Finite Group Databasebeta

← #29834659

Subgroups — #29834659

Order 15616.

546 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456333C211yesmaximal, characteristic, complemented
2#21447983C212yesmaximal, complemented
3#21447976C212yesmaximal, complemented
4#21447971C211yesmaximal, characteristic, complemented
5#21447966C211yesmaximal, characteristic, complemented
6#21447961C211yesmaximal, characteristic, complemented
7#21447956C211yesmaximal, characteristic, complemented
8#21447601C212yesmaximal, complemented
9C2×C4⋊#11758185C211yesmaximal, characteristic, complemented
10#21447526C212yesmaximal
11C4×#135×C61C211yesmaximal, characteristic
12#16840319—21no—
13C22×#11758185C2212yes—
14#16840167C2211yescharacteristic
15#16840167C2211yescharacteristic
16#16840164—21no—
17#16840164C2212yes—
18#16840160—21no—
19#16839800—21no—
20#16839800C2212yes—
21#16839589C2212yes—
22#16839589C2212yes—
23#16839587C2212yes—
24#16839587C2212yes—
25C4⋊#11758185C2211yescharacteristic
26C4⋊#11758185C2211yescharacteristic
27C4⋊#11758185C2212yes—
28#16839566C2212yes—
29#16839566C2212yes—
30#16839563C2212yes—
31#16839563C2212yes—
32C2×#135×C61C2211yescharacteristic
33C4×#41×C61—21no—
34C4×Q16×C61C2211yescharacteristic
35C4×Q16×C61C2211yescharacteristic
36C4×SD16×C61—21no—
37C4×SD16×C61—21no—
38C4×D8×C61C2211yescharacteristic
39C4×D8×C61C2211yescharacteristic
40C61×#427C2211yescharacteristic
41C61×#416—21no—
42C2×C4×C8×C61C2211yescharacteristic
43#14519777—21no—
44#14519777—21no—
45#14519769—22no—
46#14519756—41no—
47#14519756—41no—
48C4×Dic122—41no—
49C4×Dic122—41no—
50#14519736—22no—
51C2×#11758185—24no—
52C2×#11758185C2312yes—
53C2×#11758185C2312yes—
54#14519711—21no—
55#14519711—21no—
56#14519711C2312yes—
57#14519711C2312yes—
58#14519709—22no—
59#14519709—22no—
60C2×#41×C61—21no—
61#135×C61—22no—
62#135×C61Q814yescomplemented
63C2×Q16×C61C2311yescharacteristic
64C2×SD16×C61—21no—
65C2×D8×C61C2311yescharacteristic
66C22×C8×C61D411yescharacteristic
67C22×C8×C61C2311yescharacteristic
68C4×Q8×C61—21no—
69C4×Q8×C61—21no—
70C4×D4×C61—21no—
71C4×D4×C61—21no—
72C4×D4×C61—41no—
73C4×D4×C61—41no—
74#117×C61—21no—
75#117×C61—41no—
76C2×C42×C61—41no—
77C2×C42×C61D411yescharacteristic
78#107×C61C2311yescharacteristic
79#107×C61C2311yescharacteristic
80#106×C61—21no—
81#103×C61—21no—
82#103×C61—21no—
83#102×C61—21no—
84#102×C61—21no—
85C4×C8×C61D411yescharacteristic
86C4×C8×C61D411yescharacteristic
87C4×C8×C61C2311yescharacteristic
88C4×C8×C61C2311yescharacteristic
89#41×C61—24no—
90#41×C61—42no—
91C2×Q8×C61—21no—
92C2×D4×C61—21no—
93C2×D4×C61—41no—
94C22×C4×C61—41no—
95C22×C4×C61—41no—
96C22×C4×C61—41no—
97C22×C4×C61D811yescharacteristic
98C22×C4×C61D811yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C2×Dic122—41no—
101C2×Dic122—41no—
102Q16×C61—21no—
103Q16×C61C2×Q812yes—
104SD16×C61—24no—
105D8×C61—21no—
106D8×C61C2×Q812yes—
107C2×C8×C61—22no—
108C2×C8×C61C2×D411yescharacteristic
109C2×C8×C61C2×D411yescharacteristic
110C2×C8×C61C2×D411yescharacteristic
111C2×C8×C61C2×Q812yes—
112C2×C8×C61#4112yes—
113C2×C8×C61C2411yescharacteristic
114C4⋊C4×C61—21no—
115C4⋊C4×C61—21no—
116C4⋊C4×C61—41no—
117C4⋊C4×C61—41no—
118C22⋊C4×C61—41no—
119C22⋊C4×C61—41no—
120C22⋊C4×C61—41no—
121C42×C61—41no—
122C42×C61—41no—
123C42×C61—41no—
124C42×C61D811yescharacteristic
125C42×C61D811yescharacteristic
126C42×C61C2×D411yescharacteristic
127C42×C61C2×D411yescharacteristic
128C22⋊Dic61—42no—
129C4⋊Dic61—22no—
130C4⋊Dic61—22no—
131C4⋊Dic61—41no—
132C4⋊Dic61—41no—
133#11758196—42no—
134#11758196—42no—
135C4×Dic61—42no—
136#11758185—22no—
137#11758185—22no—
138C23×C61—41no—
139C23×C61C2×D811yescharacteristic
140Q8×C61—22no—
141Q8×C61—41no—
142D4×C61—22no—
143D4×C61—44no—
144D4×C61—41no—
145C2×C4×C61—22no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—44no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—82no—
152C2×C4×C61—82no—
153C2×C4×C61C22⋊Q812yes—
154C2×C4×C61C2×D811yescharacteristic
155C2×C4×C61C2×D811yescharacteristic
156C2×C4×C61C2×D811yescharacteristic
157C2×C4×C61C2×D811yescharacteristic
158C2×C4×C61C2×D811yescharacteristic
159C2×C4×C61#13712yes—
160C2×C4×C61C22×D411yescharacteristic
161C2×Dic61—42no—
162C2×Dic61—42no—
163Dic122—82no—
164Dic122—82no—
165C8×C61—21no—
166C8×C61—21no—
167C8×C61C22⋊Q812yes—
168C8×C61C22×Q811yescharacteristic
169C8×C61C2×#4111yescharacteristic
170#33626—611nomaximal, Sylow
171C22×C61—21no—
172C22×C61—21no—
173C22×C61—42no—
174C22×C61—81no—
175C22×C61—81no—
176C22×C61#47412yes—
177C22×C61C22×D811yescharacteristic
178C4×C61—21no—
179C4×C61—21no—
180C4×C61—21no—
181C4×C61—42no—
182C4×C61—81no—
183C4×C61—81no—
184C4×C61#47412yes—
185C4×C61#50812yes—
186C4×C61C2×C22⋊Q811yescharacteristic
187C4×C61C2×#13711yescharacteristic
188Dic61—82no—
189Dic61—82no—
190#3160—611no—
191C4×#135D6111yescharacteristic
192#2229—612no—
193#2228—612no—
194#2227—611no—
195#2226—611no—
196#2225—611no—
197#2224—611no—
198#2158—612no—
199C2×C4⋊C16—611no—
200#2138—612no—
201C2×C61—22no—
202C2×C61—82no—
203C2×C61C2×#47411yescharacteristic
204C2×C61C2×#50811yescharacteristic
205C2×C61#341411yescharacteristic
206C2×#135C2×D6111yescharacteristic
207#533—1221no—
208C4×#41—21no—
209C22×C16—612no—
210#500—611no—
211#500—611no—
212#499—612no—
213#499—1221no—
214#498—1221no—
215C4×Q16C2×D6111yescharacteristic
216C4×Q16C2×D6111yescharacteristic
217C4×SD16—21no—
218C4×SD16—21no—
219C4×D8C2×D6111yescharacteristic
220C4×D8C2×D6111yescharacteristic
221#427—612no—
222#427—1221no—
223#427C2×D6111yescharacteristic
224#416—21no—
225C2×C4×C8C2×D6111yescharacteristic
226#367—612no—
227#367—612no—
228#366—612no—
229#366—612no—
230C4⋊C16—611no—
231C4⋊C16—611no—
232C4⋊C16—612no—
233#358—612no—
234#358—612no—
235#357—612no—
236#357—612no—
237C61#3362611yescharacteristic, complemented, Sylow
238C2×#41—21no—
239#135—22no—
240#135Dic12214yescomplemented
241C2×Q16C22×D6111yescharacteristic
242C2×SD16—21no—
243C2×D8C22×D6111yescharacteristic
244C22×C8C22⋊D6111yescharacteristic
245C22×C8C22×D6111yescharacteristic
246C4⋊Q8—1221no—
247C4⋊Q8—1221no—
248#125—1222no—
249C22⋊Q8—2441no—
250C22⋊Q8—2441no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8—2441no—
254C4×Q8—2441no—
255C4×D4—21no—
256C4×D4—21no—
257C4×D4—41no—
258C4×D4—41no—
259#117—21no—
260#117—41no—
261#117—1222no—
262C2×C42—41no—
263C2×C42C22⋊D6111yescharacteristic
264C2×C16—612no—
265C2×C16—612no—
266C2×C16—1224no—
267#107—612no—
268#107—612no—
269#107—1221no—
270#107—1221no—
271#107C22×D6111yescharacteristic
272#107C22×D6111yescharacteristic
273#106—21no—
274#106—1222no—
275#106—1222no—
276#103—21no—
277#103—21no—
278#102—21no—
279#102—21no—
280C4×C8C22⋊D6111yescharacteristic
281C4×C8C22⋊D6111yescharacteristic
282C4×C8C22×D6111yescharacteristic
283C4×C8C22×D6111yescharacteristic
284#41—24no—
285#41—42no—
286C2×Q8—21no—
287C2×Q8—2441no—
288C2×Q8—2441no—
289C2×D4—21no—
290C2×D4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4#1175819911yescharacteristic
295C22×C4#1175819911yescharacteristic
296C22×C4C2×C22⋊D6111yescharacteristic
297Q16—21no—
298Q16C2×Dic12212yes—
299SD16—24no—
300D8—21no—
301D8C2×Dic12212yes—
302C2×C8—22no—
303C2×C8C2×Dic12212yes—
304C2×C8#1175822212yes—
305C2×C8C2×C22⋊D6111yescharacteristic
306C2×C8C2×C22⋊D6111yescharacteristic
307C2×C8C2×C22⋊D6111yescharacteristic
308C2×C8C23×D6111yescharacteristic
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—1222no—
314C4⋊C4—1222no—
315C4⋊C4—2442no—
316C4⋊C4—2442no—
317C4⋊C4—2441no—
318C4⋊C4—2441no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—2442no—
323C42—41no—
324C42—41no—
325C42—41no—
326C42—2442no—
327C42#1175819911yescharacteristic
328C42#1175819911yescharacteristic
329C42C2×C22⋊D6111yescharacteristic
330C42C2×C22⋊D6111yescharacteristic
331C16—1222no—
332C16—1222no—
333C23—41no—
334C23C2×#1175819911yescharacteristic
335Q8—22no—
336Q8—41no—
337Q8—4882no—
338Q8—4882no—
339D4—22no—
340D4—44no—
341D4—41no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—44no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—2442no—
351C2×C4—2442no—
352C2×C4#1451975612yes—
353C2×C4C2×#1175819911yescharacteristic
354C2×C4C2×#1175819911yescharacteristic
355C2×C4C2×#1175819911yescharacteristic
356C2×C4C2×#1175819911yescharacteristic
357C2×C4C2×#1175819911yescharacteristic
358C2×C4#1451980712yes—
359C2×C4C22×C22⋊D6111yescharacteristic
360C8—21no—
361C8—21no—
362C8#1451975612yes—
363C8C22×Dic12211yescharacteristic
364C8C2×#1175822211yescharacteristic
365C22—21no—
366C22—21no—
367C22—42no—
368C22—81no—
369C22—81no—
370C22#1684004212yes—
371C22C22×#1175819911yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—81no—
377C4—81no—
378C4—4882no—
379C4—4882no—
380C4#1684004212yes—
381C4#1684019212yes—
382C4C2×#1451975611yescharacteristic
383C4C2×#1451980711yescharacteristic
384C2—22no—
385C2—82no—
386C2C2×#1684004211yescharacteristic
387C2C2×#1684019211yescharacteristic
388C2#2145895311yescharacteristic
subgroupG/H
centerC2×C4C2×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC4×#135×C61C2
socleC22×C61C22×D8
prime radicalC2×#135×C61C22
hypercenterC4×#135D61
hypocenterC61#33626
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C4×#135

Complements

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

NH ≅ G/Ntype
1. #21456333C2semidirect
2. #21447983C2semidirect
3. #21447976C2semidirect
4. #21447971C2semidirect
5. #21447966C2semidirect
6. #21447961C2semidirect
7. #21447956C2semidirect
8. #21447601C2semidirect
9. C2×C4⋊#11758185C2semidirect
62. #135×C61Q8semidirect
237. C61#33626semidirect
240. #135Dic122semidirect

Maximal subgroups

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

subgroupindex
#214563332
#214479832
#214479762
#214479712
#214479662
#214479612
#214479562
#214476012
C2×C4⋊#117581852
#214475262
C4×#135×C612
#3362661