Finite Group Databasebeta

← #11787811

Subgroups — #11787811

Order 1152.

795 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5521C211yesmaximal, characteristic
2C2×#63860C211yesmaximal, characteristic
3#10592703C211yesmaximal, characteristic, complemented
4#10591107C211yesmaximal, characteristic, complemented
5C2×#63558C211yesmaximal, characteristic
6C3×#4532C211yesmaximal, characteristic, complemented
7#10589161C211yesmaximal, characteristic, complemented
8#71649—31nomaximal
9#68759—31nomaximal
10C22×#634C2211yescharacteristic
11C22×C3×D12C2211yescharacteristic
12C32×#117C2211yescharacteristic
13C2×#3748C2211yescharacteristic
14C3×#926C2211yescharacteristic
15C2×C3×C4×Dic3C2211yescharacteristic
16C2×C3×#208C2211yescharacteristic
17C3×#893—22no—
18C3×D3×C22⋊C4—22no—
19C3×C4×D12—22no—
20C2×(C2×D3)⋊Dic3—22no—
21#63860—22no—
22#63860C414yescomplemented
23#63637—22no—
24#63567—22no—
25#63558—22no—
26C3×#518—31no—
27C2×#936—31no—
28#5521D311yescharacteristic
29C2×#875—31no—
30C2×#833—31no—
31C2×#823—31no—
32#5120—31no—
33#5004—31no—
34#5004—31no—
35#5004—31no—
36#5004—61no—
37C3×#339—31no—
38#4532—31no—
39#4532D311yescharacteristic, complemented
40#4523—31no—
41C23×C3×D3D412yes—
42C22×C32×C4C2311yescharacteristic
43C22×C3⋊Dic3—21no—
44C2×#634—21no—
45C2×#634—21no—
46C2×#634—21no—
47C2×#634—21no—
48C2×#634—21no—
49C2×#634C2×C412yescomplemented
50C22×C3×Dic3—21no—
51C2×C3×D12—21no—
52C2×C3×D12—21no—
53C2×C3×D12—21no—
54C2×C3×D12—21no—
55C2×C3×D12C2×C412yes—
56C2×C3×D12D412yes—
57C2×C3×C4×D3—24no—
58C32×C4⋊C4D412yes—
59C32×C22⋊C4—21no—
60C32×C42—21no—
61#3748—21no—
62#3748—21no—
63C2×C3⋊(C3⋊C8)—21no—
64C3×#217—21no—
65C3×(C2×D3)⋊C4—24no—
66C3×C4⋊Dic3D412yes—
67C3×C4×Dic3—21no—
68C3×C4×Dic3—21no—
69C3×C4×Dic3C2×C412yescomplemented
70C3×#208—21no—
71C3×#208—21no—
72C2×C3×C3⋊C8—21no—
73(C2×D3)⋊Dic3—28no—
74#1885—91noSylow
75C22×C3×D4—31no—
76C22×D12C2×D311yescharacteristic
77C22×Dic6—31no—
78C22×Dic6—31no—
79C22×Dic6—61no—
80C2×C3×M16—31no—
81C3×C4×D4—62no—
82C3×C4×D4—62no—
83C3×#117—21no—
84C3×#117—31no—
85C3×#117C2×D311yescharacteristic
86C3×#117C2×D311yescharacteristic
87C2×C3×C22⋊C4—62no—
88C2×C3×C42—31no—
89C2×#217—62no—
90#936—34no—
91#936—62no—
92C2×(C2×D3)⋊C4—62no—
93#926C2×D311yescharacteristic
94C2×C4×Dic3C2×D311yescharacteristic
95C2×#208—31no—
96C2×#208—31no—
97C2×#208—61no—
98C2×#208C2×D311yescharacteristic
99#893—22no—
100D3×C22⋊C4—22no—
101#875—34no—
102#875—62no—
103C4×D12—22no—
104#833—62no—
105#823—62no—
106#816—62no—
107#810—62no—
108#810—64no—
109#810—62no—
110#810—62no—
111C23×C32C2×D411yescharacteristic
112C22×C3×D3—22no—
113C22×C3×D3—22no—
114C22×C3×D3—44no—
115C22×C3×D3C22⋊C414yes—
116C2×C32×C4—22no—
117C2×C32×C4C22×C411yescharacteristic
118C2×C32×C4C2×D411yescharacteristic
119C2×C32×C4C2×D411yescharacteristic
120C2×C32×C4C2×D411yescharacteristic
121C2×C32×C4#4111yescharacteristic
122C2×C32×C4#4111yescharacteristic
123C2×C3⋊Dic3—22no—
124C2×C3⋊Dic3—41no—
125C2×C3⋊Dic3—41no—
126#634—22no—
127#634—22no—
128#634—41no—
129#634—41no—
130C2×C3×Dic3—22no—
131C2×C3×Dic3—22no—
132C2×C3×Dic3—42no—
133C3×D12—22no—
134C3×D12—22no—
135C3×D12—22no—
136C3×D12—22no—
137C3×C4×D3—84no—
138C3⋊(C3⋊C8)—41no—
139C3×C3⋊C8—41no—
140C2×#124—91no—
141#518—31no—
142#419—91no—
143#419—91no—
144C2×#100—91no—
145#339—31no—
146#339—91no—
147C24×C3—32no—
148C23×D3#6712yes—
149C2×C3×D4—32no—
150C2×C3×D4—32no—
151C2×C3×D4—61no—
152C2×C3×D4—61no—
153C2×C3×D4—61no—
154C2×C3×D4—61no—
155C22×C3×C4—21no—
156C22×C3×C4—64no—
157C22×C3×C4—61no—
158C22×C3×C4C22×D311yescharacteristic
159C22×C3×C4C22×D311yescharacteristic
160C22×Dic3—21no—
161C22×Dic3—61no—
162C22×Dic3—61no—
163C22×Dic3—121no—
164C2×D12—21no—
165C2×D12—21no—
166C2×D12—21no—
167C2×D12—21no—
168C2×D12C4×D312yes—
169C2×D12D1212yes—
170C2×C4×D3—24no—
171C2×Dic6—32no—
172C2×Dic6—32no—
173C2×Dic6—62no—
174C2×Dic6—62no—
175C2×Dic6—62no—
176C2×Dic6—61no—
177C2×Dic6—61no—
178C2×Dic6—61no—
179C2×Dic6—61no—
180C2×Dic6—61no—
181C2×Dic6—61no—
182C2×Dic6—61no—
183C2×Dic6—61no—
184C2×Dic6—61no—
185C2×Dic6—61no—
186C2×Dic6—121no—
187C2×Dic6—121no—
188C2×Dic6—121no—
189C3×M16—61no—
190C3×M16—61no—
191C2×C3×C8—61no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—32no—
194C3×C4⋊C4D1212yes—
195C3×C4⋊C4#6712yes—
196C3×C22⋊C4—21no—
197C3×C22⋊C4—21no—
198C3×C22⋊C4—41no—
199C3×C22⋊C4—64no—
200C3×C22⋊C4—61no—
201C3×C42—21no—
202C3×C42—21no—
203C3×C42—32no—
204C3×C42—41no—
205C3×C42—61no—
206C3×C42—61no—
207#217—21no—
208#217—68no—
209(C2×D3)⋊C4—24no—
210(C2×D3)⋊C4—68no—
211C4⋊Dic3#6712yes—
212C4×Dic3—21no—
213C4×Dic3—21no—
214C4×Dic3C4×D312yescomplemented
215#208—21no—
216#208—21no—
217#208—61no—
218#208—61no—
219#208—61no—
220#208—61no—
221#208—121no—
222#208—121no—
223C2×C3⋊C8—21no—
224C2×C3⋊C8—62no—
225C2×C3⋊C8—61no—
226C2×C3⋊C8—61no—
227C22×C32—22no—
228C22×C32C2×C22⋊C411yescharacteristic
229C22×C32#12011yescharacteristic
230C22×C32#12311yescharacteristic
231C2×C3×D3—24no—
232C2×C3×D3—44no—
233C2×C3×D3—44no—
234C2×C3×D3—82no—
235C32×C4—42no—
236C32×C4C4×D411yescharacteristic
237C32×C4C4×D411yescharacteristic
238C32×C4#12111yescharacteristic
239C32×C4#12111yescharacteristic
240C3⋊Dic3—42no—
241C3×Dic3—42no—
242C3×Dic3—42no—
243C22×Q8—91no—
244C22×D4—31no—
245C2×M16—31no—
246C2×M16—91no—
247#124—94no—
248#124—182no—
249C4×D4—62no—
250C4×D4—62no—
251#117—31no—
252#117D3211yescharacteristic
253C2×C22⋊C4—62no—
254C2×C22⋊C4—182no—
255C2×C42—31no—
256#103—182no—
257#103—182no—
258#100—182no—
259C23×C3—21no—
260C23×C3—34no—
261C23×C3—62no—
262C23×C3—62no—
263C23×C3—124no—
264C23×C3D3×D411yescharacteristic
265C23×C3C2×#6711yescharacteristic
266C22×D3—22no—
267C22×D3—22no—
268C22×D3—44no—
269C22×D3(C2×D3)⋊C414yes—
270C3×D4—62no—
271C3×D4—62no—
272C3×D4—62no—
273C3×D4—62no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—42no—
283C2×C3×C4—62no—
284C2×C3×C4—62no—
285C2×C3×C4—122no—
286C2×C3×C4—244no—
287C2×C3×C4C2×C4×D311yescharacteristic
288C2×C3×C4C2×C4×D311yescharacteristic
289C2×C3×C4C2×D1211yescharacteristic
290C2×C3×C4C2×D1211yescharacteristic
291C2×C3×C4#23511yescharacteristic
292C2×C3×C4#23511yescharacteristic
293C2×C3×C4#23511yescharacteristic
294C2×C3×C4D3×D411yescharacteristic
295C2×C3×C4D3×D411yescharacteristic
296C2×C3×C4#23911yescharacteristic
297C2×C3×C4C2×#6711yescharacteristic
298C2×C3×C4C2×#6711yescharacteristic
299C2×Dic3—22no—
300C2×Dic3—22no—
301C2×Dic3—42no—
302C2×Dic3—62no—
303C2×Dic3—62no—
304C2×Dic3—122no—
305C2×Dic3—122no—
306C2×Dic3—122no—
307C2×Dic3—121no—
308C2×Dic3—121no—
309C2×Dic3—121no—
310C2×Dic3—121no—
311D12—22no—
312D12—22no—
313D12—22no—
314D12—22no—
315C4×D3—84no—
316Dic6—64no—
317Dic6—62no—
318Dic6—62no—
319Dic6—62no—
320Dic6—62no—
321Dic6—122no—
322Dic6—122no—
323Dic6—122no—
324Dic6—121no—
325Dic6—121no—
326Dic6—121no—
327Dic6—121no—
328C3×C8—121no—
329C3⋊C8—41no—
330C3⋊C8—122no—
331C3⋊C8—121no—
332C3⋊C8—121no—
333C2×C32—21no—
334C2×C32—21no—
335C2×C32#38611yescharacteristic
336C2×C32#45512yes—
337C3×D3—44no—
338C24—32no—
339C2×Q8—92no—
340C2×Q8—181no—
341C2×Q8—181no—
342C2×Q8—181no—
343C2×Q8—181no—
344C2×Q8—181no—
345C2×D4—32no—
346C2×D4—32no—
347C2×D4—61no—
348C2×D4—61no—
349C2×D4—61no—
350C2×D4—61no—
351C22×C4—64no—
352C22×C4—61no—
353C22×C4—181no—
354C22×C4C2×D3211yescharacteristic
355M16—61no—
356M16—61no—
357M16—181no—
358M16—181no—
359C2×C8—61no—
360C2×C8—181no—
361C4⋊C4—32no—
362C4⋊C4#62612yes—
363C22⋊C4—21no—
364C22⋊C4—64no—
365C22⋊C4—61no—
366C22⋊C4—188no—
367C42—21no—
368C42—32no—
369C42—61no—
370C42—61no—
371C22×C3—22no—
372C22×C3—22no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—42no—
377C22×C3—64no—
378C22×C3—124no—
379C22×C3—124no—
380C22×C3—242no—
381C22×C3D3×C22⋊C411yescharacteristic
382C22×C3#88211yescharacteristic
383C22×C3#89411yescharacteristic
384C22×C3C2×(C2×D3)⋊C411yescharacteristic
385C22×C3#92911yescharacteristic
386C22×C3#95311yescharacteristic
387C2×D3—24no—
388C2×D3—44no—
389C2×D3—44no—
390C2×D3—82no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—42no—
396C3×C4—42no—
397C3×C4—82no—
398C3×C4—122no—
399C3×C4—122no—
400C3×C4C4×D1211yescharacteristic
401C3×C4#88411yescharacteristic
402C3×C4#89311yescharacteristic
403C3×C4#89511yescharacteristic
404C3×C4C4×#6711yescharacteristic
405C3×C4C4×#6711yescharacteristic
406C3×C4#93011yescharacteristic
407C3×C4#93011yescharacteristic
408Dic3—42no—
409Dic3—42no—
410Dic3—124no—
411Dic3—122no—
412Dic3—122no—
413C32#188511yescharacteristic, complemented, Sylow
414C23—34no—
415C23—62no—
416C23—62no—
417C23—124no—
418C23D3×#6711yescharacteristic
419Q8—182no—
420Q8—182no—
421Q8—361no—
422Q8—361no—
423D4—62no—
424D4—62no—
425D4—62no—
426D4—62no—
427C2×C4—22no—
428C2×C4—62no—
429C2×C4—62no—
430C2×C4—122no—
431C2×C4—182no—
432C2×C4—244no—
433C2×C4—361no—
434C2×C4—361no—
435C2×C4#379811yescharacteristic
436C2×C4#379911yescharacteristic
437C2×C4C4×D3211yescharacteristic
438C2×C4D3×D1211yescharacteristic
439C2×C4C2×#62611yescharacteristic
440C2×C4D3×#6711yescharacteristic
441C8—121no—
442C8—361no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—22no—
449C2×C3—41no—
450C2×C3—41no—
451C2×C3—124no—
452C2×C3#466511yescharacteristic
453C2×C3#482512yes—
454C2×C3#521311yescharacteristic
455C2×C3#524212yes—
456D3—44no—
457C22—22no—
458C22—64no—
459C22—124no—
460C22—124no—
461C22—242no—
462C22#6385011yescharacteristic
463C22D3×(C2×D3)⋊C411yescharacteristic
464C22#6397711yescharacteristic
465C4—42no—
466C4—122no—
467C4—122no—
468C4—362no—
469C4C4×#62611yescharacteristic
470C4#6390411yescharacteristic
471C4#6390611yescharacteristic
472C4#6390911yescharacteristic
473C3—21no—
474C3#6875911yescharacteristic, complemented
475C3#7164911yescharacteristic, complemented
476C2—21no—
477C2—21no—
478C2—124no—
479C2#1059138211yescharacteristic
480C2#1059168812yes—
subgroupG/H
centerC22#63977
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC32×#117C22
socleC22×C32#120
prime radicalC22×C3×D12C22
cosocleC22×C32×C4C23
hypercenter#117D32
hypocenterC32#1885
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
3. #10592703C2semidirect
4. #10591107C2semidirect
6. C3×#4532C2semidirect
7. #10589161C2semidirect
22. #63860C4semidirect
39. #4532D3semidirect
49. C2×#634C2×C4semidirect
69. C3×C4×Dic3C2×C4semidirect
214. C4×Dic3C4×D3semidirect
413. C32#1885semidirect
474. C3#68759semidirect
475. C3#71649semidirect

Maximal subgroups

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

subgroupindex
C3×#55212
C2×#638602
#105927032
#105911072
C2×#635582
C3×#45322
#105891612
#716493
#687593