Finite Group Databasebeta

← #49161761

Subgroups — #49161761

Order 37504.

928 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627594C211yesmaximal, characteristic
2C2×#22627586C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D293C211yesmaximal, characteristic, complemented
4C22×#17989779C211yesmaximal, characteristic, complemented
5C2×C22⋊C4×D293C211yesmaximal, characteristic, complemented
6#34036030C211yesmaximal, characteristic, complemented
7#34036019C211yesmaximal, characteristic, complemented
8#34036018C211yesmaximal, characteristic, complemented
9#34036007C211yesmaximal, characteristic, complemented
10#34036007C211yesmaximal, characteristic, complemented
11#34036006C211yesmaximal, characteristic, complemented
12#34036000C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic293C211yesmaximal, characteristic, complemented
14#34035976C211yesmaximal, characteristic, complemented
15#385×C293C211yesmaximal, characteristic, complemented
16C24×D293C2211yescharacteristic
17C22×D1172C2211yescharacteristic
18C22×D1172C2211yescharacteristic
19C22×C22⋊D293C2211yescharacteristic
20C23×Dic293C2211yescharacteristic
21C22×C4×D293C2211yescharacteristic
22C22×C4×D293C2211yescharacteristic
23C22×C4×D293C2211yescharacteristic, complemented
24#22627594C418yescomplemented
25#22627586—24no—
26C4×C22⋊D293—24no—
27C2×#17989781C2211yescharacteristic, complemented
28C2×C22⋊Dic293C2211yescharacteristic
29C2×#17989779—24no—
30C2×#17989779C2211yescharacteristic
31C2×#17989779C2211yescharacteristic
32C2×#17989779C2211yescharacteristic
33C2×#17989779C2211yescharacteristic
34C2×#17989779C2211yescharacteristic
35C2×#17989779C2211yescharacteristic
36C2×#17989779C2211yescharacteristic
37C2×#17989779C2211yescharacteristic
38C2×#17989779C2211yescharacteristic, complemented
39C2×#17989779C2211yescharacteristic, complemented
40C2×#17989779C2211yescharacteristic, complemented
41C22⋊C4×D293—24no—
42C2×C4×Dic293C2211yescharacteristic, complemented
43C2×C4×Dic293C2211yescharacteristic, complemented
44C2×C4×Dic293C2211yescharacteristic, complemented
45#22627531C2211yescharacteristic, complemented
46#22627531C2211yescharacteristic, complemented
47#22627531C2211yescharacteristic, complemented
48#22627531C2211yescharacteristic, complemented
49C23×C4×C293C2211yescharacteristic
50C2×C4⋊C4×C293C2211yescharacteristic, complemented
51C2×C4⋊C4×C293C2211yescharacteristic, complemented
52C2×C22⋊C4×C293C2211yescharacteristic
53C2×C22⋊C4×C293C2211yescharacteristic
54#95×C293C2211yescharacteristic, complemented
55#95×C293C2211yescharacteristic, complemented
56C23×D293—22no—
57C23×D293—22no—
58C23×D293—22no—
59C23×D293—22no—
60C23×D293—41no—
61C23×D293—41no—
62C23×D293C2×C414yescomplemented
63C23×D293C2311yescharacteristic
64C23×D293C2311yescharacteristic
65C23×D293C2311yescharacteristic
66C2×D1172—22no—
67C2×D1172—22no—
68C2×D1172—41no—
69C2×D1172—41no—
70C2×D1172C2×C414yescomplemented
71C2×D1172C2×C414yescomplemented
72C2×C22⋊D293—22no—
73C2×C22⋊D293—22no—
74C2×C22⋊D293C2×C414yescomplemented
75C22×Dic293—22no—
76C22×Dic293—22no—
77C22×Dic293—41no—
78C22×Dic293C2311yescharacteristic
79C22×Dic293C2311yescharacteristic
80C22×Dic293C2311yescharacteristic
81C22×Dic293C2311yescharacteristic
82C22×Dic293C2311yescharacteristic
83C2×C4×D293—22no—
84C2×C4×D293—22no—
85C2×C4×D293—22no—
86C2×C4×D293—22no—
87C2×C4×D293—22no—
88C2×C4×D293—22no—
89C2×C4×D293—41no—
90C2×C4×D293—41no—
91C2×C4×D293—41no—
92#17989781—22no—
93C22⋊Dic293D414yescomplemented
94#17989779—22no—
95#17989779—22no—
96#17989779—22no—
97#17989779—22no—
98#17989779—22no—
99#17989779—22no—
100#17989779—22no—
101#17989779—41no—
102#17989779C2×C414yescomplemented
103#17989779C2×C414yescomplemented
104#17989779D414yescomplemented
105C4×Dic293—22no—
106C4×Dic293—22no—
107C4×Dic293—41no—
108C24×C293C2311yescharacteristic
109C22×C4×C293—22no—
110C22×C4×C293—22no—
111C22×C4×C293—41no—
112C22×C4×C293C2311yescharacteristic
113C22×C4×C293C2311yescharacteristic
114C22×C4×C293C2311yescharacteristic
115C22×C4×C293C2311yescharacteristic
116C22×C4×C293C2311yescharacteristic
117C22×C4×C293C2311yescharacteristic
118C4⋊C4×C293—22no—
119C4⋊C4×C293—41no—
120C22⋊C4×C293—22no—
121C22⋊C4×C293C2×C414yescomplemented
122C22×D293—21no—
123C22×D293—21no—
124C22×D293—21no—
125C22×D293—21no—
126C22×D293—28no—
127C22×D293—21no—
128C22×D293—21no—
129C22×D293—21no—
130C22×D293—21no—
131C22×D293—21no—
132C22×D293—21no—
133C22×D293—21no—
134C22×D293—21no—
135C22×D293—21no—
136C22×D293—21no—
137C22×D293—21no—
138C22×D293—21no—
139C22×D293—28no—
140C22×D293—44no—
141C22×D293—44no—
142C22×D293—44no—
143C22×D293—44no—
144C22×D293—44no—
145C22×D293C22×C412yes—
146C22×D293C22×C412yes—
147C22×D293C22×C412yes—
148C22×D293C2×D412yes—
149C22×D293#4112yes—
150D1172—28no—
151D1172—44no—
152C22⋊D293—28no—
153C2×Dic293—21no—
154C2×Dic293—21no—
155C2×Dic293—21no—
156C2×Dic293—21no—
157C2×Dic293—21no—
158C2×Dic293—21no—
159C2×Dic293—21no—
160C2×Dic293—21no—
161C2×Dic293—21no—
162C2×Dic293—21no—
163C2×Dic293—21no—
164C2×Dic293—44no—
165C2×Dic293C22×C412yes—
166C2×Dic293C2×D412yes—
167C2×Dic293C2×D412yes—
168C2×Dic293C2×D412yes—
169C4×D293—44no—
170C4×D293—44no—
171C4×D293—44no—
172C23×C293—21no—
173C23×C293—21no—
174C23×C293—21no—
175C23×C293—21no—
176C23×C293C22×C412yes—
177C23×C293C2×D412yes—
178C23×C293#4112yes—
179C23×C293C2411yescharacteristic
180C2×C4×C293—21no—
181C2×C4×C293—21no—
182C2×C4×C293—21no—
183C2×C4×C293—21no—
184C2×C4×C293—21no—
185C2×C4×C293—21no—
186C2×C4×C293—21no—
187C2×C4×C293—21no—
188C2×C4×C293—21no—
189C2×C4×C293—21no—
190C2×C4×C293—21no—
191C2×C4×C293—21no—
192C2×C4×C293—21no—
193C2×C4×C293—21no—
194C2×C4×C293—21no—
195C2×C4×C293—44no—
196C2×C4×C293C22×C412yes—
197C2×C4×C293C22×C412yes—
198C2×C4×C293C2×D412yes—
199C22×C293—22no—
200C22×C293—22no—
201C22×C293—22no—
202C22×C293—22no—
203C22×C293—22no—
204C22×C293—22no—
205C22×C293C4×D414yescomplemented
206C22×C293C23×C411yescharacteristic
207C22×C293C22×D411yescharacteristic
208C22×C293C22×D411yescharacteristic
209C22×C293C2×#4111yescharacteristic
210C22×C293C2×#4111yescharacteristic
211C22×C293H(5,𝔽2)11yescharacteristic
212C22×C293H(5,𝔽2)11yescharacteristic
213C2×D293—22no—
214C2×D293—22no—
215C2×D293—22no—
216C2×D293—22no—
217C2×D293—22no—
218C2×D293—22no—
219C2×D293—22no—
220C2×D293—22no—
221C2×D293—22no—
222C2×D293—22no—
223C2×D293—22no—
224C2×D293—22no—
225C2×D293—416no—
226C2×D293—41no—
227C2×D293—41no—
228C2×D293—41no—
229C2×D293—41no—
230C2×D293—41no—
231C2×D293—41no—
232C2×D293—41no—
233C2×D293—41no—
234C2×D293C4×D414yescomplemented
235C4×C293—22no—
236C4×C293—22no—
237C4×C293—22no—
238C4×C293—41no—
239C4×C293—41no—
240C4×C293—41no—
241Dic293—22no—
242Dic293—22no—
243Dic293—22no—
244Dic293—22no—
245Dic293—41no—
246C2×C293—24no—
247C2×C293C2×C4×D411yescharacteristic
248C2×C293C2×C4×D411yescharacteristic
249C2×C293#51811yescharacteristic
250C2×C293D4211yescharacteristic
251C2×C293#54611yescharacteristic
252C2×C293#54611yescharacteristic
253C2×C293#55111yescharacteristic
254D293—24no—
255D293—42no—
256D293—42no—
257C293#235811yescharacteristic, complemented, Sylow
258#2358—2931nomaximal, Sylow
259C2×#120—2931no—
260C2×C4×D4—2931no—
261C2×C4×D4—2931no—
262C22×C22⋊C4—2931no—
263C22×C22⋊C4—2931no—
264#390—2931no—
265#388—2931no—
266#388—2931no—
267#386—2931no—
268#386—2931no—
269#386—2931no—
270#385—2931no—
271#385D29311yescharacteristic, complemented
272C4×C22⋊C4—2931no—
273C4×C22⋊C4—2931no—
274C25—2931no—
275C22×D4—2931no—
276C22×D4—2931no—
277C22×D4—2931no—
278C23×C4—2931no—
279C23×C4—2931no—
280C23×C4—2931no—
281C23×C4—2931no—
282C23×C4C2×D29311yescharacteristic
283#120—2938no—
284C4×D4—5864no—
285C4×D4—5864no—
286C2×C4⋊C4—2931no—
287C2×C4⋊C4C2×D29311yescharacteristic, complemented
288C2×C4⋊C4C2×D29311yescharacteristic, complemented
289C2×C22⋊C4—2931no—
290C2×C22⋊C4—2931no—
291C2×C22⋊C4—2931no—
292C2×C22⋊C4—2931no—
293C2×C22⋊C4—2931no—
294C2×C22⋊C4—2931no—
295C2×C22⋊C4—2931no—
296C2×C22⋊C4—2931no—
297C2×C22⋊C4—2931no—
298C2×C22⋊C4—2931no—
299C2×C22⋊C4—2931no—
300C2×C22⋊C4—2931no—
301C2×C22⋊C4—5864no—
302C2×C22⋊C4—5864no—
303C2×C22⋊C4C2×D29311yescharacteristic
304C2×C22⋊C4C2×D29311yescharacteristic
305C2×C42—2931no—
306C2×C42—2931no—
307C2×C42—2931no—
308#95—2931no—
309#95—2931no—
310#95—2931no—
311#95—2931no—
312#95C2×D29311yescharacteristic, complemented
313#95C2×D29311yescharacteristic, complemented
314C24—2931no—
315C24—2931no—
316C24—2931no—
317C24—2934no—
318C24—5862no—
319C24—5862no—
320C24—5862no—
321C24—5862no—
322C24—11721no—
323C24—11721no—
324C24C22×D29311yescharacteristic
325C2×D4—2934no—
326C2×D4—2934no—
327C2×D4—2934no—
328C2×D4—5862no—
329C2×D4—5862no—
330C2×D4—5862no—
331C2×D4—5862no—
332C2×D4—11721no—
333C2×D4—11721no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—41no—
337C22×C4—2931no—
338C22×C4—2931no—
339C22×C4—2931no—
340C22×C4—2931no—
341C22×C4—2931no—
342C22×C4—5862no—
343C22×C4—5862no—
344C22×C4—5862no—
345C22×C4—5862no—
346C22×C4—5862no—
347C22×C4—5862no—
348C22×C4—5862no—
349C22×C4—5862no—
350C22×C4—11721no—
351C22×C4—11721no—
352C22×C4—11721no—
353C22×C4—11721no—
354C22×C4C22×D29311yescharacteristic
355C22×C4C22×D29311yescharacteristic
356C22×C4C22×D29311yescharacteristic
357C22×C4C22×D29311yescharacteristic
358C22×C4C22×D29311yescharacteristic
359C22×C4C22×D29311yescharacteristic
360C4⋊C4—22no—
361C4⋊C4—41no—
362C4⋊C4—5862no—
363C22⋊C4—22no—
364C22⋊C4—2934no—
365C22⋊C4—2934no—
366C22⋊C4—2934no—
367C22⋊C4—2934no—
368C22⋊C4—5862no—
369C22⋊C4—5862no—
370C22⋊C4—5862no—
371C22⋊C4—5862no—
372C22⋊C4—5862no—
373C22⋊C4—5862no—
374C22⋊C4—5862no—
375C22⋊C4—11721no—
376C22⋊C4C4×D29314yescomplemented
377C42—5862no—
378C42—5862no—
379C42—11721no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—2932no—
385C23—2932no—
386C23—2932no—
387C23—2932no—
388C23—2932no—
389C23—5868no—
390C23—5861no—
391C23—5861no—
392C23—5861no—
393C23—5861no—
394C23—5868no—
395C23—5861no—
396C23—5861no—
397C23—5861no—
398C23—5861no—
399C23—5861no—
400C23—5861no—
401C23—5861no—
402C23—5861no—
403C23—5861no—
404C23—5861no—
405C23—5861no—
406C23—5861no—
407C23—11724no—
408C23—11724no—
409C23—11724no—
410C23—11724no—
411C23—11724no—
412C23C2×C4×D29312yes—
413C23D4×D29312yes—
414C23#1798980312yes—
415C23C23×D29311yescharacteristic
416D4—5868no—
417D4—5868no—
418D4—11724no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—44no—
435C2×C4—2932no—
436C2×C4—2932no—
437C2×C4—2932no—
438C2×C4—2932no—
439C2×C4—5861no—
440C2×C4—5861no—
441C2×C4—5861no—
442C2×C4—5861no—
443C2×C4—5861no—
444C2×C4—5861no—
445C2×C4—5861no—
446C2×C4—5861no—
447C2×C4—5861no—
448C2×C4—5861no—
449C2×C4—5861no—
450C2×C4—11724no—
451C2×C4—11724no—
452C2×C4—11724no—
453C2×C4—11724no—
454C2×C4C2×C4×D29312yes—
455C2×C4C2×C4×D29312yes—
456C2×C4D4×D29312yes—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—2934no—
464C22—5862no—
465C22—5862no—
466C22—5862no—
467C22—5862no—
468C22—5862no—
469C22—5862no—
470C22—5862no—
471C22—5862no—
472C22—5862no—
473C22—5862no—
474C22—5862no—
475C22—5862no—
476C22—117216no—
477C22—11721no—
478C22—11721no—
479C22—11721no—
480C22—11721no—
481C22—11721no—
482C22—11721no—
483C22—11721no—
484C22—11721no—
485C22#2262758614yescomplemented
486C22C22×C4×D29311yescharacteristic
487C22C2×D4×D29311yescharacteristic
488C22C2×D4×D29311yescharacteristic
489C22#41×D29311yescharacteristic
490C22C2×#1798980311yescharacteristic
491C22#2262766911yescharacteristic
492C22#2262767011yescharacteristic
493C4—22no—
494C4—22no—
495C4—22no—
496C4—41no—
497C4—41no—
498C4—41no—
499C4—5862no—
500C4—5862no—
501C4—5862no—
502C4—5862no—
503C4—11721no—
504C2—24no—
505C2—5864no—
506C2—11722no—
507C2—11722no—
508C2C4×D4×D29311yescharacteristic
509C2C2×#2262758611yescharacteristic
510C2#3403660111yescharacteristic
511C2#3403676111yescharacteristic
512C2#3403677011yescharacteristic
513C2#3403677111yescharacteristic
514C2#3403680611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D293
commutatorC22×C293C23×C4
Fitting#385×C293C2
socle, cosocleC23×C293C24
prime radicalC2×#22627594C2
hypercenter#385D293
hypocenterC293#2358
LCSG ▷ C22×C293 ▷ C293
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#22627586C2semidirect
3. C2×C4×C22⋊D293C2semidirect
4. C22×#17989779C2semidirect
5. C2×C22⋊C4×D293C2semidirect
6. #34036030C2semidirect
7. #34036019C2semidirect
8. #34036018C2semidirect
9. #34036007C2semidirect
10. #34036007C2semidirect
11. #34036006C2semidirect
12. #34036000C2semidirect
13. C22⋊C4×Dic293C2semidirect
14. #34035976C2semidirect
15. #385×C293C2semidirect
23. C22×C4×D293C22semidirect
24. #22627594C4semidirect
27. C2×#17989781C22semidirect
38. C2×#17989779C22semidirect
39. C2×#17989779C22semidirect
40. C2×#17989779C22semidirect
42. C2×C4×Dic293C22semidirect
43. C2×C4×Dic293C22semidirect
44. C2×C4×Dic293C22semidirect
45. #22627531C22semidirect
46. #22627531C22semidirect
47. #22627531C22semidirect
48. #22627531C22semidirect
50. C2×C4⋊C4×C293C22semidirect
51. C2×C4⋊C4×C293C22semidirect
54. #95×C293C22semidirect
55. #95×C293C22semidirect
62. C23×D293C2×C4semidirect
70. C2×D1172C2×C4semidirect
71. C2×D1172C2×C4semidirect
74. C2×C22⋊D293C2×C4semidirect
93. C22⋊Dic293D4semidirect
102. #17989779C2×C4semidirect
103. #17989779C2×C4semidirect
104. #17989779D4semidirect
121. C22⋊C4×C293C2×C4semidirect
205. C22×C293C4×D4semidirect
234. C2×D293C4×D4semidirect
257. C293#2358semidirect
271. #385D293semidirect
287. C2×C4⋊C4C2×D293semidirect
288. C2×C4⋊C4C2×D293semidirect
312. #95C2×D293semidirect
313. #95C2×D293semidirect
376. C22⋊C4C4×D293semidirect
485. C22#22627586semidirect

Maximal subgroups

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

subgroupindex
C2×#226275942
C2×#226275862
C2×C4×C22⋊D2932
C22×#179897792
C2×C22⋊C4×D2932
#340360302
#340360192
#340360182
#340360072
#340360062
#340360002
C22⋊C4×Dic2932
#340359762
#385×C2932
#2358293