Finite Group Databasebeta

← #29834609

Subgroups — #29834609

Order 15616.

636 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840205C211yesmaximal, characteristic, complemented
2#21458921C211yesmaximal, characteristic, complemented
3#21455540C211yesmaximal, characteristic
4C2×#16840038C211yesmaximal, characteristic, complemented
5#21447950C214yesmaximal, complemented
6#21447945C214yesmaximal, complemented
7#21447566C211yesmaximal, characteristic, complemented
8C2×#16839577C211yesmaximal, characteristic, complemented
9C61×#2161C211yesmaximal, characteristic, complemented
10#16840562—21no—
11C2×#14519810—21no—
12C2×#14519799C2211yescharacteristic
13C22×Dic244C2211yescharacteristic
14C2×#14519783C2211yescharacteristic
15#16840205—22no—
16#16840205—22no—
17C2×Dic488C2212yescomplemented
18C2×#14519722C2212yescomplemented
19#16840180—22no—
20#16840178C2214yes—
21C2×#14519717C2211yescharacteristic
22M32×D61—22no—
23#16840038—22no—
24#16840038C2212yes—
25#16840038C2214yes—
26#16840033C2212yes—
27#16840026—22no—
28#16839744—21no—
29C2×#14519695—21no—
30#16839577C2212yescomplemented
31#16839577C2214yescomplemented
32#16839574C2212yescomplemented
33C2×M32×C61C2211yescharacteristic
34C2×#108×C61C2211yescharacteristic
35C2×#108×C61C2211yescharacteristic
36C61×#368C2214yescomplemented
37C2×#11758222—21no—
38C2×#11758222D411yescharacteristic
39C22×Dic122—21no—
40#14519810—22no—
41#14519810—24no—
42#14519810—42no—
43#14519810—41no—
44#14519805—22no—
45#14519805—41no—
46#14519799—21no—
47#14519799—21no—
48#14519799—21no—
49#14519799—21no—
50#14519799—42no—
51C2×Dic244—21no—
52C2×Dic244—21no—
53C2×Dic244—21no—
54C2×Dic244—21no—
55C2×Dic244—21no—
56C2×Dic244—41no—
57C2×Dic244C2312yes—
58C2×Dic244C2311yescharacteristic
59C2×#11758190—21no—
60C2×#11758190—41no—
61C2×D488C2311yescharacteristic
62#14519783—21no—
63#14519783—21no—
64#14519783D412yes—
65#14519783C2312yes—
66C2×#11758194D411yescharacteristic
67C2×#11758189—21no—
68C2×C8×D61—21no—
69Dic488—24no—
70#14519722—24no—
71#14519717—21no—
72#14519717C2312yes—
73#14519716—24no—
74C2×#11758185—21no—
75C16×D61—24no—
76#14519695—24no—
77#14519695—22no—
78#14519693—22no—
79C2×M16×C61—21no—
80C2×M16×C61—21no—
81C22×C8×C61C2311yescharacteristic
82M32×C61D412yes—
83M32×C61D412yes—
84C2×C16×C61C2312yes—
85#108×C61—21no—
86#108×C61—21no—
87#108×C61C2312yes—
88#108×C61C2312yes—
89C22×C4×C61C2×D411yescharacteristic
90C2×C22⋊D61—21no—
91C2×C22⋊D61—41no—
92C22×Dic61—41no—
93#11758222—21no—
94#11758222—21no—
95#11758222—41no—
96#11758222—41no—
97#11758222—41no—
98#11758222—41no—
99#11758222D812yes—
100#11758222C2×D412yes—
101C2×D244—21no—
102C2×D244D811yescharacteristic
103C2×C4×D61—21no—
104C2×C4×D61—41no—
105C2×Dic122—22no—
106C2×Dic122—21no—
107C2×Dic122—41no—
108C2×Dic122—41no—
109C2×Dic122—41no—
110C2×Dic122—41no—
111C2×Dic122—41no—
112C2×Dic122D811yescharacteristic
113M16×C61—22no—
114M16×C61—22no—
115M16×C61—41no—
116M16×C61—41no—
117C2×C8×C61—41no—
118C2×C8×C61—41no—
119C2×C8×C61C2×D411yescharacteristic
120C2×C8×C61C2×D411yescharacteristic
121C2×C8×C61C2×D411yescharacteristic
122C2×C8×C61#4111yescharacteristic
123C2×C8×C61#4111yescharacteristic
124C2×C8×C61C2411yescharacteristic
125#11758194—21no—
126#11758194C2×D412yes—
127C2×#92418—21no—
128Dic244—22no—
129Dic244—21no—
130Dic244—22no—
131Dic244—21no—
132Dic244—44no—
133Dic244—42no—
134Dic244—41no—
135Dic244—41no—
136D488—21no—
137D488—21no—
138D488—42no—
139#11758190—44no—
140#11758190—42no—
141#11758190—42no—
142#11758190—41no—
143#11758190—41no—
144#11758189—22no—
145#11758189—22no—
146C8×D61—22no—
147C8×D61—22no—
148C16×C61C2×D414yes—
149#11758185—22no—
150C23×C61#13511yescharacteristic
151C22×D61—21no—
152C22×D61—41no—
153C2×C4×C61#12111yescharacteristic
154C2×C4×C61#12111yescharacteristic
155C2×C4×C61C2×D811yescharacteristic
156C2×C4×C61C2×D811yescharacteristic
157C2×C4×C61#13511yescharacteristic
158C2×C4×C61C22×D411yescharacteristic
159C22⋊D61—22no—
160C22⋊D61—22no—
161C22⋊D61—81no—
162C22⋊D61—81no—
163C2×Dic61—21no—
164C2×Dic61—42no—
165C2×Dic61—41no—
166C2×Dic61—81no—
167C2×Dic61—81no—
168D244—21no—
169D244—41no—
170D244—41no—
171D244C2×D812yes—
172C4×D61—22no—
173C4×D61—22no—
174C4×D61—81no—
175C4×D61—81no—
176Dic122—21no—
177Dic122—41no—
178Dic122—42no—
179Dic122—42no—
180Dic122—41no—
181Dic122—81no—
182Dic122—81no—
183Dic122C2×D812yes—
184C8×C61—42no—
185C8×C61—42no—
186C8×C61#12111yescharacteristic
187C8×C61#12111yescharacteristic
188C8×C61C22×D411yescharacteristic
189C8×C61C2×#4111yescharacteristic
190#92418—22no—
191#33619—611nomaximal, Sylow
192C22×C61—41no—
193C22×C61#46611yescharacteristic
194C22×C61#46611yescharacteristic
195C22×C61C2×#13511yescharacteristic
196C2×D61—22no—
197C2×D61—81no—
198C2×D61—81no—
199C2×D61—161no—
200C4×C61#46611yescharacteristic
201C4×C61#46611yescharacteristic
202C4×C61C2×#12111yescharacteristic
203C4×C61C22×D811yescharacteristic
204Dic61—22no—
205Dic61—82no—
206Dic61—81no—
207C2×#510—611no—
208#3407—611no—
209#3067—611no—
210C2×#473—611no—
211#2223—614no—
212#2222—614no—
213#2161D6111yescharacteristic, complemented
214#2148—611no—
215C2×#362—611no—
216C2×C61—21no—
217C2×C61—21no—
218C2×C61—21no—
219C2×C61C2×#46611yescharacteristic
220D61—42no—
221D61—161no—
222#575—1221no—
223C2×#137—1221no—
224C2×#135—611no—
225C22×Q16—611no—
226C2×#131—611no—
227#510—1222no—
228#510—1222no—
229C2×Q32—612no—
230C2×SD32—612no—
231#504—614no—
232#504—1222no—
233C2×M32—611no—
234C2×M32—1222no—
235C2×M32C2×D6111yescharacteristic
236#473—612no—
237#473—614no—
238#473—1222no—
239#472—612no—
240#471—1222no—
241C2×#108C2×D6111yescharacteristic
242C2×#108C2×D6111yescharacteristic
243#413—1221no—
244C2×#101—1221no—
245#368C2×D6114yescomplemented
246#362—614no—
247#362—612no—
248#361—612no—
249C61#3361911yescharacteristic, complemented, Sylow
250C2×#41—611no—
251C2×#41—1221no—
252C22×Q8—1221no—
253#137—1224no—
254#137—1222no—
255#137—2441no—
256#137—2442no—
257AGL(1,ℤ/8)—1222no—
258AGL(1,ℤ/8)—2441no—
259#135—1221no—
260#135—1221no—
261#135—1221no—
262#135—1221no—
263#135—2442no—
264C2×Q16—612no—
265C2×Q16—611no—
266C2×Q16—1221no—
267C2×Q16—1221no—
268C2×Q16—1221no—
269C2×Q16—1221no—
270C2×Q16—1221no—
271C2×Q16—2441no—
272C2×SD16—1221no—
273C2×SD16—2441no—
274C2×D8—611no—
275#131—612no—
276#131—612no—
277#131—1221no—
278#131—1221no—
279C2×M16—21no—
280C2×M16—21no—
281C2×M16—611no—
282C2×M16—1221no—
283C22×C8—1221no—
284C22×C8C22×D6111yescharacteristic
285Q32—1224no—
286SD32—1224no—
287M32—612no—
288M32—1221no—
289M32—1224no—
290M32D24412yes—
291M32D24412yes—
292C2×C16—1221no—
293C2×C16—1224no—
294C2×C16C22×D6112yes—
295#108—21no—
296#108—21no—
297#108C22×D6112yes—
298#108C22×D6112yes—
299#101—1224no—
300#101—1222no—
301#100—1222no—
302#41—612no—
303#41—612no—
304#41—1221no—
305#41—1221no—
306#41—2441no—
307#41—2441no—
308#41—2441no—
309#41—2441no—
310C2×Q8—611no—
311C2×Q8—1221no—
312C2×Q8—1222no—
313C2×Q8—2441no—
314C2×Q8—2441no—
315C2×Q8—2441no—
316C2×Q8—2441no—
317C2×Q8—2441no—
318C2×D4—611no—
319C2×D4—1221no—
320C2×D4—1221no—
321C2×D4—2441no—
322C22×C4—1221no—
323C22×C4—2441no—
324C22×C4—2441no—
325C22×C4D4×D6111yescharacteristic
326Q16—1222no—
327Q16—1221no—
328Q16—1222no—
329Q16—1221no—
330Q16—2444no—
331Q16—2442no—
332Q16—2441no—
333Q16—2441no—
334SD16—2444no—
335SD16—2442no—
336SD16—2442no—
337SD16—2441no—
338SD16—2441no—
339D8—1221no—
340D8—1221no—
341D8—2442no—
342M16—22no—
343M16—22no—
344M16—41no—
345M16—41no—
346M16—612no—
347M16—1221no—
348M16—1222no—
349M16—1222no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—1221no—
353C2×C8—1222no—
354C2×C8—1222no—
355C2×C8C2×D24411yescharacteristic
356C2×C8C2×D24411yescharacteristic
357C2×C8D4×D6111yescharacteristic
358C2×C8#1175822611yescharacteristic
359C2×C8#1175822611yescharacteristic
360C2×C8C23×D6111yescharacteristic
361C16—1222no—
362C16C2×D24414yes—
363C23—1221no—
364C23—2441no—
365C23#1451980011yescharacteristic
366Q8—612no—
367Q8—1221no—
368Q8—4881no—
369Q8—4881no—
370Q8—2441no—
371Q8—2442no—
372Q8—2442no—
373Q8—2441no—
374D4—612no—
375D4—1222no—
376D4—1222no—
377D4—1221no—
378D4—4881no—
379D4—4881no—
380D4—2441no—
381D4—2441no—
382C2×C4—1221no—
383C2×C4—1222no—
384C2×C4—1222no—
385C2×C4—4881no—
386C2×C4—4881no—
387C2×C4—4881no—
388C2×C4—4881no—
389C2×C4—2441no—
390C2×C4—2442no—
391C2×C4#1451975211yescharacteristic
392C2×C4#1451975211yescharacteristic
393C2×C4D8×D6111yescharacteristic
394C2×C4D8×D6111yescharacteristic
395C2×C4#1451980011yescharacteristic
396C2×C4C2×D4×D6111yescharacteristic
397C8—42no—
398C8—42no—
399C8—1222no—
400C8#1451975211yescharacteristic
401C8#1451975211yescharacteristic
402C8C22×D24411yescharacteristic
403C8C2×#1175822611yescharacteristic
404C22—41no—
405C22—1222no—
406C22—9761no—
407C22—4881no—
408C22—4881no—
409C22#1683999911yescharacteristic
410C22#1683999911yescharacteristic
411C22C2×#1451980011yescharacteristic
412C4—1222no—
413C4—4881no—
414C4—4882no—
415C4#1683999911yescharacteristic
416C4#1683999911yescharacteristic
417C4C2×#1451975211yescharacteristic
418C4C2×D8×D6111yescharacteristic
419C2—21no—
420C2—21no—
421C2—21no—
422C2—9761no—
423C2—2442no—
424C2C2×#1683999911yescharacteristic
subgroupG/H
centerC2C2×#16839999
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2161C2
socleC2×C61C2×#466
prime radical#21455540C2
hypercenter#2161D61
hypocenterC61#33619
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ #2161

Complements

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

NH ≅ G/Ntype
1. C2×#16840205C2semidirect
2. #21458921C2semidirect
4. C2×#16840038C2semidirect
5. #21447950C2semidirect
6. #21447945C2semidirect
7. #21447566C2semidirect
8. C2×#16839577C2semidirect
9. C61×#2161C2semidirect
17. C2×Dic488C22semidirect
18. C2×#14519722C22semidirect
30. #16839577C22semidirect
31. #16839577C22semidirect
32. #16839574C22semidirect
36. C61×#368C22semidirect
213. #2161D61semidirect
245. #368C2×D61semidirect
249. C61#33619semidirect

Maximal subgroups

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

subgroupindex
C2×#168402052
#214589212
#214555402
C2×#168400382
#214479502
#214479452
#214475662
C2×#168395772
C61×#21612
#3361961