Finite Group Databasebeta

← #11778222

Subgroups — #11778222

Order 1152.

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

normal subgroups only

This group has 717 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
1C3×#5493C211yesmaximal, characteristic
2#10592646C211yesmaximal, characteristic, complemented
3#10591181C212yesmaximal, complemented
4C3×#4525C211yesmaximal, characteristic, complemented
5#10589150C212yesmaximal, complemented
6#71846—31nomaximal
7#68108—31nomaximal
8C22×#634C2211yescharacteristic, complemented
9C22×C3×#67—21no—
10C2×C32×C22⋊C4C2211yescharacteristic
11C2×#3757—21no—
12C3×#930—21no—
13C3×#930C2212yes—
14C3×#886—22no—
15C3×#882—22no—
16C2×#3716—22no—
17#63658—21no—
18#63658C2211yescharacteristic, complemented
19C3×#830C2212yescomplemented
20#63567—22no—
21C3×#534—31no—
22#5493D311yescharacteristic
23#5040—32no—
24#4947—31no—
25#4947—31no—
26#4947—61no—
27#4738—32no—
28C3×#323—31no—
29#4537—32no—
30#4525D311yescharacteristic, complemented
31#4470—32no—
32C24×C32D411yescharacteristic
33C23×C3×D3—21no—
34C22×C32×C4D411yescharacteristic
35C22×C32×C4C2311yescharacteristic
36C22×C3⋊Dic3—21no—
37C22×C3⋊Dic3—21no—
38C2×#634—22no—
39C2×#634—42no—
40C2×C3×#67—22no—
41C2×C3×#67—22no—
42C2×C3×#67—42no—
43C2×C3×#67—42no—
44C22×C3×Dic3—21no—
45C2×C3×D12—21no—
46C2×C3×D12D412yes—
47C32×C22⋊C4—22no—
48#3757—42no—
49C3×(C2×D3)⋊C4—22no—
50C3×(C2×D3)⋊C4—22no—
51C3×C4⋊Dic3—21no—
52C3×C4⋊Dic3D412yes—
53C2×C3×C3⋊C8—22no—
54#3716—44no—
55#1601—91noSylow
56C22×C3×D4—61no—
57C22×#67—21no—
58C22×Dic6—31no—
59C22×Dic6—31no—
60C22×Dic6—61no—
61C3×#123—62no—
62C3×#121—32no—
63C3×#121—61no—
64C3×#120—62no—
65C2×C3×C22⋊C4—21no—
66C2×C3×C22⋊C4C2×D311yescharacteristic
67C2×C3×C22⋊C4C2×D311yescharacteristic
68C2×#217—62no—
69C2×#217—61no—
70C2×#217—61no—
71C2×#214—62no—
72#930—21no—
73#930C2×D312yes—
74C2×#204—62no—
75#886—22no—
76#882—22no—
77C3×C22⋊C8—32no—
78C3⋊#95—31no—
79C3⋊#95—31no—
80C3⋊#95—61no—
81C3⋊#95—62no—
82C3⋊#95—61no—
83C3⋊#95—61no—
84#830C2×D312yescomplemented
85#816—62no—
86#810—62no—
87C23×C32—21no—
88C23×C32—21no—
89C23×C32—41no—
90C23×C32—41no—
91C23×C32SD1612yescomplemented
92C23×C32C2×D411yescharacteristic
93C22×C3×D3—22no—
94C22×C3×D3—42no—
95C22×C3×D3—81no—
96C22×C3×D3—81no—
97C2×C32×C4—21no—
98C2×C32×C4—41no—
99C2×C32×C4—41no—
100C2×C32×C4C2×D412yes—
101C2×C3⋊Dic3—22no—
102C2×C3⋊Dic3—41no—
103C2×C3⋊Dic3—41no—
104C2×C3⋊Dic3—41no—
105C2×C3⋊Dic3—41no—
106C2×C3⋊Dic3—41no—
107#634—24no—
108#634—42no—
109C3×#67—44no—
110C3×#67—42no—
111C3×#67—42no—
112C2×C3×Dic3—22no—
113C2×C3×Dic3—42no—
114C3×D12—42no—
115C3×D12—81no—
116C3×C3⋊C8—42no—
117#534—31no—
118#448—92no—
119#393—91no—
120#327—92no—
121#323—31no—
122C24×C3—21no—
123C24×C3—61no—
124C24×C3D1211yescharacteristic
125C24×C3#6711yescharacteristic
126C23×D3—21no—
127C2×C3×D4—32no—
128C2×C3×D4—61no—
129C2×C3×D4—62no—
130C2×C3×D4—62no—
131C2×C3×D4—122no—
132C2×C3×D4—122no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—61no—
136C22×C3×C4D1211yescharacteristic
137C22×C3×C4#6711yescharacteristic
138C22×C3×C4C22×D311yescharacteristic
139C22×C3×C4C22×D311yescharacteristic
140C2×#67—22no—
141C2×#67—22no—
142C2×#67—42no—
143C2×#67—42no—
144C22×Dic3—21no—
145C22×Dic3—62no—
146C22×Dic3—61no—
147C22×Dic3—61no—
148C22×Dic3—62no—
149C22×Dic3—61no—
150C22×Dic3—61no—
151C2×D12—21no—
152C2×D12#6712yes—
153C2×Dic6—62no—
154C2×Dic6—62no—
155C2×Dic6—122no—
156C2×Dic6—122no—
157C2×Dic6—122no—
158C2×Dic6—124no—
159C2×C3×C8—62no—
160C3×C4⋊C4—32no—
161C3×C4⋊C4—61no—
162C3×C22⋊C4—22no—
163C3×C22⋊C4—22no—
164C3×C22⋊C4—42no—
165C3×C22⋊C4—62no—
166C3×C22⋊C4—62no—
167#217—122no—
168#217—122no—
169#217—124no—
170#214—124no—
171(C2×D3)⋊C4—22no—
172(C2×D3)⋊C4—22no—
173C4⋊Dic3—21no—
174C4⋊Dic3#6712yes—
175C2×C3⋊C8—22no—
176#204—124no—
177C22×C32—21no—
178C22×C32—21no—
179C22×C32—22no—
180C22×C32—42no—
181C22×C32—41no—
182C22×C32—41no—
183C22×C32—81no—
184C22×C32#12011yescharacteristic
185C22×C32C2×SD1611yescharacteristic
186C22×C32AGL(1,ℤ/8)11yescharacteristic
187C2×C3×D3—42no—
188C2×C3×D3—42no—
189C2×C3×D3—42no—
190C2×C3×D3—84no—
191C32×C4—22no—
192C32×C4—41no—
193C3⋊Dic3—42no—
194C3⋊Dic3—81no—
195C3×Dic3—42no—
196C22×Q8—91no—
197C22×D4—61no—
198C2×SD16—182no—
199#123—62no—
200#121—32no—
201#121—61no—
202#120—62no—
203C2×C22⋊C4—181no—
204C2×C22⋊C4D3211yescharacteristic
205#103—182no—
206C22⋊C8—32no—
207#95—91no—
208#95—181no—
209C23×C3—21no—
210C23×C3—21no—
211C23×C3—21no—
212C23×C3—21no—
213C23×C3—22no—
214C23×C3—21no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3—42no—
222C23×C3—42no—
223C23×C3—62no—
224C23×C3—122no—
225C23×C3—241no—
226C23×C3—241no—
227C23×C3#20412yescomplemented
228C23×C3#21412yescomplemented
229C23×C3C2×D1211yescharacteristic
230C23×C3C2×#6711yescharacteristic
231C22×D3—22no—
232C22×D3—42no—
233C22×D3—81no—
234C22×D3—81no—
235C3×D4—122no—
236C3×D4—124no—
237C3×D4—122no—
238C3×D4—122no—
239C3×D4—241no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—22no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—42no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—42no—
250C2×C3×C4—62no—
251C2×C3×C4—122no—
252C2×C3×C4D3×D412yes—
253C2×C3×C4C2×#6712yes—
254#67—44no—
255#67—42no—
256#67—42no—
257C2×Dic3—22no—
258C2×Dic3—42no—
259C2×Dic3—64no—
260C2×Dic3—62no—
261C2×Dic3—62no—
262C2×Dic3—122no—
263C2×Dic3—121no—
264C2×Dic3—121no—
265C2×Dic3—122no—
266C2×Dic3—121no—
267C2×Dic3—121no—
268C2×Dic3—122no—
269C2×Dic3—121no—
270C2×Dic3—121no—
271C2×Dic3—122no—
272C2×Dic3—121no—
273C2×Dic3—121no—
274C2×Dic3—122no—
275C2×Dic3—121no—
276C2×Dic3—121no—
277D12—42no—
278D12—81no—
279Dic6—64no—
280Dic6—64no—
281Dic6—124no—
282Dic6—122no—
283Dic6—122no—
284Dic6—124no—
285C3×C8—122no—
286C3⋊C8—42no—
287C2×C32—21no—
288C2×C32—21no—
289C2×C32—42no—
290C2×C32#45011yescharacteristic
291C2×C32#45511yescharacteristic
292C2×C32B(4,𝔽2)11yescharacteristic
293C3×D3—82no—
294C24—61no—
295C24#62611yescharacteristic
296C2×Q8—182no—
297C2×Q8—362no—
298C2×D4—32no—
299C2×D4—61no—
300C2×D4—62no—
301C2×D4—62no—
302C2×D4—122no—
303C2×D4—122no—
304C22×C4—61no—
305C22×C4—181no—
306C22×C4—181no—
307C22×C4#62611yescharacteristic
308C22×C4C2×D3211yescharacteristic
309SD16—364no—
310C2×C8—62no—
311C4⋊C4—32no—
312C4⋊C4—61no—
313C22⋊C4—22no—
314C22⋊C4—62no—
315C22⋊C4—62no—
316C22⋊C4—362no—
317C22×C3—22no—
318C22×C3—22no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—22no—
324C22×C3—22no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—42no—
329C22×C3—42no—
330C22×C3—42no—
331C22×C3—44no—
332C22×C3—41no—
333C22×C3—41no—
334C22×C3—41no—
335C22×C3—41no—
336C22×C3—81no—
337C22×C3—81no—
338C22×C3—81no—
339C22×C3—81no—
340C22×C3—82no—
341C22×C3—122no—
342C22×C3—122no—
343C22×C3—122no—
344C22×C3—244no—
345C22×C3#88211yescharacteristic
346C22×C3C2×#20411yescharacteristic
347C22×C3#90811yescharacteristic
348C22×C3#93211yescharacteristic
349C22×C3C2×#21411yescharacteristic
350C22×C3#95311yescharacteristic
351C2×D3—42no—
352C2×D3—42no—
353C2×D3—42no—
354C2×D3—84no—
355C3×C4—22no—
356C3×C4—22no—
357C3×C4—42no—
358C3×C4—41no—
359C3×C4—41no—
360C3×C4—81no—
361C3×C4—122no—
362Dic3—42no—
363Dic3—124no—
364Dic3—122no—
365Dic3—122no—
366Dic3—242no—
367Dic3—241no—
368Dic3—241no—
369C32#160111yescharacteristic, complemented, Sylow
370C23—21no—
371C23—21no—
372C23—41no—
373C23—41no—
374C23—62no—
375C23—122no—
376C23—241no—
377C23—241no—
378C23#371612yescomplemented
379C23C2×#62611yescharacteristic
380Q8—184no—
381Q8—362no—
382D4—122no—
383D4—124no—
384D4—122no—
385D4—122no—
386D4—241no—
387C2×C4—21no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—62no—
391C2×C4—122no—
392C2×C4—182no—
393C2×C4—361no—
394C2×C4—361no—
395C2×C4—361no—
396C2×C4—361no—
397C2×C4—361no—
398C2×C4D3×#6712yes—
399C8—122no—
400C2×C3—22no—
401C2×C3—22no—
402C2×C3—21no—
403C2×C3—21no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3—42no—
410C2×C3—42no—
411C2×C3—82no—
412C2×C3—242no—
413C2×C3#473111yescharacteristic
414C2×C3#474111yescharacteristic
415C2×C3#474611yescharacteristic
416C2×C3#521811yescharacteristic
417C2×C3#524211yescharacteristic
418C2×C3#524411yescharacteristic
419D3—82no—
420C22—21no—
421C22—21no—
422C22—22no—
423C22—42no—
424C22—41no—
425C22—41no—
426C22—81no—
427C22—122no—
428C22—122no—
429C22—122no—
430C22—244no—
431C22C2×#371611yescharacteristic
432C22#6383011yescharacteristic
433C22#6397711yescharacteristic
434C4—22no—
435C4—41no—
436C4—122no—
437C4—362no—
438C4—721no—
439C3—21no—
440C3#6810811yescharacteristic, complemented
441C3#7184611yescharacteristic, complemented
442C2—21no—
443C2—21no—
444C2—42no—
445C2—242no—
446C2#1059117211yescharacteristic
447C2#1059151311yescharacteristic
448C2#1059153111yescharacteristic
subgroupG/H
centerC22#63977
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC2×C32×C22⋊C4C22
socleC22×C32#120
prime radicalC3×#5493C2
hypercenterC2×C22⋊C4D32
hypocenterC32#1601
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #10592646C2semidirect
3. #10591181C2semidirect
4. C3×#4525C2semidirect
5. #10589150C2semidirect
8. C22×#634C22semidirect
18. #63658C22semidirect
19. C3×#830C22semidirect
30. #4525D3semidirect
84. #830C2×D3semidirect
91. C23×C32SD16semidirect
227. C23×C3#204semidirect
228. C23×C3#214semidirect
369. C32#1601semidirect
378. C23#3716semidirect
440. C3#68108semidirect
441. C3#71846semidirect

Maximal subgroups

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

subgroupindex
C3×#54932
#105926462
#105911812
C3×#45252
#105891502
#718463
#681083