Finite Group Databasebeta

← #11805447

Subgroups — #11805447

Order 1152.

1765 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2C2×#63900C212yesmaximal, complemented
3C2×#63894C212yesmaximal
4C3×#4935C211yesmaximal, characteristic, complemented
5C3×C4×C4⋊Dic3C211yesmaximal, characteristic, complemented
6#10591378C214yesmaximal, complemented
7#10591325C212yesmaximal, complemented
8#10591212C212yesmaximal, complemented
9#74360—31nomaximal
10C4⋊C4×D12—31nomaximal
11C22×C4⋊(C3⋊D3)C2211yescharacteristic
12C22×C4×C3⋊D3C2212yes—
13C22×#624C2214yes—
14C2×C32×C42C2211yescharacteristic
15C2×#3752C2212yes—
16C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C4×C4⋊(C3⋊D3)C418yescomplemented
18C2×C3×C4⋊Dic3C2211yescharacteristic
19C2×C3×C4⋊Dic3C2212yes—
20C2×C3×C4⋊Dic3C2212yes—
21C2×C3×C4⋊Dic3C2212yes—
22C2×C3×C4⋊Dic3C2211yescharacteristic
23C2×C3×C4×Dic3C2212yes—
24C2×#3724C2214yes—
25C2×#3722C2214yes—
26#63900—28no—
27#63894—28no—
28C3×C3⋊#95C2212yes—
29#63579C2214yes—
30C2×#893—32no—
31C2×D3×C4⋊C4—32no—
32C2×D3×C4⋊C4—32no—
33C2×C4×D12—31no—
34C2×C4×D12—31no—
35C2×C4×D12—32no—
36C2×C4×D12—61no—
37C3×#384—31no—
38C3×C4×C4⋊C4—31no—
39#4989—34no—
40#4987—32no—
41#4987—32no—
42#4972—32no—
43#4935D311yescharacteristic, complemented
44C4×C4⋊Dic3D311yescharacteristic, complemented
45#4928—32no—
46#4668—34no—
47C23×C3⋊D3C2312yes—
48C22×C32×C4C2311yescharacteristic
49C22×C32×C4C2311yescharacteristic
50C22×C32×C4C2311yescharacteristic
51C22×C3⋊Dic3C2312yes—
52C2×C4⋊(C3⋊D3)C2×C414yes—
53C2×C4⋊(C3⋊D3)D414yes—
54C2×C4⋊(C3⋊D3)Q814yes—
55C2×C4×C3⋊D3—24no—
56C2×C4×C3⋊D3—24no—
57C2×C4×C3⋊D3C2×C418yescomplemented
58C22×C3×Dic3C2312yes—
59C22×C3×Dic3C2314yes—
60C22×C3×Dic3C2312yes—
61C2×#624—28no—
62C2×#624—28no—
63C2×#624—44no—
64C32×C42C2×C414yescomplemented
65#3752C2×C418yescomplemented
66C4⋊(C3⋊Dic3)C2×C414yescomplemented
67C3×C4⋊Dic3—24no—
68C3×C4⋊Dic3—24no—
69C3×C4⋊Dic3—22no—
70C3×C4⋊Dic3—42no—
71C3×C4⋊Dic3D414yescomplemented
72C3×C4×Dic3—24no—
73#3724—28no—
74#3722—28no—
75D4×C4⋊C4—91noSylow
76C22×D12—31no—
77C22×D12—31no—
78C22×D12—61no—
79C22×C4×D3—32no—
80C22×C4×D3—32no—
81C22×C4×D3—34no—
82C22×C4×D3—34no—
83C22×C4×D3—62no—
84C2×C3×C4⋊C4—31no—
85C2×C3×C4⋊C4—32no—
86C2×C3×C4⋊C4—31no—
87C2×C3×C4⋊C4—32no—
88C2×C3×C4⋊C4—32no—
89C2×C3×C42—21no—
90C2×C3×C42—32no—
91C2×C3×C42C2×D311yescharacteristic
92C2×C3×C42C2×D311yescharacteristic
93C2×(C2×D3)⋊C4—32no—
94C2×(C2×D3)⋊C4—32no—
95C2×(C2×D3)⋊C4—34no—
96C2×(C2×D3)⋊C4—34no—
97C2×(C2×D3)⋊C4—62no—
98C2×C4⋊Dic3—31no—
99C2×C4⋊Dic3—31no—
100C2×C4⋊Dic3—61no—
101C2×C4⋊Dic3C2×D312yes—
102C2×C4⋊Dic3C2×D311yescharacteristic
103C2×C4⋊Dic3C2×D312yes—
104C2×C4⋊Dic3C2×D312yes—
105C2×C4⋊Dic3C2×D311yescharacteristic
106C2×Dic3⋊C4—34no—
107C2×Dic3⋊C4—34no—
108C2×C4×Dic3C2×D312yes—
109#893—68no—
110D3×C4⋊C4—68no—
111D3×C4⋊C4—68no—
112C4×D12—38no—
113C4×D12—38no—
114C4×D12—68no—
115C4×D12—68no—
116C3×#95—32no—
117C3⋊#95—34no—
118C3⋊#95—34no—
119C3⋊#95C2×D312yes—
120C23×C32C2411yescharacteristic
121C22×C3⋊D3—22no—
122C22×C3⋊D3—22no—
123C22×C3⋊D3—22no—
124C22×C3⋊D3—22no—
125C22×C3⋊D3C22×C414yes—
126C22×C3⋊D3C2×D414yes—
127C22×C3⋊D3C2×Q814yes—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4C22×C412yes—
131C2×C32×C4C22×C412yes—
132C2×C32×C4C22×C412yes—
133C2×C32×C4C2×D412yes—
134C2×C32×C4C2×D412yes—
135C2×C32×C4C2×Q812yes—
136C2×C32×C4#4112yes—
137C2×C3⋊Dic3—22no—
138C2×C3⋊Dic3—22no—
139C2×C3⋊Dic3C22×C414yes—
140C4⋊(C3⋊D3)C4⋊C4116yescomplemented
141C4×C3⋊D3—216no—
142C2×C3×Dic3—22no—
143C2×C3×Dic3—22no—
144C2×C3×Dic3—22no—
145C2×C3×Dic3—24no—
146C2×C3×Dic3—24no—
147C2×C3×Dic3—22no—
148C2×C3×Dic3—24no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3C2×D414yes—
151#624—416no—
152C2×C4×D4—92no—
153C2×C4×D4—91no—
154C22×C4⋊C4—92no—
155#389—92no—
156#385—94no—
157#384—31no—
158#380—92no—
159C4×C4⋊C4—31no—
160C23×D3—32no—
161C23×D3—32no—
162C23×D3—62no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—32no—
167C22×C3×C4—32no—
168C22×C3×C4—34no—
169C22×C3×C4C22×D311yescharacteristic
170C22×C3×C4C22×D311yescharacteristic
171C22×C3×C4C22×D311yescharacteristic
172C22×C3×C4C22×D311yescharacteristic
173C22×C3×C4C22×D311yescharacteristic
174C22×C3×C4C22×D311yescharacteristic
175C22×Dic3—32no—
176C22×Dic3—32no—
177C22×Dic3—62no—
178C22×Dic3C22×D312yes—
179C22×Dic3C22×D314yes—
180C22×Dic3C22×D312yes—
181C2×D12—34no—
182C2×D12—34no—
183C2×D12—34no—
184C2×D12—34no—
185C2×D12—34no—
186C2×D12—34no—
187C2×D12—64no—
188C2×D12—64no—
189C2×D12—64no—
190C2×C4×D3—38no—
191C2×C4×D3—38no—
192C2×C4×D3—68no—
193C2×C4×D3—64no—
194C2×C4×D3—64no—
195C2×C4×D3—64no—
196C2×C4×D3—64no—
197C2×C4×D3—68no—
198C2×C4×D3—68no—
199C2×C4×D3—68no—
200C2×C4×D3—68no—
201C2×C4×D3—124no—
202C2×C4×D3—124no—
203C2×C4×D3—124no—
204C2×C4×D3—124no—
205C3×C4⋊C4—34no—
206C3×C4⋊C4—64no—
207C3×C4⋊C4—64no—
208C3×C4⋊C4—62no—
209C3×C4⋊C4—122no—
210C3×C42—24no—
211C3×C42—64no—
212C3×C42C4×D314yescomplemented
213C3×C42C4×D314yescomplemented
214(C2×D3)⋊C4—38no—
215(C2×D3)⋊C4—38no—
216(C2×D3)⋊C4—68no—
217(C2×D3)⋊C4—68no—
218(C2×D3)⋊C4—68no—
219C4⋊Dic3—22no—
220C4⋊Dic3—24no—
221C4⋊Dic3—24no—
222C4⋊Dic3—34no—
223C4⋊Dic3—34no—
224C4⋊Dic3—42no—
225C4⋊Dic3—64no—
226C4⋊Dic3D1214yescomplemented
227Dic3⋊C4—68no—
228Dic3⋊C4—68no—
229C4×Dic3—24no—
230C22×C32C23×C411yescharacteristic
231C22×C32C22×D411yescharacteristic
232C22×C32C22×D411yescharacteristic
233C22×C32C22×Q811yescharacteristic
234C22×C32C2×#4111yescharacteristic
235C22×C32H(5,𝔽2)11yescharacteristic
236C22×C322−1+411yescharacteristic
237C2×C3⋊D3—24no—
238C2×C3⋊D3—24no—
239C2×C3⋊D3—24no—
240C2×C3⋊D3—24no—
241C2×C3⋊D3—24no—
242C2×C3⋊D3—24no—
243C2×C3⋊D3C2×C4⋊C418yescomplemented
244C32×C4—22no—
245C32×C4C2×C4⋊C414yescomplemented
246C32×C4C4×D414yescomplemented
247C3⋊Dic3—24no—
248C3×Dic3—24no—
249C3×Dic3—44no—
250C3×Dic3—42no—
251C22×D4—91no—
252C23×C4—94no—
253C23×C4—92no—
254C4×D4—98no—
255C4×D4—188no—
256C2×C4⋊C4—31no—
257C2×C4⋊C4—32no—
258C2×C4⋊C4—32no—
259C2×C4⋊C4—31no—
260C2×C4⋊C4—32no—
261C2×C4⋊C4—91no—
262C2×C4⋊C4—94no—
263C2×C4⋊C4—188no—
264C2×C22⋊C4—92no—
265C2×C22⋊C4—94no—
266C2×C42—32no—
267C2×C42D3211yescharacteristic
268#95—32no—
269#95—94no—
270C23×C3—21no—
271C23×C3C23×D311yescharacteristic
272C23×C3C23×D311yescharacteristic
273C22×D3—34no—
274C22×D3—34no—
275C22×D3—34no—
276C22×D3—34no—
277C22×D3—34no—
278C22×D3—34no—
279C22×D3—64no—
280C22×D3—64no—
281C22×D3—64no—
282C22×D3—62no—
283C22×D3—62no—
284C22×D3—62no—
285C22×D3—62no—
286C22×D3—62no—
287C22×D3—62no—
288C22×D3—62no—
289C22×D3—62no—
290C22×D3—122no—
291C22×D3—122no—
292C22×D3—122no—
293C22×D3—122no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—34no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—62no—
309C2×C3×C4—62no—
310C2×C3×C4—64no—
311C2×C3×C4—62no—
312C2×C3×C4—64no—
313C2×C3×C4—62no—
314C2×C3×C4—62no—
315C2×C3×C4—64no—
316C2×C3×C4C2×C4×D312yes—
317C2×C3×C4C2×C4×D312yes—
318C2×C3×C4C2×C4×D312yes—
319C2×C3×C4C2×C4×D312yes—
320C2×C3×C4C2×C4×D312yes—
321C2×C3×C4C2×C4×D312yes—
322C2×C3×C4C2×D1212yes—
323C2×C3×C4#23512yes—
324C2×C3×C4D3×D412yes—
325C2×C3×C4D3×D412yes—
326C2×C3×C4D3×D412yes—
327C2×C3×C4D3×Q812yes—
328C2×C3×C4D3×Q812yes—
329C2×C3×C4#23912yes—
330C2×Dic3—22no—
331C2×Dic3—22no—
332C2×Dic3—22no—
333C2×Dic3—22no—
334C2×Dic3—24no—
335C2×Dic3—24no—
336C2×Dic3—24no—
337C2×Dic3—22no—
338C2×Dic3—34no—
339C2×Dic3—34no—
340C2×Dic3—64no—
341C2×Dic3—62no—
342C2×Dic3—62no—
343C2×Dic3—62no—
344C2×Dic3—62no—
345C2×Dic3—122no—
346C2×Dic3—122no—
347C2×Dic3C2×D1214yes—
348D12—316no—
349D12—316no—
350D12—616no—
351C4×D3—616no—
352C4×D3—616no—
353C4×D3—1216no—
354C4×D3—1216no—
355C4×D3—1216no—
356C2×C32C22×C4⋊C411yescharacteristic
357C2×C32C2×C4×D411yescharacteristic
358C2×C32#52011yescharacteristic
359C2×C32D4211yescharacteristic
360C2×C32#54711yescharacteristic
361C2×C32D4×Q811yescharacteristic
362C2×C32#55411yescharacteristic
363C3⋊D3—28no—
364C24—92no—
365C2×D4—94no—
366C2×D4—94no—
367C2×D4—94no—
368C22×C4—32no—
369C22×C4—32no—
370C22×C4—34no—
371C22×C4—92no—
372C22×C4—98no—
373C22×C4—188no—
374C22×C4—188no—
375C22×C4—184no—
376C22×C4—184no—
377C22×C4—364no—
378C22×C4C2×D3211yescharacteristic
379C22×C4C2×D3211yescharacteristic
380C22×C4C2×D3211yescharacteristic
381C4⋊C4—34no—
382C4⋊C4—62no—
383C4⋊C4—64no—
384C4⋊C4—64no—
385C4⋊C4—94no—
386C4⋊C4—122no—
387C4⋊C4—188no—
388C22⋊C4—98no—
389C22⋊C4—188no—
390C42—64no—
391C42#62414yescomplemented
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3C22×C4×D311yescharacteristic
400C22×C3C22×C4×D311yescharacteristic
401C22×C3C22×D1211yescharacteristic
402C22×C3C2×#23511yescharacteristic
403C22×C3C2×D3×D411yescharacteristic
404C22×C3C2×D3×D411yescharacteristic
405C22×C3C2×D3×D411yescharacteristic
406C22×C3#100411yescharacteristic
407C22×C3C2×D3×Q811yescharacteristic
408C22×C3C2×D3×Q811yescharacteristic
409C22×C3C2×#23911yescharacteristic
410C22×C3#100711yescharacteristic
411C22×C3#100911yescharacteristic
412C22×C3#101011yescharacteristic
413C2×D3—38no—
414C2×D3—38no—
415C2×D3—68no—
416C2×D3—64no—
417C2×D3—64no—
418C2×D3—64no—
419C2×D3—64no—
420C2×D3—64no—
421C2×D3—64no—
422C2×D3—64no—
423C2×D3—64no—
424C2×D3—64no—
425C2×D3—64no—
426C2×D3—64no—
427C2×D3—64no—
428C2×D3—124no—
429C2×D3—124no—
430C2×D3—124no—
431C2×D3—124no—
432C2×D3—124no—
433C2×D3—124no—
434C3×C4—22no—
435C3×C4—22no—
436C3×C4—24no—
437C3×C4—24no—
438C3×C4—42no—
439C3×C4—64no—
440C3×C4—122no—
441C3×C4—124no—
442C3×C4C4×D1214yescomplemented
443C3×C4D3×C4⋊C414yescomplemented
444C3×C4D3×C4⋊C414yescomplemented
445C3×C4#89314yescomplemented
446Dic3—24no—
447Dic3—44no—
448Dic3—42no—
449Dic3—64no—
450Dic3—64no—
451Dic3—124no—
452C32D4×C4⋊C411yescharacteristic, complemented, Sylow
453C23—94no—
454C23—94no—
455C23—94no—
456C23—182no—
457C23—182no—
458C23—182no—
459C23—182no—
460C23C22×D3211yescharacteristic
461D4—916no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—34no—
465C2×C4—62no—
466C2×C4—62no—
467C2×C4—62no—
468C2×C4—62no—
469C2×C4—64no—
470C2×C4—62no—
471C2×C4—64no—
472C2×C4—64no—
473C2×C4—94no—
474C2×C4—182no—
475C2×C4—182no—
476C2×C4—1816no—
477C2×C4—3616no—
478C2×C4#379712yes—
479C2×C4#379912yes—
480C2×C4D3×D1212yes—
481C2×C4#380212yes—
482C2×C4C2×#62412yes—
483C2×C4C2×#62412yes—
484C2×C4C2×#62412yes—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3C2×C4×D1211yescharacteristic
493C2×C3C2×D3×C4⋊C411yescharacteristic
494C2×C3C2×D3×C4⋊C411yescharacteristic
495C2×C3C2×#89311yescharacteristic
496C2×C3#550411yescharacteristic
497C2×C3#552311yescharacteristic
498C2×C3D4×D1211yescharacteristic
499C2×C3#555111yescharacteristic
500C2×C3Q8×D1211yescharacteristic
501C2×C3#557911yescharacteristic
502C2×C3#571711yescharacteristic
503C2×C3#572611yescharacteristic
504C2×C3#572711yescharacteristic
505C2×C3#573011yescharacteristic
506D3—68no—
507D3—68no—
508D3—128no—
509C22—98no—
510C22—184no—
511C22—184no—
512C22—184no—
513C22—184no—
514C22—184no—
515C22—184no—
516C22#6429711yescharacteristic
517C22C2×#379711yescharacteristic
518C22C2×#379911yescharacteristic
519C22C2×D3×D1211yescharacteristic
520C22C2×#380211yescharacteristic
521C22#6430711yescharacteristic
522C22C22×#62411yescharacteristic
523C4—22no—
524C4—64no—
525C4—122no—
526C4—124no—
527C4—184no—
528C4#6389414yescomplemented
529C4#6390014yescomplemented
530C3—21no—
531C3C4⋊C4×D1211yescharacteristic, complemented
532C3#7436011yescharacteristic, complemented
533C2—188no—
534C2#1059386911yescharacteristic
535C2#1059390311yescharacteristic
536C2#1059395811yescharacteristic
537C2#1059400411yescharacteristic
538C2C2×#6389411yescharacteristic
539C2C2×#6390011yescharacteristic
540C2#1059499011yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutatorC22×C32C23×C4
FittingC2×C32×C42C22
socle, cosocleC23×C32C24
prime radicalC22×C4⋊(C3⋊D3)C22
hypercenterC2×C42D32
hypocenterC32D4×C4⋊C4
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
2. C2×#63900C2semidirect
4. C3×#4935C2semidirect
5. C3×C4×C4⋊Dic3C2semidirect
6. #10591378C2semidirect
7. #10591325C2semidirect
8. #10591212C2semidirect
17. C4×C4⋊(C3⋊D3)C4semidirect
43. #4935D3semidirect
44. C4×C4⋊Dic3D3semidirect
57. C2×C4×C3⋊D3C2×C4semidirect
64. C32×C42C2×C4semidirect
65. #3752C2×C4semidirect
66. C4⋊(C3⋊Dic3)C2×C4semidirect
71. C3×C4⋊Dic3D4semidirect
140. C4⋊(C3⋊D3)C4⋊C4semidirect
212. C3×C42C4×D3semidirect
213. C3×C42C4×D3semidirect
226. C4⋊Dic3D12semidirect
243. C2×C3⋊D3C2×C4⋊C4semidirect
245. C32×C4C2×C4⋊C4semidirect
246. C32×C4C4×D4semidirect
391. C42#624semidirect
442. C3×C4C4×D12semidirect
443. C3×C4D3×C4⋊C4semidirect
444. C3×C4D3×C4⋊C4semidirect
445. C3×C4#893semidirect
452. C32D4×C4⋊C4semidirect
528. C4#63894semidirect
529. C4#63900semidirect
531. C3C4⋊C4×D12semidirect
532. C3#74360semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊(C3⋊D3)2
C2×#639002
C2×#638942
C3×#49352
C3×C4×C4⋊Dic32
#105913782
#105913252
#105912122
#743603
C4⋊C4×D123