Finite Group Databasebeta

← #11920412

Subgroups — #11920412

Order 1152.

604 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 56 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#41×C22⋊C9C211yesmaximal, characteristic, complemented
2C2×C4⋊((C22⋊C9)⋊C2)C211yesmaximal, characteristic, complemented
3#10589519C212yesmaximal, complemented
4#10589517C212yesmaximal, complemented
5#10589511C211yesmaximal, characteristic, complemented
6#84082—31nomaximal
7C22×(C22⋊C9)⋊C2—21no—
8Q8×C22⋊C9C2211yescharacteristic, complemented
9D4×C22⋊C9—21no—
10D4×C22⋊C9C2211yescharacteristic, complemented
11C2×C4×C22⋊C9—21no—
12C2×C4×C22⋊C9C2211yescharacteristic
13C4⋊((C22⋊C9)⋊C2)—21no—
14C4⋊((C22⋊C9)⋊C2)C2212yes—
15#63512—41nomaximal
16(C22⋊C9)⋊C8C2212yescomplemented
17C22×C3×#41D311yescharacteristic
18C2×#949—31no—
19C2×#949—31no—
20C2×#930—31no—
21#5238—32no—
22#5238—32no—
23#5232—31no—
24#5232—31no—
25#5227—32no—
26#5217—32no—
27#5207—31no—
28C22×C22⋊C9—21no—
29C22×C22⋊C9D411yescharacteristic
30C2×(C22⋊C9)⋊C2—22no—
31C2×(C22⋊C9)⋊C2—42no—
32C9×#41—41no—
33C2×D36—41no—
34C4×C22⋊C9—21no—
35C4×C22⋊C9D411yescharacteristic
36C4×C22⋊C9C2311yescharacteristic
37#3675—42no—
38#3673—42no—
39#3667—41no—
40#3006—91noSylow
41C2×C3×#41—31no—
42C2×C3×#41—31no—
43C2×C3×#41—31no—
44C2×C3×#41—31no—
45C2×C3×#41—61no—
46C2×C3×#41—61no—
47C22×C3×Q8C2×D311yescharacteristic
48C22×C3×D4—21no—
49C22×C3×D4C2×D311yescharacteristic
50C23×C3×C4—21no—
51C23×C3×C4C2×D311yescharacteristic
52C22×#67—61no—
53C22×D12—31no—
54#949—62no—
55#949—62no—
56#949—61no—
57#949—61no—
58#949—61no—
59#949—61no—
60C2×#215—32no—
61C2×#215—32no—
62C2×#213—32no—
63C2×#213—32no—
64#930—32no—
65#930—32no—
66#930—61no—
67#930—61no—
68C2×(C2×D3)⋊C4—61no—
69C2×C4⋊Dic3—31no—
70C2×#208—31no—
71C2×#208—31no—
72#835—32no—
73#835—32no—
74#832—32no—
75#832—32no—
76#830—32no—
77#830—32no—
78C22×D9—81no—
79C2×C22⋊C9—21no—
80C2×C22⋊C9—41no—
81C2×C22⋊C9C2×D411yescharacteristic
82(C22⋊C9)⋊C2—42no—
83Q8×C9—41no—
84D4×C9—41no—
85D4×C9—81no—
86C2×C4×C9—41no—
87C2×C4×C9—81no—
88D36—42no—
89D36—81no—
90C9⋊C8—42no—
91C22×#41D911yescharacteristic, complemented
92C2×AGL(1,ℤ/8)—91no—
93C2×AGL(1,ℤ/8)—91no—
94C2×#121—91no—
95#449—92no—
96#449—92no—
97#448—92no—
98#447—92no—
99#417—91no—
100#417—91no—
101#407—91no—
102C24×C3—21no—
103C24×C3#6711yescharacteristic
104C23×D3—61no—
105C3×#41—32no—
106C3×#41—32no—
107C3×#41—31no—
108C3×#41—31no—
109C3×#41—31no—
110C3×#41—61no—
111C3×#41—61no—
112C3×#41—61no—
113C3×#41—62no—
114C3×#41—62no—
115C3×#41S411yescharacteristic
116C2×C3×Q8—31no—
117C2×C3×Q8—31no—
118C2×C3×Q8—61no—
119C2×C3×D4—31no—
120C2×C3×D4—31no—
121C2×C3×D4—61no—
122C2×C3×D4—61no—
123C2×C3×D4—61no—
124C2×C3×D4—61no—
125C2×C3×D4—61no—
126C22×C3×C4—21no—
127C22×C3×C4—31no—
128C22×C3×C4—31no—
129C22×C3×C4—31no—
130C22×C3×C4—31no—
131C22×C3×C4—61no—
132C22×C3×C4—61no—
133C22×C3×C4—61no—
134C22×C3×C4—61no—
135C22×C3×C4#6711yescharacteristic
136C22×C3×C4C22×D311yescharacteristic
137C2×#67—62no—
138C2×#67—62no—
139C2×#67—122no—
140C2×#67—122no—
141C22×Dic3—61no—
142C2×D12—32no—
143C2×D12—61no—
144C2×D12—61no—
145C2×D12—61no—
146C2×D12—61no—
147C2×D12—61no—
148#215—64no—
149#215—62no—
150#215—62no—
151#213—64no—
152#213—62no—
153#213—62no—
154(C2×D3)⋊C4—62no—
155(C2×D3)⋊C4—62no—
156C4⋊Dic3—32no—
157C4⋊Dic3—61no—
158#208—61no—
159#208—61no—
160#208—61no—
161#208—61no—
162C2×C3⋊C8—32no—
163C2×C3⋊C8—32no—
164C22×C9—41no—
165C22×C9—81no—
166C2×D9—82no—
167C2×D9—162no—
168C22⋊C9AGL(1,ℤ/8)11yescharacteristic, complemented
169C4×C9—41no—
170C4×C9—41no—
171C4×C9—81no—
172C2×#41—31no—
173C2×#41—31no—
174C2×#41—31no—
175C2×#41—31no—
176C2×#41—61no—
177C2×#41—61no—
178C22×Q8C2×D911yescharacteristic, complemented
179C22×D4—21no—
180C22×D4—91no—
181C22×D4—181no—
182C22×D4C2×D911yescharacteristic, complemented
183C23×C4—21no—
184C23×C4C2×D911yescharacteristic
185AGL(1,ℤ/8)—182no—
186AGL(1,ℤ/8)—182no—
187AGL(1,ℤ/8)—181no—
188AGL(1,ℤ/8)—181no—
189AGL(1,ℤ/8)—181no—
190AGL(1,ℤ/8)—181no—
191C2×SD16—92no—
192C2×SD16—92no—
193C2×D8—92no—
194C2×D8—92no—
195C2×M16—91no—
196C2×M16—91no—
197#121—92no—
198#121—92no—
199#121—181no—
200#121—181no—
201C2×C4⋊C4—91no—
202C2×C22⋊C4—181no—
203#103—92no—
204#103—92no—
205#102—92no—
206#102—92no—
207C22⋊C8—92no—
208C22⋊C8—92no—
209C23×C3—21no—
210C23×C3—31no—
211C23×C3—31no—
212C23×C3—41no—
213C23×C3—61no—
214C23×C3—61no—
215C23×C3—61no—
216C23×C3—121no—
217C23×C3C2×#6711yescharacteristic
218C22×D3—62no—
219C22×D3—121no—
220C22×D3—122no—
221C22×D3—122no—
222C22×D3—121no—
223C3×Q8—32no—
224C3×Q8—31no—
225C3×Q8—61no—
226C3×Q8C2×S411yescharacteristic
227C3×D4—21no—
228C3×D4—32no—
229C3×D4—31no—
230C3×D4—61no—
231C3×D4—61no—
232C3×D4—61no—
233C3×D4—62no—
234C3×D4—61no—
235C3×D4C2×S411yescharacteristic
236C2×C3×C4—21no—
237C2×C3×C4—31no—
238C2×C3×C4—31no—
239C2×C3×C4—31no—
240C2×C3×C4—31no—
241C2×C3×C4—31no—
242C2×C3×C4—31no—
243C2×C3×C4—31no—
244C2×C3×C4—61no—
245C2×C3×C4—61no—
246C2×C3×C4—61no—
247C2×C3×C4—61no—
248C2×C3×C4—62no—
249C2×C3×C4—61no—
250C2×C3×C4—61no—
251C2×C3×C4C2×S411yescharacteristic
252#67—122no—
253#67—124no—
254#67—122no—
255C2×Dic3—62no—
256C2×Dic3—122no—
257D12—62no—
258D12—62no—
259D12—121no—
260D12—121no—
261C3⋊C8—62no—
262C3⋊C8—62no—
263C2×C9—41no—
264C2×C9—81no—
265C2×C9—161no—
266D9—162no—
267C24—21no—
268C24—181no—
269C24#61111yescharacteristic
270#41—32no—
271#41—32no—
272#41—31no—
273#41—31no—
274#41—31no—
275#41—61no—
276#41—61no—
277#41—61no—
278#41—62no—
279#41—62no—
280#41(C22⋊C9)⋊C211yescharacteristic, complemented
281C2×Q8—31no—
282C2×Q8—31no—
283C2×Q8—61no—
284C2×D4—31no—
285C2×D4—31no—
286C2×D4—61no—
287C2×D4—61no—
288C2×D4—61no—
289C2×D4—61no—
290C2×D4—61no—
291C2×D4—92no—
292C2×D4—181no—
293C2×D4—181no—
294C2×D4—181no—
295C2×D4—181no—
296C2×D4—182no—
297C2×D4—182no—
298C2×D4—181no—
299C2×D4—362no—
300C2×D4—362no—
301C22×C4—21no—
302C22×C4—31no—
303C22×C4—31no—
304C22×C4—31no—
305C22×C4—31no—
306C22×C4—61no—
307C22×C4—61no—
308C22×C4—61no—
309C22×C4—61no—
310C22×C4—181no—
311C22×C4#61111yescharacteristic
312C22×C4C22×D911yescharacteristic
313SD16—184no—
314SD16—182no—
315SD16—182no—
316D8—184no—
317D8—182no—
318D8—182no—
319M16—181no—
320M16—181no—
321M16—181no—
322M16—181no—
323C2×C8—92no—
324C2×C8—92no—
325C4⋊C4—92no—
326C4⋊C4—181no—
327C22⋊C4—182no—
328C22⋊C4—182no—
329C22×C3—21no—
330C22×C3—31no—
331C22×C3—31no—
332C22×C3—31no—
333C22×C3—61no—
334C22×C3—61no—
335C22×C3—61no—
336C22×C3—61no—
337C22×C3—61no—
338C22×C3—121no—
339C22×C3—121no—
340C22×C3—121no—
341C22×C3—121no—
342C22×C3#94911yescharacteristic
343C22×C3GL(2,ℤ/4)11yescharacteristic
344C2×D3—122no—
345C2×D3—122no—
346C2×D3—122no—
347C2×D3—242no—
348C2×D3—242no—
349C3×C4—21no—
350C3×C4—31no—
351C3×C4—31no—
352C3×C4—61no—
353C3×C4GL(2,ℤ/4)11yescharacteristic
354C3×C4C22×S411yescharacteristic
355Dic3—122no—
356C9—41noSylow
357C23—21no—
358C23—31no—
359C23—31no—
360C23—41no—
361C23—61no—
362C23—61no—
363C23—61no—
364C23—121no—
365C23—182no—
366C23—361no—
367C23—362no—
368C23—362no—
369C23—361no—
370C23C2×#61111yescharacteristic
371Q8—32no—
372Q8—31no—
373Q8—61no—
374Q8C2×(C22⋊C9)⋊C211yescharacteristic, complemented
375D4—21no—
376D4—32no—
377D4—31no—
378D4—61no—
379D4—61no—
380D4—61no—
381D4—62no—
382D4—61no—
383D4—182no—
384D4—182no—
385D4—361no—
386D4—362no—
387D4—364no—
388D4—362no—
389D4—361no—
390D4C2×(C22⋊C9)⋊C211yescharacteristic, complemented
391C2×C4—21no—
392C2×C4—31no—
393C2×C4—31no—
394C2×C4—31no—
395C2×C4—31no—
396C2×C4—31no—
397C2×C4—31no—
398C2×C4—31no—
399C2×C4—61no—
400C2×C4—61no—
401C2×C4—61no—
402C2×C4—61no—
403C2×C4—62no—
404C2×C4—61no—
405C2×C4—61no—
406C2×C4—182no—
407C2×C4—362no—
408C2×C4C2×(C22⋊C9)⋊C211yescharacteristic
409C8—182no—
410C8—182no—
411C2×C3—21no—
412C2×C3—31no—
413C2×C3—31no—
414C2×C3—41no—
415C2×C3—61no—
416C2×C3—121no—
417C2×C3C2×GL(2,ℤ/4)11yescharacteristic
418D3—242no—
419C22—21no—
420C22—31no—
421C22—31no—
422C22—31no—
423C22—61no—
424C22—61no—
425C22—61no—
426C22—61no—
427C22—61no—
428C22—121no—
429C22—121no—
430C22—121no—
431C22—121no—
432C22—362no—
433C22—362no—
434C22—362no—
435C22—722no—
436C22—722no—
437C22#6351211yescharacteristic, complemented
438C22#6369411yescharacteristic
439C4—21no—
440C4—31no—
441C4—31no—
442C4—61no—
443C4—362no—
444C4#6369411yescharacteristic
445C4C22×(C22⋊C9)⋊C211yescharacteristic
446C3#8551311yescharacteristic
447C2—21no—
448C2—31no—
449C2—31no—
450C2—41no—
451C2—61no—
452C2—121no—
453C2—722no—
454C2C2×#6369411yescharacteristic
subgroupG/H
centerC2C2×#63694
commutator, cosocleC4×C22⋊C9C23
FrattiniC3×C4C22×S4
FittingC22×C3×#41D3
socleC23×C3C2×#67
hypercenter#41(C22⋊C9)⋊C2
hypocenterC22⋊C9AGL(1,ℤ/8)
LCSG ▷ C4×C22⋊C9 ▷ C2×C22⋊C9 ▷ C22⋊C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #41×C22⋊C9C2semidirect
2. C2×C4⋊((C22⋊C9)⋊C2)C2semidirect
3. #10589519C2semidirect
4. #10589517C2semidirect
5. #10589511C2semidirect
8. Q8×C22⋊C9C22semidirect
10. D4×C22⋊C9C22semidirect
16. (C22⋊C9)⋊C8C22semidirect
91. C22×#41D9semidirect
168. C22⋊C9AGL(1,ℤ/8)semidirect
178. C22×Q8C2×D9semidirect
182. C22×D4C2×D9semidirect
280. #41(C22⋊C9)⋊C2semidirect
374. Q8C2×(C22⋊C9)⋊C2semidirect
390. D4C2×(C22⋊C9)⋊C2semidirect
437. C22#63512semidirect

Maximal subgroups

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

subgroupindex
#41×C22⋊C92
C2×C4⋊((C22⋊C9)⋊C2)2
#105895192
#105895172
#105895112
#840823
#635124