Finite Group Databasebeta

← #44402713

Subgroups — #44402713

Order 29056.

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×#20300257C211yesmaximal, characteristic
2C2×C4×C22⋊D227C211yesmaximal, characteristic, complemented
3C2×(C22⋊D227)⋊C4C211yesmaximal, characteristic, complemented
4C22×C22⋊Dic227C211yesmaximal, characteristic, complemented
5#28408932C211yesmaximal, characteristic, complemented
6#28408926C211yesmaximal, characteristic, complemented
7#28408920C211yesmaximal, characteristic, complemented
8#28408911C211yesmaximal, characteristic, complemented
9#28408909C211yesmaximal, characteristic, complemented
10#28408908C211yesmaximal, characteristic, complemented
11#28408905C211yesmaximal, characteristic, complemented
12#28408904C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic227C211yesmaximal, characteristic, complemented
14C4×C22⋊Dic227C211yesmaximal, characteristic, complemented
15C22×C22⋊C4×C227C211yesmaximal, characteristic, complemented
16C22×C22⋊D227C2211yescharacteristic
17C22×C22⋊D227C2211yescharacteristic
18C22×C22⋊D227C2211yescharacteristic
19C23×Dic227C2211yescharacteristic
20C23×Dic227C2211yescharacteristic
21C22×C4×D227C2211yescharacteristic, complemented
22#20300257C418yescomplemented
23C4×C22⋊D227—24no—
24(C22⋊D227)⋊C4—24no—
25C2×Dic227⋊C4C2211yescharacteristic, complemented
26C2×Dic227⋊C4C2211yescharacteristic, complemented
27C2×Dic227⋊C4C2211yescharacteristic, complemented
28C2×C22⋊Dic227—24no—
29C2×C22⋊Dic227C2211yescharacteristic
30C2×C22⋊Dic227C2211yescharacteristic
31C2×C22⋊Dic227C2211yescharacteristic
32C2×C22⋊Dic227C2211yescharacteristic
33C2×C22⋊Dic227C2211yescharacteristic
34C2×C22⋊Dic227C2211yescharacteristic
35C2×C22⋊Dic227C2211yescharacteristic
36C2×(C2×D227)⋊C4C2211yescharacteristic, complemented
37C2×(C2×D227)⋊C4C2211yescharacteristic, complemented
38C2×(C2×D227)⋊C4C2211yescharacteristic, complemented
39C2×C4×Dic227C2211yescharacteristic, complemented
40C2×C4×Dic227C2211yescharacteristic, complemented
41C2×C4×Dic227C2211yescharacteristic, complemented
42C227⋊#95C2211yescharacteristic, complemented
43C227⋊#95C2211yescharacteristic, complemented
44C227⋊#95C2211yescharacteristic, complemented
45C227⋊#95C2211yescharacteristic, complemented
46C227⋊#95C2211yescharacteristic, complemented
47C227⋊#95C2211yescharacteristic, complemented
48C25×C227C2211yescharacteristic
49C23×C4×C227C2211yescharacteristic
50C23×C4×C227C2211yescharacteristic
51C2×C22⋊C4×C227—24no—
52C2×C22⋊C4×C227C2211yescharacteristic
53C2×C22⋊C4×C227C2211yescharacteristic
54C2×C22⋊C4×C227C2211yescharacteristic
55C2×C22⋊C4×C227C2211yescharacteristic
56C23×D227C2311yescharacteristic
57C2×C22⋊D227—22no—
58C2×C22⋊D227—22no—
59C2×C22⋊D227—22no—
60C2×C22⋊D227—22no—
61C2×C22⋊D227—41no—
62C2×C22⋊D227—41no—
63C2×C22⋊D227C2×C414yescomplemented
64C2×C22⋊D227C2×C414yescomplemented
65C2×C22⋊D227C2×C414yescomplemented
66C22×Dic227—22no—
67C22×Dic227—22no—
68C22×Dic227—22no—
69C22×Dic227—22no—
70C22×Dic227—41no—
71C22×Dic227—41no—
72C22×Dic227C2311yescharacteristic
73C22×Dic227C2311yescharacteristic
74C22×Dic227C2311yescharacteristic
75C22×Dic227C2311yescharacteristic
76C22×Dic227C2311yescharacteristic
77C22×Dic227C2311yescharacteristic
78C22×Dic227C2311yescharacteristic
79C2×C4×D227—22no—
80C2×C4×D227—22no—
81C2×C4×D227—41no—
82Dic227⋊C4—22no—
83Dic227⋊C4—22no—
84Dic227⋊C4—41no—
85C22⋊Dic227—22no—
86C22⋊Dic227—22no—
87C22⋊Dic227—22no—
88C22⋊Dic227C2×C414yescomplemented
89C22⋊Dic227C2×C414yescomplemented
90C22⋊Dic227C2×C414yescomplemented
91C22⋊Dic227D414yescomplemented
92(C2×D227)⋊C4—22no—
93(C2×D227)⋊C4—22no—
94(C2×D227)⋊C4—41no—
95C4×Dic227—22no—
96C4×Dic227—22no—
97C4×Dic227—41no—
98C24×C227—22no—
99C24×C227—22no—
100C24×C227—22no—
101C24×C227—22no—
102C24×C227—41no—
103C24×C227—41no—
104C24×C227C2×C414yescomplemented
105C24×C227C2311yescharacteristic
106C24×C227C2311yescharacteristic
107C24×C227C2311yescharacteristic
108C22×C4×C227—22no—
109C22×C4×C227—22no—
110C22×C4×C227—22no—
111C22×C4×C227—22no—
112C22×C4×C227—41no—
113C22×C4×C227—41no—
114C22×C4×C227C2311yescharacteristic
115C22×C4×C227C2311yescharacteristic
116C22×C4×C227C2311yescharacteristic
117C22×C4×C227C2311yescharacteristic
118C22⋊C4×C227—22no—
119C22⋊C4×C227—22no—
120C22⋊C4×C227—22no—
121C22⋊C4×C227D414yescomplemented
122C22×D227—21no—
123C22×D227—21no—
124C22×D227—21no—
125C22×D227—21no—
126C22×D227—21no—
127C22×D227—21no—
128C22×D227C22×C412yes—
129C22⋊D227—28no—
130C22⋊D227—28no—
131C22⋊D227—44no—
132C2×Dic227—21no—
133C2×Dic227—21no—
134C2×Dic227—21no—
135C2×Dic227—21no—
136C2×Dic227—21no—
137C2×Dic227—21no—
138C2×Dic227—21no—
139C2×Dic227—21no—
140C2×Dic227—21no—
141C2×Dic227—21no—
142C2×Dic227—21no—
143C2×Dic227—21no—
144C2×Dic227—21no—
145C2×Dic227—21no—
146C2×Dic227—21no—
147C2×Dic227—21no—
148C2×Dic227—44no—
149C2×Dic227—44no—
150C2×Dic227C22×C412yes—
151C2×Dic227C22×C412yes—
152C2×Dic227C22×C412yes—
153C2×Dic227C2×D412yes—
154C2×Dic227C2×D412yes—
155C4×D227—44no—
156C23×C227—21no—
157C23×C227—21no—
158C23×C227—21no—
159C23×C227—21no—
160C23×C227—21no—
161C23×C227—21no—
162C23×C227—21no—
163C23×C227—21no—
164C23×C227—28no—
165C23×C227—28no—
166C23×C227—21no—
167C23×C227—21no—
168C23×C227—21no—
169C23×C227—21no—
170C23×C227—21no—
171C23×C227—21no—
172C23×C227—44no—
173C23×C227—44no—
174C23×C227—44no—
175C23×C227—44no—
176C23×C227—44no—
177C23×C227C22×C412yes—
178C23×C227C22×C412yes—
179C23×C227C22×C412yes—
180C23×C227C2×D412yes—
181C23×C227C2×D412yes—
182C23×C227#4112yes—
183C23×C227#4112yes—
184C23×C227C2411yescharacteristic
185C2×C4×C227—21no—
186C2×C4×C227—21no—
187C2×C4×C227—21no—
188C2×C4×C227—21no—
189C2×C4×C227—21no—
190C2×C4×C227—21no—
191C2×C4×C227—21no—
192C2×C4×C227—21no—
193C2×C4×C227—21no—
194C2×C4×C227—21no—
195C2×C4×C227—44no—
196C2×C4×C227—44no—
197C2×C4×C227C2×D412yes—
198C2×C4×C227C2×D412yes—
199C22×C227—22no—
200C22×C227—22no—
201C22×C227—22no—
202C22×C227—22no—
203C22×C227—22no—
204C22×C227—22no—
205C22×C227—22no—
206C22×C227—22no—
207C22×C227—22no—
208C22×C227—22no—
209C22×C227—22no—
210C22×C227—22no—
211C22×C227—22no—
212C22×C227—22no—
213C22×C227—22no—
214C22×C227—416no—
215C22×C227—41no—
216C22×C227—41no—
217C22×C227—41no—
218C22×C227—41no—
219C22×C227C4×D414yescomplemented
220C22×C227C4×D414yescomplemented
221C22×C227C23×C411yescharacteristic
222C22×C227C22×D411yescharacteristic
223C22×C227C22×D411yescharacteristic
224C22×C227C2×#4111yescharacteristic
225C22×C227C2×#4111yescharacteristic
226C22×C227H(5,𝔽2)11yescharacteristic
227C22×C227H(5,𝔽2)11yescharacteristic
228C2×D227—22no—
229C2×D227—22no—
230C2×D227—22no—
231C2×D227—41no—
232C2×D227—41no—
233C2×D227—41no—
234C2×D227—41no—
235C4×C227—22no—
236C4×C227—22no—
237C4×C227—41no—
238C4×C227—41no—
239Dic227—22no—
240Dic227—22no—
241Dic227—22no—
242Dic227—22no—
243Dic227—22no—
244Dic227—41no—
245Dic227—41no—
246C2×C227—24no—
247C2×C227—24no—
248C2×C227—42no—
249C2×C227C2×C4×D411yescharacteristic
250C2×C227C2×C4×D411yescharacteristic
251C2×C227#51811yescharacteristic
252C2×C227D4211yescharacteristic
253C2×C227#54611yescharacteristic
254C2×C227#54611yescharacteristic
255C2×C227#55111yescharacteristic
256D227—42no—
257C227#235811yescharacteristic, complemented, Sylow
258#2358—2271nomaximal, Sylow
259C2×#120—2271no—
260C2×C4×D4—2271no—
261C2×C4×D4—2271no—
262C22×C22⋊C4—2271no—
263C22×C22⋊C4D22711yescharacteristic, complemented
264#390—2271no—
265#388—2271no—
266#388—2271no—
267#386—2271no—
268#386—2271no—
269#386—2271no—
270#385—2271no—
271#385—2271no—
272C4×C22⋊C4—2271no—
273C4×C22⋊C4—2271no—
274C25C2×D22711yescharacteristic
275C22×D4—2271no—
276C22×D4—2271no—
277C22×D4—2271no—
278C23×C4—2271no—
279C23×C4—2271no—
280C23×C4—2271no—
281C23×C4C2×D22711yescharacteristic
282C23×C4C2×D22711yescharacteristic
283#120—2278no—
284C4×D4—4544no—
285C4×D4—4544no—
286C2×C4⋊C4—2271no—
287C2×C4⋊C4—2271no—
288C2×C4⋊C4—2271no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—4544no—
291C2×C22⋊C4—2271no—
292C2×C22⋊C4—2271no—
293C2×C22⋊C4—2271no—
294C2×C22⋊C4—2271no—
295C2×C22⋊C4—2271no—
296C2×C22⋊C4—2271no—
297C2×C22⋊C4—2271no—
298C2×C22⋊C4—2271no—
299C2×C22⋊C4—2271no—
300C2×C22⋊C4—2271no—
301C2×C22⋊C4C2×D22711yescharacteristic
302C2×C22⋊C4C2×D22711yescharacteristic
303C2×C22⋊C4C2×D22711yescharacteristic
304C2×C22⋊C4C2×D22711yescharacteristic
305C2×C42—2271no—
306C2×C42—2271no—
307C2×C42—2271no—
308#95—2271no—
309#95—2271no—
310#95—2271no—
311#95—2271no—
312#95—2271no—
313#95—2271no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—41no—
319C24—41no—
320C24—2271no—
321C24C4×D22714yescomplemented
322C24C22×D22711yescharacteristic
323C24C22×D22711yescharacteristic
324C24C22×D22711yescharacteristic
325C2×D4—9081no—
326C2×D4—9081no—
327C2×D4—4542no—
328C2×D4—4542no—
329C2×D4—4542no—
330C2×D4—4542no—
331C2×D4—2274no—
332C2×D4—2274no—
333C2×D4—2274no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—9081no—
341C22×C4—9081no—
342C22×C4—9081no—
343C22×C4—4542no—
344C22×C4—4542no—
345C22×C4—4542no—
346C22×C4—4542no—
347C22×C4—4542no—
348C22×C4—4542no—
349C22×C4—2271no—
350C22×C4—2271no—
351C22×C4—2271no—
352C22×C4—2271no—
353C22×C4—2271no—
354C22×C4—2271no—
355C22×C4—2271no—
356C22×C4C22×D22711yescharacteristic
357C22×C4C22×D22711yescharacteristic
358C22×C4C22×D22711yescharacteristic
359C22×C4C22×D22711yescharacteristic
360C4⋊C4—9081no—
361C4⋊C4—4542no—
362C4⋊C4—4542no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4—9081no—
367C22⋊C4—4542no—
368C22⋊C4—4542no—
369C22⋊C4—4542no—
370C22⋊C4—4542no—
371C22⋊C4—4542no—
372C22⋊C4—2274no—
373C22⋊C4—2274no—
374C22⋊C4—2274no—
375C22⋊C4—2274no—
376C22⋊C4C22⋊D22714yescomplemented
377C42—9081no—
378C42—4542no—
379C42—4542no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—28no—
395C23—28no—
396C23—44no—
397C23—44no—
398C23—44no—
399C23—44no—
400C23—44no—
401C23—4541no—
402C23—4541no—
403C23—4541no—
404C23—4541no—
405C23—4541no—
406C23—4541no—
407C23—2272no—
408C23C2×C4×D22712yes—
409C23C2×C4×D22712yes—
410C23C2×C4×D22712yes—
411C23D4×D22712yes—
412C23C2×C22⋊D22712yes—
413C23#1681388312yes—
414C23#1681388412yes—
415C23C23×D22711yescharacteristic
416D4—9084no—
417D4—4548no—
418D4—4548no—
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—44no—
430C2×C4—44no—
431C2×C4—9084no—
432C2×C4—9084no—
433C2×C4—9084no—
434C2×C4—4541no—
435C2×C4—4541no—
436C2×C4—4541no—
437C2×C4—4541no—
438C2×C4—4541no—
439C2×C4—4541no—
440C2×C4—4541no—
441C2×C4—4541no—
442C2×C4—4541no—
443C2×C4—4541no—
444C2×C4—4541no—
445C2×C4—4541no—
446C2×C4—4541no—
447C2×C4—4541no—
448C2×C4—4541no—
449C2×C4—4541no—
450C2×C4—2272no—
451C2×C4—2272no—
452C2×C4—2272no—
453C2×C4—2272no—
454C2×C4—2272no—
455C2×C4C2×C22⋊D22712yes—
456C2×C4C2×C22⋊D22712yes—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—416no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—9081no—
478C22—9081no—
479C22—9081no—
480C22—9081no—
481C22—4542no—
482C22—4542no—
483C22—4542no—
484C22(C22⋊D227)⋊C414yescomplemented
485C22C4×C22⋊D22714yescomplemented
486C22C22×C4×D22711yescharacteristic
487C22C2×D4×D22711yescharacteristic
488C22C22×C22⋊D22711yescharacteristic
489C22C2×#1681388311yescharacteristic
490C22C2×#1681388411yescharacteristic
491C22#2030033311yescharacteristic
492C22#2030033311yescharacteristic
493C4—22no—
494C4—22no—
495C4—41no—
496C4—41no—
497C4—9081no—
498C4—9081no—
499C4—4542no—
500C4—4542no—
501C4—4542no—
502C4—4542no—
503C4—4542no—
504C2—24no—
505C2—24no—
506C2—42no—
507C2—9082no—
508C2C2×(C22⋊D227)⋊C411yescharacteristic
509C2C2×C4×C22⋊D22711yescharacteristic
510C2#2840950511yescharacteristic
511C2D4×C22⋊D22711yescharacteristic
512C2#2840967211yescharacteristic
513C2#2840967611yescharacteristic
514C2#2840970811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D227
commutatorC22×C227C23×C4
FittingC22×C22⋊C4×C227C2
socle, cosocleC23×C227C24
prime radicalC2×#20300257C2
hypercenterC22×C22⋊C4D227
hypocenterC227#2358
LCSG ▷ C22×C227 ▷ C227
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D227C2semidirect
3. C2×(C22⋊D227)⋊C4C2semidirect
4. C22×C22⋊Dic227C2semidirect
5. #28408932C2semidirect
6. #28408926C2semidirect
7. #28408920C2semidirect
8. #28408911C2semidirect
9. #28408909C2semidirect
10. #28408908C2semidirect
11. #28408905C2semidirect
12. #28408904C2semidirect
13. C22⋊C4×Dic227C2semidirect
14. C4×C22⋊Dic227C2semidirect
15. C22×C22⋊C4×C227C2semidirect
21. C22×C4×D227C22semidirect
22. #20300257C4semidirect
25. C2×Dic227⋊C4C22semidirect
26. C2×Dic227⋊C4C22semidirect
27. C2×Dic227⋊C4C22semidirect
36. C2×(C2×D227)⋊C4C22semidirect
37. C2×(C2×D227)⋊C4C22semidirect
38. C2×(C2×D227)⋊C4C22semidirect
39. C2×C4×Dic227C22semidirect
40. C2×C4×Dic227C22semidirect
41. C2×C4×Dic227C22semidirect
42. C227⋊#95C22semidirect
43. C227⋊#95C22semidirect
44. C227⋊#95C22semidirect
45. C227⋊#95C22semidirect
46. C227⋊#95C22semidirect
47. C227⋊#95C22semidirect
63. C2×C22⋊D227C2×C4semidirect
64. C2×C22⋊D227C2×C4semidirect
65. C2×C22⋊D227C2×C4semidirect
88. C22⋊Dic227C2×C4semidirect
89. C22⋊Dic227C2×C4semidirect
90. C22⋊Dic227C2×C4semidirect
91. C22⋊Dic227D4semidirect
104. C24×C227C2×C4semidirect
121. C22⋊C4×C227D4semidirect
219. C22×C227C4×D4semidirect
220. C22×C227C4×D4semidirect
257. C227#2358semidirect
263. C22×C22⋊C4D227semidirect
321. C24C4×D227semidirect
376. C22⋊C4C22⋊D227semidirect
484. C22(C22⋊D227)⋊C4semidirect
485. C22C4×C22⋊D227semidirect

Maximal subgroups

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

subgroupindex
C2×#203002572
C2×C4×C22⋊D2272
C2×(C22⋊D227)⋊C42
C22×C22⋊Dic2272
#284089322
#284089262
#284089202
#284089112
#284089092
#284089082
#284089052
#284089042
C22⋊C4×Dic2272
C4×C22⋊Dic2272
C22×C22⋊C4×C2272
#2358227