no—
317C4×D3—62no—
318C4×D3—62no—
319C4×D3—62no—
320C4×D3—62no—
321C4×D3—122no—
322C4×D3—122no—
323C4×D3—121no—
324C4×D3—121no—
325C4×D3—121no—
326C4×D3—121no—
327C4×D3—122no—
328C4×D3—122no—
329C4×D3—122no—
330C4×D3—122no—
331Dic6—62no—
332Dic6—62no—
333Dic6—122no—
334C3×C8—64no—
335C3⋊C8—24no—
336C2×C32—22no—
337C2×C32#38611yescharacteristic
338C2×C32C2×#10411yescharacteristic
339C2×C32#42111yescharacteristic
340C3⋊D3—22no—
341C3⋊D3—41no—
342C3⋊D3—81no—
343C24—91no—
344#41—182no—
345#41—182no—
346C2×Q8—91no—
347C2×D4—91no—
348C2×D4—181no—
349C2×D4—181no—
350C2×D4—181no—
351C22×C4—62no—
352C22×C4—91no—
353C22×C4—91no—
354C22×C4—91no—
355C22×C4—91no—
356C22×C4—91no—
357C22×C4—182no—
358C22×C4—182no—
359C22×C4—181no—
360C22×C4C2×D3211yescharacteristic
361M16—34no—
362M16—62no—
363M16—184no—
364C2×C8—62no—
365C2×C8—184no—
366C4⋊C4—181no—
367C4⋊C4—181no—
368C4⋊C4#62411yescharacteristic
369C4⋊C4#62411yescharacteristic
370C22⋊C4—21no—
371C22⋊C4—181no—
372C22⋊C4—181no—
373C22⋊C4—181no—
374C22⋊C4—181no—
375C22⋊C4—184no—
376C42—21no—
377C42—62no—
378C42—62no—
379C22×C3—22no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—21no—
383C22×C3—41no—
384C22×C3—41no—
385C22×C3D3×C22⋊C412yes—
386C22×C3#88412yes—
387C22×C3#89512yes—
388C2×D3—34no—
389C2×D3—62no—
390C2×D3—62no—
391C2×D3—62no—
392C2×D3—62no—
393C2×D3—62no—
394C2×D3—62no—
395C2×D3—62no—
396C2×D3—121no—
397C2×D3—121no—
398C2×D3—121no—
399C2×D3—121no—
400C2×D3—124no—
401C2×D3—124no—
402C2×D3—128no—
403C2×D3—122no—
404C2×D3—122no—
405C2×D3—122no—
406C2×D3—122no—
407C2×D3—122no—
408C2×D3—122no—
409C3×C4—22no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—42no—
413C3×C4—42no—
414C3×C4—42no—
415C3×C4—42no—
416C3×C4—124no—
417C3×C4C4×D1212yes—
418C3×C4#88212yes—
419C3×C4#89312yes—
420C3×C4#89412yes—
421Dic3—44no—
422Dic3—62no—
423Dic3—62no—
424Dic3—121no—
425Dic3—121no—
426Dic3—122no—
427Dic3—122no—
428C32#187111yescharacteristic, complemented, Sylow
429C23—91no—
430C23—91no—
431C23—182no—
432C23—182no—
433C23—181no—
434C23—361no—
435C23#381111yescharacteristic
436Q8—182no—
437D4—182no—
438D4—362no—
439D4—362no—
440D4—362no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—64no—
444C2×C4—91no—
445C2×C4—91no—
446C2×C4—122no—
447C2×C4—122no—
448C2×C4—182no—
449C2×C4—182no—
450C2×C4—182no—
451C2×C4—181no—
452C2×C4—181no—
453C2×C4—181no—
454C2×C4—181no—
455C2×C4—181no—
456C2×C4—361no—
457C2×C4—361no—
458C2×C4#379912yes—
459C2×C4D3×D1212yes—
460C2×C4C2×#62411yescharacteristic
461C2×C4#381111yescharacteristic
462C8—64no—
463C2×C3—22no—
464C2×C3—22no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—42no—
469C2×C3#466512yes—
470C2×C3D3×#10412yes—
471C2×C3#482112yes—
472D3—64no—
473D3—122no—
474D3—122no—
475D3—122no—
476D3—242no—
477D3—242no—
478C22—21no—
479C22—21no—
480C22—92no—
481C22—181no—
482C22—181no—
483C22—181no—
484C22—181no—
485C22—181no—
486C22—364no—
487C22—361no—
488C22—361no—
489C22—361no—
490C22#6391012yes—
491C22#6397411yescharacteristic
492C4—41no—
493C4—41no—
494C4—124no—
495C4—181no—
496C4—181no—
497C4—361no—
498C4#6389412yes—
499C4#6389711yescharacteristic
500C4#6392311yescharacteristic
501C3—21no—
502C3#6875512yescomplemented
503C2—22no—
504C2—182no—
505C2—361no—
506C2—721no—
507C2#1059138011yescharacteristic
508C2#1059167111yescharacteristic
509C2#1059167211yescharacteristic
subgroupG/H
centerC2×C4#3811
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC32×#117C22
socleC22×C32C2×C22⋊C4
prime radical#10595995C2
cosocleC22×C32×C4C23
hypercenter#117D32
hypocenterC32#1871
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. C2×#63818C2semidirect
3. C4×#3722C2semidirect
4. C2×#63570C2semidirect
5. C3×#4532C2semidirect
10. #64106C4semidirect
11. #64106C4semidirect
12. #64104C4semidirect
13. #64104C4semidirect
33. #4532D3semidirect
65. C3×#208D4semidirect
213. #208D12semidirect
230. C2×C3⋊D3#104semidirect
428. C32#1871semidirect
502. C3#68755semidirect

Maximal subgroups

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

subgroupindex
#105959952
C2×#638182
C4×#37222
C2×#635702
C3×#45322
#687553