Finite Group Databasebeta

← #29834375

Subgroups — #29834375

Order 15616.

694 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840202C211yesmaximal, characteristic, complemented
2C2×#16840192C211yesmaximal, characteristic
3#21456313C211yesmaximal, characteristic, complemented
4#21447881C212yesmaximal, complemented
5#21447874C212yesmaximal, complemented
6#21447869C212yesmaximal, complemented
7#21447864C212yesmaximal, complemented
8#21447578C211yesmaximal, characteristic, complemented
9#21447529C211yesmaximal, characteristic, complemented
10C2×#16839565C211yesmaximal, characteristic, complemented
11C61×#2951C211yesmaximal, characteristic, complemented
12C2×#14519810—21no—
13C2×#14519799C2211yescharacteristic
14C22×Dic244C2211yescharacteristic
15#16840338—21no—
16#16840202—24no—
17#16840192—24no—
18C2×#14519723C2212yes—
19C2×#14519723C2211yescharacteristic
20C2×#14519721C2212yes—
21C2×#14519721C2211yescharacteristic
22C2×#14519720C2211yescharacteristic
23C2×#14519718C2211yescharacteristic
24C2×#14519717C2211yescharacteristic
25C22×#11758185C2211yescharacteristic
26#16840156—22no—
27#16840146C2212yes—
28#16840142C2212yes—
29C4⋊#11758185C2212yes—
30C4⋊#11758185C2212yes—
31#16839565C2212yes—
32#16839565C2212yes—
33#16839565C2212yes—
34#16839562C2212yes—
35C2×#135×C61C2211yescharacteristic
36C61×#520—21no—
37C61×#440—22no—
38C4×Q16×C61C2212yes—
39C4×D8×C61C2212yes—
40C2×#107×C61C2211yescharacteristic
41C61×#417—21no—
42C61×#405C2211yescharacteristic
43C2×#11758222—21no—
44C22×Dic122—21no—
45#14519810—44no—
46#14519799—22no—
47#14519799—22no—
48C2×Dic244—22no—
49C2×Dic244—22no—
50C2×Dic244—21no—
51C2×Dic244—22no—
52C2×Dic244C2312yes—
53C2×Dic244C2311yescharacteristic
54C2×#11758190—21no—
55C2×#11758190—22no—
56C2×D488C2311yescharacteristic
57#14519777—22no—
58#14519765—22no—
59#14519761—42no—
60#14519757—41no—
61#14519757—41no—
62#14519723—22no—
63#14519723—24no—
64#14519721—22no—
65#14519721—24no—
66#14519720—22no—
67#14519718—22no—
68#14519717—22no—
69C2×#11758185—22no—
70C2×#11758185C2312yes—
71C2×#11758185C2311yescharacteristic
72C2×#11758185C2311yescharacteristic
73C2×#41×C61—21no—
74#135×C61—22no—
75#135×C61D414yes—
76C2×Q16×C61C2311yescharacteristic
77C2×SD16×C61—21no—
78C2×D8×C61C2311yescharacteristic
79C2×M16×C61D411yescharacteristic
80C22×C8×C61C2311yescharacteristic
81C4×Q8×C61—22no—
82C4×D4×C61—22no—
83C4×D4×C61—42no—
84#117×C61—41no—
85#117×C61D411yescharacteristic
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61—41no—
88#107×C61—21no—
89#107×C61C2312yes—
90#103×C61—22no—
91#102×C61—22no—
92#97×C61D412yes—
93C4×C8×C61C2312yes—
94#41×C61—24no—
95#41×C61—42no—
96C2×Q8×C61—21no—
97C2×D4×C61—21no—
98C2×D4×C61—41no—
99C22×C4×C61—41no—
100C22×C4×C61—41no—
101C22×C4×C61—42no—
102C22×C4×C61C2×D411yescharacteristic
103C2×C22⋊D61—41no—
104C22×Dic61—41no—
105#11758222—42no—
106#11758222—42no—
107C2×D244—21no—
108C2×C4×D61—41no—
109C2×Dic122—22no—
110C2×Dic122—21no—
111C2×Dic122—42no—
112C2×Dic122—42no—
113C2×Dic122—41no—
114Q16×C61—21no—
115Q16×C61C2×D412yes—
116SD16×C61—24no—
117D8×C61—21no—
118D8×C61C2×D412yes—
119M16×C61—22no—
120C2×C8×C61C2×D412yes—
121C2×C8×C61C2×D411yescharacteristic
122C2×C8×C61C2×D412yes—
123C2×C8×C61#4112yes—
124C2×C8×C61C2411yescharacteristic
125C4⋊C4×C61—22no—
126C4⋊C4×C61—41no—
127C4⋊C4×C61—42no—
128C4⋊C4×C61D811yescharacteristic
129C4⋊C4×C61D811yescharacteristic
130C22⋊C4×C61—42no—
131C22⋊C4×C61D812yes—
132C42×C61—42no—
133C42×C61C2×D412yes—
134#11758198—42no—
135#11758198—42no—
136C4⋊Dic61—42no—
137C4⋊Dic61—42no—
138Dic244—22no—
139Dic244—21no—
140Dic244—22no—
141Dic244—21no—
142Dic244—44no—
143Dic244—42no—
144D488—21no—
145D488—21no—
146#11758190—44no—
147#11758190—42no—
148#11758185—21no—
149#11758185—21no—
150#11758185—22no—
151C23×C61—41no—
152C23×C61C2×D811yescharacteristic
153C22×D61—41no—
154Q8×C61—22no—
155Q8×C61—41no—
156D4×C61—22no—
157D4×C61—44no—
158D4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—44no—
161C2×C4×C61—42no—
162C2×C4×C61—162no—
163C2×C4×C61—161no—
164C2×C4×C61#12112yes—
165C2×C4×C61C2×D812yes—
166C2×C4×C61C2×D812yes—
167C2×C4×C61C2×D811yescharacteristic
168C2×C4×C61AGL(1,ℤ/8)12yes—
169C2×C4×C61C22×D411yescharacteristic
170C22⋊D61—82no—
171C2×Dic61—41no—
172C2×Dic61—42no—
173C2×Dic61—82no—
174D244—41no—
175D244—41no—
176C4×D61—82no—
177Dic122—41no—
178Dic122—42no—
179Dic122—42no—
180Dic122—41no—
181Dic122—82no—
182C8×C61—22no—
183C8×C61#12112yes—
184C8×C61C22×D411yescharacteristic
185C8×C61C2×#4111yescharacteristic
186#33590—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—42no—
190C22×C61—81no—
191C22×C61—81no—
192C22×C61#45912yes—
193C22×C61C22×D811yescharacteristic
194C2×D61—81no—
195C2×D61—81no—
196C2×D61—81no—
197C4×C61—22no—
198C4×C61—22no—
199C4×C61—42no—
200C4×C61—82no—
201C4×C61#45912yes—
202C4×C61C2×#12111yescharacteristic
203C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
204Dic61—82no—
205Dic61—81no—
206C2×#510—611no—
207C2×#508—611no—
208#3156—611no—
209#2951D6111yescharacteristic, complemented
210#2211—612no—
211#2210—612no—
212#2209—612no—
213#2208—612no—
214#2151—611no—
215#2139—611no—
216C2×#358—611no—
217C2×C61—22no—
218C2×C61—82no—
219C2×C61C2×#45911yescharacteristic
220C2×C61#341611yescharacteristic
221C2×C61#341711yescharacteristic
222D61—161no—
223C2×#137—1221no—
224C2×#135—611no—
225C2×#135C2×D6111yescharacteristic
226C22×Q16—611no—
227#536—1221no—
228#520—21no—
229#510—1224no—
230#508—1224no—
231C2×Q32—611no—
232C2×Q32—612no—
233C2×SD32—611no—
234C2×SD32—611no—
235C2×SD32—612no—
236C2×D16—611no—
237C2×M32—611no—
238C22×C16—611no—
239#497—1222no—
240#495—612no—
241#494—612no—
242#440—22no—
243C4×Q16C2×D6112yes—
244C4×D8C2×D6112yes—
245C2×#107C2×D6111yescharacteristic
246#417—21no—
247#405C2×D6111yescharacteristic
248C4⋊C16—612no—
249C4⋊C16—612no—
250#358—612no—
251#358—612no—
252#358—612no—
253#357—612no—
254C61#3359011yescharacteristic, complemented, Sylow
255C2×#41—21no—
256C2×#41—1221no—
257C22×Q8—1221no—
258#137—2444no—
259#135—22no—
260#135—1222no—
261#135—1222no—
262#135D24414yes—
263C2×Q16—612no—
264C2×Q16—611no—
265C2×Q16—1222no—
266C2×Q16—1222no—
267C2×Q16—1221no—
268C2×Q16—1222no—
269C2×Q16C22×D6111yescharacteristic
270C2×SD16—21no—
271C2×SD16—1221no—
272C2×SD16—1222no—
273C2×D8—611no—
274C2×D8C22×D6111yescharacteristic
275C2×M16C22⋊D6111yescharacteristic
276C22×C8C22×D6111yescharacteristic
277C4⋊Q8—1222no—
278#124—1222no—
279#123—2442no—
280C22⋊Q8—2441no—
281C22⋊Q8—2441no—
282C4×Q8—22no—
283C4×D4—22no—
284C4×D4—42no—
285#117—41no—
286#117C22⋊D6111yescharacteristic
287C2×C4⋊C4—21no—
288C2×C4⋊C4—41no—
289Q32—1222no—
290Q32—1224no—
291SD32—1222no—
292SD32—1222no—
293SD32—1224no—
294D16—1222no—
295M32—1222no—
296C2×C16—612no—
297C2×C16—611no—
298C2×C16—611no—
299C2×C16—1222no—
300#107—21no—
301#107C22×D6112yes—
302#103—22no—
303#102—22no—
304#97C22⋊D6112yes—
305C4×C8C22×D6112yes—
306#41—24no—
307#41—42no—
308#41—2442no—
309#41—2442no—
310C2×Q8—21no—
311C2×Q8—1222no—
312C2×Q8—1221no—
313C2×Q8—2442no—
314C2×Q8—2442no—
315C2×Q8—2441no—
316C2×D4—21no—
317C2×D4—41no—
318C2×D4—1221no—
319C2×D4—2441no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—2441no—
324C22×C4—2441no—
325C22×C4C2×C22⋊D6111yescharacteristic
326Q16—21no—
327Q16—1222no—
328Q16—1221no—
329Q16—1222no—
330Q16—1221no—
331Q16—2444no—
332Q16—2442no—
333Q16C2×D24412yes—
334SD16—24no—
335SD16—2444no—
336SD16—2442no—
337D8—21no—
338D8—1221no—
339D8—1221no—
340D8C2×D24412yes—
341M16—22no—
342C2×C8C2×D24412yes—
343C2×C8#1175822212yes—
344C2×C8C2×C22⋊D6112yes—
345C2×C8C2×C22⋊D6111yescharacteristic
346C2×C8C23×D6111yescharacteristic
347C4⋊C4—22no—
348C4⋊C4—41no—
349C4⋊C4—42no—
350C4⋊C4—2442no—
351C4⋊C4—2442no—
352C4⋊C4#1175819911yescharacteristic
353C4⋊C4#1175819911yescharacteristic
354C22⋊C4—42no—
355C22⋊C4—2442no—
356C22⋊C4—2442no—
357C22⋊C4#1175819912yes—
358C42—42no—
359C42C2×C22⋊D6112yes—
360C16—1221no—
361C16—1221no—
362C16—1222no—
363C23—41no—
364C23—2441no—
365C23C2×#1175819911yescharacteristic
366Q8—22no—
367Q8—41no—
368Q8—4882no—
369Q8—2441no—
370Q8—2442no—
371Q8—2442no—
372Q8—2441no—
373D4—22no—
374D4—44no—
375D4—41no—
376D4—4882no—
377D4—2441no—
378D4—2441no—
379C2×C4—41no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—162no—
383C2×C4—161no—
384C2×C4—4882no—
385C2×C4—4882no—
386C2×C4—2442no—
387C2×C4—2441no—
388C2×C4#1451975112yes—
389C2×C4C2×#1175819912yes—
390C2×C4C2×#1175819912yes—
391C2×C4C2×#1175819911yescharacteristic
392C2×C4#1451980212yes—
393C2×C4C22×C22⋊D6111yescharacteristic
394C8—22no—
395C8#1451975112yes—
396C8C22×D24411yescharacteristic
397C8C2×#1175822211yescharacteristic
398C22—21no—
399C22—21no—
400C22—42no—
401C22—81no—
402C22—81no—
403C22—4881no—
404C22—4881no—
405C22—4881no—
406C22#1683995712yes—
407C22C22×#1175819911yescharacteristic
408C4—22no—
409C4—22no—
410C4—42no—
411C4—82no—
412C4—4881no—
413C4—4882no—
414C4#1683995712yes—
415C4C2×#1451975111yescharacteristic
416C4C2×#1451980211yescharacteristic
417C2—22no—
418C2—82no—
419C2—9761no—
420C2C2×#1683995711yescharacteristic
421C2#2145896611yescharacteristic
422C2#2145897011yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2951C2
socleC22×C61C22×D8
prime radicalC2×#16840192C2
hypercenter#2951D61
hypocenterC61#33590
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2951

Complements

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

NH ≅ G/Ntype
1. C2×#16840202C2semidirect
3. #21456313C2semidirect
4. #21447881C2semidirect
5. #21447874C2semidirect
6. #21447869C2semidirect
7. #21447864C2semidirect
8. #21447578C2semidirect
9. #21447529C2semidirect
10. C2×#16839565C2semidirect
11. C61×#2951C2semidirect
209. #2951D61semidirect
254. C61#33590semidirect

Maximal subgroups

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

subgroupindex
C2×#168402022
C2×#168401922
#214563132
#214478812
#214478742
#214478692
#214478642
#214475782
#214475292
C2×#168395652
C61×#29512
#3359061