Finite Group Databasebeta

← #11778406

Subgroups — #11778406

Order 1152.

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

normal subgroups only

This group has 565 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×#5617C211yesmaximal, characteristic
2C3×#5040C211yesmaximal, characteristic, complemented
3#10591304C211yesmaximal, characteristic, complemented
4#10591185C211yesmaximal, characteristic, complemented
5#10589403C211yesmaximal, characteristic, complemented
6#10589152C211yesmaximal, characteristic, complemented
7#10589145C211yesmaximal, characteristic, complemented
8#71885—31nomaximal
9#70973—31nomaximal
10C22×C3×Dic6C2211yescharacteristic
11C32×#121C2211yescharacteristic
12C3×#939—21no—
13C3×#937—21no—
14C3×D4×Dic3—21no—
15C2×C3×#214—21no—
16C2×C3×(C2×D3)⋊C4C2211yescharacteristic
17C3×#924C2211yescharacteristic
18C3×#898—21no—
19C3×#886—21no—
20C3×#885—21no—
21C2×(C2×D3)⋊Dic3—21no—
22C2×#3718—21no—
23#63658C2211yescharacteristic
24#63657C2211yescharacteristic
25C3×#830C2211yescharacteristic
26C3×#810—21no—
27#63579—21no—
28#63565—21no—
29C3×#538—31no—
30#5617D311yescharacteristic
31C3×#448—31no—
32#5231—31no—
33#5228—31no—
34#5050—31no—
35#5040D311yescharacteristic, complemented
36#4947—31no—
37#4538—31no—
38#4537—31no—
39#4537—31no—
40#4537—61no—
41#4525—31no—
42#4468—31no—
43#4467—31no—
44C23×C3×D3D411yescharacteristic
45C2×C32×D4—21no—
46C2×C32×D4D411yescharacteristic
47C22×C32×C4C2311yescharacteristic
48C22×C3⋊Dic3—21no—
49C2×C3×#67—21no—
50C2×C3×#67—21no—
51C22×C3×Dic3—21no—
52C22×C3×Dic3—21no—
53C22×C3×Dic3D411yescharacteristic
54C2×C3×Dic6—21no—
55C2×C3×Dic6—21no—
56C2×C3×Dic6—42no—
57C2×C3×Dic6D411yescharacteristic
58C32×C4⋊C4D411yescharacteristic
59C32×C22⋊C4—21no—
60C2×C3⋊(C3⋊C8)—21no—
61C3×#217—21no—
62C3×#217—21no—
63C3×#214—42no—
64C3×(C2×D3)⋊C4—21no—
65C3×(C2×D3)⋊C4—21no—
66C3×(C2×D3)⋊C4—21no—
67C3×(C2×D3)⋊C4—21no—
68C3×C4⋊Dic3—21no—
69C3×C4⋊Dic3D411yescharacteristic
70C3×Dic3⋊C4—21no—
71C3×C4×Dic3—21no—
72C2×C3×C3⋊C8—21no—
73(C2×D3)⋊Dic3—42no—
74#3718—42no—
75#1606—91noSylow
76C22×C3×Q8—31no—
77C22×Dic6C2×D311yescharacteristic
78C2×C3×SD16—61no—
79C3×#126—61no—
80C3×#124—61no—
81C3×#123—61no—
82C3×C22⋊Q8—31no—
83C3×#121—21no—
84C3×#121—61no—
85C3×#121C2×D311yescharacteristic
86C3×#121C2×D311yescharacteristic
87C3×#120—61no—
88C3×C4×D4—61no—
89C2×C3×C22⋊C4—31no—
90C2×#217—61no—
91C2×#216—61no—
92C2×#216—61no—
93#939—21no—
94#937—21no—
95D4×Dic3—21no—
96C2×#214—21no—
97C2×(C2×D3)⋊C4—61no—
98C2×(C2×D3)⋊C4C2×D311yescharacteristic
99#924C2×D311yescharacteristic
100#898—21no—
101#886—21no—
102#885—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×C32—21no—
118C23×C32C2×D411yescharacteristic
119C22×C3×D3—21no—
120C22×C3×D3—21no—
121C22×C3×D3—21no—
122C22×C3×D3—41no—
123C22×C3×D3—41no—
124C32×D4—41no—
125C32×D4—41no—
126C32×D4—41no—
127C2×C32×C4—21no—
128C2×C32×C4—41no—
129C2×C32×C4C2×D411yescharacteristic
130C2×C32×C4C2×D411yescharacteristic
131C2×C3⋊Dic3—41no—
132C2×C3⋊Dic3—41no—
133C2×C3⋊Dic3—41no—
134C3×#67—81no—
135C3×#67—81no—
136C2×C3×Dic3—22no—
137C2×C3×Dic3—21no—
138C2×C3×Dic3—21no—
139C2×C3×Dic3—21no—
140C2×C3×Dic3—41no—
141C2×C3×Dic3—41no—
142C2×C3×Dic3—41no—
143C2×C3×Dic3—81no—
144C3×Dic6—22no—
145C3×Dic6—22no—
146C3×Dic6—41no—
147C3×Dic6—41no—
148C3×Dic6—41no—
149C3⋊(C3⋊C8)—41no—
150C3×C3⋊C8—41no—
151#538—31no—
152#451—91no—
153#448—31no—
154#393—91no—
155#328—91no—
156#327—91no—
157#323—91no—
158C24×C3—31no—
159C23×D3#6711yescharacteristic
160C2×C3×Q8—31no—
161C2×C3×Q8—61no—
162C2×C3×Q8—61no—
163C2×C3×Q8—122no—
164C2×C3×D4—21no—
165C2×C3×D4—21no—
166C2×C3×D4—21no—
167C2×C3×D4—41no—
168C2×C3×D4—61no—
169C2×C3×D4—61no—
170C2×C3×D4#6711yescharacteristic
171C2×C3×D4#6711yescharacteristic
172C22×C3×C4—21no—
173C22×C3×C4—31no—
174C22×C3×C4—61no—
175C22×C3×C4—61no—
176C22×C3×C4C22×D311yescharacteristic
177C22×C3×C4C22×D311yescharacteristic
178C2×#67—21no—
179C2×#67—21no—
180C22×Dic3—21no—
181C22×Dic3—21no—
182C22×Dic3—61no—
183C22×Dic3—61no—
184C22×Dic3—121no—
185C22×Dic3#6711yescharacteristic
186C2×Dic6—21no—
187C2×Dic6—21no—
188C2×Dic6—42no—
189C2×Dic6#6711yescharacteristic
190C3×SD16—122no—
191C2×C3×C8—61no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—31no—
194C3×C4⋊C4—61no—
195C3×C4⋊C4—61no—
196C3×C4⋊C4#6711yescharacteristic
197C3×C4⋊C4#6711yescharacteristic
198C3×C22⋊C4—21no—
199C3×C22⋊C4—21no—
200C3×C22⋊C4—41no—
201C3×C22⋊C4—61no—
202C3×C22⋊C4—61no—
203C3×C22⋊C4—61no—
204C3×C22⋊C4—61no—
205C3×C22⋊C4—61no—
206C3×C22⋊C4—61no—
207C3×C42—61no—
208#217—21no—
209#217—21no—
210#217—122no—
211#216—122no—
212#216—122no—
213#214—42no—
214(C2×D3)⋊C4—21no—
215(C2×D3)⋊C4—21no—
216(C2×D3)⋊C4—21no—
217(C2×D3)⋊C4—21no—
218(C2×D3)⋊C4—122no—
219C4⋊Dic3—21no—
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—41no—
230C22×C32—41no—
231C22×C32—41no—
232C22×C32#12011yescharacteristic
233C22×C32#13511yescharacteristic
234C22×C32#13711yescharacteristic
235C2×C3×D3—42no—
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—
243C3⋊Dic3—81no—
244C3×Dic3—41no—
245C3×Dic3—41no—
246C3×Dic3—42no—
247C3×Dic3—41no—
248C22×Q8—31no—
249C2×Q16—181no—
250C2×SD16—61no—
251#126—61no—
252#124—61no—
253#123—61no—
254C22⋊Q8—31no—
255#121—61no—
256#121D3211yescharacteristic
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—21no—
269C23×C3—21no—
270C23×C3—41no—
271C23×C3—61no—
272C23×C3—61no—
273C23×C3—61no—
274C23×C3—121no—
275C23×C3—121no—
276C23×C3D3×D411yescharacteristic
277C23×C3C2×#6711yescharacteristic
278C22×D3—21no—
279C22×D3—21no—
280C22×D3—21no—
281C22×D3—41no—
282C22×D3—41no—
283C3×Q8—62no—
284C3×Q8—62no—
285C3×Q8—121no—
286C3×Q8—121no—
287C3×Q8—121no—
288C3×D4—41no—
289C3×D4—41no—
290C3×D4—41no—
291C3×D4—41no—
292C3×D4—41no—
293C3×D4—41no—
294C3×D4—81no—
295C3×D4—81no—
296C3×D4—81no—
297C3×D4—241no—
298C3×D4—241no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—62no—
307C2×C3×C4—61no—
308C2×C3×C4—61no—
309C2×C3×C4—61no—
310C2×C3×C4—81no—
311C2×C3×C4—121no—
312C2×C3×C4—121no—
313C2×C3×C4—121no—
314C2×C3×C4—241no—
315C2×C3×C4D3×D411yescharacteristic
316C2×C3×C4C2×#6711yescharacteristic
317C2×C3×C4C2×#6711yescharacteristic
318C2×C3×C4C2×#6711yescharacteristic
319#67—81no—
320#67—81no—
321C2×Dic3—22no—
322C2×Dic3—21no—
323C2×Dic3—21no—
324C2×Dic3—21no—
325C2×Dic3—41no—
326C2×Dic3—41no—
327C2×Dic3—41no—
328C2×Dic3—81no—
329C2×Dic3—121no—
330C2×Dic3—121no—
331C2×Dic3—121no—
332C2×Dic3—121no—
333C2×Dic3—122no—
334C2×Dic3—121no—
335C2×Dic3—121no—
336C2×Dic3—241no—
337C2×Dic3—241no—
338Dic6—22no—
339Dic6—22no—
340Dic6—41no—
341Dic6—41no—
342Dic6—41no—
343C3×C8—121no—
344C3⋊C8—41no—
345C3⋊C8—121no—
346C3⋊C8—121no—
347C3⋊C8—241no—
348C2×C32—21no—
349C2×C32—21no—
350C2×C32—81no—
351C2×C32#45211yescharacteristic
352C2×C32#45511yescharacteristic
353C2×C32B(4,𝔽2)11yescharacteristic
354C3×D3—81no—
355C24—31no—
356C2×Q8—31no—
357C2×Q8—61no—
358C2×Q8—61no—
359C2×Q8—122no—
360C2×D4—21no—
361C2×D4—61no—
362C2×D4—61no—
363C2×D4#62511yescharacteristic
364C22×C4—31no—
365C22×C4—61no—
366C22×C4—61no—
367C22×C4—181no—
368C22×C4C2×D3211yescharacteristic
369Q16—362no—
370SD16—122no—
371C2×C8—61no—
372C2×C8—181no—
373C4⋊C4—31no—
374C4⋊C4—61no—
375C4⋊C4—61no—
376C4⋊C4#62511yescharacteristic
377C22⋊C4—21no—
378C22⋊C4—61no—
379C22⋊C4—61no—
380C22⋊C4—61no—
381C22⋊C4—61no—
382C22⋊C4—61no—
383C22⋊C4—61no—
384C22⋊C4—362no—
385C42—61no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—41no—
394C22×C3—41no—
395C22×C3—41no—
396C22×C3—41no—
397C22×C3—41no—
398C22×C3—41no—
399C22×C3—41no—
400C22×C3—41no—
401C22×C3—81no—
402C22×C3—81no—
403C22×C3—81no—
404C22×C3—122no—
405C22×C3—121no—
406C22×C3—121no—
407C22×C3—121no—
408C22×C3—241no—
409C22×C3#91511yescharacteristic
410C22×C3#91611yescharacteristic
411C22×C3#93711yescharacteristic
412C22×C3#95011yescharacteristic
413C22×C3#95111yescharacteristic
414C22×C3#95311yescharacteristic
415C2×D3—42no—
416C2×D3—41no—
417C2×D3—41no—
418C2×D3—41no—
419C2×D3—81no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—41no—
425C3×C4—41no—
426C3×C4—41no—
427C3×C4—41no—
428C3×C4—81no—
429C3×C4—121no—
430C3×C4—121no—
431C3×C4—122no—
432C3×C4—121no—
433Dic3—41no—
434Dic3—41no—
435Dic3—41no—
436Dic3—42no—
437Dic3—241no—
438Dic3—241no—
439Dic3—242no—
440C32#160611yescharacteristic, complemented, Sylow
441C23—21no—
442C23—61no—
443C23—61no—
444C23—61no—
445C23—121no—
446C23—121no—
447C23D3×#6711yescharacteristic
448Q8—62no—
449Q8—62no—
450Q8—121no—
451Q8—121no—
452Q8—121no—
453D4—41no—
454D4—41no—
455D4—41no—
456D4—241no—
457D4—241no—
458C2×C4—21no—
459C2×C4—41no—
460C2×C4—62no—
461C2×C4—61no—
462C2×C4—61no—
463C2×C4—61no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—121no—
467C2×C4—241no—
468C2×C4—361no—
469C2×C4—361no—
470C2×C4—361no—
471C2×C4C2×#62511yescharacteristic
472C2×C4D3×#6711yescharacteristic
473C8—121no—
474C8—361no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—41no—
483C2×C3—41no—
484C2×C3—81no—
485C2×C3—81no—
486C2×C3—161no—
487C2×C3—241no—
488C2×C3#503211yescharacteristic
489C2×C3#517311yescharacteristic
490C2×C3#520211yescharacteristic
491C2×C3#523911yescharacteristic
492C2×C3#524211yescharacteristic
493C2×C3#524411yescharacteristic
494D3—81no—
495C22—21no—
496C22—21no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—122no—
501C22—121no—
502C22—121no—
503C22—121no—
504C22—241no—
505C22#6392911yescharacteristic
506C22#6393511yescharacteristic
507C22#6397611yescharacteristic
508C4—21no—
509C4—21no—
510C4—41no—
511C4—122no—
512C4—121no—
513C4—121no—
514C4—121no—
515C4—721no—
516C3—21no—
517C3#7097311yescharacteristic, complemented
518C3#7188511yescharacteristic, complemented
519C2—21no—
520C2—21no—
521C2—81no—
522C2—241no—
523C2#1059146111yescharacteristic
524C2#1059153011yescharacteristic
525C2#1059166411yescharacteristic
subgroupG/H
centerC22#63976
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC32×#121C22
socleC22×C32#120
prime radicalC3×#5617C2
hypercenter#121D32
hypocenterC32#1606
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
2. C3×#5040C2semidirect
3. #10591304C2semidirect
4. #10591185C2semidirect
5. #10589403C2semidirect
6. #10589152C2semidirect
7. #10589145C2semidirect
35. #5040D3semidirect
440. C32#1606semidirect
517. C3#70973semidirect
518. C3#71885semidirect

Maximal subgroups

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

subgroupindex
C3×#56172
C3×#50402
#105913042
#105911852
#105894032
#105891522
#105891452
#718853
#709733