Finite Group Databasebeta

← #29834611

Subgroups — #29834611

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×#16840202C211yesmaximal, characteristic, complemented
2#21458921C211yesmaximal, characteristic, complemented
3#21455535C211yesmaximal, characteristic
4C2×#16840038C211yesmaximal, characteristic, complemented
5#21447948C214yesmaximal, complemented
6#21447943C214yesmaximal, complemented
7#21447613C211yesmaximal, characteristic, complemented
8C2×#16839579C211yesmaximal, characteristic, complemented
9C61×#2148C211yesmaximal, characteristic, complemented
10#16840560—21no—
11C2×#14519810—21no—
12C22×Dic244C2211yescharacteristic
13C2×#14519783C2211yescharacteristic
14#16840202—22no—
15#16840202—22no—
16C2×#14519723C2212yescomplemented
17C2×#14519721C2212yescomplemented
18#16840180C2214yes—
19#16840178—22no—
20C2×#14519717C2211yescharacteristic
21M32×D61—22no—
22#16840038—22no—
23#16840038C2214yes—
24#16840034C2212yes—
25#16840029C2212yes—
26#16840022—22no—
27C2×#14519713C2211yescharacteristic
28#16839746—21no—
29C2×#14519695—21no—
30#16839591C2214yescomplemented
31#16839579C2214yescomplemented
32C2×#135×C61C2211yescharacteristic
33C2×M32×C61C2211yescharacteristic
34C2×#108×C61C2211yescharacteristic
35C61×#362C2212yescomplemented
36C61×#361C2212yescomplemented
37C2×#11758222D411yescharacteristic
38C22×Dic122—21no—
39#14519810—24no—
40#14519810—42no—
41#14519808—22no—
42#14519807—41no—
43#14519803—22no—
44#14519802—41no—
45#14519797—42no—
46C2×Dic244—21no—
47C2×Dic244—21no—
48C2×Dic244—21no—
49C2×Dic244—21no—
50C2×Dic244—21no—
51C2×Dic244—41no—
52C2×Dic244C2312yes—
53C2×#11758190—41no—
54#14519783—21no—
55#14519783—21no—
56#14519783D412yes—
57#14519783C2312yes—
58C2×#11758194—21no—
59C2×#11758194D411yescharacteristic
60C2×#11758189—21no—
61C2×C8×D61—21no—
62#14519723—24no—
63#14519721—24no—
64#14519717D412yes—
65#14519717D412yes—
66#14519716—24no—
67C2×#11758185C2312yes—
68C16×D61—24no—
69#14519713—21no—
70#14519713C2312yes—
71#14519696—22no—
72#14519695—24no—
73#14519694—22no—
74C2×#41×C61—21no—
75#135×C61—21no—
76#135×C61—21no—
77#135×C61—21no—
78#135×C61—21no—
79C2×Q16×C61C2311yescharacteristic
80C2×SD16×C61—21no—
81C2×D8×C61C2311yescharacteristic
82C2×M16×C61—21no—
83C22×C8×C61C2311yescharacteristic
84M32×C61—21no—
85M32×C61C2312yes—
86C2×C16×C61—21no—
87#108×C61—21no—
88#108×C61C2312yes—
89#41×C61—41no—
90#41×C61—41no—
91#41×C61—41no—
92#41×C61—41no—
93C2×Q8×C61—21no—
94C2×D4×C61—21no—
95C2×D4×C61—41no—
96C22×C4×C61—41no—
97C22×C4×C61C2×D411yescharacteristic
98C2×C22⋊D61—21no—
99C22×Dic61—41no—
100#11758222—21no—
101#11758222—21no—
102#11758222D812yes—
103#11758222C2×D412yes—
104C2×D244D811yescharacteristic
105C2×C4×D61—21no—
106C2×Dic122—22no—
107C2×Dic122—41no—
108C2×Dic122—41no—
109C2×Dic122—41no—
110C2×Dic122—41no—
111C2×Dic122—41no—
112C2×Dic122D811yescharacteristic
113Q16×C61—21no—
114Q16×C61—21no—
115SD16×C61—41no—
116SD16×C61—41no—
117D8×C61—21no—
118D8×C61—21no—
119M16×C61—22no—
120M16×C61—41no—
121C2×C8×C61—41no—
122C2×C8×C61C2×D411yescharacteristic
123C2×C8×C61C2×D411yescharacteristic
124C2×C8×C61C2×D411yescharacteristic
125C2×C8×C61#4111yescharacteristic
126C2×C8×C61#4111yescharacteristic
127C2×C8×C61C2411yescharacteristic
128#11758202—42no—
129#11758201—42no—
130#11758200—42no—
131#11758199—42no—
132#11758194—21no—
133#11758194—22no—
134#11758194—41no—
135#11758194C2×D412yes—
136C2×#92418—21no—
137C2×#92418—41no—
138Dic244—22no—
139Dic244—22no—
140Dic244—44no—
141Dic244—41no—
142Dic244—41no—
143#11758190—44no—
144#11758189—22no—
145#11758189—22no—
146C8×D61—22no—
147C8×D61—22no—
148C16×C61—22no—
149#11758185C2×D414yes—
150C23×C61—41no—
151C23×C61#13511yescharacteristic
152C22×D61—21no—
153Q8×C61—41no—
154Q8×C61—41no—
155D4×C61—41no—
156D4×C61—41no—
157D4×C61—81no—
158D4×C61—81no—
159C2×C4×C61—41no—
160C2×C4×C61—81no—
161C2×C4×C61—81no—
162C2×C4×C61#12111yescharacteristic
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61C2×D811yescharacteristic
165C2×C4×C61C2×D811yescharacteristic
166C2×C4×C61#13511yescharacteristic
167C2×C4×C61C22×D411yescharacteristic
168C22⋊D61—22no—
169C22⋊D61—22no—
170C2×Dic61—21no—
171C2×Dic61—42no—
172C2×Dic61—81no—
173C2×Dic61—81no—
174D244—21no—
175D244C2×D812yes—
176C4×D61—22no—
177C4×D61—22no—
178Dic122—21no—
179Dic122—42no—
180Dic122—42no—
181Dic122—81no—
182Dic122—81no—
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—81no—
194C22×C61—161no—
195C22×C61#46611yescharacteristic
196C22×C61#46611yescharacteristic
197C22×C61C2×#13511yescharacteristic
198C2×D61—22no—
199C2×D61—81no—
200C4×C61—81no—
201C4×C61#46611yescharacteristic
202C4×C61#46611yescharacteristic
203C4×C61C2×#12111yescharacteristic
204C4×C61C22×D811yescharacteristic
205Dic61—22no—
206Dic61—82no—
207C2×#510—611no—
208#3407—611no—
209#3067—611no—
210C2×#473—611no—
211#2223—614no—
212#2222—614no—
213#2161—611no—
214#2148D6111yescharacteristic, complemented
215C2×#362—611no—
216C2×C61—21no—
217C2×C61—21no—
218C2×C61—21no—
219C2×C61—161no—
220C2×C61C2×#46611yescharacteristic
221D61—42no—
222#575—1221no—
223C2×#137—1221no—
224C2×#135C2×D6111yescharacteristic
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×#108—611no—
242C2×#108C2×D6111yescharacteristic
243#413—1221no—
244C2×#101—1221no—
245#368—614no—
246#362—614no—
247#362C2×D6112yescomplemented
248#361C2×D6112yescomplemented
249C61#3361911yescharacteristic, complemented, Sylow
250C2×#41—21no—
251C2×#41—611no—
252C22×Q8—1221no—
253#137—1224no—
254#137—1222no—
255#137—2442no—
256#137—2441no—
257AGL(1,ℤ/8)—1222no—
258AGL(1,ℤ/8)—2441no—
259#135—21no—
260#135—21no—
261#135—21no—
262#135—21no—
263#135—2442no—
264C2×Q16—612no—
265C2×Q16—1221no—
266C2×Q16—1221no—
267C2×Q16—1221no—
268C2×Q16—1221no—
269C2×Q16—1221no—
270C2×Q16—2441no—
271C2×Q16C22×D6111yescharacteristic
272C2×SD16—21no—
273C2×SD16—2441no—
274C2×D8C22×D6111yescharacteristic
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—1224no—
300#101—1222no—
301#100—1222no—
302#41—41no—
303#41—41no—
304#41—41no—
305#41—41no—
306#41—612no—
307#41—612no—
308#41—1221no—
309#41—1221no—
310C2×Q8—21no—
311C2×Q8—611no—
312C2×Q8—1222no—
313C2×Q8—2441no—
314C2×Q8—2441no—
315C2×Q8—2441no—
316C2×Q8—2441no—
317C2×Q8—2441no—
318C2×D4—21no—
319C2×D4—41no—
320C2×D4—611no—
321C2×D4—1221no—
322C22×C4—41no—
323C22×C4—1221no—
324C22×C4—2441no—
325C22×C4D4×D6111yescharacteristic
326Q16—21no—
327Q16—21no—
328Q16—1222no—
329Q16—1222no—
330Q16—2444no—
331Q16—2442no—
332Q16—2441no—
333Q16—2441no—
334SD16—41no—
335SD16—41no—
336SD16—2444no—
337SD16—2442no—
338SD16—2442no—
339D8—21no—
340D8—21no—
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—41no—
364C23—1221no—
365C23#1451979811yescharacteristic
366Q8—41no—
367Q8—41no—
368Q8—612no—
369Q8—1221no—
370Q8—4881no—
371Q8—4881no—
372Q8—2442no—
373Q8—2442no—
374D4—41no—
375D4—41no—
376D4—81no—
377D4—81no—
378D4—612no—
379D4—1222no—
380D4—1222no—
381D4—1221no—
382C2×C4—41no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—1221no—
386C2×C4—1222no—
387C2×C4—1222no—
388C2×C4—4881no—
389C2×C4—4881no—
390C2×C4—2442no—
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—81no—
406C22—161no—
407C22—1222no—
408C22—4881no—
409C22#1683999711yescharacteristic
410C22#1683999711yescharacteristic
411C22C2×#1451979811yescharacteristic
412C4—81no—
413C4—1222no—
414C4—4882no—
415C4#1683999711yescharacteristic
416C4#1683999711yescharacteristic
417C4C2×#1451975011yescharacteristic
418C4C2×D8×D6111yescharacteristic
419C2—21no—
420C2—21no—
421C2—21no—
422C2—161no—
423C2—2442no—
424C2C2×#1683999711yescharacteristic
subgroupG/H
centerC2C2×#16839997
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2148C2
socleC2×C61C2×#466
prime radical#21455535C2
hypercenter#2148D61
hypocenterC61#33619
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ #2148

Complements

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

NH ≅ G/Ntype
1. C2×#16840202C2semidirect
2. #21458921C2semidirect
4. C2×#16840038C2semidirect
5. #21447948C2semidirect
6. #21447943C2semidirect
7. #21447613C2semidirect
8. C2×#16839579C2semidirect
9. C61×#2148C2semidirect
16. C2×#14519723C22semidirect
17. C2×#14519721C22semidirect
30. #16839591C22semidirect
31. #16839579C22semidirect
35. C61×#362C22semidirect
36. C61×#361C22semidirect
214. #2148D61semidirect
247. #362C2×D61semidirect
248. #361C2×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×#168402022
#214589212
#214555352
C2×#168400382
#214479482
#214479432
#214476132
C2×#168395792
C61×#21482
#3361961