Finite Group Databasebeta

← #29662255

Subgroups — #29662255

Order 15616.

802 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456247C211yesmaximal, characteristic, complemented
2#21455819C212yesmaximal, complemented
3#21445184C212yesmaximal, complemented
4#21445174C212yesmaximal, complemented
5#21445160C214yesmaximal, complemented
6#21444567C211yesmaximal, characteristic, complemented
7#21444469C212yesmaximal, complemented
8C2×C8×D4×C61C211yesmaximal, characteristic, complemented
9C2×#14519799—21no—
10C2×#14519799—21no—
11#16840312C2212yes—
12#16840146—21no—
13#16840146—21no—
14#16840142C2211yescharacteristic, complemented
15#16840139C2211yescharacteristic, complemented
16#16840135C2212yescomplemented
17#16840075—22no—
18#16840042C2214yes—
19#16839957C2214yes—
20#16839896—22no—
21C4×#11758199—24no—
22#16839815C2212yes—
23#16839515C2212yescomplemented
24#16839513C2212yescomplemented
25#16839509C2214yescomplemented
26#16839505C2214yescomplemented
27C23×C8×C61—21no—
28C2×C4×D4×C61C2211yescharacteristic
29C8×D4×C61—21no—
30C8×D4×C61—21no—
31C8×D4×C61C2212yes—
32C8×D4×C61C2212yes—
33C2×C4⋊C8×C61C2211yescharacteristic
34C2×C22⋊C8×C61—21no—
35C2×C4×C8×C61C2211yescharacteristic
36C2×#11758222—22no—
37#14519799—44no—
38#14519799—44no—
39C2×Dic244—21no—
40C2×Dic244—21no—
41C2×#11758190—22no—
42C2×#11758190—22no—
43C2×D488—21no—
44C2×D488—21no—
45C2×#11758199—24no—
46#14519777C2312yes—
47#14519774C2312yes—
48#14519765—22no—
49#14519751—22no—
50#14519751—22no—
51C4×D244—22no—
52C4×D244—22no—
53#14519710—24no—
54C4⋊#92418C2314yes—
55#14519698—24no—
56#14519697—24no—
57C22⋊#92418—22no—
58C4×#92418—22no—
59C22×D4×C61D411yescharacteristic
60C23×C4×C61—21no—
61C22×C8×C61—21no—
62C22×C8×C61—22no—
63C22×C8×C61—22no—
64C22×C8×C61—44no—
65C22×C8×C61D411yescharacteristic
66C22×C8×C61D411yescharacteristic
67C4×D4×C61—21no—
68C4×D4×C61—21no—
69C4×D4×C61D412yes—
70C4×D4×C61C2312yes—
71C2×C4⋊C4×C61D411yescharacteristic
72C2×C22⋊C4×C61—21no—
73C2×C42×C61C2311yescharacteristic
74C4⋊C8×C61—21no—
75C4⋊C8×C61C2312yes—
76C22⋊C8×C61—22no—
77C22⋊C8×C61—22no—
78C4×C8×C61D411yescharacteristic
79C4×C8×C61D411yescharacteristic
80C4×C8×C61C2311yescharacteristic
81C4×C8×C61C2311yescharacteristic
82C24×C61—21no—
83C2×D4×C61—24no—
84C2×D4×C61—21no—
85C2×D4×C61C2×D412yes—
86C22×C4×C61—21no—
87C22×C4×C61—22no—
88C22×C4×C61—22no—
89C22×C4×C61—42no—
90C22×C4×C61—42no—
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61#4111yescharacteristic
94C2×C22⋊D61—42no—
95#11758222—44no—
96#11758222—44no—
97C2×D244—22no—
98C2×D244—22no—
99C2×D244—22no—
100C2×C4×D61—42no—
101C2×Dic122—22no—
102C2×C8×C61—22no—
103C2×C8×C61—22no—
104C2×C8×C61—42no—
105C2×C8×C61—44no—
106C2×C8×C61—44no—
107C2×C8×C61D814yescomplemented
108C2×C8×C61C2×D411yescharacteristic
109C2×C8×C61C2×D411yescharacteristic
110C2×C8×C61C2×D411yescharacteristic
111C2×C8×C61C2×D411yescharacteristic
112C4⋊C4×C61—21no—
113C4⋊C4×C61C2×D412yes—
114C22⋊C4×C61—22no—
115C22⋊C4×C61—22no—
116C42×C61C2×D411yescharacteristic
117C42×C61C2×D411yescharacteristic
118C42×C61#4111yescharacteristic
119C42×C61C2411yescharacteristic
120#11758199—48no—
121#11758198—42no—
122#11758198—42no—
123C4⋊Dic61—22no—
124C4⋊Dic61—22no—
125C2×#92418—24no—
126Dic244—42no—
127Dic244—42no—
128D488—42no—
129D488—42no—
130#11758190—44no—
131#11758190—44no—
132C23×C61—24no—
133C23×C61—22no—
134C23×C61—42no—
135C23×C61—42no—
136C23×C61#12111yescharacteristic
137C22×D61—42no—
138D4×C61—24no—
139D4×C61—22no—
140D4×C61#13514yescomplemented
141C2×C4×C61—22no—
142C2×C4×C61—22no—
143C2×C4×C61—41no—
144C2×C4×C61—42no—
145C2×C4×C61—42no—
146C2×C4×C61—42no—
147C2×C4×C61—44no—
148C2×C4×C61#12111yescharacteristic
149C2×C4×C61#12111yescharacteristic
150C2×C4×C61#12111yescharacteristic
151C2×C4×C61C2×D812yes—
152C2×C4×C61AGL(1,ℤ/8)12yes—
153C2×C4×C61C22×D411yescharacteristic
154C2×C4×C61C22×D411yescharacteristic
155C2×C4×C61C2×#4111yescharacteristic
156C22⋊D61—84no—
157C2×Dic61—42no—
158D244—42no—
159D244—44no—
160D244—42no—
161D244—82no—
162C4×D61—84no—
163Dic122—42no—
164Dic122—42no—
165C8×C61—21no—
166C8×C61—21no—
167C8×C61—42no—
168C8×C61C2×D812yes—
169C8×C61C2×D812yes—
170#92418—44no—
171#19101—611nomaximal, Sylow
172C22×C61—21no—
173C22×C61—21no—
174C22×C61—24no—
175C22×C61—48no—
176C22×C61—42no—
177C22×C61—42no—
178C22×C61#45912yes—
179C22×C61C2×#12111yescharacteristic
180C2×D61—82no—
181C2×D61—82no—
182C2×D61—82no—
183C4×C61—21no—
184C4×C61—21no—
185C4×C61—21no—
186C4×C61—42no—
187C4×C61#45912yes—
188C4×C61C22×D811yescharacteristic
189C4×C61C2×#13512yes—
190C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
191Dic61—82no—
192#3147—611no—
193#3100—612no—
194C2×C8×D4D6111yescharacteristic, complemented
195#1669—612no—
196#1667—612no—
197#1665—614no—
198#1564—611no—
199#1540—612no—
200C2×C61—22no—
201C2×C61—44no—
202C2×C61C2×#45911yescharacteristic
203C2×C61#305011yescharacteristic
204C2×C61#306711yescharacteristic
205D61—162no—
206C2×#135—1221no—
207C2×#135—1221no—
208C23×C8—21no—
209#532—612no—
210C2×C4×D4C2×D6111yescharacteristic
211#495—1221no—
212#495—1221no—
213#494—611no—
214#493—611no—
215#492—612no—
216#480—1222no—
217#474—614no—
218#459—614no—
219#447—1222no—
220C4×D8—1224no—
221C8×D4—21no—
222C8×D4—21no—
223C8×D4C2×D6112yes—
224C8×D4C2×D6112yes—
225#431—612no—
226C2×C4⋊C8C2×D6111yescharacteristic
227C2×C22⋊C8—21no—
228C2×C4×C8C2×D6111yescharacteristic
229#335—612no—
230#334—612no—
231#332—614no—
232#331—614no—
233C61#1910111yescharacteristic, complemented, Sylow
234C2×#41—1222no—
235C22×D4D24411yescharacteristic
236C23×C4—21no—
237#135—2444no—
238#135—2444no—
239C2×Q16—1221no—
240C2×Q16—1221no—
241C2×SD16—1222no—
242C2×SD16—1222no—
243C2×D8—1224no—
244C2×D8—1221no—
245C2×D8—1221no—
246C22×C8—21no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—44no—
250C22×C8C22⋊D6111yescharacteristic
251C22×C8C22⋊D6111yescharacteristic
252C4⋊Q8—612no—
253#127—612no—
254#124—1222no—
255#121—1222no—
256#121—1222no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—1222no—
260C4×D4—1222no—
261C4×D4D24412yes—
262C4×D4C22×D6112yes—
263C2×C4⋊C4D24411yescharacteristic
264C2×C22⋊C4—21no—
265C2×C42C22×D6111yescharacteristic
266#107—1224no—
267C4⋊C8—21no—
268C4⋊C8—614no—
269C4⋊C8C22×D6112yes—
270#102—1224no—
271#102—1224no—
272C22⋊C8—22no—
273C22⋊C8—22no—
274C22⋊C8—1222no—
275C4×C8—1222no—
276C4×C8C22⋊D6111yescharacteristic
277C4×C8C22⋊D6111yescharacteristic
278C4×C8C22×D6111yescharacteristic
279C4×C8C22×D6111yescharacteristic
280C24—21no—
281#41—2444no—
282#41—2444no—
283C2×Q8—1222no—
284C2×D4—24no—
285C2×D4—21no—
286C2×D4—1222no—
287C2×D4—1222no—
288C2×D4—1222no—
289C2×D4—2442no—
290C2×D4C2×D24412yes—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—2442no—
297C22×C4C2×D24411yescharacteristic
298C22×C4#1175822211yescharacteristic
299C22×C4C2×C22⋊D6111yescharacteristic
300Q16—2442no—
301Q16—2442no—
302SD16—2444no—
303SD16—2444no—
304D8—2442no—
305D8—2448no—
306D8—2442no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—42no—
310C2×C8—44no—
311C2×C8—44no—
312C2×C8—1224no—
313C2×C8#1175819914yescomplemented
314C2×C8C2×C22⋊D6111yescharacteristic
315C2×C8C2×C22⋊D6111yescharacteristic
316C2×C8C2×C22⋊D6111yescharacteristic
317C2×C8C2×C22⋊D6111yescharacteristic
318C4⋊C4—21no—
319C4⋊C4—1222no—
320C4⋊C4—1222no—
321C4⋊C4C2×D24412yes—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—2442no—
325C22⋊C4—2442no—
326C42C2×D24411yescharacteristic
327C42#1175822211yescharacteristic
328C42C2×C22⋊D6111yescharacteristic
329C42C23×D6111yescharacteristic
330C23—24no—
331C23—22no—
332C23—42no—
333C23—42no—
334C23—2442no—
335C23#1451975111yescharacteristic
336Q8—2442no—
337Q8—2442no—
338D4—24no—
339D4—22no—
340D4—4882no—
341D4—4884no—
342D4—2442no—
343D4—2444no—
344D4—2442no—
345D4#1451979914yescomplemented
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—4884no—
354C2×C4—2442no—
355C2×C4#1451975111yescharacteristic
356C2×C4#1451975111yescharacteristic
357C2×C4#1451975111yescharacteristic
358C2×C4C2×#1175819912yes—
359C2×C4#1451980212yes—
360C2×C4C22×C22⋊D6111yescharacteristic
361C2×C4C22×D24411yescharacteristic
362C2×C4C2×#1175822211yescharacteristic
363C8—21no—
364C8—21no—
365C8—42no—
366C8—2444no—
367C8C2×#1175819912yes—
368C8C2×#1175819912yes—
369C22—21no—
370C22—21no—
371C22—24no—
372C22—48no—
373C22—42no—
374C22—42no—
375C22—4882no—
376C22—4882no—
377C22—4882no—
378C22#1683995712yes—
379C22C2×#1451975111yescharacteristic
380C4—21no—
381C4—21no—
382C4—21no—
383C4—42no—
384C4—4882no—
385C4#1683995712yes—
386C4C22×#1175819911yescharacteristic
387C4C2×#1451979912yes—
388C4C2×#1451980211yescharacteristic
389C2—22no—
390C2—44no—
391C2—9762no—
392C2C2×#1683995711yescharacteristic
393C2#2145540011yescharacteristic
394C2#2145553911yescharacteristic
subgroupG/H
centerC2×C4#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C8×D4×C61C2
socleC22×C61C2×#121
hypercenterC2×C8×D4D61
hypocenterC61#19101
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
1. #21456247C2semidirect
2. #21455819C2semidirect
3. #21445184C2semidirect
4. #21445174C2semidirect
5. #21445160C2semidirect
6. #21444567C2semidirect
7. #21444469C2semidirect
8. C2×C8×D4×C61C2semidirect
14. #16840142C22semidirect
15. #16840139C22semidirect
16. #16840135C22semidirect
23. #16839515C22semidirect
24. #16839513C22semidirect
25. #16839509C22semidirect
26. #16839505C22semidirect
107. C2×C8×C61D8semidirect
140. D4×C61#135semidirect
194. C2×C8×D4D61semidirect
233. C61#19101semidirect
313. C2×C8#11758199semidirect
345. D4#14519799semidirect

Maximal subgroups

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

subgroupindex
#214562472
#214558192
#214451842
#214451742
#214451602
#214445672
#214444692
C2×C8×D4×C612
#1910161