Finite Group Databasebeta

← #49161438

Subgroups — #49161438

Order 37504.

808 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627594C211yesmaximal, characteristic
2C2×#22627586C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D293C211yesmaximal, characteristic, complemented
4#34036031C211yesmaximal, characteristic, complemented
5#34036029C211yesmaximal, characteristic, complemented
6#34036022C211yesmaximal, characteristic, complemented
7#34036019C211yesmaximal, characteristic, complemented
8#34036018C211yesmaximal, characteristic, complemented
9#34036007C211yesmaximal, characteristic, complemented
10#34036007C211yesmaximal, characteristic, complemented
11#34036005C211yesmaximal, characteristic, complemented
12#34036003C211yesmaximal, characteristic, complemented
13#34035984C211yesmaximal, characteristic, complemented
14C4×#17989779C211yesmaximal, characteristic, complemented
15#388×C293C211yesmaximal, characteristic, complemented
16C24×D293C2211yescharacteristic
17C22×D1172C2211yescharacteristic
18C22×D1172C2211yescharacteristic
19C22×C22⋊D293C2211yescharacteristic
20C22×C4×D293C2211yescharacteristic
21C22×C4×D293C2211yescharacteristic
22C22×C4×D293C2211yescharacteristic, complemented
23#22627594C418yescomplemented
24#22627586—24no—
25C2×C4⋊Dic293C2211yescharacteristic, complemented
26C2×#17989781C2211yescharacteristic, complemented
27C2×C22⋊Dic293C2211yescharacteristic
28C2×C22⋊Dic293C2211yescharacteristic
29C2×#17989779C2211yescharacteristic
30C2×#17989779C2211yescharacteristic
31C2×#17989779C2211yescharacteristic
32C2×#17989779C2211yescharacteristic
33C2×#17989779C2211yescharacteristic
34C2×#17989779C2211yescharacteristic
35C2×#17989779C2211yescharacteristic
36C2×#17989779C2211yescharacteristic
37C2×#17989779C2211yescharacteristic, complemented
38C2×#17989779C2211yescharacteristic, complemented
39C2×#17989779C2211yescharacteristic, complemented
40C22⋊C4×D293—24no—
41C2×C4×Dic293C2211yescharacteristic, complemented
42C2×C4×Dic293C2211yescharacteristic, complemented
43#22627531C2211yescharacteristic, complemented
44#22627531C2211yescharacteristic, complemented
45#22627531C2211yescharacteristic, complemented
46#22627531C2211yescharacteristic, complemented
47C2×C4⋊C4×C293C2211yescharacteristic, complemented
48C2×C22⋊C4×C293C2211yescharacteristic
49C2×C22⋊C4×C293C2211yescharacteristic
50C2×C22⋊C4×C293C2211yescharacteristic
51C2×C42×C293C2211yescharacteristic, complemented
52#95×C293C2211yescharacteristic, complemented
53#95×C293C2211yescharacteristic, complemented
54C23×D293—22no—
55C23×D293—22no—
56C23×D293—41no—
57C23×D293—41no—
58C23×D293—41no—
59C23×D293—41no—
60C23×D293C2×C414yescomplemented
61C23×D293C2311yescharacteristic
62C23×D293C2311yescharacteristic
63C23×D293C2311yescharacteristic
64C2×D1172—22no—
65C2×D1172—22no—
66C2×D1172—41no—
67C2×D1172—41no—
68C2×D1172C2×C414yes—
69C2×D1172C2×C414yescomplemented
70C2×C22⋊D293—41no—
71C2×C22⋊D293—41no—
72C2×C22⋊D293C2×C414yescomplemented
73C22×Dic293C2311yescharacteristic
74C22×Dic293C2311yescharacteristic
75C22×Dic293C2311yescharacteristic
76C22×Dic293C2311yescharacteristic
77C22×Dic293C2311yescharacteristic
78C2×C4×D293—22no—
79C2×C4×D293—22no—
80C2×C4×D293—41no—
81C2×C4×D293—41no—
82C2×C4×D293—41no—
83C2×C4×D293—41no—
84C2×C4×D293—41no—
85C2×C4×D293—41no—
86C2×C4×D293D414yescomplemented
87C4⋊Dic293—41no—
88#17989781—41no—
89C22⋊Dic293—22no—
90C22⋊Dic293—41no—
91#17989779—22no—
92#17989779—22no—
93#17989779—22no—
94#17989779—22no—
95#17989779—22no—
96#17989779—41no—
97#17989779—41no—
98#17989779—41no—
99#17989779—41no—
100#17989779C2×C414yescomplemented
101#17989779C2×C414yescomplemented
102C4×Dic293—22no—
103C4×Dic293—41no—
104C24×C293C2311yescharacteristic
105C22×C4×C293C2311yescharacteristic
106C22×C4×C293C2311yescharacteristic
107C22×C4×C293C2311yescharacteristic
108C22×C4×C293C2311yescharacteristic
109C22×C4×C293C2311yescharacteristic
110C22×C4×C293C2311yescharacteristic
111C4⋊C4×C293—22no—
112C22⋊C4×C293—22no—
113C22⋊C4×C293—41no—
114C22⋊C4×C293C2×C414yescomplemented
115C42×C293—41no—
116C22×D293—21no—
117C22×D293—21no—
118C22×D293—21no—
119C22×D293—21no—
120C22×D293—21no—
121C22×D293—21no—
122C22×D293—21no—
123C22×D293—21no—
124C22×D293—21no—
125C22×D293—21no—
126C22×D293—21no—
127C22×D293—21no—
128C22×D293—21no—
129C22×D293—21no—
130C22×D293—21no—
131C22×D293—21no—
132C22×D293—28no—
133C22×D293—44no—
134C22×D293—44no—
135C22×D293—44no—
136C22×D293—44no—
137C22×D293—44no—
138C22×D293—44no—
139C22×D293C22×C412yes—
140C22×D293C22×C412yes—
141C22×D293C22×C412yes—
142C22×D293C2×D412yes—
143C22×D293#4112yes—
144D1172—28no—
145D1172—44no—
146C22⋊D293—44no—
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×Dic293C22×C412yes—
161C2×Dic293C2×D412yes—
162C4×D293—44no—
163C4×D293—44no—
164C4×D293—44no—
165C23×C293—21no—
166C23×C293—21no—
167C23×C293—21no—
168C23×C293—21no—
169C23×C293—21no—
170C23×C293—21no—
171C23×C293C22×C412yes—
172C23×C293C2411yescharacteristic
173C2×C4×C293—21no—
174C2×C4×C293—21no—
175C2×C4×C293—21no—
176C2×C4×C293—21no—
177C2×C4×C293—21no—
178C2×C4×C293—21no—
179C2×C4×C293—21no—
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×C293C22×C412yes—
189C2×C4×C293C22×C412yes—
190C2×C4×C293C2×D412yes—
191C22×C293—22no—
192C22×C293—22no—
193C22×C293—22no—
194C22×C293—41no—
195C22×C293—41no—
196C22×C293—41no—
197C22×C293—41no—
198C22×C293C23×C411yescharacteristic
199C22×C293C22×D411yescharacteristic
200C22×C293C2×#4111yescharacteristic
201C22×C293H(5,𝔽2)11yescharacteristic
202C22×C293H(5,𝔽2)11yescharacteristic
203C22×C293H(5,𝔽2)11yescharacteristic
204C22×C293H(5,𝔽2)11yescharacteristic
205C2×D293—22no—
206C2×D293—22no—
207C2×D293—22no—
208C2×D293—22no—
209C2×D293—22no—
210C2×D293—22no—
211C2×D293—22no—
212C2×D293—22no—
213C2×D293—22no—
214C2×D293—22no—
215C2×D293—22no—
216C2×D293—22no—
217C2×D293—416no—
218C2×D293—41no—
219C2×D293—41no—
220C2×D293—41no—
221C2×D293—41no—
222C2×D293—41no—
223C2×D293—41no—
224C2×D293—41no—
225C2×D293—41no—
226C2×D293C4×D414yes—
227C4×C293—22no—
228C4×C293—22no—
229C4×C293—22no—
230C4×C293—41no—
231C4×C293—41no—
232C4×C293—41no—
233Dic293—22no—
234Dic293—22no—
235Dic293—41no—
236Dic293—41no—
237Dic293—41no—
238C2×C293—42no—
239C2×C293C2×C4×D411yescharacteristic
240C2×C293#51811yescharacteristic
241C2×C293#51811yescharacteristic
242C2×C293#53411yescharacteristic
243C2×C293#53511yescharacteristic
244C2×C293#53811yescharacteristic
245C2×C293#53811yescharacteristic
246D293—24no—
247D293—42no—
248D293—42no—
249C293#232111yescharacteristic, complemented, Sylow
250#2321—2931nomaximal, Sylow
251C2×#120—2931no—
252C2×C4×D4—2931no—
253C22×C22⋊C4—2931no—
254#390—2931no—
255#390—2931no—
256#388—2931no—
257#388—2931no—
258#388—2931no—
259#388D29311yescharacteristic, complemented
260#386—2931no—
261#386—2931no—
262#386—2931no—
263#385—2931no—
264#379—2931no—
265C4×C22⋊C4—2931no—
266C25—2931no—
267C22×D4—2931no—
268C22×D4—2931no—
269C22×D4—2931no—
270C23×C4—2931no—
271C23×C4—2931no—
272C23×C4—2931no—
273#120—2938no—
274C4×D4—5864no—
275C2×C4⋊C4—2931no—
276C2×C4⋊C4—2931no—
277C2×C4⋊C4C2×D29311yescharacteristic, complemented
278C2×C22⋊C4—2931no—
279C2×C22⋊C4—2931no—
280C2×C22⋊C4—2931no—
281C2×C22⋊C4—2931no—
282C2×C22⋊C4—2931no—
283C2×C22⋊C4—2931no—
284C2×C22⋊C4—2931no—
285C2×C22⋊C4—2931no—
286C2×C22⋊C4—2931no—
287C2×C22⋊C4—2931no—
288C2×C22⋊C4—2931no—
289C2×C22⋊C4—2931no—
290C2×C22⋊C4—2931no—
291C2×C22⋊C4—5864no—
292C2×C22⋊C4C2×D29311yescharacteristic
293C2×C22⋊C4C2×D29311yescharacteristic
294C2×C22⋊C4C2×D29311yescharacteristic
295C2×C42—2931no—
296C2×C42—2931no—
297C2×C42C2×D29311yescharacteristic, complemented
298#95—2931no—
299#95—2931no—
300#95—2931no—
301#95—2931no—
302#95C2×D29311yescharacteristic, complemented
303#95C2×D29311yescharacteristic, complemented
304C24—2931no—
305C24—2931no—
306C24—2931no—
307C24—2934no—
308C24—5862no—
309C24—5862no—
310C24—11721no—
311C24—11721no—
312C24—11721no—
313C24—11721no—
314C24C22×D29311yescharacteristic
315C2×D4—2934no—
316C2×D4—2934no—
317C2×D4—2934no—
318C2×D4—5862no—
319C2×D4—5862no—
320C2×D4—11721no—
321C2×D4—11721no—
322C2×D4—11721no—
323C2×D4—11721no—
324C22×C4—2931no—
325C22×C4—2931no—
326C22×C4—2934no—
327C22×C4—2931no—
328C22×C4—2931no—
329C22×C4—2931no—
330C22×C4—5862no—
331C22×C4—5862no—
332C22×C4—11721no—
333C22×C4—11721no—
334C22×C4—11721no—
335C22×C4—11721no—
336C22×C4—11721no—
337C22×C4—11721no—
338C22×C4C22×D29311yescharacteristic
339C22×C4C22×D29311yescharacteristic
340C22×C4C22×D29311yescharacteristic
341C22×C4C22×D29311yescharacteristic
342C22×C4C22×D29311yescharacteristic
343C22×C4C22×D29311yescharacteristic
344C4⋊C4—22no—
345C4⋊C4—11721no—
346C4⋊C4—11721no—
347C22⋊C4—22no—
348C22⋊C4—41no—
349C22⋊C4—2934no—
350C22⋊C4—2934no—
351C22⋊C4—5862no—
352C22⋊C4—5862no—
353C22⋊C4—5862no—
354C22⋊C4—5862no—
355C22⋊C4—5862no—
356C22⋊C4—5862no—
357C22⋊C4—11721no—
358C22⋊C4—11721no—
359C22⋊C4—11721no—
360C22⋊C4—11721no—
361C22⋊C4—11721no—
362C22⋊C4C4×D29314yescomplemented
363C42—41no—
364C42—5862no—
365C42—11721no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—2932no—
373C23—2932no—
374C23—2932no—
375C23—2932no—
376C23—2932no—
377C23—5868no—
378C23—5861no—
379C23—5861no—
380C23—5861no—
381C23—5861no—
382C23—5861no—
383C23—5861no—
384C23—5861no—
385C23—5861no—
386C23—5861no—
387C23—5861no—
388C23—5861no—
389C23—5861no—
390C23—5861no—
391C23—5861no—
392C23—5861no—
393C23—5861no—
394C23—11724no—
395C23—11724no—
396C23—11724no—
397C23—11724no—
398C23—11724no—
399C23—11724no—
400C23C2×C4×D29312yes—
401C23C23×D29311yescharacteristic
402D4—5868no—
403D4—11724no—
404D4—11724no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—2932no—
421C2×C4—2932no—
422C2×C4—5861no—
423C2×C4—5861no—
424C2×C4—5861no—
425C2×C4—5861no—
426C2×C4—5861no—
427C2×C4—5861no—
428C2×C4—5861no—
429C2×C4—5861no—
430C2×C4—5861no—
431C2×C4—5861no—
432C2×C4—5861no—
433C2×C4—5861no—
434C2×C4—5861no—
435C2×C4—11724no—
436C2×C4—11724no—
437C2×C4—11724no—
438C2×C4C2×C4×D29312yes—
439C2×C4C2×C4×D29312yes—
440C2×C4D4×D29312yes—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—2934no—
449C22—5862no—
450C22—5862no—
451C22—5862no—
452C22—5862no—
453C22—5862no—
454C22—5862no—
455C22—5862no—
456C22—5862no—
457C22—5862no—
458C22—5862no—
459C22—5862no—
460C22—5862no—
461C22—117216no—
462C22—11721no—
463C22—11721no—
464C22—11721no—
465C22—11721no—
466C22—11721no—
467C22—11721no—
468C22—11721no—
469C22—11721no—
470C22C22×C4×D29311yescharacteristic
471C22C2×D4×D29311yescharacteristic
472C22#41×D29311yescharacteristic
473C22#2262766911yescharacteristic
474C22#2262767011yescharacteristic
475C22#2262767011yescharacteristic
476C22#2262767011yescharacteristic
477C4—22no—
478C4—22no—
479C4—22no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—5862no—
484C4—5862no—
485C4—11721no—
486C4—11721no—
487C4—11721no—
488C2—42no—
489C2—5864no—
490C2—11722no—
491C2—11722no—
492C2C4×D4×D29311yescharacteristic
493C2#3403660111yescharacteristic
494C2#3403660311yescharacteristic
495C2#3403668711yescharacteristic
496C2#3403669711yescharacteristic
497C2#3403671511yescharacteristic
498C2#3403671811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D293
commutatorC22×C293C23×C4
Fitting#388×C293C2
socle, cosocleC23×C293C24
prime radicalC2×#22627594C2
hypercenter#388D293
hypocenterC293#2321
LCSG ▷ C22×C293 ▷ C293
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×#22627586C2semidirect
3. C2×C22⋊C4×D293C2semidirect
4. #34036031C2semidirect
5. #34036029C2semidirect
6. #34036022C2semidirect
7. #34036019C2semidirect
8. #34036018C2semidirect
9. #34036007C2semidirect
10. #34036007C2semidirect
11. #34036005C2semidirect
12. #34036003C2semidirect
13. #34035984C2semidirect
14. C4×#17989779C2semidirect
15. #388×C293C2semidirect
22. C22×C4×D293C22semidirect
23. #22627594C4semidirect
25. C2×C4⋊Dic293C22semidirect
26. C2×#17989781C22semidirect
37. C2×#17989779C22semidirect
38. C2×#17989779C22semidirect
39. C2×#17989779C22semidirect
41. C2×C4×Dic293C22semidirect
42. C2×C4×Dic293C22semidirect
43. #22627531C22semidirect
44. #22627531C22semidirect
45. #22627531C22semidirect
46. #22627531C22semidirect
47. C2×C4⋊C4×C293C22semidirect
51. C2×C42×C293C22semidirect
52. #95×C293C22semidirect
53. #95×C293C22semidirect
60. C23×D293C2×C4semidirect
69. C2×D1172C2×C4semidirect
72. C2×C22⋊D293C2×C4semidirect
86. C2×C4×D293D4semidirect
100. #17989779C2×C4semidirect
101. #17989779C2×C4semidirect
114. C22⋊C4×C293C2×C4semidirect
249. C293#2321semidirect
259. #388D293semidirect
277. C2×C4⋊C4C2×D293semidirect
297. C2×C42C2×D293semidirect
302. #95C2×D293semidirect
303. #95C2×D293semidirect
362. C22⋊C4C4×D293semidirect

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×C22⋊C4×D2932
#340360312
#340360292
#340360222
#340360192
#340360182
#340360072
#340360052
#340360032
#340359842
C4×#179897792
#388×C2932
#2321293