Finite Group Databasebeta

← #11788722

Subgroups — #11788722

Order 1152.

671 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5745C211yesmaximal, characteristic
2#10594855C211yesmaximal, characteristic
3C3×#5120C211yesmaximal, characteristic, complemented
4#10591231C211yesmaximal, characteristic
5#10591100C211yesmaximal, characteristic, complemented
6#10589423C211yesmaximal, characteristic, complemented
7#10589158C211yesmaximal, characteristic, complemented
8#72024—31nomaximal
9#70350—31nomaximal
10C2×C3×#235C2211yescharacteristic
11C22×C3×Dic6C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13#64135C2211yescharacteristic
14#64135C2211yescharacteristic
15C3×#926C2211yescharacteristic
16C2×C3×#208C2211yescharacteristic
17C3×#907—22no—
18C3×D3×M16—22no—
19C3×#901—22no—
20#63961—22no—
21#63873C412yescomplemented
22#63863C412yescomplemented
23#63856—22no—
24C3×#816—22no—
25#63580—22no—
26#63566—22no—
27C3×#568—31no—
28#5825—31no—
29#5816—31no—
30#5745D311yescharacteristic
31C3×#419—31no—
32#5240—31no—
33#5225—31no—
34#5120—31no—
35#5120D311yescharacteristic, complemented
36#5006—31no—
37#4557—31no—
38#4557—31no—
39#4557—31no—
40#4557—61no—
41#4531—31no—
42C22×C32×C4C2311yescharacteristic
43C2×C3×#67D412yes—
44C22×C3×Dic3—21no—
45C3×#235—21no—
46C3×#235—21no—
47C3×#235—21no—
48C3×#235—21no—
49C2×C3×D12C2×C411yescharacteristic
50C2×C3×C4×D3D412yes—
51C2×C3×Dic6—21no—
52C2×C3×Dic6—21no—
53C2×C3×Dic6—21no—
54C2×C3×Dic6—21no—
55C2×C3×Dic6—21no—
56C2×C3×Dic6C2×C412yes—
57C2×C3×Dic6C2×C411yescharacteristic
58C32×M16—21no—
59C32×M16—21no—
60C2×C32×C8D412yes—
61#3757—21no—
62#3757—21no—
63C4⋊(C3⋊Dic3)—21no—
64C4⋊(C3⋊Dic3)—21no—
65C4×C3⋊Dic3—21no—
66C4×C3⋊Dic3C2×C412yes—
67C3×#217—21no—
68C3×C4⋊Dic3—21no—
69C3×C4×Dic3—21no—
70C3×#208—21no—
71C3×#208—21no—
72C2×C3×C3⋊C8D412yes—
73C3×ΓL(1,𝔽25)—24no—
74C3×D3×C8—24no—
75#3724—24no—
76(C2×D3)⋊Dic3—24no—
77#1909—91noSylow
78C2×C3×#41—31no—
79C22×C3×Q8—31no—
80C2×#235C2×D311yescharacteristic
81C22×Dic6C2×D311yescharacteristic
82C3×#131—62no—
83C3×#131—62no—
84C2×C3×M16—21no—
85C2×C3×M16—31no—
86C2×C3×M16—62no—
87C2×C3×M16C2×D311yescharacteristic
88C2×C3×M16C2×D311yescharacteristic
89C3×#117—31no—
90#947—32no—
91#946—62no—
92#944—32no—
93#944—32no—
94#936—32no—
95#935—62no—
96#929—62no—
97#926—31no—
98#926—31no—
99#926—31no—
100#926—31no—
101#926—61no—
102#926—61no—
103#926C2×D311yescharacteristic
104#924—62no—
105C2×#208C2×D311yescharacteristic
106#907—22no—
107D3×M16—22no—
108#901—22no—
109C3×#103—62no—
110#835—62no—
111#834—62no—
112#816—22no—
113#816—62no—
114#806—62no—
115C23×C32C2×D411yescharacteristic
116C22×C3×D3C22⋊C412yes—
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—
125C3×#67—22no—
126C3×#67—22no—
127C2×C3×Dic3—22no—
128C2×C3×Dic3—41no—
129C2×C3×Dic3—41no—
130C2×C3×Dic3—41no—
131C2×C3×Dic3C22⋊C412yes—
132C3×D12—21no—
133C3×D12—21no—
134C3×C4×D3—22no—
135C3×C4×D3—22no—
136C3×Dic6—21no—
137C3×Dic6—21no—
138C3×Dic6—22no—
139C3×Dic6—22no—
140C3×Dic6—41no—
141C3×Dic6—41no—
142C32×C8—22no—
143C3×C3⋊C8—22no—
144#568—31no—
145#536—91no—
146#419—31no—
147#419—91no—
148#410—91no—
149H(3,ℤ/4)—91no—
150H(3,ℤ/4)—91no—
151C3×#41—61no—
152C3×#41—61no—
153C3×#41—61no—
154C3×#41—61no—
155C2×C3×Q8—31no—
156C2×C3×Q8—32no—
157C2×C3×Q8—61no—
158C2×C3×Q8—61no—
159C2×C3×Q8—61no—
160C2×C3×Q8—61no—
161C2×C3×Q8—61no—
162C2×C3×D4—32no—
163C2×C3×D4—31no—
164C22×C3×C4—21no—
165C22×C3×C4—32no—
166C22×C3×C4—61no—
167C22×C3×C4C22×D311yescharacteristic
168C22×C3×C4C22×D311yescharacteristic
169C2×#67D1212yes—
170C22×Dic3—21no—
171#235—21no—
172#235—21no—
173#235—21no—
174#235—21no—
175C2×D12C4×D311yescharacteristic
176C2×C4×D3#6712yes—
177C2×Dic6—21no—
178C2×Dic6—21no—
179C2×Dic6—21no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6C4×D312yes—
183C2×Dic6C4×D311yescharacteristic
184C3×M16—21no—
185C3×M16—21no—
186C3×M16—21no—
187C3×M16—21no—
188C3×M16—41no—
189C3×M16—41no—
190C3×M16—64no—
191C3×M16—61no—
192C3×M16—61no—
193C2×C3×C8—22no—
194C2×C3×C8—32no—
195C2×C3×C8—64no—
196C2×C3×C8#6712yes—
197C2×C3×C8#6712yes—
198C3×C4⋊C4—61no—
199C3×C22⋊C4—61no—
200C3×C42—61no—
201#217—21no—
202#217—64no—
203#217—61no—
204#217—61no—
205#217—61no—
206#217—61no—
207#217—121no—
208#217—121no—
209(C2×D3)⋊C4—64no—
210C4⋊Dic3—21no—
211C4⋊Dic3—61no—
212C4⋊Dic3—61no—
213C4⋊Dic3—61no—
214C4⋊Dic3—61no—
215C4⋊Dic3—121no—
216C4⋊Dic3—121no—
217Dic3⋊C4—64no—
218Dic3⋊C4—64no—
219C4×Dic3—21no—
220C4×Dic3—32no—
221C4×Dic3—32no—
222C4×Dic3—62no—
223C4×Dic3—62no—
224C4×Dic3—61no—
225C4×Dic3—61no—
226#208—21no—
227#208—21no—
228C2×C3⋊C8#6712yes—
229ΓL(1,𝔽25)—24no—
230D3×C8—24no—
231C22×C32—41no—
232C22×C32C2×C22⋊C411yescharacteristic
233C22×C32#12111yescharacteristic
234C22×C32#12111yescharacteristic
235C2×C3×D3—22no—
236C2×C3×D3—42no—
237C32×C4C4×D411yescharacteristic
238C32×C4C4×D411yescharacteristic
239C32×C4#12011yescharacteristic
240C32×C4#12311yescharacteristic
241C3⋊Dic3—42no—
242C3⋊Dic3—81no—
243C3×Dic3—22no—
244C3×Dic3—42no—
245C3×Dic3—81no—
246C2×#41—31no—
247C22×Q8—31no—
248#131—62no—
249#131—62no—
250C2×M16—31no—
251C2×M16—62no—
252C2×M16D3211yescharacteristic
253C4⋊Q8—92no—
254#124—92no—
255#123—182no—
256C22⋊Q8—182no—
257#117—31no—
258#117—91no—
259#117—91no—
260#103—62no—
261#103—182no—
262#99—182no—
263C23×C3—21no—
264C23×C3—32no—
265C23×C3C2×D1211yescharacteristic
266C23×C3D3×D411yescharacteristic
267C22×D3(C2×D3)⋊C412yes—
268C3×Q8—61no—
269C3×Q8—61no—
270C3×Q8—62no—
271C3×Q8—62no—
272C3×Q8—121no—
273C3×Q8—121no—
274C3×D4—62no—
275C3×D4—62no—
276C3×D4—61no—
277C3×D4—61no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—32no—
285C2×C3×C4—62no—
286C2×C3×C4—62no—
287C2×C3×C4—62no—
288C2×C3×C4—121no—
289C2×C3×C4—121no—
290C2×C3×C4—121no—
291C2×C3×C4C2×C4×D311yescharacteristic
292C2×C3×C4C2×C4×D311yescharacteristic
293C2×C3×C4#23511yescharacteristic
294C2×C3×C4#23511yescharacteristic
295C2×C3×C4#23511yescharacteristic
296C2×C3×C4D3×D411yescharacteristic
297C2×C3×C4D3×D411yescharacteristic
298C2×C3×C4#23711yescharacteristic
299C2×C3×C4C2×#6711yescharacteristic
300C2×C3×C4C2×#6711yescharacteristic
301C2×C3×C4C2×#6711yescharacteristic
302C2×C3×C4C2×#6711yescharacteristic
303#67—22no—
304#67—22no—
305C2×Dic3—22no—
306C2×Dic3—41no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309C2×Dic3—62no—
310C2×Dic3—62no—
311C2×Dic3—122no—
312C2×Dic3—122no—
313C2×Dic3—121no—
314C2×Dic3—121no—
315C2×Dic3(C2×D3)⋊C412yes—
316D12—21no—
317D12—21no—
318C4×D3—22no—
319C4×D3—22no—
320Dic6—21no—
321Dic6—21no—
322Dic6—22no—
323Dic6—22no—
324Dic6—41no—
325Dic6—41no—
326C3×C8—22no—
327C3×C8—22no—
328C3×C8—42no—
329C3×C8—62no—
330C3⋊C8—22no—
331C2×C32—21no—
332C2×C32—21no—
333C2×C32—21no—
334C2×C32#38611yescharacteristic
335C3×D3—42no—
336#41—61no—
337#41—61no—
338#41—61no—
339#41—61no—
340C2×Q8—31no—
341C2×Q8—32no—
342C2×Q8—61no—
343C2×Q8—61no—
344C2×Q8—61no—
345C2×Q8—61no—
346C2×Q8—61no—
347C2×D4—32no—
348C2×D4—31no—
349C22×C4—32no—
350C22×C4—61no—
351C22×C4C2×D3211yescharacteristic
352M16—21no—
353M16—21no—
354M16—64no—
355M16—61no—
356M16—61no—
357C2×C8—32no—
358C2×C8—64no—
359C2×C8#62512yes—
360C4⋊C4—61no—
361C4⋊C4—184no—
362C4⋊C4—181no—
363C4⋊C4—181no—
364C22⋊C4—61no—
365C22⋊C4—184no—
366C22⋊C4—181no—
367C22⋊C4—181no—
368C42—61no—
369C42—92no—
370C42—181no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—41no—
375C22×C3—41no—
376C22×C3—62no—
377C22×C3—81no—
378C22×C3—122no—
379C22×C3D3×C22⋊C411yescharacteristic
380C22×C3#88411yescharacteristic
381C22×C3C2×(C2×D3)⋊C411yescharacteristic
382C22×C3#93011yescharacteristic
383C22×C3#93011yescharacteristic
384C22×C3#93811yescharacteristic
385C2×D3—22no—
386C2×D3—42no—
387C3×C4—21no—
388C3×C4—21no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—62no—
392C3×C4—122no—
393C3×C4—241no—
394C3×C4#67⋊C411yescharacteristic
395C3×C4#88311yescharacteristic
396C3×C4C4×#6711yescharacteristic
397C3×C4C4×#6711yescharacteristic
398C3×C4C4×#6711yescharacteristic
399C3×C4#92911yescharacteristic
400C3×C4#93711yescharacteristic
401C3×C4#95311yescharacteristic
402Dic3—22no—
403Dic3—42no—
404Dic3—81no—
405Dic3—122no—
406Dic3—122no—
407Dic3—242no—
408Dic3—241no—
409Dic3—241no—
410Dic3—242no—
411C32#190911yescharacteristic, complemented, Sylow
412C23—32no—
413C23D3×D1211yescharacteristic
414Q8—61no—
415Q8—61no—
416Q8—62no—
417Q8—62no—
418Q8—121no—
419Q8—121no—
420D4—62no—
421D4—62no—
422D4—61no—
423D4—61no—
424C2×C4—32no—
425C2×C4—62no—
426C2×C4—62no—
427C2×C4—62no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—182no—
432C2×C4—361no—
433C2×C4#379511yescharacteristic
434C2×C4#379811yescharacteristic
435C2×C4C4×D3211yescharacteristic
436C2×C4C2×#62511yescharacteristic
437C2×C4D3×#6711yescharacteristic
438C2×C4D3×#6711yescharacteristic
439C8—22no—
440C8—62no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—41no—
449C2×C3—41no—
450C2×C3—41no—
451C2×C3—122no—
452C2×C3#495611yescharacteristic
453C2×C3#521311yescharacteristic
454D3—42no—
455C22—41no—
456C22—62no—
457C22—122no—
458C22#6390611yescharacteristic
459C22#6390811yescharacteristic
460C22D3×(C2×D3)⋊C411yescharacteristic
461C4—62no—
462C4—122no—
463C4—241no—
464C4—362no—
465C4—721no—
466C4#6387911yescharacteristic
467C4C4×#62511yescharacteristic
468C4#6390511yescharacteristic
469C4#6397611yescharacteristic
470C3—21no—
471C3#7035011yescharacteristic, complemented
472C3#7202411yescharacteristic, complemented
473C2—21no—
474C2—21no—
475C2—21no—
476C2—122no—
477C2#1059138111yescharacteristic
subgroupG/H
centerC2#10591381
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC2×C32#386
prime radicalC2×C3×#235C22
cosocleC22×C32×C4C23
hypercenterC2×M16D32
hypocenterC32#1909
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
3. C3×#5120C2semidirect
5. #10591100C2semidirect
6. #10589423C2semidirect
7. #10589158C2semidirect
21. #63873C4semidirect
22. #63863C4semidirect
35. #5120D3semidirect
411. C32#1909semidirect
471. C3#70350semidirect
472. C3#72024semidirect

Maximal subgroups

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

subgroupindex
C3×#57452
#105948552
C3×#51202
#105912312
#105911002
#105894232
#105891582
#720243
#703503