Finite Group Databasebeta

← #11859345

Subgroups — #11859345

Order 1152.

1135 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5791C211yesmaximal, characteristic
2C23×C3×C3⋊C8C211yesmaximal, characteristic, complemented
3C2×#63814C211yesmaximal, characteristic, complemented
4#10592906C211yesmaximal, characteristic, complemented
5#10591155C214yesmaximal, complemented
6C3⋊C8×#67C214yesmaximal, complemented
7#10591116C211yesmaximal, characteristic, complemented
8#10591070C211yesmaximal, characteristic, complemented
9#10591055C211yesmaximal, characteristic, complemented
10#83734—31nomaximal
11#82916—31nomaximal
12C23×C32×C4C2211yescharacteristic
13C2×C3×#235C2211yescharacteristic
14C2×#3748C2211yescharacteristic
15C2×#3748C2211yescharacteristic
16C3×#953C412yes—
17C3×#930C412yes—
18C3×#929C412yes—
19C3×C4×#67C2214yes—
20C3×#926C2211yescharacteristic
21C3×#924C412yes—
22C22×C3×C3⋊C8—24no—
23C22×C3×C3⋊C8C2211yescharacteristic
24C22×C3×C3⋊C8C2212yes—
25C22×C3×C3⋊C8C2212yes—
26C22×C3×C3⋊C8C2211yescharacteristic
27C2×#3711C2212yescomplemented
28#63814—24no—
29C2×D3×C3⋊C8C2212yescomplemented
30#63657C2214yes—
31Dic3⋊(C3⋊C8)C2214yescomplemented
32(C2×D3)⋊(C3⋊C8)C2214yescomplemented
33#63554C2212yescomplemented
34Dic3×C3⋊C8C2212yescomplemented
35C23×C3×C8—31no—
36C3×#525—31no—
37C2×#948—31no—
38#5791D311yescharacteristic
39C23×C3⋊C8D311yescharacteristic, complemented
40C2×#901—31no—
41#5234—31no—
42#5207—31no—
43#5207—31no—
44#5207—61no—
45#5111—34no—
46#5110—31no—
47C8×#67—34no—
48#5102—31no—
49#5101—31no—
50#5011—34no—
51D4×C3⋊C8—34no—
52#4999—31no—
53#4997—31no—
54C24×C32C2×C411yescharacteristic
55C22×C32×C4—22no—
56C22×C32×C4—22no—
57C22×C32×C4C2×C412yes—
58C22×C32×C4C2×C411yescharacteristic
59C22×C32×C4C2312yes—
60C22×C32×C4C2311yescharacteristic
61C2×C3×#67C2×C414yes—
62C2×C3×#67C2×C412yes—
63C3×#235—22no—
64C3×#235—22no—
65C2×C3×D12C2×C411yescharacteristic
66C2×C3×C4×D3C2312yes—
67C2×C3×Dic6C2×C411yescharacteristic
68#3748—22no—
69#3748—22no—
70C2×C3⋊(C3⋊C8)C2312yes—
71C2×C3⋊(C3⋊C8)C2312yes—
72C3×#217C2×C414yes—
73C3×#217C2×C412yes—
74C3×(C2×D3)⋊C4C2×C414yes—
75C3×C4⋊Dic3C2×C412yes—
76C3×Dic3⋊C4C2×C414yes—
77C3×C4×Dic3C2312yes—
78C2×C3×C3⋊C8—22no—
79C2×C3×C3⋊C8—24no—
80C2×C3×C3⋊C8—24no—
81C2×C3×C3⋊C8D414yescomplemented
82C2×C3×C3⋊C8C2312yes—
83C2×C3×C3⋊C8C2311yescharacteristic
84C2×C3×C3⋊C8C2311yescharacteristic
85#3711—24no—
86D3×C3⋊C8—24no—
87#2929—91noSylow
88C2×C3×#41—31no—
89C23×C3×C4—21no—
90C23×C3×C4C2×D311yescharacteristic
91C23×C3×C4C2×D311yescharacteristic
92C2×#235C2×D311yescharacteristic
93C22×C3×C8—31no—
94C22×C3×C8—31no—
95C22×C3×C8—32no—
96C22×C3×C8—32no—
97C22×C3×C8—64no—
98C3×#123—32no—
99C3×C22⋊Q8—32no—
100C3×#121—32no—
101C3×#120—32no—
102C3×C4×D4—34no—
103C3×#117—31no—
104#953Dic312yes—
105#948—64no—
106#930Dic312yes—
107#929Dic312yes—
108C4×#67C2×D314yes—
109#926C2×D311yescharacteristic
110#924Dic312yes—
111C2×#208—31no—
112C2×#208—31no—
113C2×#208—31no—
114C2×#208—31no—
115C2×#208—32no—
116C2×#208—61no—
117C2×#208—61no—
118C22×C3⋊C8—24no—
119C22×C3⋊C8—32no—
120C22×C3⋊C8C2×D311yescharacteristic
121C22×C3⋊C8C2×D312yes—
122C22×C3⋊C8C2×D312yes—
123C22×C3⋊C8C2×D311yescharacteristic
124#901—64no—
125C2×ΓL(1,𝔽25)—32no—
126C2×D3×C8—32no—
127#830—34no—
128#830—34no—
129#830—34no—
130#830—64no—
131(C2×D3)⋊C8—34no—
132#815—32no—
133Dic3⋊C8—34no—
134C8×Dic3—32no—
135C4⋊(C3⋊C8)—34no—
136#803—32no—
137C4×C3⋊C8—32no—
138C23×C32—22no—
139C23×C32—24no—
140C23×C32C22×C411yescharacteristic
141C23×C32C22×C412yes—
142C22×C3×D3C22×C412yes—
143C2×C32×C4—22no—
144C2×C32×C4—24no—
145C2×C32×C4—22no—
146C2×C32×C4—22no—
147C2×C32×C4C22×C412yes—
148C2×C32×C4C22×C411yescharacteristic
149C2×C32×C4C2×D412yes—
150C2×C32×C4#4112yes—
151C2×C32×C4C2411yescharacteristic
152C3×#67—24no—
153C3×#67—28no—
154C2×C3×Dic3C22×C414yes—
155C2×C3×Dic3C22×C412yes—
156C3×D12—22no—
157C3×C4×D3—22no—
158C3×C4×D3—22no—
159C3×Dic6—22no—
160C3⋊(C3⋊C8)—22no—
161C3⋊(C3⋊C8)—22no—
162C3×C3⋊C8—22no—
163C3×C3⋊C8C2×D412yes—
164C3×C3⋊C8C2×D412yes—
165C2×#131—91no—
166C23×C8—31no—
167#525—31no—
168#435—94no—
169C8×D4—94no—
170#424—91no—
171#408—91no—
172#407—91no—
173#405—91no—
174C24×C3—21no—
175C24×C3C4×D311yescharacteristic
176C24×C3C2×Dic311yescharacteristic
177C3×#41—62no—
178C3×#41—62no—
179C2×C3×Q8—31no—
180C2×C3×D4—34no—
181C2×C3×D4—32no—
182C2×C3×D4—31no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—22no—
191C22×C3×C4—32no—
192C22×C3×C4—42no—
193C22×C3×C4—42no—
194C22×C3×C4C4×D312yes—
195C22×C3×C4C4×D311yescharacteristic
196C22×C3×C4C2×Dic312yes—
197C22×C3×C4C2×Dic311yescharacteristic
198C22×C3×C4C22×D312yes—
199C22×C3×C4C22×D312yes—
200C22×C3×C4C22×D311yescharacteristic
201C22×C3×C4C22×D311yescharacteristic
202C2×#67C2×Dic314yes—
203C2×#67C2×Dic312yes—
204#235—22no—
205#235—22no—
206C2×D12C2×Dic311yescharacteristic
207C2×C4×D3C22×D312yes—
208C2×Dic6C2×Dic311yescharacteristic
209C2×C3×C8—34no—
210C2×C3×C8—31no—
211C2×C3×C8—31no—
212C2×C3×C8—32no—
213C2×C3×C8—64no—
214C2×C3×C8—64no—
215C2×C3×C8—62no—
216C3×C4⋊C4—34no—
217C3×C4⋊C4—32no—
218C3×C22⋊C4—32no—
219C3×C22⋊C4—34no—
220C3×C22⋊C4—34no—
221C3×C42—32no—
222#217C2×Dic314yes—
223#217C2×Dic312yes—
224(C2×D3)⋊C4C2×Dic314yes—
225C4⋊Dic3C2×Dic312yes—
226Dic3⋊C4C2×Dic314yes—
227C4×Dic3C22×D312yes—
228#208—62no—
229#208—62no—
230#208—62no—
231#208—62no—
232#208—64no—
233#208—122no—
234#208—122no—
235C2×C3⋊C8—22no—
236C2×C3⋊C8—24no—
237C2×C3⋊C8—24no—
238C2×C3⋊C8—32no—
239C2×C3⋊C8—32no—
240C2×C3⋊C8—32no—
241C2×C3⋊C8—32no—
242C2×C3⋊C8—62no—
243C2×C3⋊C8—62no—
244C2×C3⋊C8—64no—
245C2×C3⋊C8#6714yescomplemented
246C2×C3⋊C8C22×D312yes—
247C2×C3⋊C8C22×D311yescharacteristic
248C2×C3⋊C8C22×D311yescharacteristic
249ΓL(1,𝔽25)—64no—
250D3×C8—64no—
251C22×C32—28no—
252C22×C32—22no—
253C22×C32—22no—
254C22×C32—22no—
255C22×C32C4×D412yes—
256C22×C32#13114yescomplemented
257C22×C32C23×C411yescharacteristic
258C2×C3×D3—22no—
259C2×C3×D3—24no—
260C32×C4—22no—
261C32×C4C4×D412yes—
262C32×C4C22×D411yescharacteristic
263C32×C4C2×#4111yescharacteristic
264C3×Dic3—22no—
265C3×Dic3—24no—
266C2×#41—31no—
267C23×C4D3211yescharacteristic
268#131—184no—
269C2×M16—92no—
270C2×M16—91no—
271C2×M16—91no—
272C22×C8—31no—
273C22×C8—31no—
274C22×C8—32no—
275C22×C8—32no—
276C22×C8—64no—
277C22×C8—92no—
278#123—32no—
279C22⋊Q8—32no—
280#121—32no—
281#120—32no—
282C4×D4—34no—
283#117—31no—
284C4⋊C8—94no—
285C22⋊C8—94no—
286C22⋊C8—94no—
287#97—92no—
288C4×C8—92no—
289C23×C3—24no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—24no—
293C23×C3—22no—
294C23×C3—21no—
295C23×C3—32no—
296C23×C3—42no—
297C23×C3—44no—
298C23×C3C2×C4×D312yes—
299C23×C3C2×C4×D311yescharacteristic
300C23×C3C22×Dic311yescharacteristic
301C23×C3C22×Dic312yes—
302C22×D3C22×Dic312yes—
303C3×Q8—62no—
304C3×D4—68no—
305C3×D4—64no—
306C3×D4—62no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—34no—
321C2×C3×C4—32no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—44no—
325C2×C3×C4—42no—
326C2×C3×C4—62no—
327C2×C3×C4—62no—
328C2×C3×C4C2×C4×D312yes—
329C2×C3×C4C2×C4×D311yescharacteristic
330C2×C3×C4#23512yes—
331C2×C3×C4D3×D412yes—
332C2×C3×C4#23712yes—
333C2×C3×C4C22×Dic312yes—
334C2×C3×C4C22×Dic311yescharacteristic
335C2×C3×C4C2×#6712yes—
336C2×C3×C4C23×D311yescharacteristic
337C2×C3×C4C23×D311yescharacteristic
338#67—24no—
339#67—28no—
340C2×Dic3C22×Dic314yes—
341C2×Dic3C22×Dic312yes—
342D12—22no—
343C4×D3—22no—
344C4×D3—22no—
345Dic6—22no—
346C3×C8—32no—
347C3×C8—32no—
348C3×C8—62no—
349C3⋊C8—22no—
350C3⋊C8—62no—
351C3⋊C8—62no—
352C3⋊C8—62no—
353C3⋊C8—62no—
354C3⋊C8—122no—
355C3⋊C8—122no—
356C3⋊C8C2×#6712yes—
357C3⋊C8C2×#6712yes—
358C2×C32—22no—
359C2×C32—24no—
360C2×C32C2×C4×D411yescharacteristic
361C2×C32C2×#13112yes—
362C3×D3—42no—
363C24D3×Dic311yescharacteristic
364#41—62no—
365#41—62no—
366C2×Q8—31no—
367C2×D4—34no—
368C2×D4—32no—
369C2×D4—31no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—32no—
373C22×C4D3×Dic312yes—
374C22×C4D3×Dic311yescharacteristic
375C22×C4C2×D3212yes—
376C22×C4C2×D3211yescharacteristic
377M16—184no—
378M16—182no—
379M16—182no—
380C2×C8—31no—
381C2×C8—31no—
382C2×C8—34no—
383C2×C8—32no—
384C2×C8—64no—
385C2×C8—64no—
386C2×C8—62no—
387C2×C8—92no—
388C2×C8—92no—
389C2×C8—184no—
390C4⋊C4—32no—
391C4⋊C4—34no—
392C22⋊C4—32no—
393C22⋊C4—34no—
394C22⋊C4—34no—
395C42—32no—
396C22×C3—28no—
397C22×C3—28no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—22no—
404C22×C3—21no—
405C22×C3—24no—
406C22×C3—22no—
407C22×C3—42no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—48no—
411C22×C3—64no—
412C22×C3—62no—
413C22×C3#90114yescomplemented
414C22×C3C4×#6712yes—
415C22×C3D4×Dic312yes—
416C22×C3#94814yescomplemented
417C22×C3C22×C4×D311yescharacteristic
418C22×C3C23×Dic311yescharacteristic
419C2×D3—22no—
420C2×D3—24no—
421C3×C4—22no—
422C3×C4—22no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—22no—
426C3×C4—42no—
427C3×C4—64no—
428C3×C4—62no—
429C3×C4C4×#6712yes—
430C3×C4D4×Dic312yes—
431C3×C4C2×#23511yescharacteristic
432C3×C4C2×D3×D411yescharacteristic
433C3×C4C2×#23711yescharacteristic
434C3×C4C22×#6711yescharacteristic
435Dic3—22no—
436Dic3—24no—
437C32#292911yescharacteristic, complemented, Sylow
438C23—24no—
439C23—22no—
440C23—32no—
441C23C2×D3×Dic312yes—
442C23C2×D3×Dic311yescharacteristic
443Q8—62no—
444D4—68no—
445D4—64no—
446D4—62no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—34no—
452C2×C4—32no—
453C2×C4—62no—
454C2×C4—62no—
455C2×C4C2×D3×Dic312yes—
456C2×C4C2×D3×Dic311yescharacteristic
457C2×C4#380412yes—
458C2×C4D3×#6712yes—
459C2×C4C22×D3211yescharacteristic
460C8—32no—
461C8—32no—
462C8—62no—
463C8—182no—
464C8—182no—
465C2×C3—24no—
466C2×C3—22no—
467C2×C3—22no—
468C2×C3—24no—
469C2×C3—21no—
470C2×C3—22no—
471C2×C3—44no—
472C2×C3—42no—
473C2×C3—122no—
474C2×C3C2×#90112yes—
475C2×C3C2×C4×#6711yescharacteristic
476C2×C3C2×D4×Dic311yescharacteristic
477C2×C3C2×#94812yes—
478D3—42no—
479C22—28no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—62no—
484C22—64no—
485C22#6381414yescomplemented
486C22Dic3×#6712yes—
487C22C22×D3×Dic311yescharacteristic
488C4—22no—
489C4—62no—
490C4—64no—
491C4Dic3×#6712yes—
492C4C2×#380411yescharacteristic
493C4C2×D3×#6711yescharacteristic
494C3—21no—
495C3#8291611yescharacteristic, complemented
496C3#8373411yescharacteristic, complemented
497C2—24no—
498C2—22no—
499C2—122no—
500C2C2×#6381412yes—
501C2C2×Dic3×#6711yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×D32
commutator, socleC22×C32C23×C4
FittingC23×C32×C4C22
prime radicalC3×#5791C2
cosocleC2×C32×C4C24
hypercenterC23×C4D32
hypocenterC32#2929
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C23×C3×C3⋊C8C2semidirect
3. C2×#63814C2semidirect
4. #10592906C2semidirect
5. #10591155C2semidirect
6. C3⋊C8×#67C2semidirect
7. #10591116C2semidirect
8. #10591070C2semidirect
9. #10591055C2semidirect
27. C2×#3711C22semidirect
29. C2×D3×C3⋊C8C22semidirect
31. Dic3⋊(C3⋊C8)C22semidirect
32. (C2×D3)⋊(C3⋊C8)C22semidirect
33. #63554C22semidirect
34. Dic3×C3⋊C8C22semidirect
39. C23×C3⋊C8D3semidirect
81. C2×C3×C3⋊C8D4semidirect
245. C2×C3⋊C8#67semidirect
256. C22×C32#131semidirect
413. C22×C3#901semidirect
416. C22×C3#948semidirect
437. C32#2929semidirect
485. C22#63814semidirect
495. C3#82916semidirect
496. C3#83734semidirect

Maximal subgroups

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

subgroupindex
C3×#57912
C23×C3×C3⋊C82
C2×#638142
#105929062
#105911552
C3⋊C8×#672
#105911162
#105910702
#105910552
#837343
#829163