Finite Group Databasebeta

← #11859491

Subgroups — #11859491

Order 1152.

1079 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5832C211yesmaximal, characteristic
2C2×C3×#948C211yesmaximal, characteristic, complemented
3C2×#63814C211yesmaximal, characteristic, complemented
4#10592933C211yesmaximal, characteristic, complemented
5#10591156C214yesmaximal, complemented
6C3⋊C8×#67C214yesmaximal, complemented
7#10591115C211yesmaximal, characteristic, complemented
8#10591071C211yesmaximal, characteristic, complemented
9C2×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
10#83786—31nomaximal
11#83275—31nomaximal
12C2×C32×#41C2211yescharacteristic
13C2×C3×#235C2211yescharacteristic
14C2×#3748C2211yescharacteristic
15C22×C3⋊(C3⋊C8)C2211yescharacteristic
16C3×#948—24no—
17C3×#947C411yescharacteristic
18C3×#944C411yescharacteristic
19C3×#940C411yescharacteristic
20C3×#939C414yes—
21C3×#936C411yescharacteristic
22C3×C4×#67C2214yes—
23C2×C3×C4×Dic3C2211yescharacteristic
24C2×C3×#208C2211yescharacteristic
25C2×C3×#208C2212yes—
26C22×C3×C3⋊C8C2212yes—
27C22×C3×C3⋊C8C2211yescharacteristic
28C2×#3711C2212yes—
29#63814—24no—
30C2×D3×C3⋊C8C2212yes—
31#63657C2214yes—
32Dic3⋊(C3⋊C8)C2214yes—
33(C2×D3)⋊(C3⋊C8)C2214yes—
34Dic3×C3⋊C8C2212yes—
35Dic3×C3⋊C8C2212yes—
36C2×C3×#131—31no—
37C3×#532—31no—
38#5832D311yescharacteristic
39C2×#948—31no—
40C2×#948D311yescharacteristic, complemented
41C2×#901—31no—
42#5234—31no—
43#5234—31no—
44#5234—31no—
45#5234—61no—
46#5129—31no—
47#5128—34no—
48#5118—31no—
49C8×#67—34no—
50C2×C8×Dic3—31no—
51#5012—34no—
52D4×C3⋊C8—34no—
53#4924—31no—
54C2×C4×C3⋊C8—31no—
55C32×#41—22no—
56C32×#41—22no—
57C2×C32×Q8C2×C411yescharacteristic
58C2×C32×D4C2×C411yescharacteristic
59C2×C32×D4C2×C412yes—
60C22×C32×C4C2312yes—
61C22×C32×C4C2311yescharacteristic
62C2×C3×#67C2×C414yes—
63C2×C3×#67C2×C412yes—
64C22×C3×Dic3C2×C412yes—
65C3×#235—22no—
66C3×#235—22no—
67C2×C3×D12C2×C411yescharacteristic
68C2×C3×C4×D3C2312yes—
69C2×C3×Dic6C2×C411yescharacteristic
70#3748—22no—
71C2×C3⋊(C3⋊C8)—22no—
72C2×C3⋊(C3⋊C8)C2312yes—
73C2×C3⋊(C3⋊C8)C2312yes—
74C3×#217C2×C414yes—
75C3×(C2×D3)⋊C4C2×C414yes—
76C3×Dic3⋊C4C2×C414yes—
77C3×C4×Dic3C2×C411yescharacteristic
78C3×C4×Dic3C2×C411yescharacteristic
79C3×C4×Dic3C2312yes—
80C3×#208—22no—
81C3×#208—24no—
82C2×C3×C3⋊C8—24no—
83C2×C3×C3⋊C8D414yescomplemented
84C2×C3×C3⋊C8C2312yes—
85C2×C3×C3⋊C8C2311yescharacteristic
86C2×C3×C3⋊C8C2311yescharacteristic
87#3711—24no—
88D3×C3⋊C8—24no—
89#2931—91noSylow
90C2×C3×#41—21no—
91C2×C3×#41—31no—
92C2×C3×#41C2×D311yescharacteristic
93C2×C3×#41C2×D311yescharacteristic
94C2×#235C2×D311yescharacteristic
95C3×#131—64no—
96C2×C3×M16—32no—
97C2×C3×M16—31no—
98C22×C3×C8—31no—
99C22×C3×C8—32no—
100C3×C4⋊Q8—31no—
101C3×#127—31no—
102C3×#124—31no—
103C3×#124—31no—
104C3×#121—34no—
105C3×C4×D4—34no—
106C2×C3×C42—31no—
107#948—24no—
108#948—64no—
109#947Dic311yescharacteristic
110#944Dic311yescharacteristic
111#940Dic311yescharacteristic
112#939Dic314yes—
113#936Dic311yescharacteristic
114C4×#67C2×D314yes—
115C2×C4×Dic3C2×D311yescharacteristic
116C2×#208—31no—
117C2×#208—31no—
118C2×#208—32no—
119C2×#208—61no—
120C2×#208C2×D311yescharacteristic
121C2×#208C2×D312yes—
122C22×C3⋊C8—31no—
123C22×C3⋊C8—31no—
124C22×C3⋊C8—32no—
125C22×C3⋊C8—61no—
126C22×C3⋊C8C2×D312yes—
127C22×C3⋊C8C2×D311yescharacteristic
128#901—64no—
129C2×ΓL(1,𝔽25)—32no—
130C2×D3×C8—32no—
131#830—34no—
132#830—34no—
133#830—34no—
134#830—64no—
135(C2×D3)⋊C8—34no—
136Dic3⋊C8—34no—
137C8×Dic3—32no—
138C8×Dic3—32no—
139C4⋊(C3⋊C8)—34no—
140C4×C3⋊C8—32no—
141C4×C3⋊C8—32no—
142C23×C32C22×C412yes—
143C23×C32C22×C411yescharacteristic
144C22×C3×D3C22×C412yes—
145C32×Q8—22no—
146C32×D4—24no—
147C32×D4—22no—
148C2×C32×C4—22no—
149C2×C32×C4—22no—
150C2×C32×C4C22×C412yes—
151C2×C32×C4C22×C411yescharacteristic
152C2×C32×C4C2×D412yes—
153C2×C32×C4#4112yes—
154C2×C32×C4C2411yescharacteristic
155C3×#67—24no—
156C3×#67—28no—
157C2×C3×Dic3—24no—
158C2×C3×Dic3C22×C412yes—
159C2×C3×Dic3C22×C412yes—
160C2×C3×Dic3C22×C412yes—
161C3×D12—22no—
162C3×C4×D3—22no—
163C3×C4×D3—22no—
164C3×Dic6—22no—
165C3⋊(C3⋊C8)—22no—
166C3⋊(C3⋊C8)—22no—
167C3×C3⋊C8—22no—
168C3×C3⋊C8C2×D412yes—
169C3×C3⋊C8C2×D412yes—
170C2×#131—31no—
171C2×#131—91no—
172#532—31no—
173#436—94no—
174C8×D4—94no—
175#423—91no—
176#408—91no—
177#408—91no—
178C2×C4×C8—91no—
179C3×#41—22no—
180C3×#41—22no—
181C3×#41—22no—
182C3×#41—22no—
183C3×#41—42no—
184C3×#41—42no—
185C3×#41—62no—
186C3×#41—62no—
187C2×C3×Q8—21no—
188C2×C3×Q8—31no—
189C2×C3×Q8C4×D311yescharacteristic
190C2×C3×Q8C2×Dic311yescharacteristic
191C2×C3×D4—22no—
192C2×C3×D4—21no—
193C2×C3×D4—34no—
194C2×C3×D4—32no—
195C2×C3×D4—31no—
196C2×C3×D4C4×D312yes—
197C2×C3×D4C4×D311yescharacteristic
198C2×C3×D4C2×Dic311yescharacteristic
199C2×C3×D4C2×Dic312yes—
200C22×C3×C4—21no—
201C22×C3×C4—22no—
202C22×C3×C4—32no—
203C22×C3×C4—32no—
204C22×C3×C4C22×D312yes—
205C22×C3×C4C22×D312yes—
206C22×C3×C4C22×D311yescharacteristic
207C22×C3×C4C22×D311yescharacteristic
208C2×#67C2×Dic314yes—
209C2×#67C2×Dic312yes—
210C22×Dic3C2×Dic312yes—
211#235—22no—
212#235—22no—
213C2×D12C2×Dic311yescharacteristic
214C2×C4×D3C22×D312yes—
215C2×Dic6C2×Dic311yescharacteristic
216C3×M16—64no—
217C3×M16—62no—
218C2×C3×C8—34no—
219C2×C3×C8—31no—
220C2×C3×C8—31no—
221C2×C3×C8—32no—
222C2×C3×C8—64no—
223C3×C4⋊C4—34no—
224C3×C22⋊C4—34no—
225C3×C22⋊C4—34no—
226C3×C42—31no—
227C3×C42—31no—
228C3×C42—32no—
229#217C2×Dic314yes—
230(C2×D3)⋊C4C2×Dic314yes—
231Dic3⋊C4C2×Dic314yes—
232C4×Dic3C2×Dic311yescharacteristic
233C4×Dic3C2×Dic311yescharacteristic
234C4×Dic3C22×D312yes—
235#208—22no—
236#208—24no—
237#208—62no—
238#208—62no—
239#208—64no—
240#208—122no—
241C2×C3⋊C8—24no—
242C2×C3⋊C8—32no—
243C2×C3⋊C8—32no—
244C2×C3⋊C8—32no—
245C2×C3⋊C8—32no—
246C2×C3⋊C8—62no—
247C2×C3⋊C8—62no—
248C2×C3⋊C8—62no—
249C2×C3⋊C8—62no—
250C2×C3⋊C8—64no—
251C2×C3⋊C8—122no—
252C2×C3⋊C8#6714yescomplemented
253C2×C3⋊C8C22×D312yes—
254C2×C3⋊C8C22×D311yescharacteristic
255C2×C3⋊C8C22×D311yescharacteristic
256ΓL(1,𝔽25)—64no—
257D3×C8—64no—
258C22×C32—22no—
259C22×C32—22no—
260C22×C32—24no—
261C22×C32C4×D412yes—
262C22×C32C23×C411yescharacteristic
263C2×C3×D3—22no—
264C2×C3×D3—24no—
265C32×C4—22no—
266C32×C4C4×D412yes—
267C32×C4C22×D411yescharacteristic
268C32×C4C2×#4111yescharacteristic
269C3×Dic3—22no—
270C3×Dic3—22no—
271C3×Dic3#13114yescomplemented
272C2×#41—31no—
273C2×#41D3211yescharacteristic
274#131—64no—
275#131—184no—
276C2×M16—31no—
277C2×M16—32no—
278C2×M16—92no—
279C2×M16—91no—
280C22×C8—31no—
281C22×C8—32no—
282C22×C8—92no—
283C22×C8—91no—
284C4⋊Q8—31no—
285#127—31no—
286#124—31no—
287#124—31no—
288#121—34no—
289C4×D4—34no—
290C2×C42—31no—
291C4⋊C8—94no—
292C22⋊C8—94no—
293C22⋊C8—94no—
294C4×C8—92no—
295C4×C8—92no—
296C23×C3—21no—
297C23×C3—22no—
298C23×C3—32no—
299C23×C3C2×C4×D312yes—
300C23×C3C2×C4×D311yescharacteristic
301C23×C3C22×Dic312yes—
302C23×C3C22×Dic311yescharacteristic
303C22×D3C22×Dic312yes—
304C3×Q8—22no—
305C3×Q8—22no—
306C3×Q8—42no—
307C3×Q8—62no—
308C3×D4—24no—
309C3×D4—24no—
310C3×D4—22no—
311C3×D4—22no—
312C3×D4—42no—
313C3×D4—44no—
314C3×D4—68no—
315C3×D4—64no—
316C3×D4—62no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—32no—
327C2×C3×C4—32no—
328C2×C3×C4—32no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—64no—
332C2×C3×C4—62no—
333C2×C3×C4—62no—
334C2×C3×C4C2×C4×D312yes—
335C2×C3×C4C2×C4×D311yescharacteristic
336C2×C3×C4#23512yes—
337C2×C3×C4D3×D412yes—
338C2×C3×C4#23712yes—
339C2×C3×C4C22×Dic312yes—
340C2×C3×C4C22×Dic311yescharacteristic
341C2×C3×C4C2×#6712yes—
342C2×C3×C4C23×D311yescharacteristic
343C2×C3×C4C23×D311yescharacteristic
344#67—24no—
345#67—28no—
346C2×Dic3—24no—
347C2×Dic3C22×Dic312yes—
348C2×Dic3C22×Dic312yes—
349C2×Dic3C22×Dic312yes—
350D12—22no—
351C4×D3—22no—
352C4×D3—22no—
353Dic6—22no—
354C3×C8—32no—
355C3×C8—32no—
356C3×C8—62no—
357C3⋊C8—22no—
358C3⋊C8—62no—
359C3⋊C8—62no—
360C3⋊C8—62no—
361C3⋊C8—62no—
362C3⋊C8—122no—
363C3⋊C8—122no—
364C3⋊C8C2×#6712yes—
365C3⋊C8C2×#6712yes—
366C2×C32—22no—
367C2×C32—42no—
368C2×C32C2×C4×D411yescharacteristic
369C2×C32C2×#13112yes—
370C3×D3—42no—
371#41—22no—
372#41—22no—
373#41—62no—
374#41—62no—
375C2×Q8—31no—
376C2×Q8D3×Dic311yescharacteristic
377C2×D4—34no—
378C2×D4—32no—
379C2×D4—31no—
380C2×D4D3×Dic312yes—
381C2×D4D3×Dic311yescharacteristic
382C22×C4—32no—
383C22×C4—32no—
384C22×C4C2×D3212yes—
385C22×C4C2×D3211yescharacteristic
386M16—64no—
387M16—62no—
388M16—184no—
389M16—182no—
390C2×C8—31no—
391C2×C8—31no—
392C2×C8—34no—
393C2×C8—32no—
394C2×C8—64no—
395C2×C8—92no—
396C2×C8—92no—
397C2×C8—184no—
398C2×C8—182no—
399C4⋊C4—34no—
400C22⋊C4—34no—
401C22⋊C4—34no—
402C42—32no—
403C42—31no—
404C42—31no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—24no—
408C22×C3—22no—
409C22×C3—22no—
410C22×C3—24no—
411C22×C3—21no—
412C22×C3—22no—
413C22×C3—42no—
414C22×C3—42no—
415C22×C3—44no—
416C22×C3—64no—
417C22×C3—62no—
418C22×C3C4×#6712yes—
419C22×C3D4×Dic312yes—
420C22×C3C22×C4×D311yescharacteristic
421C22×C3C23×Dic311yescharacteristic
422C2×D3—22no—
423C2×D3—24no—
424C3×C4—22no—
425C3×C4—22no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—22no—
429C3×C4—34no—
430C3×C4—42no—
431C3×C4—62no—
432C3×C4—62no—
433C3×C4C4×#6712yes—
434C3×C4D4×Dic312yes—
435C3×C4C2×#23511yescharacteristic
436C3×C4C2×D3×D411yescharacteristic
437C3×C4C2×#23711yescharacteristic
438C3×C4C22×#6711yescharacteristic
439Dic3—22no—
440Dic3—22no—
441Dic3#94814yescomplemented
442C32#293111yescharacteristic, complemented, Sylow
443C23—32no—
444C23C2×D3×Dic312yes—
445C23C2×D3×Dic311yescharacteristic
446Q8—22no—
447Q8—62no—
448D4—24no—
449D4—22no—
450D4—68no—
451D4—64no—
452D4—62no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—32no—
456C2×C4—32no—
457C2×C4—32no—
458C2×C4—64no—
459C2×C4—62no—
460C2×C4—62no—
461C2×C4C2×D3×Dic312yes—
462C2×C4C2×D3×Dic311yescharacteristic
463C2×C4#380412yes—
464C2×C4D3×#6712yes—
465C2×C4C22×D3211yescharacteristic
466C8—32no—
467C8—32no—
468C8—62no—
469C8—182no—
470C8—182no—
471C2×C3—22no—
472C2×C3—22no—
473C2×C3—22no—
474C2×C3—21no—
475C2×C3—42no—
476C2×C3—42no—
477C2×C3—42no—
478C2×C3—82no—
479C2×C3—122no—
480C2×C3D3×#13112yes—
481C2×C3C2×C4×#6711yescharacteristic
482C2×C3C2×D4×Dic311yescharacteristic
483C2×C3C2×#94812yes—
484D3—42no—
485C22—22no—
486C22—24no—
487C22—22no—
488C22—64no—
489C22—62no—
490C22Dic3×#6712yes—
491C22C22×D3×Dic311yescharacteristic
492C4—22no—
493C4—34no—
494C4—62no—
495C4—62no—
496C4Dic3×#6712yes—
497C4C2×#380411yescharacteristic
498C4C2×D3×#6711yescharacteristic
499C3—21no—
500C3#8327511yescharacteristic, complemented
501C3#8378611yescharacteristic, complemented
502C2—22no—
503C2—42no—
504C2—122no—
505C2D3×#94812yes—
506C2C2×Dic3×#6711yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×D32
commutator, socleC22×C32C23×C4
FittingC2×C32×#41C22
prime radicalC3×#5832C2
cosocleC2×C32×C4C24
hypercenterC2×#41D32
hypocenterC32#2931
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. C2×C3×#948C2semidirect
3. C2×#63814C2semidirect
4. #10592933C2semidirect
5. #10591156C2semidirect
6. C3⋊C8×#67C2semidirect
7. #10591115C2semidirect
8. #10591071C2semidirect
9. C2×Dic3×C3⋊C8C2semidirect
40. C2×#948D3semidirect
83. C2×C3×C3⋊C8D4semidirect
252. C2×C3⋊C8#67semidirect
271. C3×Dic3#131semidirect
441. Dic3#948semidirect
442. C32#2931semidirect
500. C3#83275semidirect
501. C3#83786semidirect

Maximal subgroups

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

subgroupindex
C3×#58322
C2×C3×#9482
C2×#638142
#105929332
#105911562
C3⋊C8×#672
#105911152
#105910712
C2×Dic3×C3⋊C82
#837863
#832753