Finite Group Databasebeta

← #11778418

Subgroups — #11778418

Order 1152.

560 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 84 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×#5644C211yesmaximal, characteristic
2C3×#5050C211yesmaximal, characteristic, complemented
3#10591304C211yesmaximal, characteristic, complemented
4#10591171C211yesmaximal, characteristic, complemented
5#10589404C211yesmaximal, characteristic, complemented
6#10589150C211yesmaximal, characteristic, complemented
7#10589145C211yesmaximal, characteristic, complemented
8#71885—31nomaximal
9#70981—31nomaximal
10C22×C3×Dic6C2211yescharacteristic
11C32×C22⋊Q8C2211yescharacteristic
12C3×#947—21no—
13C2×C3×#216—21no—
14C3×#930C2211yescharacteristic
15C2×C3×(C2×D3)⋊C4C2211yescharacteristic
16C3×#898—21no—
17C3×#895—21no—
18C3×#894—21no—
19C3×#882—21no—
20C3×#67⋊C4—21no—
21C2×(C2×D3)⋊Dic3—21no—
22C2×#3715—21no—
23#63658C2211yescharacteristic
24#63657C2211yescharacteristic
25C3×#830C2211yescharacteristic
26C3×#810—21no—
27#63579—21no—
28#63565—21no—
29C3×#538—31no—
30#5644D311yescharacteristic
31C3×#451—31no—
32#5231—31no—
33#5227—31no—
34#5050D311yescharacteristic, complemented
35#5040—31no—
36#4947—31no—
37#4538—31no—
38#4538—31no—
39#4538—61no—
40#4537—31no—
41#4525—31no—
42#4470—31no—
43#4468—31no—
44C23×C3×D3D411yescharacteristic
45C2×C32×Q8D411yescharacteristic
46C22×C32×C4C2311yescharacteristic
47C22×C3⋊Dic3—21no—
48C2×C3×#67—21no—
49C22×C3×Dic3—21no—
50C22×C3×Dic3D411yescharacteristic
51C2×C3×D12—21no—
52C2×C3×D12—21no—
53C2×C3×D12D411yescharacteristic
54C2×C3×C4×D3—21no—
55C2×C3×Dic6—21no—
56C2×C3×Dic6—21no—
57C2×C3×Dic6—42no—
58C32×C4⋊C4—21no—
59C32×C4⋊C4D411yescharacteristic
60C32×C22⋊C4—21no—
61C2×C3⋊(C3⋊C8)—21no—
62C3×#216—42no—
63C3×(C2×D3)⋊C4—21no—
64C3×(C2×D3)⋊C4—21no—
65C3×(C2×D3)⋊C4—21no—
66C3×(C2×D3)⋊C4—21no—
67C3×(C2×D3)⋊C4—21no—
68C3×(C2×D3)⋊C4—21no—
69C3×C4⋊Dic3D411yescharacteristic
70C3×Dic3⋊C4—21no—
71C3×C4×Dic3—21no—
72C2×C3×C3⋊C8—21no—
73(C2×D3)⋊Dic3—42no—
74#3715—42no—
75#1606—91noSylow
76C22×C3×Q8—31no—
77C22×Dic6C2×D311yescharacteristic
78C2×C3×Q16—61no—
79C3×#126—61no—
80C3×#124—61no—
81C3×#123—61no—
82C3×C22⋊Q8—21no—
83C3×C22⋊Q8C2×D311yescharacteristic
84C3×C22⋊Q8C2×D311yescharacteristic
85C3×#121—31no—
86C3×#121—61no—
87C3×#120—61no—
88C3×C4×D4—61no—
89C2×C3×C22⋊C4—31no—
90C2×#217—61no—
91#947—21no—
92C2×#216—21no—
93C2×#215—61no—
94C2×#214—61no—
95#930C2×D311yescharacteristic
96C2×(C2×D3)⋊C4—61no—
97C2×(C2×D3)⋊C4C2×D311yescharacteristic
98#898—21no—
99#895—21no—
100#894—21no—
101#882—21no—
102#67⋊C4—21no—
103C3×#103—61no—
104C3×C22⋊C8—31no—
105#835—61no—
106C3⋊#95—31no—
107C3⋊#95—31no—
108C3⋊#95—61no—
109C3⋊#95—61no—
110C3⋊#95—61no—
111#830—31no—
112#830—31no—
113#830—61no—
114#830C2×D311yescharacteristic
115#810—21no—
116#810—61no—
117C23×C32C2×D411yescharacteristic
118C22×C3×D3—21no—
119C22×C3×D3—21no—
120C22×C3×D3—21no—
121C22×C3×D3—21no—
122C22×C3×D3—41no—
123C22×C3×D3—41no—
124C32×Q8—41no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—41no—
128C2×C32×C4C2×D411yescharacteristic
129C2×C32×C4C2×D411yescharacteristic
130C2×C3⋊Dic3—41no—
131C2×C3⋊Dic3—41no—
132C2×C3⋊Dic3—41no—
133C3×#67—41no—
134C3×#67—41no—
135C2×C3×Dic3—22no—
136C2×C3×Dic3—21no—
137C2×C3×Dic3—21no—
138C2×C3×Dic3—41no—
139C2×C3×Dic3—41no—
140C2×C3×Dic3—41no—
141C3×D12—41no—
142C3×D12—81no—
143C3×D12—81no—
144C3×C4×D3—81no—
145C3×Dic6—22no—
146C3×Dic6—22no—
147C3×Dic6—41no—
148C3×Dic6—41no—
149C3⋊(C3⋊C8)—41no—
150C3×C3⋊C8—41no—
151#538—31no—
152#451—31no—
153#448—91no—
154#393—91no—
155#328—91no—
156#327—91no—
157#323—91no—
158C24×C3—31no—
159C23×D3#6711yescharacteristic
160C2×C3×Q8—21no—
161C2×C3×Q8—61no—
162C2×C3×Q8—61no—
163C2×C3×Q8—122no—
164C2×C3×Q8#6711yescharacteristic
165C2×C3×Q8#6711yescharacteristic
166C2×C3×D4—31no—
167C2×C3×D4—61no—
168C2×C3×D4—61no—
169C2×C3×D4—61no—
170C22×C3×C4—21no—
171C22×C3×C4—31no—
172C22×C3×C4—61no—
173C22×C3×C4—61no—
174C22×C3×C4C22×D311yescharacteristic
175C22×C3×C4C22×D311yescharacteristic
176C2×#67—21no—
177C22×Dic3—21no—
178C22×Dic3—61no—
179C22×Dic3—61no—
180C22×Dic3—121no—
181C22×Dic3#6711yescharacteristic
182C2×D12—21no—
183C2×D12—21no—
184C2×D12#6711yescharacteristic
185C2×C4×D3—21no—
186C2×Dic6—21no—
187C2×Dic6—21no—
188C2×Dic6—42no—
189C3×Q16—122no—
190C2×C3×C8—61no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—31no—
195C3×C4⋊C4—41no—
196C3×C4⋊C4—61no—
197C3×C4⋊C4#6711yescharacteristic
198C3×C4⋊C4#6711yescharacteristic
199C3×C22⋊C4—21no—
200C3×C22⋊C4—21no—
201C3×C22⋊C4—41no—
202C3×C22⋊C4—61no—
203C3×C22⋊C4—61no—
204C3×C22⋊C4—61no—
205C3×C22⋊C4—61no—
206C3×C22⋊C4—61no—
207C3×C22⋊C4—61no—
208C3×C42—61no—
209#217—122no—
210#216—42no—
211#215—122no—
212#214—122no—
213(C2×D3)⋊C4—21no—
214(C2×D3)⋊C4—21no—
215(C2×D3)⋊C4—21no—
216(C2×D3)⋊C4—21no—
217(C2×D3)⋊C4—21no—
218(C2×D3)⋊C4—21no—
219(C2×D3)⋊C4—122no—
220C4⋊Dic3#6711yescharacteristic
221Dic3⋊C4—21no—
222C4×Dic3—21no—
223C2×C3⋊C8—21no—
224C2×C3⋊C8—61no—
225C2×C3⋊C8—61no—
226C2×C3⋊C8—121no—
227C22×C32—21no—
228C22×C32—21no—
229C22×C32#12011yescharacteristic
230C22×C32#13511yescharacteristic
231C22×C32#13711yescharacteristic
232C2×C3×D3—42no—
233C2×C3×D3—41no—
234C2×C3×D3—41no—
235C2×C3×D3—41no—
236C2×C3×D3—41no—
237C2×C3×D3—41no—
238C2×C3×D3—41no—
239C2×C3×D3—81no—
240C32×C4—21no—
241C32×C4—21no—
242C32×C4—41no—
243C32×C4—41no—
244C3⋊Dic3—81no—
245C3×Dic3—41no—
246C3×Dic3—41no—
247C3×Dic3—42no—
248C22×Q8—31no—
249C2×Q16—61no—
250C2×SD16—181no—
251#126—61no—
252#124—61no—
253#123—61no—
254C22⋊Q8D3211yescharacteristic
255#121—31no—
256#121—61no—
257#120—61no—
258C4×D4—61no—
259C2×C22⋊C4—31no—
260C2×C22⋊C4—181no—
261#103—61no—
262#103—181no—
263C22⋊C8—31no—
264C22⋊C8—91no—
265#95—91no—
266#95—181no—
267C23×C3—21no—
268C23×C3—61no—
269C23×C3—61no—
270C23×C3—61no—
271C23×C3—61no—
272C23×C3—121no—
273C23×C3—121no—
274C23×C3D3×D411yescharacteristic
275C23×C3C2×#6711yescharacteristic
276C22×D3—21no—
277C22×D3—21no—
278C22×D3—21no—
279C22×D3—21no—
280C22×D3—41no—
281C22×D3—41no—
282C3×Q8—41no—
283C3×Q8—41no—
284C3×Q8—62no—
285C3×Q8—62no—
286C3×Q8—81no—
287C3×Q8—121no—
288C3×Q8—121no—
289C3×D4—121no—
290C3×D4—121no—
291C3×D4—121no—
292C3×D4—241no—
293C3×D4—241no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—62no—
305C2×C3×C4—61no—
306C2×C3×C4—61no—
307C2×C3×C4—81no—
308C2×C3×C4—121no—
309C2×C3×C4—121no—
310C2×C3×C4—121no—
311C2×C3×C4—241no—
312C2×C3×C4D3×D411yescharacteristic
313C2×C3×C4C2×#6711yescharacteristic
314C2×C3×C4C2×#6711yescharacteristic
315C2×C3×C4C2×#6711yescharacteristic
316#67—41no—
317#67—41no—
318C2×Dic3—22no—
319C2×Dic3—21no—
320C2×Dic3—21no—
321C2×Dic3—41no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3—121no—
325C2×Dic3—121no—
326C2×Dic3—121no—
327C2×Dic3—121no—
328C2×Dic3—122no—
329C2×Dic3—121no—
330C2×Dic3—121no—
331C2×Dic3—241no—
332C2×Dic3—241no—
333D12—41no—
334D12—81no—
335D12—81no—
336C4×D3—81no—
337Dic6—22no—
338Dic6—22no—
339Dic6—41no—
340Dic6—41no—
341C3×C8—121no—
342C3⋊C8—41no—
343C3⋊C8—121no—
344C3⋊C8—121no—
345C3⋊C8—241no—
346C2×C32—21no—
347C2×C32—21no—
348C2×C32#45211yescharacteristic
349C2×C32#45511yescharacteristic
350C2×C32B(4,𝔽2)11yescharacteristic
351C3×D3—81no—
352C3×D3—81no—
353C24—31no—
354C2×Q8—61no—
355C2×Q8—61no—
356C2×Q8—122no—
357C2×Q8#62511yescharacteristic
358C2×D4—31no—
359C2×D4—61no—
360C2×D4—61no—
361C2×D4—61no—
362C22×C4—31no—
363C22×C4—61no—
364C22×C4—61no—
365C22×C4—181no—
366C22×C4C2×D3211yescharacteristic
367Q16—122no—
368SD16—362no—
369C2×C8—61no—
370C2×C8—181no—
371C4⋊C4—21no—
372C4⋊C4—31no—
373C4⋊C4—61no—
374C4⋊C4#62511yescharacteristic
375C22⋊C4—21no—
376C22⋊C4—61no—
377C22⋊C4—61no—
378C22⋊C4—61no—
379C22⋊C4—61no—
380C22⋊C4—61no—
381C22⋊C4—61no—
382C22⋊C4—362no—
383C42—61no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—41no—
392C22×C3—41no—
393C22×C3—122no—
394C22×C3—121no—
395C22×C3—121no—
396C22×C3—121no—
397C22×C3—121no—
398C22×C3—121no—
399C22×C3—121no—
400C22×C3—241no—
401C22×C3#91811yescharacteristic
402C22×C3#91911yescharacteristic
403C22×C3#93711yescharacteristic
404C22×C3#95011yescharacteristic
405C22×C3#95111yescharacteristic
406C22×C3#95311yescharacteristic
407C2×D3—42no—
408C2×D3—41no—
409C2×D3—41no—
410C2×D3—41no—
411C2×D3—41no—
412C2×D3—41no—
413C2×D3—41no—
414C2×D3—81no—
415C3×C4—21no—
416C3×C4—21no—
417C3×C4—21no—
418C3×C4—21no—
419C3×C4—41no—
420C3×C4—41no—
421C3×C4—41no—
422C3×C4—41no—
423C3×C4—41no—
424C3×C4—41no—
425C3×C4—81no—
426C3×C4—81no—
427C3×C4—121no—
428C3×C4—121no—
429C3×C4—122no—
430Dic3—41no—
431Dic3—41no—
432Dic3—42no—
433Dic3—241no—
434Dic3—241no—
435Dic3—242no—
436C32#160611yescharacteristic, complemented, Sylow
437C23—61no—
438C23—61no—
439C23—61no—
440C23—61no—
441C23—121no—
442C23—121no—
443C23D3×#6711yescharacteristic
444Q8—41no—
445Q8—62no—
446Q8—62no—
447Q8—121no—
448Q8—121no—
449D4—121no—
450D4—121no—
451D4—121no—
452D4—241no—
453D4—241no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—41no—
457C2×C4—62no—
458C2×C4—61no—
459C2×C4—61no—
460C2×C4—121no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4—241no—
464C2×C4—361no—
465C2×C4—361no—
466C2×C4—361no—
467C2×C4C2×#62511yescharacteristic
468C2×C4D3×#6711yescharacteristic
469C8—121no—
470C8—361no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—41no—
479C2×C3—41no—
480C2×C3—241no—
481C2×C3—241no—
482C2×C3#503211yescharacteristic
483C2×C3#518711yescharacteristic
484C2×C3#520211yescharacteristic
485C2×C3#523911yescharacteristic
486C2×C3#524211yescharacteristic
487C2×C3#524411yescharacteristic
488D3—81no—
489D3—81no—
490C22—21no—
491C22—21no—
492C22—122no—
493C22—121no—
494C22—121no—
495C22—121no—
496C22—121no—
497C22—121no—
498C22—121no—
499C22—241no—
500C22#6394311yescharacteristic
501C22#6394911yescharacteristic
502C22#6397611yescharacteristic
503C4—21no—
504C4—21no—
505C4—41no—
506C4—41no—
507C4—122no—
508C4—121no—
509C4—121no—
510C4—721no—
511C3—21no—
512C3#7098111yescharacteristic, complemented
513C3#7188511yescharacteristic, complemented
514C2—21no—
515C2—21no—
516C2—241no—
517C2—241no—
518C2#1059153011yescharacteristic
519C2#1059159711yescharacteristic
520C2#1059166411yescharacteristic
subgroupG/H
centerC22#63976
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC32×C22⋊Q8C22
socleC22×C32#120
prime radicalC3×#5644C2
hypercenterC22⋊Q8D32
hypocenterC32#1606
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C3×#5050C2semidirect
3. #10591304C2semidirect
4. #10591171C2semidirect
5. #10589404C2semidirect
6. #10589150C2semidirect
7. #10589145C2semidirect
34. #5050D3semidirect
436. C32#1606semidirect
512. C3#70981semidirect
513. C3#71885semidirect

Maximal subgroups

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

subgroupindex
C3×#56442
C3×#50502
#105913042
#105911712
#105894042
#105891502
#105891452
#718853
#709813