Finite Group Databasebeta

← #46839831

Subgroups — #46839831

Order 34688.

800 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576142C211yesmaximal, characteristic, complemented
2C2×#22576130C211yesmaximal, characteristic
3C2×C4×D1084C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D271C211yesmaximal, characteristic, complemented
5#31773834C211yesmaximal, characteristic, complemented
6#31773822C211yesmaximal, characteristic, complemented
7#31773814C211yesmaximal, characteristic, complemented
8#31773813C211yesmaximal, characteristic, complemented
9#31773788C211yesmaximal, characteristic, complemented
10#31773775C211yesmaximal, characteristic, complemented
11#31773775C211yesmaximal, characteristic, complemented
12#31773771C211yesmaximal, characteristic, complemented
13#31773754C211yesmaximal, characteristic, complemented
14#31773752C211yesmaximal, characteristic, complemented
15#388×C271C211yesmaximal, characteristic, complemented
16C24×D271C2211yescharacteristic
17C22×D1084C2211yescharacteristic
18C22×D1084C2211yescharacteristic
19C22×C22⋊D271C2211yescharacteristic
20C22×C4×D271C2211yescharacteristic
21C22×C4×D271C2211yescharacteristic
22C22×C4×D271C2211yescharacteristic, complemented
23#22576142—24no—
24#22576130—24no—
25C4×D1084—24no—
26C2×C4⋊Dic271C2211yescharacteristic, complemented
27C2×C4⋊Dic271C2211yescharacteristic, complemented
28C2×C4⋊Dic271C2211yescharacteristic, complemented
29C2×Dic271⋊C4C2211yescharacteristic, complemented
30C2×C22⋊Dic271C2211yescharacteristic
31C2×C22⋊Dic271C2211yescharacteristic
32C2×(C2×D271)⋊C4C2211yescharacteristic
33C2×(C2×D271)⋊C4C2211yescharacteristic
34C2×(C2×D271)⋊C4C2211yescharacteristic
35C2×(C2×D271)⋊C4C2211yescharacteristic
36C2×(C2×D271)⋊C4C2211yescharacteristic
37C2×(C2×D271)⋊C4C2211yescharacteristic
38C2×(C2×D271)⋊C4C2211yescharacteristic
39C2×(C2×D271)⋊C4C2211yescharacteristic
40C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
41C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
42C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
43C22⋊C4×D271—24no—
44C271⋊#95C2211yescharacteristic, complemented
45C271⋊#95C2211yescharacteristic, complemented
46C271⋊#95C2211yescharacteristic, complemented
47C271⋊#95C2211yescharacteristic, complemented
48C2×C4⋊C4×C271C2211yescharacteristic, complemented
49C2×C22⋊C4×C271C2211yescharacteristic
50C2×C22⋊C4×C271C2211yescharacteristic
51C2×C22⋊C4×C271C2211yescharacteristic
52C2×C42×C271C2211yescharacteristic, complemented
53#95×C271C2211yescharacteristic, complemented
54#95×C271C2211yescharacteristic, complemented
55C23×D271—22no—
56C23×D271—22no—
57C23×D271—22no—
58C23×D271—22no—
59C23×D271—22no—
60C23×D271—41no—
61C23×D271—41no—
62C23×D271C2311yescharacteristic
63C23×D271C2311yescharacteristic
64C23×D271C2311yescharacteristic
65C2×D1084—22no—
66C2×D1084—22no—
67C2×D1084—22no—
68C2×D1084—22no—
69C2×D1084—41no—
70C2×D1084D414yes—
71C2×C22⋊D271—22no—
72C2×C22⋊D271—22no—
73C2×C22⋊D271—41no—
74C22×Dic271C2311yescharacteristic
75C22×Dic271C2311yescharacteristic
76C22×Dic271C2311yescharacteristic
77C22×Dic271C2311yescharacteristic
78C22×Dic271C2311yescharacteristic
79C2×C4×D271—22no—
80C2×C4×D271—22no—
81C2×C4×D271—22no—
82C2×C4×D271—22no—
83C2×C4×D271—22no—
84C2×C4×D271—22no—
85C2×C4×D271—41no—
86C2×C4×D271—41no—
87C2×C4×D271—41no—
88C4⋊Dic271—22no—
89C4⋊Dic271—22no—
90C4⋊Dic271—41no—
91Dic271⋊C4—22no—
92C22⋊Dic271—22no—
93C22⋊Dic271—22no—
94(C2×D271)⋊C4—22no—
95(C2×D271)⋊C4—22no—
96(C2×D271)⋊C4—22no—
97(C2×D271)⋊C4—22no—
98(C2×D271)⋊C4—22no—
99(C2×D271)⋊C4—22no—
100(C2×D271)⋊C4—22no—
101(C2×D271)⋊C4—22no—
102(C2×D271)⋊C4—22no—
103(C2×D271)⋊C4—41no—
104(C2×D271)⋊C4—41no—
105C24×C271C2311yescharacteristic
106C22×C4×C271C2311yescharacteristic
107C22×C4×C271C2311yescharacteristic
108C22×C4×C271C2311yescharacteristic
109C22×C4×C271C2311yescharacteristic
110C22×C4×C271C2311yescharacteristic
111C22×C4×C271C2311yescharacteristic
112C4⋊C4×C271—41no—
113C22⋊C4×C271—22no—
114C22⋊C4×C271—41no—
115C22⋊C4×C271D414yescomplemented
116C42×C271—22no—
117C22×D271—21no—
118C22×D271—21no—
119C22×D271—21no—
120C22×D271—21no—
121C22×D271—28no—
122C22×D271—21no—
123C22×D271—21no—
124C22×D271—21no—
125C22×D271—21no—
126C22×D271—21no—
127C22×D271—21no—
128C22×D271—21no—
129C22×D271—21no—
130C22×D271—21no—
131C22×D271—21no—
132C22×D271—21no—
133C22×D271—21no—
134C22×D271—44no—
135C22×D271—44no—
136C22×D271—44no—
137C22×D271—44no—
138C22×D271—44no—
139C22×D271—44no—
140C22×D271C2×D412yes—
141C22×D271C2×D412yes—
142C22×D271#4112yes—
143C22×D271#4112yes—
144C22×D271#4112yes—
145D1084—28no—
146D1084—44no—
147C22⋊D271—44no—
148C2×Dic271—21no—
149C2×Dic271—21no—
150C2×Dic271—21no—
151C2×Dic271—21no—
152C2×Dic271—21no—
153C2×Dic271—21no—
154C2×Dic271—21no—
155C2×Dic271—21no—
156C2×Dic271—21no—
157C2×Dic271—21no—
158C2×Dic271—21no—
159C2×Dic271—21no—
160C2×Dic271—21no—
161C2×Dic271—21no—
162C2×Dic271—21no—
163C4×D271—44no—
164C4×D271—44no—
165C4×D271—44no—
166C23×C271—21no—
167C23×C271—21no—
168C23×C271—21no—
169C23×C271—21no—
170C23×C271—21no—
171C23×C271—21no—
172C23×C271C2×D412yes—
173C23×C271C2411yescharacteristic
174C2×C4×C271—21no—
175C2×C4×C271—21no—
176C2×C4×C271—21no—
177C2×C4×C271—21no—
178C2×C4×C271—21no—
179C2×C4×C271—21no—
180C2×C4×C271—21no—
181C2×C4×C271—21no—
182C2×C4×C271—21no—
183C2×C4×C271—21no—
184C2×C4×C271—21no—
185C2×C4×C271—21no—
186C2×C4×C271—21no—
187C2×C4×C271—21no—
188C2×C4×C271—21no—
189C2×C4×C271C2×D412yes—
190C2×C4×C271C2×D412yes—
191C2×C4×C271C2×D412yes—
192C22×C271—22no—
193C22×C271—22no—
194C22×C271—22no—
195C22×C271—41no—
196C22×C271—41no—
197C22×C271—41no—
198C22×C271—41no—
199C22×C271C22×D411yescharacteristic
200C22×C271C22×D411yescharacteristic
201C22×C271C2×#4111yescharacteristic
202C22×C271C2×#4111yescharacteristic
203C22×C271C2×#4111yescharacteristic
204C22×C271H(5,𝔽2)11yescharacteristic
205C22×C271H(5,𝔽2)11yescharacteristic
206C2×D271—22no—
207C2×D271—22no—
208C2×D271—22no—
209C2×D271—22no—
210C2×D271—22no—
211C2×D271—22no—
212C2×D271—22no—
213C2×D271—22no—
214C2×D271—22no—
215C2×D271—22no—
216C2×D271—22no—
217C2×D271—22no—
218C2×D271—416no—
219C2×D271—41no—
220C2×D271—41no—
221C2×D271—41no—
222C2×D271—41no—
223C2×D271—41no—
224C2×D271—41no—
225C2×D271—41no—
226C2×D271—41no—
227C2×D271#12314yescomplemented
228C4×C271—22no—
229C4×C271—22no—
230C4×C271—22no—
231C4×C271—41no—
232C4×C271—41no—
233C4×C271—41no—
234Dic271—41no—
235Dic271—41no—
236Dic271—41no—
237Dic271—41no—
238Dic271—41no—
239C2×C271—42no—
240C2×C271C2×#12311yescharacteristic
241C2×C271#52511yescharacteristic
242C2×C271#53811yescharacteristic
243C2×C271D4211yescharacteristic
244C2×C271#54611yescharacteristic
245C2×C271#54611yescharacteristic
246C2×C271#55111yescharacteristic
247D271—24no—
248D271—42no—
249D271—42no—
250C271#241811yescharacteristic, complemented, Sylow
251#2418—2711nomaximal, Sylow
252C2×#123—2711no—
253C2×#120—2711no—
254C2×C4×D4—2711no—
255C22×C22⋊C4—2711no—
256#399—2711no—
257#397—2711no—
258#394—2711no—
259#394—2711no—
260#388—2711no—
261#388D27111yescharacteristic, complemented
262#386—2711no—
263#386—2711no—
264#385—2711no—
265#380—2711no—
266#379—2711no—
267C25—2711no—
268C22×D4—2711no—
269C22×D4—2711no—
270C22×D4—2711no—
271C23×C4—2711no—
272C23×C4—2711no—
273C23×C4—2711no—
274#123—5424no—
275#120—5424no—
276C4×D4—5424no—
277C2×C4⋊C4—2711no—
278C2×C4⋊C4—2711no—
279C2×C4⋊C4—2711no—
280C2×C4⋊C4—2711no—
281C2×C4⋊C4C2×D27111yescharacteristic, complemented
282C2×C22⋊C4—2711no—
283C2×C22⋊C4—2711no—
284C2×C22⋊C4—2711no—
285C2×C22⋊C4—2711no—
286C2×C22⋊C4—2711no—
287C2×C22⋊C4—2711no—
288C2×C22⋊C4—2711no—
289C2×C22⋊C4—2711no—
290C2×C22⋊C4—2711no—
291C2×C22⋊C4—2711no—
292C2×C22⋊C4—2711no—
293C2×C22⋊C4—2711no—
294C2×C22⋊C4—2711no—
295C2×C22⋊C4—5424no—
296C2×C22⋊C4C2×D27111yescharacteristic
297C2×C22⋊C4C2×D27111yescharacteristic
298C2×C22⋊C4C2×D27111yescharacteristic
299C2×C42C2×D27111yescharacteristic, complemented
300#95—2711no—
301#95—2711no—
302#95—2711no—
303#95—2711no—
304#95C2×D27111yescharacteristic, complemented
305#95C2×D27111yescharacteristic, complemented
306C24—2711no—
307C24—2711no—
308C24—2711no—
309C24—5422no—
310C24—5422no—
311C24—5422no—
312C24—5422no—
313C24—5422no—
314C24—10841no—
315C24—10841no—
316C24C22×D27111yescharacteristic
317C2×D4—2714no—
318C2×D4—5422no—
319C2×D4—5422no—
320C2×D4—5422no—
321C2×D4—5422no—
322C2×D4—5422no—
323C2×D4—5422no—
324C2×D4—10841no—
325C2×D4—10841no—
326C22×C4—2711no—
327C22×C4—2711no—
328C22×C4—2711no—
329C22×C4—2711no—
330C22×C4—2711no—
331C22×C4—5422no—
332C22×C4—5422no—
333C22×C4—5422no—
334C22×C4—5422no—
335C22×C4—5422no—
336C22×C4—5422no—
337C22×C4—10841no—
338C22×C4—10841no—
339C22×C4—10841no—
340C22×C4C22×D27111yescharacteristic
341C22×C4C22×D27111yescharacteristic
342C22×C4C22×D27111yescharacteristic
343C22×C4C22×D27111yescharacteristic
344C22×C4C22×D27111yescharacteristic
345C22×C4C22×D27111yescharacteristic
346C4⋊C4—41no—
347C4⋊C4—5422no—
348C4⋊C4—5422no—
349C4⋊C4—5422no—
350C4⋊C4—10841no—
351C22⋊C4—22no—
352C22⋊C4—41no—
353C22⋊C4—5422no—
354C22⋊C4—5422no—
355C22⋊C4—5422no—
356C22⋊C4—5422no—
357C22⋊C4—5422no—
358C22⋊C4—5422no—
359C22⋊C4—5422no—
360C22⋊C4—5422no—
361C22⋊C4—5422no—
362C22⋊C4—5422no—
363C22⋊C4—5422no—
364C22⋊C4—10841no—
365C22⋊C4—10841no—
366C22⋊C4D108414yescomplemented
367C42—22no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—2712no—
375C23—2712no—
376C23—2712no—
377C23—2712no—
378C23—2712no—
379C23—5428no—
380C23—5421no—
381C23—5421no—
382C23—5421no—
383C23—5421no—
384C23—5421no—
385C23—5421no—
386C23—5421no—
387C23—5421no—
388C23—5421no—
389C23—5421no—
390C23—5421no—
391C23—5421no—
392C23—5421no—
393C23—5421no—
394C23—5421no—
395C23—5421no—
396C23—10844no—
397C23—10844no—
398C23—10844no—
399C23—10844no—
400C23—10844no—
401C23—10844no—
402C23C2×D108412yes—
403C23C23×D27111yescharacteristic
404D4—5428no—
405D4—10844no—
406D4—10844no—
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—21no—
421C2×C4—21no—
422C2×C4—5421no—
423C2×C4—5421no—
424C2×C4—5421no—
425C2×C4—5421no—
426C2×C4—5421no—
427C2×C4—5421no—
428C2×C4—5421no—
429C2×C4—5421no—
430C2×C4—5421no—
431C2×C4—5421no—
432C2×C4—5421no—
433C2×C4—5421no—
434C2×C4—5421no—
435C2×C4—5421no—
436C2×C4—5421no—
437C2×C4—10844no—
438C2×C4—10844no—
439C2×C4—10844no—
440C2×C4D4×D27112yes—
441C2×C4C2×D108412yes—
442C2×C4C2×D108412yes—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—2714no—
451C22—5422no—
452C22—5422no—
453C22—5422no—
454C22—5422no—
455C22—5422no—
456C22—5422no—
457C22—5422no—
458C22—5422no—
459C22—5422no—
460C22—5422no—
461C22—5422no—
462C22—5422no—
463C22—108416no—
464C22—10841no—
465C22—10841no—
466C22—10841no—
467C22—10841no—
468C22—10841no—
469C22—10841no—
470C22—10841no—
471C22—10841no—
472C22C2×D4×D27111yescharacteristic
473C22C22×D108411yescharacteristic
474C22#41×D27111yescharacteristic
475C22#41×D27111yescharacteristic
476C22#41×D27111yescharacteristic
477C22#2257620511yescharacteristic
478C22#2257620611yescharacteristic
479C4—22no—
480C4—22no—
481C4—22no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—10841no—
486C4—10841no—
487C4—10841no—
488C4—10841no—
489C4—10841no—
490C2—42no—
491C2—5424no—
492C2—10842no—
493C2—10842no—
494C2#123×D27111yescharacteristic
495C2#3177440911yescharacteristic
496C2#3177448111yescharacteristic
497C2D4×D108411yescharacteristic
498C2#3177454211yescharacteristic
499C2#3177454411yescharacteristic
500C2#3177457211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#388×C271C2
prime radicalC2×#22576130C2
hypercenter#388D271
hypocenterC271#2418
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#22576142C2semidirect
3. C2×C4×D1084C2semidirect
4. C2×C22⋊C4×D271C2semidirect
5. #31773834C2semidirect
6. #31773822C2semidirect
7. #31773814C2semidirect
8. #31773813C2semidirect
9. #31773788C2semidirect
10. #31773775C2semidirect
11. #31773775C2semidirect
12. #31773771C2semidirect
13. #31773754C2semidirect
14. #31773752C2semidirect
15. #388×C271C2semidirect
22. C22×C4×D271C22semidirect
26. C2×C4⋊Dic271C22semidirect
27. C2×C4⋊Dic271C22semidirect
28. C2×C4⋊Dic271C22semidirect
29. C2×Dic271⋊C4C22semidirect
40. C2×(C2×D271)⋊C4C22semidirect
41. C2×(C2×D271)⋊C4C22semidirect
42. C2×(C2×D271)⋊C4C22semidirect
44. C271⋊#95C22semidirect
45. C271⋊#95C22semidirect
46. C271⋊#95C22semidirect
47. C271⋊#95C22semidirect
48. C2×C4⋊C4×C271C22semidirect
52. C2×C42×C271C22semidirect
53. #95×C271C22semidirect
54. #95×C271C22semidirect
115. C22⋊C4×C271D4semidirect
227. C2×D271#123semidirect
250. C271#2418semidirect
261. #388D271semidirect
281. C2×C4⋊C4C2×D271semidirect
299. C2×C42C2×D271semidirect
304. #95C2×D271semidirect
305. #95C2×D271semidirect
366. C22⋊C4D1084semidirect

Maximal subgroups

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

subgroupindex
C2×#225761422
C2×#225761302
C2×C4×D10842
C2×C22⋊C4×D2712
#317738342
#317738222
#317738142
#317738132
#317737882
#317737752
#317737712
#317737542
#317737522
#388×C2712
#2418271