Finite Group Databasebeta

← #11779217

Subgroups — #11779217

Order 1152.

564 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 92 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#10593904C211yesmaximal, characteristic, complemented
2#10592774C211yesmaximal, characteristic, complemented
3C3×#5072C211yesmaximal, characteristic
4C3×#5070C211yesmaximal, characteristic, complemented
5#10589092C211yesmaximal, characteristic, complemented
6#10589085C211yesmaximal, characteristic, complemented
7#10589081C211yesmaximal, characteristic, complemented
8#71205—31nomaximal
9#71014—31nomaximal
10C2×D3×Dic6—21no—
11C32×#127C2211yescharacteristic
12C2×#3754—21no—
13C4×#634C2211yescharacteristic
14C2×C3×#213—21no—
15C3×C4×D12C2211yescharacteristic
16C3×#869C2211yescharacteristic
17#63919—21no—
18#63899—22no—
19#63898—21no—
20#63856—21no—
21#63659—21no—
22C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
23C3×#832—22no—
24C3×#832—21no—
25C3×C4⋊(C3⋊C8)C2211yescharacteristic
26C3×C4×C3⋊C8C2211yescharacteristic
27#5727—31no—
28D4×Dic6—31no—
29C3×#486—31no—
30C3×#459—31no—
31#5075—31no—
32#5075—31no—
33#5075—61no—
34#5072D311yescharacteristic
35#5070D311yescharacteristic, complemented
36#4547—31no—
37D4⋊(C3⋊C8)—31no—
38#4486—31no—
39#4484—31no—
40#4459—31no—
41#4456—31no—
42C2×C32×D4—21no—
43C2×C32×D4—21no—
44C2×C32×D4—21no—
45C2×#634D411yescharacteristic
46C2×C3×D12D411yescharacteristic
47C2×C3×C4×D3—21no—
48C2×C3×Dic6—21no—
49C2×#627—22no—
50C2×D3×Dic3—22no—
51D3×Dic6—42no—
52D3×Dic6—42no—
53C32×C42C2311yescharacteristic
54#3754—42no—
55C4⋊(C3⋊Dic3)D411yescharacteristic
56#3750—21no—
57C4×C3⋊Dic3—21no—
58C2×C3⋊(C3⋊C8)—21no—
59C3×#213—42no—
60C3×(C2×D3)⋊C4—21no—
61C3×C4⋊Dic3—21no—
62C3×C4⋊Dic3D412yes—
63C3×C4⋊Dic3D411yescharacteristic
64C2×C3×C3⋊C8—21no—
65C2×C3×C3⋊C8—21no—
66#3724—22no—
67(C2×D3)⋊Dic3—22no—
68#1627—91noSylow
69C2×D3×Q8—61no—
70C22×Dic6—61no—
71C2×C3×D8—61no—
72C3×C4⋊Q8—31no—
73C3×#127—21no—
74C3×#127C2×D311yescharacteristic
75C3×#127C2×D311yescharacteristic
76C3×C4×D4—31no—
77#946—61no—
78D4×Dic3—61no—
79C2×#214—61no—
80C2×#214—61no—
81C2×#214—62no—
82C2×#213—21no—
83#924—61no—
84#896—62no—
85#893—61no—
86#888—61no—
87#888—61no—
88#878—62no—
89C4×D12C2×D311yescharacteristic
90#869C2×D311yescharacteristic
91C4×Dic6—31no—
92C4×Dic6—31no—
93C4×Dic6—61no—
94C3×C4⋊C8—31no—
95C3×#102—61no—
96C3×#102—62no—
97C3×C4×C8—31no—
98#832—22no—
99#832—21no—
100#832—61no—
101#832—61no—
102#832—62no—
103C4⋊(C3⋊C8)—31no—
104C4⋊(C3⋊C8)—31no—
105C4⋊(C3⋊C8)—61no—
106C4⋊(C3⋊C8)C2×D311yescharacteristic
107C4×C3⋊C8C2×D311yescharacteristic
108C23×C32—41no—
109C22×C3×D3—21no—
110C32×D4—41no—
111C32×D4—41no—
112C32×D4—42no—
113C32×D4—41no—
114C32×D4—41no—
115C2×C32×C4C2×D411yescharacteristic
116C2×C32×C4C2×D411yescharacteristic
117C2×C32×C4C2×D411yescharacteristic
118C2×C3⋊Dic3—21no—
119C2×C3⋊Dic3—21no—
120#634—21no—
121#634D812yescomplemented
122C2×C3×Dic3—22no—
123C2×C3×Dic3—21no—
124C3×D12—21no—
125C3×D12SD1612yescomplemented
126C3×C4×D3—41no—
127C3×C4×D3—41no—
128C3×Dic6—41no—
129C3×Dic6—41no—
130#627—44no—
131D3×Dic3—44no—
132C3⋊(C3⋊C8)—41no—
133C3×C3⋊C8—41no—
134C3×C3⋊C8—41no—
135D4×Q8—91no—
136#486—31no—
137#460—91no—
138#459—31no—
139#332—91no—
140Q8⋊C8—91no—
141D4⋊C8—91no—
142C2×C3×Q8—61no—
143C2×C3×D4—22no—
144C2×C3×D4—21no—
145C2×C3×D4—21no—
146C2×C3×D4—21no—
147C2×C3×D4—21no—
148C2×C3×D4—21no—
149C2×C3×D4—21no—
150C2×C3×D4—31no—
151C2×C3×D4—41no—
152C2×C3×D4—41no—
153C22×C3×C4—61no—
154C22×Dic3—62no—
155D3×Q8—122no—
156D3×Q8—122no—
157C2×D12#6711yescharacteristic
158C2×C4×D3—21no—
159C2×C4×D3—62no—
160C2×Dic6—21no—
161C2×Dic6—31no—
162C2×Dic6—31no—
163C2×Dic6—61no—
164C2×Dic6—62no—
165C2×Dic6—62no—
166C2×Dic6—122no—
167C2×Dic6—122no—
168C3×D8—122no—
169C2×C3×C8—61no—
170C2×C3×C8—61no—
171C3×C4⋊C4—31no—
172C3×C4⋊C4—32no—
173C3×C4⋊C4—61no—
174C3×C22⋊C4—61no—
175C3×C42—21no—
176C3×C42C22×D311yescharacteristic
177C3×C42C22×D311yescharacteristic
178#217—62no—
179#214—122no—
180#214—122no—
181#214—124no—
182#213—42no—
183(C2×D3)⋊C4—21no—
184(C2×D3)⋊C4—62no—
185C4⋊Dic3—21no—
186C4⋊Dic3—31no—
187C4⋊Dic3—31no—
188C4⋊Dic3—61no—
189C4⋊Dic3#6712yes—
190C4⋊Dic3#6711yescharacteristic
191Dic3⋊C4—62no—
192Dic3⋊C4—62no—
193Dic3⋊C4—61no—
194Dic3⋊C4—61no—
195Dic3⋊C4—121no—
196C4×Dic3—61no—
197C4×Dic3—61no—
198C4×Dic3—121no—
199C2×C3⋊C8—21no—
200C2×C3⋊C8—21no—
201C2×C3⋊C8—62no—
202C2×C3⋊C8—61no—
203C2×C3⋊C8—61no—
204C22×C32—81no—
205C22×C32—81no—
206C22×C32—81no—
207C22×C32#12011yescharacteristic
208C2×C3×D3—22no—
209C2×C3×D3—41no—
210C2×C3×D3—41no—
211C32×C4—21no—
212C32×C4C2×D811yescharacteristic
213C32×C4C2×SD1611yescharacteristic
214C32×C4AGL(1,ℤ/8)11yescharacteristic
215C32×C4#13711yescharacteristic
216C3⋊Dic3—22no—
217C3⋊Dic3—41no—
218C3×Dic3—42no—
219C3×Dic3—41no—
220C22×Q8—181no—
221C2×SD16—181no—
222C2×D8—61no—
223C4⋊Q8—31no—
224C4⋊Q8—181no—
225#127D3211yescharacteristic
226C22⋊Q8—182no—
227C22⋊Q8—181no—
228C4×Q8—91no—
229C4×D4—31no—
230C4×D4—181no—
231C4⋊C8—31no—
232C4⋊C8—91no—
233#102—61no—
234#102—62no—
235#102—181no—
236C4×C8—31no—
237C23×C3—42no—
238C23×C3—41no—
239C23×C3—41no—
240C23×C3—61no—
241C22×D3—21no—
242C3×Q8—121no—
243C3×Q8—121no—
244C3×D4—32no—
245C3×D4—42no—
246C3×D4—42no—
247C3×D4—42no—
248C3×D4—42no—
249C3×D4—41no—
250C3×D4—41no—
251C3×D4—41no—
252C3×D4—41no—
253C3×D4—41no—
254C3×D4—41no—
255C3×D4—41no—
256C3×D4—41no—
257C3×D4—61no—
258C3×D4—81no—
259C3×D4—81no—
260C3×D4—82no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—61no—
265C2×C3×C4—62no—
266C2×C3×C4—121no—
267C2×C3×C4—121no—
268C2×C3×C4D3×D411yescharacteristic
269C2×C3×C4D3×D411yescharacteristic
270C2×C3×C4D3×D411yescharacteristic
271C2×C3×C4D3×D411yescharacteristic
272C2×C3×C4C2×#6711yescharacteristic
273C2×C3×C4C2×#6711yescharacteristic
274C2×Dic3—21no—
275C2×Dic3—22no—
276C2×Dic3—61no—
277C2×Dic3—61no—
278C2×Dic3—61no—
279C2×Dic3—61no—
280C2×Dic3—121no—
281C2×Dic3—121no—
282C2×Dic3—124no—
283D12—21no—
284D12#21412yescomplemented
285C4×D3—41no—
286C4×D3—41no—
287C4×D3—124no—
288Dic6—32no—
289Dic6—32no—
290Dic6—41no—
291Dic6—41no—
292Dic6—62no—
293Dic6—61no—
294Dic6—61no—
295Dic6—121no—
296Dic6—124no—
297Dic6—124no—
298C3×C8—121no—
299C3×C8—121no—
300C3⋊C8—41no—
301C3⋊C8—41no—
302C3⋊C8—122no—
303C3⋊C8—121no—
304C3⋊C8—121no—
305C2×C32—161no—
306C2×C32#44711yescharacteristic
307C2×C32#44811yescharacteristic
308C2×C32#45411yescharacteristic
309C3×D3—42no—
310C2×Q8—61no—
311C2×Q8—91no—
312C2×Q8—182no—
313C2×Q8—362no—
314C2×Q8—362no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—31no—
319C22×C4—61no—
320C22×C4—182no—
321SD16—362no—
322D8—122no—
323C2×C8—61no—
324C2×C8—61no—
325C2×C8—181no—
326C4⋊C4—31no—
327C4⋊C4—32no—
328C4⋊C4—61no—
329C4⋊C4—91no—
330C4⋊C4—182no—
331C4⋊C4—181no—
332C22⋊C4—61no—
333C22⋊C4—182no—
334C42—181no—
335C42C2×D3211yescharacteristic
336C22×C3—21no—
337C22×C3—62no—
338C22×C3—82no—
339C22×C3—82no—
340C22×C3—82no—
341C22×C3—81no—
342C22×C3—81no—
343C22×C3—81no—
344C22×C3—81no—
345C22×C3—81no—
346C22×C3—81no—
347C22×C3—121no—
348C22×C3—121no—
349C22×C3#93711yescharacteristic
350C22×C3#93711yescharacteristic
351C2×D3—22no—
352C2×D3—41no—
353C2×D3—41no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—41no—
361C3×C4—121no—
362C3×C4—122no—
363C3×C4D3×D811yescharacteristic
364C3×C4D3×D811yescharacteristic
365C3×C4#91111yescharacteristic
366C3×C4#91111yescharacteristic
367C3×C4D3×SD1611yescharacteristic
368C3×C4#91511yescharacteristic
369C3×C4C2×#21411yescharacteristic
370C3×C4#95111yescharacteristic
371Dic3—42no—
372Dic3—41no—
373Dic3—62no—
374Dic3—62no—
375Dic3—122no—
376Dic3—121no—
377Dic3—121no—
378Dic3—241no—
379C32#162711yescharacteristic, complemented, Sylow
380C23—41no—
381C23—61no—
382Q8—92no—
383Q8—121no—
384Q8—121no—
385Q8—181no—
386Q8—364no—
387D4—32no—
388D4—42no—
389D4—41no—
390D4—41no—
391D4—41no—
392D4—41no—
393D4—61no—
394C2×C4—61no—
395C2×C4—62no—
396C2×C4—121no—
397C2×C4—121no—
398C2×C4—181no—
399C2×C4—181no—
400C2×C4—364no—
401C2×C4D3×#6711yescharacteristic
402C2×C4D3×#6711yescharacteristic
403C2×C4#381111yescharacteristic
404C8—121no—
405C8—121no—
406C8—361no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3—21no—
410C2×C3—122no—
411C2×C3—162no—
412C2×C3—161no—
413C2×C3—161no—
414C2×C3#504011yescharacteristic
415C2×C3#506211yescharacteristic
416C2×C3#515611yescharacteristic
417C2×C3#515611yescharacteristic
418C2×C3#517011yescharacteristic
419C2×C3#520111yescharacteristic
420D3—42no—
421C22—62no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—121no—
426C22—121no—
427C22#6398311yescharacteristic
428C4—21no—
429C4—121no—
430C4—122no—
431C4—182no—
432C4—361no—
433C4#6392611yescharacteristic
434C4#6392711yescharacteristic
435C4D3×#21411yescharacteristic
436C4#6392911yescharacteristic
437C3—21no—
438C3#7101411yescharacteristic, complemented
439C3#7120511yescharacteristic, complemented
440C2—122no—
441C2—161no—
442C2#1059151611yescharacteristic
443C2#1059151711yescharacteristic
444C2#1059152511yescharacteristic
subgroupG/H
centerC22#63983
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×#127C22
socleC22×C32#120
prime radicalC3×#5072C2
hypercenter#127D32
hypocenterC32#1627
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #10593904C2semidirect
2. #10592774C2semidirect
4. C3×#5070C2semidirect
5. #10589092C2semidirect
6. #10589085C2semidirect
7. #10589081C2semidirect
35. #5070D3semidirect
121. #634D8semidirect
125. C3×D12SD16semidirect
284. D12#214semidirect
379. C32#1627semidirect
438. C3#71014semidirect
439. C3#71205semidirect

Maximal subgroups

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

subgroupindex
#105939042
#105927742
C3×#50722
C3×#50702
#105890922
#105890852
#105890812
#712053
#710143