Finite Group Databasebeta

← #20286226

Subgroups — #20286226

Order 6784.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133248C211yesmaximal, characteristic, complemented
2C22×#11719924C211yesmaximal, characteristic, complemented
3#16811269C211yesmaximal, characteristic
4#16811015C211yesmaximal, characteristic, complemented
5#16810871C211yesmaximal, characteristic, complemented
6#16810862C211yesmaximal, characteristic, complemented
7#16810845C211yesmaximal, characteristic, complemented
8#16810825C211yesmaximal, characteristic, complemented
9#16810783C211yesmaximal, characteristic, complemented
10#16810779C211yesmaximal, characteristic, complemented
11#16810773C211yesmaximal, characteristic, complemented
12#16810762C211yesmaximal, characteristic, complemented
13#16810721C211yesmaximal, characteristic, complemented
14#16810695C211yesmaximal, characteristic, complemented
15D4×Q8×C53C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C22×D212C2211yescharacteristic
18#13133248—24no—
19C2×#11719925C2211yescharacteristic, complemented
20C2×#11719925C2211yescharacteristic, complemented
21C2×#11719925C2211yescharacteristic, complemented
22C2×#11719924—24no—
23C2×#11719924C2211yescharacteristic
24C2×#11719924C2211yescharacteristic
25C2×#11719924C2211yescharacteristic
26C2×#11719924C2211yescharacteristic
27C2×#11719924C2211yescharacteristic
28C2×#11719924C2211yescharacteristic
29C2×#11719917C2211yescharacteristic, complemented
30C22×#88322C2211yescharacteristic
31#13133205C2211yescharacteristic
32#13133202—21no—
33#13133196C2211yescharacteristic
34#13133191C2211yescharacteristic
35#13133190—21no—
36#13133185—21no—
37C4×D212C2211yescharacteristic
38C22⋊C4×D53—21no—
39#13133150C2211yescharacteristic, complemented
40C4⋊#88322C2211yescharacteristic, complemented
41#13133141C2211yescharacteristic, complemented
42#13133141C2211yescharacteristic, complemented
43#13133141C2211yescharacteristic, complemented
44#13133140C2211yescharacteristic, complemented
45#13133137C2211yescharacteristic
46#13133137C2211yescharacteristic
47C22⋊#88322C2211yescharacteristic
48C22⋊#88322C2211yescharacteristic, complemented
49#13133128C2211yescharacteristic, complemented
50C22×Q8×C53C2211yescharacteristic
51C22×Q8×C53C2211yescharacteristic, complemented
52C4⋊Q8×C53—21no—
53C4⋊Q8×C53C2211yescharacteristic, complemented
54C22⋊Q8×C53—21no—
55C22⋊Q8×C53—21no—
56C22⋊Q8×C53C2211yescharacteristic
57C22⋊Q8×C53C2211yescharacteristic, complemented
58C4×Q8×C53C2211yescharacteristic, complemented
59C4×D4×C53—21no—
60C4×D4×C53C2211yescharacteristic
61C23×D53—21no—
62C2×Q8×C53—21no—
63C2×Q8×C53—21no—
64C2×Q8×C53—22no—
65C2×Q8×C53—22no—
66C2×Q8×C53—41no—
67C2×Q8×C53—41no—
68C2×Q8×C53C2311yescharacteristic
69C2×Q8×C53C2311yescharacteristic
70C2×Q8×C53C2311yescharacteristic
71C2×D4×C53D411yescharacteristic
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53C2311yescharacteristic
75C22×C4×C53C2311yescharacteristic
76C2×C22⋊D53—21no—
77C2×C22⋊D53—21no—
78#11719945—22no—
79#11719945—41no—
80C2×D212—22no—
81C2×D212—21no—
82C2×D212—41no—
83C2×D212C2311yescharacteristic
84C2×D212C2311yescharacteristic
85C2×C4×D53—21no—
86C2×C4×D53—21no—
87C2×Dic106C2311yescharacteristic
88C4⋊C4×C53—21no—
89C4⋊C4×C53—21no—
90C4⋊C4×C53—21no—
91C4⋊C4×C53—21no—
92C4⋊C4×C53—21no—
93C4⋊C4×C53D411yescharacteristic
94C4⋊C4×C53C2311yescharacteristic
95C22⋊C4×C53—21no—
96C22⋊C4×C53—21no—
97C22⋊C4×C53D411yescharacteristic
98C22⋊C4×C53D411yescharacteristic
99C42×C53—21no—
100C42×C53C2311yescharacteristic
101C22⋊Dic53—21no—
102#11719925—22no—
103#11719925—22no—
104#11719925—41no—
105#11719924—22no—
106#11719924—22no—
107#11719924—22no—
108#11719924—22no—
109#11719924—41no—
110#11719924D414yescomplemented
111#11719921—21no—
112#11719921—21no—
113#11719921—21no—
114#11719921—21no—
115C4⋊Dic53C2311yescharacteristic
116#11719919—21no—
117#11719917—22no—
118C2×#88322—22no—
119C2×#88322C2311yescharacteristic
120C2×#88322C2311yescharacteristic
121C2×#88322C2311yescharacteristic
122C2×#88322C2311yescharacteristic
123C23×C53C2×D411yescharacteristic
124C23×C53C2×D411yescharacteristic
125C22×D53—21no—
126C22×D53—21no—
127C22×D53—42no—
128C22×D53—42no—
129C22×D53—81no—
130Q8×C53—21no—
131Q8×C53—21no—
132Q8×C53—21no—
133Q8×C53—21no—
134Q8×C53—21no—
135Q8×C53—42no—
136Q8×C53—42no—
137Q8×C53C2×D412yes—
138D4×C53—22no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—42no—
148C2×C4×C53—42no—
149C2×C4×C53C2×D411yescharacteristic
150C2×C4×C53C2×D411yescharacteristic
151C2×C4×C53C2×D411yescharacteristic
152C2×C4×C53C2×D411yescharacteristic
153C2×C4×C53C2411yescharacteristic
154C22⋊D53—42no—
155C22⋊D53—81no—
156C2×Dic53—21no—
157C2×Dic53—21no—
158D212—21no—
159D212—21no—
160D212—21no—
161D212—42no—
162D212C2×D412yes—
163C4×D53—42no—
164C4×D53—81no—
165Dic106—21no—
166Dic106—21no—
167#88322—21no—
168#88322—21no—
169#88322—21no—
170#88322C2×D412yes—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—21no—
176C22×C53#13712yescomplemented
177C22×C53C22×D411yescharacteristic
178C2×D53—22no—
179C2×D53—41no—
180C2×D53—41no—
181C2×D53—41no—
182C2×D53—41no—
183C2×D53—41no—
184C2×D53—84no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—22no—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—41no—
191C4×C53—41no—
192C4×C53—41no—
193C4×C53C22×D411yescharacteristic
194C4×C53H(5,𝔽2)11yescharacteristic
195Dic53—41no—
196Dic53—41no—
197#3266—531nomaximal, Sylow
198C2×C53—22no—
199C2×C53—41no—
200C2×C53D4211yescharacteristic
201C2×C53C2×#13711yescharacteristic
202C2×C53#57711yescharacteristic
203D53—42no—
204D53—81no—
205C2×#137—531no—
206C22×SD16—531no—
207D4×Q8D5311yescharacteristic, complemented
208#546—531no—
209#497—531no—
210#468—531no—
211#467—531no—
212#464—531no—
213#461—531no—
214#452—531no—
215#451—531no—
216#450—531no—
217#448—531no—
218#440—531no—
219#435—531no—
220C53#326611yescharacteristic, complemented, Sylow
221C2×#41—531no—
222C22×Q8C2×D5311yescharacteristic
223C22×Q8C2×D5311yescharacteristic, complemented
224C22×D4—531no—
225#137—1064no—
226C2×Q16—531no—
227C2×Q16—531no—
228C2×Q16—531no—
229C2×SD16—531no—
230C2×SD16—531no—
231C2×SD16—531no—
232C2×SD16—531no—
233C2×SD16—531no—
234C2×SD16—531no—
235C2×SD16—1064no—
236C2×M16—531no—
237C22×C8—531no—
238C4⋊Q8—21no—
239C4⋊Q8C2×D5311yescharacteristic, complemented
240#124—531no—
241#123—1061no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—531no—
245C22⋊Q8C2×D5311yescharacteristic
246C22⋊Q8C2×D5311yescharacteristic, complemented
247#121—531no—
248#121—1061no—
249#120—1061no—
250C4×Q8C2×D5311yescharacteristic, complemented
251C4×D4—21no—
252C4×D4—531no—
253C4×D4C2×D5311yescharacteristic
254C2×C22⋊C4—1061no—
255#107—531no—
256C4⋊C8—531no—
257#103—531no—
258#103—531no—
259#103—531no—
260#103—531no—
261#102—531no—
262#102—531no—
263C22⋊C8—531no—
264C22⋊C8—531no—
265#97—531no—
266C24—1061no—
267#41—1062no—
268#41—2121no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—531no—
276C2×Q8C22×D5311yescharacteristic
277C2×Q8C22×D5311yescharacteristic
278C2×Q8C22×D5311yescharacteristic
279C2×D4—531no—
280C2×D4—531no—
281C2×D4—1062no—
282C2×D4—1061no—
283C2×D4—1061no—
284C2×D4—1061no—
285C2×D4—2121no—
286C2×D4C22⋊D5311yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—1061no—
290C22×C4—1061no—
291C22×C4C22×D5311yescharacteristic
292C22×C4C22×D5311yescharacteristic
293Q16—1062no—
294Q16—1062no—
295Q16—2121no—
296SD16—534no—
297SD16—1062no—
298SD16—1062no—
299SD16—1062no—
300SD16—1062no—
301SD16—2121no—
302M16—1062no—
303C2×C8—531no—
304C2×C8—531no—
305C2×C8—531no—
306C2×C8—531no—
307C2×C8—1062no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—531no—
314C4⋊C4—1061no—
315C4⋊C4C22⋊D5311yescharacteristic
316C4⋊C4C22×D5311yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—1061no—
320C22⋊C4—1061no—
321C22⋊C4—1061no—
322C22⋊C4—1061no—
323C22⋊C4—1061no—
324C22⋊C4C22⋊D5311yescharacteristic
325C22⋊C4C22⋊D5311yescharacteristic
326C42—21no—
327C42C22×D5311yescharacteristic
328C23—1061no—
329C23—1061no—
330C23—4241no—
331C23—2122no—
332C23—2122no—
333C23C2×C22⋊D5311yescharacteristic
334C23C2×C22⋊D5311yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—42no—
342Q8—1061no—
343Q8—1061no—
344Q8D4×D5312yes—
345D4—22no—
346D4—532no—
347D4—1061no—
348D4—1061no—
349D4—1061no—
350D4—4241no—
351D4—2122no—
352D4—2122no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—1061no—
364C2×C4—1061no—
365C2×C4—4241no—
366C2×C4—2122no—
367C2×C4C2×C22⋊D5311yescharacteristic
368C2×C4C2×C22⋊D5311yescharacteristic
369C2×C4C2×C22⋊D5311yescharacteristic
370C2×C4C2×C22⋊D5311yescharacteristic
371C2×C4C23×D5311yescharacteristic
372C8—532no—
373C8—1061no—
374C8—1061no—
375C8—1061no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—1062no—
382C22—4244no—
383C22—2121no—
384C22—2121no—
385C22—2121no—
386C22—2121no—
387C22—2121no—
388C22#1313324812yescomplemented
389C22C22×C22⋊D5311yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—2121no—
399C4—2121no—
400C4C2×D4×D5311yescharacteristic
401C4#1313326211yescharacteristic
402C2—22no—
403C2—41no—
404C2—4241no—
405C2—2122no—
406C2D4×C22⋊D5311yescharacteristic
407C2C2×#1313324811yescharacteristic
408C2#1681143111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingD4×Q8×C53C2
socleC22×C53C22×D4
hypercenterD4×Q8D53
hypocenterC53#3266
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#13133248C2semidirect
2. C22×#11719924C2semidirect
4. #16811015C2semidirect
5. #16810871C2semidirect
6. #16810862C2semidirect
7. #16810845C2semidirect
8. #16810825C2semidirect
9. #16810783C2semidirect
10. #16810779C2semidirect
11. #16810773C2semidirect
12. #16810762C2semidirect
13. #16810721C2semidirect
14. #16810695C2semidirect
15. D4×Q8×C53C2semidirect
19. C2×#11719925C22semidirect
20. C2×#11719925C22semidirect
21. C2×#11719925C22semidirect
29. C2×#11719917C22semidirect
39. #13133150C22semidirect
40. C4⋊#88322C22semidirect
41. #13133141C22semidirect
42. #13133141C22semidirect
43. #13133141C22semidirect
44. #13133140C22semidirect
48. C22⋊#88322C22semidirect
49. #13133128C22semidirect
51. C22×Q8×C53C22semidirect
53. C4⋊Q8×C53C22semidirect
57. C22⋊Q8×C53C22semidirect
58. C4×Q8×C53C22semidirect
110. #11719924D4semidirect
176. C22×C53#137semidirect
207. D4×Q8D53semidirect
220. C53#3266semidirect
223. C22×Q8C2×D53semidirect
239. C4⋊Q8C2×D53semidirect
246. C22⋊Q8C2×D53semidirect
250. C4×Q8C2×D53semidirect
388. C22#13133248semidirect