Finite Group Databasebeta

← #29834607

Subgroups — #29834607

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×#16840203C211yesmaximal, characteristic, complemented
2#21458921C211yesmaximal, characteristic, complemented
3#21455540C211yesmaximal, characteristic
4C2×#16840034C211yesmaximal, characteristic, complemented
5#21447952C214yesmaximal, complemented
6#21447943C214yesmaximal, complemented
7#21447613C211yesmaximal, characteristic, complemented
8#21447569C211yesmaximal, characteristic, complemented
9C2×C61×#362C211yesmaximal, characteristic, complemented
10#16840562—21no—
11C2×#14519808—21no—
12C2×#14519799C2211yescharacteristic
13C2×#14519783C2211yescharacteristic
14#16840203—22no—
15#16840203—22no—
16C2×#14519723C2212yescomplemented
17C2×#14519720C2212yescomplemented
18#16840180C2214yes—
19#16840178—22no—
20C2×#14519717C2211yescharacteristic
21M32×D61—22no—
22#16840038C2212yes—
23#16840034—22no—
24#16840034C2214yes—
25#16840033C2212yes—
26#16840026—22no—
27C2×#14519713C2211yescharacteristic
28#16839744—21no—
29C2×#14519696—21no—
30#16839591C2214yescomplemented
31#16839579C2212yescomplemented
32#16839576C2212yescomplemented
33C22×Q16×C61C2211yescharacteristic
34C2×M32×C61C2211yescharacteristic
35C2×#108×C61C2211yescharacteristic
36C61×#362C2214yescomplemented
37C2×#11758222—21no—
38C2×#11758222D411yescharacteristic
39#14519810—22no—
40#14519810—41no—
41#14519808—24no—
42#14519808—42no—
43#14519805—22no—
44#14519805—41no—
45#14519799—21no—
46#14519799—21no—
47#14519799—21no—
48#14519799—21no—
49#14519799—42no—
50C2×Dic244C2311yescharacteristic
51C2×#11758202—41no—
52C2×#11758190—21no—
53C2×#11758201—41no—
54C2×D488C2311yescharacteristic
55#14519783—21no—
56#14519783—21no—
57#14519783D412yes—
58#14519783C2312yes—
59C2×#11758194—21no—
60C2×#11758194D411yescharacteristic
61C2×#11758189—21no—
62C2×C8×D61—21no—
63#14519723—24no—
64#14519720—24no—
65#14519717D412yes—
66#14519717D412yes—
67#14519716—24no—
68C2×#11758185C2312yes—
69C16×D61—24no—
70#14519713—21no—
71#14519713C2312yes—
72#14519696—24no—
73#14519695—22no—
74#14519693—22no—
75C22×Q8×C61—21no—
76C2×Q16×C61—21no—
77C2×Q16×C61—21no—
78C2×Q16×C61—21no—
79C2×Q16×C61—21no—
80C2×Q16×C61—21no—
81C2×Q16×C61C2312yes—
82C2×M16×C61—21no—
83C22×C8×C61C2311yescharacteristic
84M32×C61—21no—
85M32×C61C2312yes—
86C2×C16×C61—21no—
87#108×C61—21no—
88#108×C61C2312yes—
89C2×Q8×C61—22no—
90C2×Q8×C61—41no—
91C2×Q8×C61—41no—
92C2×Q8×C61—41no—
93C2×Q8×C61—41no—
94C2×Q8×C61—41no—
95C22×C4×C61—41no—
96C22×C4×C61C2×D411yescharacteristic
97C2×C22⋊D61—21no—
98C2×C22⋊D61—41no—
99#11758222—21no—
100#11758222—21no—
101#11758222—41no—
102#11758222—41no—
103#11758222—41no—
104#11758222—41no—
105#11758222D812yes—
106#11758222C2×D412yes—
107C2×D244—21no—
108C2×D244D811yescharacteristic
109C2×C4×D61—21no—
110C2×C4×D61—41no—
111C2×Dic122—21no—
112C2×Dic122D811yescharacteristic
113Q16×C61—22no—
114Q16×C61—22no—
115Q16×C61—41no—
116Q16×C61—41no—
117M16×C61—22no—
118M16×C61—41no—
119C2×C8×C61—41no—
120C2×C8×C61C2×D411yescharacteristic
121C2×C8×C61C2×D411yescharacteristic
122C2×C8×C61C2×D411yescharacteristic
123C2×C8×C61#4111yescharacteristic
124C2×C8×C61#4111yescharacteristic
125C2×C8×C61C2411yescharacteristic
126#11758202—44no—
127#11758201—44no—
128#11758194—21no—
129#11758194—22no—
130#11758194—41no—
131#11758194C2×D412yes—
132C2×#92418—21no—
133C2×#92418—41no—
134Dic244—21no—
135Dic244—21no—
136Dic244—42no—
137D488—21no—
138D488—21no—
139D488—42no—
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×C61—22no—
149#11758185C2×D414yes—
150C23×C61#13511yescharacteristic
151C22×D61—21no—
152C22×D61—41no—
153Q8×C61—42no—
154Q8×C61—42no—
155Q8×C61—81no—
156Q8×C61—81no—
157C2×C4×C61—42no—
158C2×C4×C61—81no—
159C2×C4×C61—81no—
160C2×C4×C61#12111yescharacteristic
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61C2×D811yescharacteristic
163C2×C4×C61C2×D811yescharacteristic
164C2×C4×C61#13511yescharacteristic
165C2×C4×C61C22×D411yescharacteristic
166C22⋊D61—22no—
167C22⋊D61—22no—
168C22⋊D61—81no—
169C22⋊D61—81no—
170C2×Dic61—21no—
171C2×Dic61—41no—
172D244—21no—
173D244—41no—
174D244—41no—
175D244C2×D812yes—
176C4×D61—22no—
177C4×D61—22no—
178C4×D61—81no—
179C4×D61—81no—
180Dic122—21no—
181Dic122—41no—
182Dic122—41no—
183Dic122C2×D812yes—
184C8×C61—42no—
185C8×C61#12111yescharacteristic
186C8×C61#12111yescharacteristic
187C8×C61C22×D411yescharacteristic
188C8×C61C2×#4111yescharacteristic
189#92418—22no—
190#92418—42no—
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—82no—
201C4×C61#46611yescharacteristic
202C4×C61#46611yescharacteristic
203C4×C61C2×#12111yescharacteristic
204C4×C61C22×D811yescharacteristic
205Dic61—22no—
206Dic61—81no—
207C2×#510—611no—
208#3407—611no—
209#3067—611no—
210C2×#473—611no—
211#2223—614no—
212#2222—614no—
213#2161—611no—
214#2148—611no—
215C2×#362D6111yescharacteristic, complemented
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×Q16C2×D6111yescharacteristic
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—614no—
237#473—612no—
238#473—1222no—
239#472—612no—
240#471—1222no—
241C2×#108—611no—
242C2×#108C2×D6111yescharacteristic
243#413—1221no—
244C2×#101—1221no—
245#368—614no—
246#362—612no—
247#362C2×D6114yescomplemented
248#361—612no—
249C61#3361911yescharacteristic, complemented, Sylow
250C2×#41—611no—
251C2×#41—1221no—
252C22×Q8—21no—
253#137—1222no—
254#137—1224no—
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—21no—
265C2×Q16—21no—
266C2×Q16—21no—
267C2×Q16—21no—
268C2×Q16—21no—
269C2×Q16—611no—
270C2×Q16—2441no—
271C2×Q16C22×D6112yes—
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—611no—
281C2×M16—1221no—
282C2×M16—1221no—
283C22×C8—1221no—
284C22×C8C22×D6111yescharacteristic
285Q32—1224no—
286SD32—1224no—
287M32—21no—
288M32—612no—
289M32—612no—
290M32—1224no—
291M32C22×D6112yes—
292C2×C16—21no—
293C2×C16—612no—
294C2×C16—1224no—
295#108—21no—
296#108—612no—
297#108—1221no—
298#108C22×D6112yes—
299#101—1222no—
300#101—1224no—
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—22no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—611no—
317C2×Q8—1221no—
318C2×D4—611no—
319C2×D4—1221no—
320C2×D4—1221no—
321C2×D4—2441no—
322C22×C4—41no—
323C22×C4—1221no—
324C22×C4—2441no—
325C22×C4D4×D6111yescharacteristic
326Q16—22no—
327Q16—22no—
328Q16—41no—
329Q16—41no—
330Q16—1221no—
331Q16—1221no—
332Q16—2442no—
333Q16—2444no—
334SD16—2442no—
335SD16—2442no—
336SD16—2444no—
337SD16—2441no—
338SD16—2441no—
339D8—1221no—
340D8—1221no—
341D8—2442no—
342M16—22no—
343M16—41no—
344M16—612no—
345M16—1221no—
346M16—1222no—
347M16—1222no—
348M16—1222no—
349M16—2441no—
350C2×C8—41no—
351C2×C8—1221no—
352C2×C8—1222no—
353C2×C8—1222no—
354C2×C8—2441no—
355C2×C8#1175822211yescharacteristic
356C2×C8#1175822211yescharacteristic
357C2×C8D4×D6111yescharacteristic
358C2×C8D4×D6111yescharacteristic
359C2×C8D4×D6111yescharacteristic
360C2×C8C23×D6111yescharacteristic
361C16—22no—
362C16—614no—
363C23—1221no—
364C23—2441no—
365C23#1451979811yescharacteristic
366Q8—42no—
367Q8—42no—
368Q8—81no—
369Q8—81no—
370Q8—612no—
371Q8—1221no—
372Q8—2441no—
373Q8—2441no—
374D4—612no—
375D4—1222no—
376D4—1222no—
377D4—1221no—
378D4—4881no—
379D4—4881no—
380D4—2441no—
381D4—2441no—
382C2×C4—42no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—1221no—
386C2×C4—1222no—
387C2×C4—1222no—
388C2×C4—4881no—
389C2×C4—4881no—
390C2×C4—2441no—
391C2×C4#1451975011yescharacteristic
392C2×C4#1451975011yescharacteristic
393C2×C4D8×D6111yescharacteristic
394C2×C4D8×D6111yescharacteristic
395C2×C4#1451979811yescharacteristic
396C2×C4C2×D4×D6111yescharacteristic
397C8—42no—
398C8—1222no—
399C8—2442no—
400C8#1451975011yescharacteristic
401C8#1451975011yescharacteristic
402C8C2×D4×D6111yescharacteristic
403C8C2×#1175822211yescharacteristic
404C22—41no—
405C22—1222no—
406C22—9761no—
407C22—4881no—
408C22—4881no—
409C22#1683999711yescharacteristic
410C22#1683999711yescharacteristic
411C22C2×#1451979811yescharacteristic
412C4—82no—
413C4—1222no—
414C4—4881no—
415C4#1683999711yescharacteristic
416C4#1683999711yescharacteristic
417C4C2×#1451975011yescharacteristic
418C4C2×D8×D6111yescharacteristic
419C2—21no—
420C2—21no—
421C2—21no—
422C2—9761no—
423C2—2442no—
424C2C2×#1683999711yescharacteristic
subgroupG/H
centerC2C2×#16839997
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#362C2
socleC2×C61C2×#466
prime radical#21455540C2
hypercenterC2×#362D61
hypocenterC61#33619
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ C2×#362

Complements

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

NH ≅ G/Ntype
1. C2×#16840203C2semidirect
2. #21458921C2semidirect
4. C2×#16840034C2semidirect
5. #21447952C2semidirect
6. #21447943C2semidirect
7. #21447613C2semidirect
8. #21447569C2semidirect
9. C2×C61×#362C2semidirect
16. C2×#14519723C22semidirect
17. C2×#14519720C22semidirect
30. #16839591C22semidirect
31. #16839579C22semidirect
32. #16839576C22semidirect
36. C61×#362C22semidirect
215. C2×#362D61semidirect
247. #362C2×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×#168402032
#214589212
#214555402
C2×#168400342
#214479522
#214479432
#214476132
#214475692
C2×C61×#3622
#3361961