Finite Group Databasebeta

← #21455093

Subgroups — #21455093

Order 7808.

752 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14519823C211yesmaximal, characteristic
2C2×#14519809C211yesmaximal, characteristic, complemented
3C2×SD16×D61C211yesmaximal, characteristic, complemented
4#16840335C211yesmaximal, characteristic, complemented
5#16839927C211yesmaximal, characteristic, complemented
6#16839924C211yesmaximal, characteristic, complemented
7#16839923C211yesmaximal, characteristic, complemented
8#16839915C211yesmaximal, characteristic, complemented
9#16839914C211yesmaximal, characteristic, complemented
10#16839912C211yesmaximal, characteristic, complemented
11#16839911C211yesmaximal, characteristic, complemented
12#16839765C211yesmaximal, characteristic, complemented
13#16839757C211yesmaximal, characteristic, complemented
14#16839727C211yesmaximal, characteristic, complemented
15C61×#452C211yesmaximal, characteristic, complemented
16#14519823—24no—
17#14519823—24no—
18C2×#11758226C2211yescharacteristic
19C2×#11758226C2211yescharacteristic
20C2×#11758222—21no—
21C2×#11758222C2211yescharacteristic
22C2×Q8×D61C2211yescharacteristic, complemented
23C22×D244—21no—
24C22×D244C2211yescharacteristic
25C2×D4×D61—21no—
26C2×D4×D61—21no—
27C2×D4×D61C2211yescharacteristic
28C2×D4×D61C2211yescharacteristic
29#14519809—24no—
30C2×#11758202C2211yescharacteristic, complemented
31C2×#11758190C2211yescharacteristic, complemented
32C2×#11758190C2211yescharacteristic, complemented
33C2×#11758200C2211yescharacteristic, complemented
34C2×#11758201C2211yescharacteristic, complemented
35C2×#11758201C2211yescharacteristic, complemented
36SD16×D61—24no—
37C2×#11758189C2211yescharacteristic, complemented
38C2×C8×D61C2211yescharacteristic, complemented
39#14519776—21no—
40#14519763—21no—
41#14519762—21no—
42#14519759—21no—
43#14519757C2211yescharacteristic, complemented
44#14519756C2211yescharacteristic, complemented
45#14519753C2211yescharacteristic, complemented
46#14519735C2211yescharacteristic, complemented
47#14519701C2211yescharacteristic, complemented
48#14519699C2211yescharacteristic, complemented
49#14519699C2211yescharacteristic, complemented
50#14519698C2211yescharacteristic, complemented
51#14519697C2211yescharacteristic, complemented
52#14519697C2211yescharacteristic, complemented
53C22⋊#92418C2211yescharacteristic, complemented
54#14519689C2211yescharacteristic, complemented
55#14519689C2211yescharacteristic, complemented
56C2×#41×C61C2211yescharacteristic
57C2×Q16×C61C2211yescharacteristic, complemented
58C2×SD16×C61C2211yescharacteristic, complemented
59C22⋊Q8×C61C2211yescharacteristic, complemented
60#103×C61C2211yescharacteristic, complemented
61#102×C61C2211yescharacteristic, complemented
62C22⋊C8×C61C2211yescharacteristic, complemented
63C23×D61—21no—
64C23×D61—21no—
65C23×D61—21no—
66#41×C61—22no—
67#41×C61—22no—
68C2×Q8×C61C2311yescharacteristic
69C2×Q8×C61C2311yescharacteristic
70C2×D4×C61—21no—
71C2×D4×C61C2311yescharacteristic
72C22×C4×C61—21no—
73C22×C4×C61C2311yescharacteristic
74C2×C22⋊D61—21no—
75C2×C22⋊D61—21no—
76C2×C22⋊D61D411yescharacteristic
77C2×C22⋊D61D411yescharacteristic
78#11758226—22no—
79#11758226—22no—
80#11758226—22no—
81#11758226D414yes—
82Q8×D61—22no—
83Q8×D61—22no—
84D4×D61—28no—
85D4×D61—28no—
86D4×D61—22no—
87D4×D61—22no—
88D4×D61—22no—
89D4×D61D414yes—
90#11758222—28no—
91#11758222—22no—
92#11758222—22no—
93C2×D244—28no—
94C2×D244—21no—
95C2×D244—21no—
96C2×D244—21no—
97C2×D244—22no—
98C2×D244—22no—
99C2×D244D411yescharacteristic
100C2×D244C2311yescharacteristic
101C2×D244C2311yescharacteristic
102C2×C4×D61—21no—
103C2×C4×D61—21no—
104C2×C4×D61—21no—
105C2×C4×D61C2311yescharacteristic
106C2×C4×D61C2311yescharacteristic
107C2×Dic122—21no—
108C2×Dic122D411yescharacteristic
109C2×Dic122C2311yescharacteristic
110Q16×C61—22no—
111SD16×C61—22no—
112C2×C8×C61C2311yescharacteristic
113C2×C8×C61C2311yescharacteristic
114C4⋊C4×C61—21no—
115C4⋊C4×C61C2311yescharacteristic
116C22⋊C4×C61—21no—
117C22⋊Dic61—21no—
118#11758202—22no—
119#11758201—22no—
120#11758201—22no—
121#11758200—22no—
122#11758198—21no—
123#11758198—21no—
124#11758198—21no—
125C4⋊Dic61—21no—
126C4⋊Dic61C2311yescharacteristic
127#11758196—21no—
128#11758196—21no—
129C4×Dic61—21no—
130C2×#92418C2311yescharacteristic
131C2×#92418C2311yescharacteristic
132#11758190—22no—
133#11758190—22no—
134#11758189—22no—
135C8×D61—22no—
136C23×C61—21no—
137C23×C61C2×D411yescharacteristic
138C22×D61—21no—
139C22×D61—21no—
140C22×D61—24no—
141C22×D61—24no—
142C22×D61—24no—
143C22×D61—81no—
144C22×D61—81no—
145C22×D61—81no—
146C22×D61—81no—
147C22×D61—81no—
148C22×D61—81no—
149C22×D61C2×D411yescharacteristic
150C22×D61C2×D411yescharacteristic
151Q8×C61—21no—
152Q8×C61—21no—
153Q8×C61—21no—
154Q8×C61C2×D412yes—
155D4×C61—24no—
156D4×C61—21no—
157D4×C61C2×D412yes—
158C2×C4×C61—24no—
159C2×C4×C61—21no—
160C2×C4×C61—21no—
161C2×C4×C61—21no—
162C2×C4×C61—21no—
163C2×C4×C61—21no—
164C2×C4×C61C2×D411yescharacteristic
165C2×C4×C61C2411yescharacteristic
166C22⋊D61—24no—
167C22⋊D61—22no—
168C22⋊D61—22no—
169C22⋊D61—24no—
170C2×Dic61—21no—
171C2×Dic61—21no—
172C2×Dic61C2×D411yescharacteristic
173C2×Dic61C2×D411yescharacteristic
174D244—24no—
175D244—24no—
176D244—22no—
177D244—24no—
178D244—21no—
179D244—21no—
180D244C2×D412yes—
181D244C2×D412yes—
182C4×D61—24no—
183C4×D61—21no—
184C4×D61—21no—
185C4×D61—24no—
186C4×D61—42no—
187C4×D61C2×D412yes—
188C4×D61C2×D412yes—
189Dic122—22no—
190Dic122—21no—
191Dic122—21no—
192Dic122—42no—
193C8×C61—21no—
194C8×C61—21no—
195#92418—21no—
196#92418—21no—
197C22×C61—21no—
198C22×C61—22no—
199C22×C61—21no—
200C22×C61—21no—
201C22×C61—41no—
202C22×C61—41no—
203C22×C61C22×D411yescharacteristic
204C2×D61—21no—
205C2×D61—21no—
206C2×D61—21no—
207C2×D61—22no—
208C2×D61—22no—
209C2×D61—21no—
210C2×D61—21no—
211C2×D61—41no—
212C2×D61—41no—
213C2×D61—41no—
214C2×D61—41no—
215C2×D61—84no—
216C2×D61—84no—
217C2×D61—84no—
218C2×D61#12012yes—
219C4×C61—21no—
220C4×C61—22no—
221C4×C61—41no—
222C4×C61—41no—
223C4×C61C22×D411yescharacteristic
224C4×C61C22×D411yescharacteristic
225Dic61—21no—
226Dic61—41no—
227Dic61—41no—
228Dic61#12012yes—
229#3009—611nomaximal, Sylow
230C2×C61—41no—
231C2×C61—42no—
232C2×C61C2×#12011yescharacteristic
233C2×C61#57711yescharacteristic
234C2×C61#57711yescharacteristic
235D61—22no—
236D61—41no—
237D61—42no—
238D61—42no—
239C2×H(5,𝔽2)—611no—
240C2×#137—611no—
241C22×SD16—611no—
242#536—611no—
243#452—611no—
244#452—611no—
245#452—611no—
246#452D6111yescharacteristic, complemented
247#450—611no—
248#450—611no—
249#450—611no—
250#450—611no—
251#418—611no—
252#416—611no—
253#408—611no—
254C61#300911yescharacteristic, complemented, Sylow
255H(5,𝔽2)—1224no—
256H(5,𝔽2)—1224no—
257C2×#41—611no—
258C2×#41—611no—
259C2×#41—611no—
260C2×#41—1221no—
261C2×#41C2×D6111yescharacteristic
262C22×Q8—611no—
263C22×D4—611no—
264C22×D4—611no—
265C22×D4—611no—
266C22×D4—1221no—
267C22×D4—1221no—
268C22×D4—1221no—
269#137—1224no—
270C2×Q16—611no—
271C2×Q16C2×D6111yescharacteristic, complemented
272C2×SD16—611no—
273C2×SD16—611no—
274C2×SD16—611no—
275C2×SD16—611no—
276C2×SD16—611no—
277C2×SD16—1224no—
278C2×SD16C2×D6111yescharacteristic, complemented
279C2×M16—611no—
280C22×C8—611no—
281C4⋊Q8—1221no—
282#124—1221no—
283#123—1221no—
284#123—1221no—
285C22⋊Q8—611no—
286C22⋊Q8—611no—
287C22⋊Q8—611no—
288C22⋊Q8C2×D6111yescharacteristic, complemented
289#117—611no—
290#103—611no—
291#103C2×D6111yescharacteristic, complemented
292#102—611no—
293#102—611no—
294#102—611no—
295#102—611no—
296#102—611no—
297#102C2×D6111yescharacteristic, complemented
298C22⋊C8—611no—
299C22⋊C8—611no—
300C22⋊C8—611no—
301C22⋊C8C2×D6111yescharacteristic, complemented
302C24—1221no—
303C24—1221no—
304C24—1221no—
305#41—22no—
306#41—22no—
307#41—614no—
308#41—1222no—
309#41—1222no—
310#41—1228no—
311#41—1222no—
312#41—1222no—
313#41—1222no—
314C2×Q8—611no—
315C2×Q8—611no—
316C2×Q8—1221no—
317C2×Q8—1222no—
318C2×Q8—1222no—
319C2×Q8C22×D6111yescharacteristic
320C2×Q8C22×D6111yescharacteristic
321C2×D4—21no—
322C2×D4—614no—
323C2×D4—611no—
324C2×D4—611no—
325C2×D4—611no—
326C2×D4—611no—
327C2×D4—611no—
328C2×D4—1222no—
329C2×D4—1222no—
330C2×D4—1221no—
331C2×D4—1221no—
332C2×D4—1228no—
333C2×D4—1222no—
334C2×D4—1221no—
335C2×D4—1221no—
336C2×D4—1222no—
337C2×D4—1228no—
338C2×D4—1228no—
339C2×D4—1222no—
340C2×D4—1221no—
341C2×D4C22×D6111yescharacteristic
342C22×C4—21no—
343C22×C4—611no—
344C22×C4—611no—
345C22×C4—1221no—
346C22×C4—1221no—
347C22×C4—1221no—
348C22×C4C22×D6111yescharacteristic
349Q16—22no—
350Q16—1222no—
351SD16—22no—
352SD16—1222no—
353SD16—1222no—
354SD16—1222no—
355SD16—1222no—
356SD16—1222no—
357M16—1222no—
358C2×C8—611no—
359C2×C8—611no—
360C2×C8—1222no—
361C2×C8C22×D6111yescharacteristic
362C2×C8C22×D6111yescharacteristic
363C4⋊C4—21no—
364C4⋊C4—611no—
365C4⋊C4—1221no—
366C4⋊C4—1221no—
367C4⋊C4—1221no—
368C4⋊C4C22×D6111yescharacteristic
369C22⋊C4—21no—
370C22⋊C4—1221no—
371C22⋊C4—1221no—
372C22⋊C4—1221no—
373C22⋊C4—1221no—
374C42—1221no—
375C23—21no—
376C23—611no—
377C23—611no—
378C23—1224no—
379C23—1224no—
380C23—1224no—
381C23—1221no—
382C23—1221no—
383C23—4881no—
384C23—4881no—
385C23—4881no—
386C23—4881no—
387C23—4881no—
388C23—4881no—
389C23D4×D6111yescharacteristic
390Q8—21no—
391Q8—21no—
392Q8—21no—
393Q8—1222no—
394Q8—1221no—
395Q8—1221no—
396Q8—2442no—
397Q8D4×D6112yes—
398D4—24no—
399D4—21no—
400D4—612no—
401D4—612no—
402D4—1222no—
403D4—1224no—
404D4—1224no—
405D4—1224no—
406D4—1222no—
407D4—1224no—
408D4—1222no—
409D4—1224no—
410D4—1221no—
411D4—1221no—
412D4D4×D6112yes—
413C2×C4—24no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—611no—
420C2×C4—612no—
421C2×C4—612no—
422C2×C4—611no—
423C2×C4—1224no—
424C2×C4—1224no—
425C2×C4—1221no—
426C2×C4—1221no—
427C2×C4—1221no—
428C2×C4—1221no—
429C2×C4—2442no—
430C2×C4D4×D6111yescharacteristic
431C2×C4C23×D6111yescharacteristic
432C8—21no—
433C8—21no—
434C8—1221no—
435C8—1221no—
436C22—21no—
437C22—22no—
438C22—21no—
439C22—21no—
440C22—41no—
441C22—41no—
442C22—612no—
443C22—1221no—
444C22—1221no—
445C22—1222no—
446C22—1222no—
447C22—1221no—
448C22—1221no—
449C22—1221no—
450C22—4884no—
451C22—4884no—
452C22—4884no—
453C22—2441no—
454C22—2441no—
455C22—2441no—
456C22—2441no—
457C22C2×D4×D6111yescharacteristic
458C4—21no—
459C4—22no—
460C4—41no—
461C4—41no—
462C4—612no—
463C4—1221no—
464C4—2441no—
465C4—2441no—
466C4C2×D4×D6111yescharacteristic
467C4C2×D4×D6111yescharacteristic
468C2—41no—
469C2—42no—
470C2—1222no—
471C2—2441no—
472C2—2442no—
473C2—2442no—
474C2#120×D6111yescharacteristic
475C2#1684056911yescharacteristic
476C2#1684057111yescharacteristic
subgroupG/H
centerC22C2×D4×D61
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC61×#452C2
socleC22×C61C22×D4
hypercenter#452D61
hypocenterC61#3009
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. C2×#14519809C2semidirect
3. C2×SD16×D61C2semidirect
4. #16840335C2semidirect
5. #16839927C2semidirect
6. #16839924C2semidirect
7. #16839923C2semidirect
8. #16839915C2semidirect
9. #16839914C2semidirect
10. #16839912C2semidirect
11. #16839911C2semidirect
12. #16839765C2semidirect
13. #16839757C2semidirect
14. #16839727C2semidirect
15. C61×#452C2semidirect
22. C2×Q8×D61C22semidirect
30. C2×#11758202C22semidirect
31. C2×#11758190C22semidirect
32. C2×#11758190C22semidirect
33. C2×#11758200C22semidirect
34. C2×#11758201C22semidirect
35. C2×#11758201C22semidirect
37. C2×#11758189C22semidirect
38. C2×C8×D61C22semidirect
43. #14519757C22semidirect
44. #14519756C22semidirect
45. #14519753C22semidirect
46. #14519735C22semidirect
47. #14519701C22semidirect
48. #14519699C22semidirect
49. #14519699C22semidirect
50. #14519698C22semidirect
51. #14519697C22semidirect
52. #14519697C22semidirect
53. C22⋊#92418C22semidirect
54. #14519689C22semidirect
55. #14519689C22semidirect
57. C2×Q16×C61C22semidirect
58. C2×SD16×C61C22semidirect
59. C22⋊Q8×C61C22semidirect
60. #103×C61C22semidirect
61. #102×C61C22semidirect
62. C22⋊C8×C61C22semidirect
246. #452D61semidirect
254. C61#3009semidirect
271. C2×Q16C2×D61semidirect
278. C2×SD16C2×D61semidirect
288. C22⋊Q8C2×D61semidirect
291. #103C2×D61semidirect
297. #102C2×D61semidirect
301. C22⋊C8C2×D61semidirect