Finite Group Databasebeta

← #11784939

Subgroups — #11784939

Order 1152.

1175 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5964C211yesmaximal, characteristic
2#10592933C211yesmaximal, characteristic, complemented
3C3×#5240C211yesmaximal, characteristic, complemented
4#10591111C212yesmaximal, complemented
5#10591083C212yesmaximal, complemented
6#72146—31nomaximal
7#70054—31nomaximal
8C2×C32×#41C2211yescharacteristic
9C3×#1010—22no—
10C3×#1009—22no—
11C3×D3×#41—24no—
12C3×#1007C412yescomplemented
13C3×#1004—22no—
14C3×#1004C412yescomplemented
15C2×C3×#235—22no—
16C2×C3×#235C2212yes—
17C2×#3748C2211yescharacteristic, complemented
18C22×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
19C3×#926C2212yescomplemented
20#63657—22no—
21C3×#834—22no—
22#63569—24no—
23#63565—22no—
24#63562—22no—
25C3×#585—31no—
26#5964D311yescharacteristic
27C3×#410—31no—
28#5240D311yescharacteristic, complemented
29#5236—32no—
30#5234—31no—
31#5234—31no—
32#5234—61no—
33#5233—32no—
34#5005—32no—
35#5002—32no—
36C32×#41—22no—
37C32×#41—22no—
38C2×C32×Q8C2×C411yescharacteristic
39C2×C32×D4C2×C411yescharacteristic
40C2×C32×D4D412yes—
41C22×C32×C4D412yes—
42C22×C32×C4C2311yescharacteristic
43C2×C3×#67—24no—
44C2×C3×#67—22no—
45C3×#239—24no—
46C3×D3×Q8—24no—
47C3×#237—28no—
48C3×#237—24no—
49C3×D3×D4—28no—
50C3×D3×D4—24no—
51C3×#235—24no—
52C3×#235—24no—
53C3×#235—24no—
54C3×#235—22no—
55C3×#235—22no—
56C3×#235C2×C414yes—
57C3×#235D414yes—
58C2×C3×D12—22no—
59C2×C3×D12D412yes—
60C2×C3×C4×D3—24no—
61C2×C3×C4×D3—22no—
62C2×C3×Dic6—22no—
63C2×C3×Dic6D412yes—
64#3748—22no—
65C2×C3⋊(C3⋊C8)—21no—
66C2×C3⋊(C3⋊C8)—21no—
67C2×C3⋊(C3⋊C8)—22no—
68C3×#217—22no—
69C3×C4⋊Dic3—22no—
70C3×C4×Dic3—22no—
71#1794—91noSylow
72C3×2−1+4—32no—
73C3×2−1+4—62no—
74C3×H(5,𝔽2)—32no—
75C3×H(5,𝔽2)—62no—
76C3×H(5,𝔽2)—62no—
77C2×C3×#41—21no—
78C2×C3×#41—32no—
79C2×C3×#41—64no—
80C2×C3×#41—62no—
81C2×C3×#41C2×D311yescharacteristic
82C2×C3×#41C2×D311yescharacteristic
83#1010—22no—
84#1009—22no—
85D3×#41—24no—
86#1007Dic312yescomplemented
87#1004—22no—
88#1004Dic312yescomplemented
89C2×#235—22no—
90C2×#235C2×D312yes—
91C3×#117—32no—
92#926C2×D312yescomplemented
93C2×#208—31no—
94C2×#208—31no—
95C2×#208—61no—
96C22×C3⋊C8—31no—
97C22×C3⋊C8—31no—
98C22×C3⋊C8—61no—
99C3×#99—62no—
100#837—64no—
101#835—62no—
102#834—22no—
103#832—62no—
104#830—64no—
105#830—62no—
106#830—62no—
107#810—62no—
108#809—62no—
109#805—64no—
110C23×C32C22⋊C412yes—
111C23×C32C2×D411yescharacteristic
112C22×C3×D3—24no—
113C22×C3×D3—22no—
114C32×Q8—22no—
115C32×D4—24no—
116C32×D4—22no—
117C2×C32×C4—22no—
118C2×C32×C4—22no—
119C2×C32×C4C22⋊C412yes—
120C2×C32×C4C22×C411yescharacteristic
121C2×C32×C4C2×D411yescharacteristic
122C2×C32×C4C2×D412yes—
123C2×C32×C4C2×D412yes—
124C3×#67—28no—
125C3×#67—24no—
126C3×#67—24no—
127C2×C3×Dic3—24no—
128C2×C3×Dic3—22no—
129C2×C3×Dic3—42no—
130C3×D12—24no—
131C3×D12—22no—
132C3×D12C22⋊C414yes—
133C3×C4×D3—28no—
134C3×C4×D3—24no—
135C3×C4×D3—24no—
136C3×Dic6—24no—
137C3×Dic6—22no—
138C3×Dic6C22⋊C414yes—
139C3⋊(C3⋊C8)—41no—
140C3⋊(C3⋊C8)—41no—
141#585—31no—
142#421—92no—
143#416—92no—
144#410—31no—
145#408—91no—
146C3×#41—22no—
147C3×#41—22no—
148C3×#41—22no—
149C3×#41—22no—
150C3×#41—34no—
151C3×#41—34no—
152C3×#41—42no—
153C3×#41—42no—
154C3×#41—64no—
155C3×#41—64no—
156C3×#41—64no—
157C3×#41—68no—
158C3×#41—64no—
159C3×#41—64no—
160C3×#41—62no—
161C3×#41—62no—
162C2×C3×Q8—21no—
163C2×C3×Q8—32no—
164C2×C3×Q8—62no—
165C2×C3×Q8—64no—
166C2×C3×Q8C4×D311yescharacteristic
167C2×C3×Q8C2×Dic311yescharacteristic
168C2×C3×D4—21no—
169C2×C3×D4—22no—
170C2×C3×D4—32no—
171C2×C3×D4—62no—
172C2×C3×D4—64no—
173C2×C3×D4—68no—
174C2×C3×D4—64no—
175C2×C3×D4—62no—
176C2×C3×D4C4×D311yescharacteristic
177C2×C3×D4C2×Dic311yescharacteristic
178C2×C3×D4#6712yes—
179C2×C3×D4#6712yes—
180C22×C3×C4—22no—
181C22×C3×C4—21no—
182C22×C3×C4—64no—
183C22×C3×C4—62no—
184C22×C3×C4D1212yes—
185C22×C3×C4#6712yes—
186C22×C3×C4C22×D311yescharacteristic
187C22×C3×C4C22×D311yescharacteristic
188C2×#67—24no—
189C2×#67—22no—
190#239—24no—
191D3×Q8—24no—
192#237—28no—
193#237—24no—
194D3×D4—28no—
195D3×D4—24no—
196#235—24no—
197#235—24no—
198#235—24no—
199#235—22no—
200#235—22no—
201#235C2×Dic314yes—
202#235#6714yes—
203C2×D12—22no—
204C2×D12#6712yes—
205C2×C4×D3—24no—
206C2×C4×D3—22no—
207C2×Dic6—22no—
208C2×Dic6#6712yes—
209C3×C4⋊C4—62no—
210C3×C22⋊C4—62no—
211C3×C42—62no—
212#217—22no—
213C4⋊Dic3—22no—
214C4×Dic3—22no—
215#208—62no—
216#208—62no—
217#208—122no—
218C2×C3⋊C8—62no—
219C2×C3⋊C8—62no—
220C2×C3⋊C8—61no—
221C2×C3⋊C8—61no—
222C2×C3⋊C8—62no—
223C2×C3⋊C8—62no—
224C2×C3⋊C8—61no—
225C2×C3⋊C8—61no—
226C2×C3⋊C8—122no—
227C22×C32—22no—
228C22×C32—41no—
229C22×C32—42no—
230C22×C32C2×C22⋊C411yescharacteristic
231C22×C32#12012yes—
232C2×C3×D3—24no—
233C2×C3×D3—82no—
234C2×C3×D3—84no—
235C32×C4—22no—
236C32×C4C2×C22⋊C412yes—
237C32×C4#12011yescharacteristic
238C32×C4#12011yescharacteristic
239C3×Dic3—24no—
240C3×Dic3—82no—
2412−1+4—32no—
2422−1+4—62no—
243H(5,𝔽2)—32no—
244H(5,𝔽2)—62no—
245H(5,𝔽2)—62no—
246C2×#41—32no—
247C2×#41—64no—
248C2×#41—62no—
249C2×#41D3211yescharacteristic
250C2×M16—91no—
251C22×C8—91no—
252#117—32no—
253#104—184no—
254#103—182no—
255#102—182no—
256#99—62no—
257C22⋊C8—182no—
258C23×C3—22no—
259C23×C3—21no—
260C23×C3—64no—
261C23×C3—62no—
262C23×C3(C2×D3)⋊C412yes—
263C23×C3#21712yes—
264C23×C3C2×#6711yescharacteristic
265C23×C3C2×#6711yescharacteristic
266C22×D3—24no—
267C22×D3—22no—
268C3×Q8—22no—
269C3×Q8—22no—
270C3×Q8—34no—
271C3×Q8—42no—
272C3×Q8—64no—
273C3×Q8—62no—
274C3×D4—24no—
275C3×D4—24no—
276C3×D4—22no—
277C3×D4—22no—
278C3×D4—34no—
279C3×D4—42no—
280C3×D4—44no—
281C3×D4—64no—
282C3×D4—68no—
283C3×D4—64no—
284C3×D4—64no—
285C3×D4—62no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—68no—
298C2×C3×C4—64no—
299C2×C3×C4—62no—
300C2×C3×C4—64no—
301C2×C3×C4—64no—
302C2×C3×C4—122no—
303C2×C3×C4(C2×D3)⋊C412yes—
304C2×C3×C4#21712yes—
305C2×C3×C4C2×C4×D311yescharacteristic
306C2×C3×C4C2×D1211yescharacteristic
307C2×C3×C4D3×D412yes—
308C2×C3×C4D3×D412yes—
309C2×C3×C4C22×Dic311yescharacteristic
310C2×C3×C4C2×#6711yescharacteristic
311C2×C3×C4C2×#6712yes—
312C2×C3×C4C2×#6712yes—
313#67—28no—
314#67—24no—
315#67—24no—
316C2×Dic3—24no—
317C2×Dic3—22no—
318C2×Dic3—42no—
319D12—24no—
320D12—22no—
321D12#21714yes—
322C4×D3—28no—
323C4×D3—24no—
324C4×D3—24no—
325Dic6—24no—
326Dic6—22no—
327Dic6#21714yes—
328C3⋊C8—122no—
329C3⋊C8—122no—
330C3⋊C8—121no—
331C3⋊C8—121no—
332C3⋊C8—121no—
333C3⋊C8—121no—
334C2×C32—21no—
335C2×C32—22no—
336C2×C32—42no—
337C2×C32#37911yescharacteristic
338C3×D3—44no—
339#41—22no—
340#41—22no—
341#41—34no—
342#41—34no—
343#41—64no—
344#41—64no—
345#41—64no—
346#41—68no—
347#41—64no—
348#41—64no—
349#41—62no—
350#41—62no—
351C2×Q8—32no—
352C2×Q8—62no—
353C2×Q8—64no—
354C2×Q8D3×Dic311yescharacteristic
355C2×D4—32no—
356C2×D4—62no—
357C2×D4—64no—
358C2×D4—68no—
359C2×D4—64no—
360C2×D4—62no—
361C2×D4D3×Dic311yescharacteristic
362C2×D4#62512yes—
363C22×C4—64no—
364C22×C4—62no—
365C22×C4#62612yes—
366C22×C4C2×D3211yescharacteristic
367M16—182no—
368C2×C8—181no—
369C2×C8—181no—
370C2×C8—182no—
371C4⋊C4—62no—
372C22⋊C4—62no—
373C42—62no—
374C22×C3—22no—
375C22×C3—22no—
376C22×C3—21no—
377C22×C3—22no—
378C22×C3—42no—
379C22×C3—41no—
380C22×C3—41no—
381C22×C3—42no—
382C22×C3—42no—
383C22×C3—42no—
384C22×C3—44no—
385C22×C3—64no—
386C22×C3—242no—
387C22×C3—244no—
388C22×C3C2×(C2×D3)⋊C411yescharacteristic
389C22×C3#93712yes—
390C22×C3C2×#21711yescharacteristic
391C22×C3#95312yes—
392C2×D3—24no—
393C2×D3—82no—
394C2×D3—84no—
395C3×C4—22no—
396C3×C4—22no—
397C3×C4—22no—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—42no—
401C3×C4—64no—
402C3×C4—242no—
403C3×C4D3×C22⋊C412yes—
404C3×C4#88211yescharacteristic
405C3×C4#88211yescharacteristic
406C3×C4C2×#21712yes—
407C3×C4#95311yescharacteristic
408C3×C4#95311yescharacteristic
409Dic3—24no—
410Dic3—82no—
411C32#179411yescharacteristic, complemented, Sylow
412C23—64no—
413C23—62no—
414C23(C2×D3)⋊Dic312yes—
415C23C2×#62511yescharacteristic
416Q8—22no—
417Q8—34no—
418Q8—64no—
419Q8—62no—
420D4—24no—
421D4—22no—
422D4—34no—
423D4—64no—
424D4—68no—
425D4—64no—
426D4—64no—
427D4—62no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—68no—
431C2×C4—64no—
432C2×C4—62no—
433C2×C4—64no—
434C2×C4—64no—
435C2×C4—122no—
436C2×C4(C2×D3)⋊Dic312yes—
437C2×C4C2×D3×Dic311yescharacteristic
438C2×C4C2×#62611yescharacteristic
439C2×C4D3×#6712yes—
440C2×C4D3×#6712yes—
441C8—361no—
442C8—361no—
443C2×C3—22no—
444C2×C3—22no—
445C2×C3—21no—
446C2×C3—22no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—42no—
450C2×C3—42no—
451C2×C3—42no—
452C2×C3—82no—
453C2×C3—124no—
454C2×C3#495511yescharacteristic
455C2×C3#524711yescharacteristic
456D3—44no—
457C22—22no—
458C22—41no—
459C22—42no—
460C22—64no—
461C22—242no—
462C22—244no—
463C22C2×(C2×D3)⋊Dic311yescharacteristic
464C22#6397612yes—
465C4—22no—
466C4—64no—
467C4—242no—
468C4D3×#21712yes—
469C4#6397711yescharacteristic
470C4#6397711yescharacteristic
471C3—21no—
472C3#7005411yescharacteristic, complemented
473C3#7214611yescharacteristic, complemented
474C2—22no—
475C2—21no—
476C2—42no—
477C2—124no—
478C2#1059170611yescharacteristic
subgroupG/H
centerC4#63977
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×#41C22
socleC2×C32#379
prime radicalC3×#5964C2
cosocleC22×C32×C4C23
hypercenterC2×#41D32
hypocenterC32#1794
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. #10592933C2semidirect
3. C3×#5240C2semidirect
4. #10591111C2semidirect
5. #10591083C2semidirect
12. C3×#1007C4semidirect
14. C3×#1004C4semidirect
17. C2×#3748C22semidirect
18. C22×C3⋊(C3⋊C8)C22semidirect
19. C3×#926C22semidirect
28. #5240D3semidirect
86. #1007Dic3semidirect
88. #1004Dic3semidirect
92. #926C2×D3semidirect
411. C32#1794semidirect
472. C3#70054semidirect
473. C3#72146semidirect

Maximal subgroups

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

subgroupindex
C3×#59642
#105929332
C3×#52402
#105911112
#105910832
#721463
#700543