Finite Group Databasebeta

← #49161762

Subgroups — #49161762

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×#22627592C211yesmaximal, characteristic
2C2×C4×C22⋊D293C211yesmaximal, characteristic, complemented
3C22×#17989779C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D293C211yesmaximal, characteristic, complemented
5#34036029C211yesmaximal, characteristic, complemented
6#34036020C211yesmaximal, characteristic, complemented
7#34036019C211yesmaximal, characteristic, complemented
8#34036009C211yesmaximal, characteristic, complemented
9#34036008C211yesmaximal, characteristic, complemented
10#34036005C211yesmaximal, characteristic, complemented
11#34036002C211yesmaximal, characteristic, complemented
12#34036000C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic293C211yesmaximal, characteristic, complemented
14C4×#17989779C211yesmaximal, characteristic, complemented
15C2×C4×D4×C293C211yesmaximal, characteristic, complemented
16C24×D293C2211yescharacteristic
17C22×C22⋊D293C2211yescharacteristic
18C22×C22⋊D293C2211yescharacteristic
19C23×Dic293C2211yescharacteristic
20C22×C4×D293C2211yescharacteristic
21C22×C4×D293C2211yescharacteristic
22#22627592C418yescomplemented
23C4×C22⋊D293—24no—
24C2×#17989781C2211yescharacteristic, complemented
25C2×#17989781C2211yescharacteristic, complemented
26C2×C22⋊Dic293C2211yescharacteristic
27C2×C22⋊Dic293C2211yescharacteristic
28C2×C22⋊Dic293C2211yescharacteristic, complemented
29C2×C22⋊Dic293C2211yescharacteristic, complemented
30C2×#17989779—24no—
31C2×#17989779C2211yescharacteristic
32C2×#17989779C2211yescharacteristic
33C2×#17989779C2211yescharacteristic
34C2×#17989779C2211yescharacteristic
35C2×#17989779C2211yescharacteristic
36C2×#17989779C2211yescharacteristic
37C2×#17989779C2211yescharacteristic
38C2×#17989779C2211yescharacteristic
39C22⋊C4×D293—24no—
40C2×C4×Dic293C2211yescharacteristic, complemented
41C2×C4×Dic293C2211yescharacteristic, complemented
42#22627531C2211yescharacteristic, complemented
43#22627531C2211yescharacteristic, complemented
44#22627531C2211yescharacteristic, complemented
45#22627531C2211yescharacteristic, complemented
46#22627531C2211yescharacteristic, complemented
47#22627531C2211yescharacteristic, complemented
48C22×D4×C293C2211yescharacteristic
49C23×C4×C293C2211yescharacteristic
50C23×C4×C293C2211yescharacteristic, complemented
51C4×D4×C293—24no—
52C2×C4⋊C4×C293C2211yescharacteristic, complemented
53C2×C22⋊C4×C293C2211yescharacteristic
54C2×C22⋊C4×C293C2211yescharacteristic, complemented
55C2×C42×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
65C2×C22⋊D293—22no—
66C2×C22⋊D293—22no—
67C2×C22⋊D293—41no—
68C2×C22⋊D293—41no—
69C2×C22⋊D293C2×C414yescomplemented
70C2×C22⋊D293C2×C414yescomplemented
71C22×Dic293—22no—
72C22×Dic293—22no—
73C22×Dic293—41no—
74C22×Dic293C2311yescharacteristic
75C22×Dic293C2311yescharacteristic
76C22×Dic293C2311yescharacteristic
77C22×Dic293C2311yescharacteristic
78C22×Dic293C2311yescharacteristic
79C22×Dic293C2311yescharacteristic
80C2×C4×D293—22no—
81C2×C4×D293—22no—
82C2×C4×D293—22no—
83C2×C4×D293—22no—
84C2×C4×D293—41no—
85C2×C4×D293—41no—
86#17989781—22no—
87#17989781—41no—
88C22⋊Dic293—22no—
89C22⋊Dic293—22no—
90C22⋊Dic293—41no—
91C22⋊Dic293C2×C414yescomplemented
92#17989779—22no—
93#17989779—22no—
94#17989779—22no—
95#17989779—22no—
96#17989779—22no—
97#17989779C2×C414yescomplemented
98#17989779C2×C414yescomplemented
99#17989779D414yescomplemented
100C4×Dic293—22no—
101C4×Dic293—41no—
102C24×C293C2311yescharacteristic
103C24×C293C2311yescharacteristic
104C2×D4×C293—22no—
105C2×D4×C293—22no—
106C2×D4×C293C2×C414yescomplemented
107C22×C4×C293—22no—
108C22×C4×C293—22no—
109C22×C4×C293—22no—
110C22×C4×C293—22no—
111C22×C4×C293—41no—
112C22×C4×C293—41no—
113C22×C4×C293C2311yescharacteristic
114C22×C4×C293C2311yescharacteristic
115C22×C4×C293C2311yescharacteristic
116C22×C4×C293C2311yescharacteristic
117C22×C4×C293C2311yescharacteristic
118C4⋊C4×C293—22no—
119C22⋊C4×C293—22no—
120C22⋊C4×C293D414yescomplemented
121C42×C293—22no—
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—28no—
134C22×D293—44no—
135C22×D293—44no—
136C22×D293—44no—
137C22×D293—44no—
138C22×D293—44no—
139C22×D293C22×C412yes—
140C22×D293C22×C412yes—
141C22×D293C2×D412yes—
142C22×D293#4112yes—
143C22⋊D293—28no—
144C22⋊D293—44no—
145C2×Dic293—21no—
146C2×Dic293—21no—
147C2×Dic293—21no—
148C2×Dic293—21no—
149C2×Dic293—21no—
150C2×Dic293—21no—
151C2×Dic293—21no—
152C2×Dic293—21no—
153C2×Dic293—21no—
154C2×Dic293—21no—
155C2×Dic293—21no—
156C2×Dic293—21no—
157C2×Dic293—21no—
158C2×Dic293—21no—
159C2×Dic293—21no—
160C2×Dic293—44no—
161C2×Dic293C22×C412yes—
162C2×Dic293C22×C412yes—
163C2×Dic293C2×D412yes—
164C4×D293—44no—
165C4×D293—44no—
166C23×C293—21no—
167C23×C293—21no—
168C23×C293—21no—
169C23×C293—21no—
170C23×C293—21no—
171C23×C293—21no—
172C23×C293—21no—
173C23×C293—21no—
174C23×C293—21no—
175C23×C293—21no—
176C23×C293C22×C412yes—
177C23×C293C22×C412yes—
178C23×C293C2×D412yes—
179C23×C293#4112yes—
180C23×C293C2411yescharacteristic
181D4×C293—28no—
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—44no—
194C2×C4×C293—44no—
195C2×C4×C293C22×C412yes—
196C2×C4×C293C2×D412yes—
197C2×C4×C293C2×D412yes—
198C2×C4×C293C2×D412yes—
199C22×C293—22no—
200C22×C293—22no—
201C22×C293—22no—
202C22×C293—22no—
203C22×C293—22no—
204C22×C293—22no—
205C22×C293—22no—
206C22×C293—22no—
207C22×C293—22no—
208C22×C293—41no—
209C22×C293—41no—
210C22×C293—41no—
211C22×C293—41no—
212C22×C293C4×D414yescomplemented
213C22×C293C23×C411yescharacteristic
214C22×C293C22×D411yescharacteristic
215C22×C293C22×D411yescharacteristic
216C22×C293C2×#4111yescharacteristic
217C22×C293C2×#4111yescharacteristic
218C22×C293H(5,𝔽2)11yescharacteristic
219C22×C293H(5,𝔽2)11yescharacteristic
220C2×D293—22no—
221C2×D293—22no—
222C2×D293—22no—
223C2×D293—22no—
224C2×D293—22no—
225C2×D293—22no—
226C2×D293—22no—
227C2×D293—22no—
228C2×D293—22no—
229C2×D293—416no—
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—22no—
239C4×C293—41no—
240Dic293—22no—
241Dic293—22no—
242Dic293—22no—
243Dic293—41no—
244Dic293—41no—
245Dic293—41no—
246C2×C293—24no—
247C2×C293—42no—
248C2×C293C2×C4×D411yescharacteristic
249C2×C293C2×C4×D411yescharacteristic
250C2×C293#51811yescharacteristic
251C2×C293D4211yescharacteristic
252C2×C293#54611yescharacteristic
253C2×C293#54611yescharacteristic
254C2×C293#55111yescharacteristic
255D293—24no—
256D293—42no—
257C293#235811yescharacteristic, complemented, Sylow
258#2358—2931nomaximal, Sylow
259C2×#120—2931no—
260C2×C4×D4—2931no—
261C2×C4×D4D29311yescharacteristic, complemented
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#385—2931no—
272C4×C22⋊C4—2931no—
273C4×C22⋊C4—2931no—
274C25—2931no—
275C22×D4—2931no—
276C22×D4—2931no—
277C22×D4C2×D29311yescharacteristic
278C23×C4—2931no—
279C23×C4—2931no—
280C23×C4—2931no—
281C23×C4C2×D29311yescharacteristic
282C23×C4C2×D29311yescharacteristic, complemented
283#120—2938no—
284C4×D4—24no—
285C4×D4—5864no—
286C2×C4⋊C4—2931no—
287C2×C4⋊C4—2931no—
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, complemented
305C2×C42—2931no—
306C2×C42—2931no—
307C2×C42C2×D29311yescharacteristic, complemented
308#95—2931no—
309#95—2931no—
310#95—2931no—
311#95—2931no—
312#95—2931no—
313#95—2931no—
314C24—2931no—
315C24—2931no—
316C24—2934no—
317C24—5862no—
318C24—5862no—
319C24—5862no—
320C24—5862no—
321C24—11721no—
322C24—11721no—
323C24C22×D29311yescharacteristic
324C24C22×D29311yescharacteristic
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—2934no—
328C2×D4—2934no—
329C2×D4—5862no—
330C2×D4—5862no—
331C2×D4—11721no—
332C2×D4—11721no—
333C2×D4C4×D29314yescomplemented
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—2931no—
341C22×C4—2931no—
342C22×C4—2931no—
343C22×C4—2931no—
344C22×C4—2931no—
345C22×C4—2931no—
346C22×C4—5862no—
347C22×C4—5862no—
348C22×C4—5862no—
349C22×C4—5862no—
350C22×C4—5862no—
351C22×C4—5862no—
352C22×C4—11721no—
353C22×C4—11721no—
354C22×C4—11721no—
355C22×C4C22×D29311yescharacteristic
356C22×C4C22×D29311yescharacteristic
357C22×C4C22×D29311yescharacteristic
358C22×C4C22×D29311yescharacteristic
359C22×C4C22×D29311yescharacteristic
360C4⋊C4—22no—
361C4⋊C4—5862no—
362C4⋊C4—11721no—
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⋊C4C22⋊D29314yescomplemented
377C42—22no—
378C42—5862no—
379C42—11721no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—2932no—
391C23—2932no—
392C23—2932no—
393C23—2932no—
394C23—5868no—
395C23—5861no—
396C23—5861no—
397C23—5861no—
398C23—5861no—
399C23—5868no—
400C23—5861no—
401C23—5861no—
402C23—5861no—
403C23—5861no—
404C23—5861no—
405C23—5861no—
406C23—11724no—
407C23—11724no—
408C23—11724no—
409C23—11724no—
410C23—11724no—
411C23C2×C4×D29312yes—
412C23C2×C4×D29312yes—
413C23C2×C22⋊D29312yes—
414C23#1798980212yes—
415C23C23×D29311yescharacteristic
416D4—28no—
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—44no—
431C2×C4—44no—
432C2×C4—2932no—
433C2×C4—2932no—
434C2×C4—2932no—
435C2×C4—5861no—
436C2×C4—5861no—
437C2×C4—5861no—
438C2×C4—5861no—
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×C4C2×C4×D29312yes—
454C2×C4D4×D29312yes—
455C2×C4C2×C22⋊D29312yes—
456C2×C4C2×C22⋊D29312yes—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—2934no—
471C22—5862no—
472C22—5862no—
473C22—5862no—
474C22—5862no—
475C22—5862no—
476C22—5862no—
477C22—5862no—
478C22—5862no—
479C22—5862no—
480C22—117216no—
481C22—11721no—
482C22—11721no—
483C22—11721no—
484C22—11721no—
485C22C4×C22⋊D29314yescomplemented
486C22C22×C4×D29311yescharacteristic
487C22C2×D4×D29311yescharacteristic
488C22C22×C22⋊D29311yescharacteristic
489C22#41×D29311yescharacteristic
490C22C2×#1798980211yescharacteristic
491C22#2262766911yescharacteristic
492C22#2262767011yescharacteristic
493C4—22no—
494C4—22no—
495C4—22no—
496C4—22no—
497C4—41no—
498C4—5862no—
499C4—5862no—
500C4—5862no—
501C4—11721no—
502C4—11721no—
503C4—11721no—
504C2—24no—
505C2—42no—
506C2—5864no—
507C2—11722no—
508C2C4×D4×D29311yescharacteristic
509C2C2×C4×C22⋊D29311yescharacteristic
510C2#3403660111yescharacteristic
511C2D4×C22⋊D29311yescharacteristic
512C2#3403676711yescharacteristic
513C2#3403677511yescharacteristic
514C2#3403680811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D293
commutatorC22×C293C23×C4
FittingC2×C4×D4×C293C2
socle, cosocleC23×C293C24
prime radicalC2×#22627592C2
hypercenterC2×C4×D4D293
hypocenterC293#2358
LCSG ▷ C22×C293 ▷ C293
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D293C2semidirect
3. C22×#17989779C2semidirect
4. C2×C22⋊C4×D293C2semidirect
5. #34036029C2semidirect
6. #34036020C2semidirect
7. #34036019C2semidirect
8. #34036009C2semidirect
9. #34036008C2semidirect
10. #34036005C2semidirect
11. #34036002C2semidirect
12. #34036000C2semidirect
13. C22⋊C4×Dic293C2semidirect
14. C4×#17989779C2semidirect
15. C2×C4×D4×C293C2semidirect
22. #22627592C4semidirect
24. C2×#17989781C22semidirect
25. C2×#17989781C22semidirect
28. C2×C22⋊Dic293C22semidirect
29. C2×C22⋊Dic293C22semidirect
40. C2×C4×Dic293C22semidirect
41. C2×C4×Dic293C22semidirect
42. #22627531C22semidirect
43. #22627531C22semidirect
44. #22627531C22semidirect
45. #22627531C22semidirect
46. #22627531C22semidirect
47. #22627531C22semidirect
50. C23×C4×C293C22semidirect
52. C2×C4⋊C4×C293C22semidirect
54. C2×C22⋊C4×C293C22semidirect
55. C2×C42×C293C22semidirect
62. C23×D293C2×C4semidirect
69. C2×C22⋊D293C2×C4semidirect
70. C2×C22⋊D293C2×C4semidirect
91. C22⋊Dic293C2×C4semidirect
97. #17989779C2×C4semidirect
98. #17989779C2×C4semidirect
99. #17989779D4semidirect
106. C2×D4×C293C2×C4semidirect
120. C22⋊C4×C293D4semidirect
212. C22×C293C4×D4semidirect
234. C2×D293C4×D4semidirect
257. C293#2358semidirect
261. C2×C4×D4D293semidirect
282. C23×C4C2×D293semidirect
288. C2×C4⋊C4C2×D293semidirect
304. C2×C22⋊C4C2×D293semidirect
307. C2×C42C2×D293semidirect
333. C2×D4C4×D293semidirect
376. C22⋊C4C22⋊D293semidirect
485. C22C4×C22⋊D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226275922
C2×C4×C22⋊D2932
C22×#179897792
C2×C22⋊C4×D2932
#340360292
#340360202
#340360192
#340360092
#340360082
#340360052
#340360022
#340360002
C22⋊C4×Dic2932
C4×#179897792
C2×C4×D4×C2932
#2358293