Finite Group Databasebeta

← #29834605

Subgroups — #29834605

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#21458922C211yesmaximal, characteristic, complemented
3#21455539C211yesmaximal, characteristic
4#21447951C214yesmaximal, complemented
5#21447942C214yesmaximal, complemented
6#21447613C211yesmaximal, characteristic, complemented
7#21447566C211yesmaximal, characteristic, complemented
8C2×#16839578C211yesmaximal, characteristic, complemented
9C2×C61×#473C211yesmaximal, characteristic, complemented
10#16840560—21no—
11C2×#14519799C2211yescharacteristic
12#16840203—22no—
13#16840203—22no—
14C2×#14519723C2212yescomplemented
15C2×#14519720C2212yescomplemented
16#16840179—22no—
17#16840179C2214yes—
18C2×#14519717—22no—
19C2×#14519717C2211yescharacteristic
20C2×#14519717C2211yescharacteristic
21#16840037C2212yes—
22#16840032C2212yes—
23#16840025—22no—
24C2×#14519713C2211yescharacteristic
25#16839744—21no—
26#16839591C2214yescomplemented
27#16839578C2214yescomplemented
28#16839577C2212yescomplemented
29#16839574C2212yescomplemented
30C2×#137×C61—21no—
31C22×Q16×C61C2211yescharacteristic
32C2×#131×C61C2211yescharacteristic
33C61×#473—22no—
34C61×#473C2214yes—
35C2×#108×C61C2211yescharacteristic
36C2×#101×C61—21no—
37C2×#11758222—21no—
38#14519808—41no—
39#14519807—22no—
40#14519803—41no—
41#14519802—22no—
42#14519799—21no—
43#14519799—21no—
44#14519799—21no—
45#14519799—21no—
46#14519797—42no—
47C2×Dic244C2311yescharacteristic
48C2×#11758190—21no—
49C2×D488C2311yescharacteristic
50C2×#11758194—21no—
51#14519723—24no—
52#14519720—24no—
53#14519717—21no—
54#14519717—24no—
55#14519717D412yes—
56#14519717D412yes—
57#14519717C2312yes—
58C2×#11758185—21no—
59C2×#11758185—24no—
60C2×#11758185C2312yes—
61#14519713—21no—
62#14519713C2312yes—
63#14519695—22no—
64#14519693—22no—
65C2×#41×C61D411yescharacteristic
66C22×Q8×C61—21no—
67#137×C61—24no—
68#137×C61—42no—
69C2×Q16×C61—21no—
70C2×Q16×C61—21no—
71C2×Q16×C61—21no—
72C2×Q16×C61—21no—
73C2×Q16×C61—21no—
74C2×Q16×C61—41no—
75C2×Q16×C61C2312yes—
76C2×SD16×C61—41no—
77#131×C61—21no—
78#131×C61—21no—
79#131×C61D412yes—
80#131×C61C2312yes—
81C2×M16×C61—21no—
82C2×M16×C61—21no—
83C2×M16×C61D411yescharacteristic
84C22×C8×C61—21no—
85C22×C8×C61C2311yescharacteristic
86#108×C61—21no—
87#108×C61C2312yes—
88#101×C61—24no—
89#41×C61—21no—
90#41×C61—21no—
91#41×C61D812yes—
92#41×C61C2×D412yes—
93C2×Q8×C61—22no—
94C2×Q8×C61—41no—
95C2×Q8×C61—41no—
96C2×Q8×C61—41no—
97C2×Q8×C61—41no—
98C2×Q8×C61—41no—
99C2×Q8×C61D811yescharacteristic
100C2×D4×C61—21no—
101C2×D4×C61D811yescharacteristic
102C22×C4×C61—21no—
103C22×C4×C61—41no—
104C22×C4×C61C2×D411yescharacteristic
105C2×C22⋊D61—41no—
106#11758222—41no—
107#11758222—41no—
108#11758222—41no—
109#11758222—41no—
110C2×D244—21no—
111C2×C4×D61—41no—
112C2×Dic122—21no—
113Q16×C61—22no—
114Q16×C61—22no—
115Q16×C61—44no—
116Q16×C61—41no—
117Q16×C61—41no—
118SD16×C61—44no—
119M16×C61—21no—
120M16×C61—22no—
121M16×C61—22no—
122M16×C61—22no—
123M16×C61—41no—
124M16×C61C2×D412yes—
125C2×C8×C61—21no—
126C2×C8×C61—22no—
127C2×C8×C61—22no—
128C2×C8×C61—41no—
129C2×C8×C61C2×D411yescharacteristic
130C2×C8×C61C2×D411yescharacteristic
131C2×C8×C61C2×D411yescharacteristic
132C2×C8×C61#4111yescharacteristic
133C2×C8×C61#4111yescharacteristic
134C2×C8×C61C2411yescharacteristic
135#11758202—42no—
136#11758201—42no—
137#11758200—42no—
138#11758199—42no—
139#11758194—22no—
140#11758194—41no—
141C2×#92418—41no—
142Dic244—21no—
143Dic244—21no—
144D488—21no—
145D488—21no—
146#11758190—41no—
147#11758190—41no—
148#11758185—22no—
149#11758185C2×D414yes—
150C23×C61—21no—
151C23×C61#13511yescharacteristic
152C22×D61—41no—
153Q8×C61—21no—
154Q8×C61—42no—
155Q8×C61—42no—
156Q8×C61—81no—
157Q8×C61—81no—
158Q8×C61C2×D812yes—
159D4×C61—22no—
160D4×C61—22no—
161D4×C61—21no—
162D4×C61C2×D812yes—
163C2×C4×C61—21no—
164C2×C4×C61—22no—
165C2×C4×C61—22no—
166C2×C4×C61—42no—
167C2×C4×C61—81no—
168C2×C4×C61—81no—
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61C2×D811yescharacteristic
172C2×C4×C61C2×D811yescharacteristic
173C2×C4×C61#13511yescharacteristic
174C2×C4×C61C22×D411yescharacteristic
175C22⋊D61—81no—
176C22⋊D61—81no—
177C2×Dic61—41no—
178D244—41no—
179D244—41no—
180C4×D61—81no—
181C4×D61—81no—
182Dic122—41no—
183Dic122—41no—
184C8×C61—22no—
185C8×C61—42no—
186C8×C61#12111yescharacteristic
187C8×C61#12111yescharacteristic
188C8×C61C22×D411yescharacteristic
189C8×C61C2×#4111yescharacteristic
190#92418—42no—
191#33619—611nomaximal, Sylow
192C22×C61—22no—
193C22×C61—41no—
194C22×C61—81no—
195C22×C61#46611yescharacteristic
196C22×C61#46611yescharacteristic
197C22×C61C2×#13511yescharacteristic
198C2×D61—81no—
199C2×D61—161no—
200C4×C61—22no—
201C4×C61—82no—
202C4×C61#46611yescharacteristic
203C4×C61#46611yescharacteristic
204C4×C61C2×#12111yescharacteristic
205C4×C61C22×D811yescharacteristic
206Dic61—81no—
207C2×#510—611no—
208#3407—611no—
209#3067—611no—
210C2×#473D6111yescharacteristic, complemented
211#2223—614no—
212#2222—614no—
213#2161—611no—
214#2148—611no—
215C2×#362—611no—
216C2×C61—21no—
217C2×C61—21no—
218C2×C61—21no—
219C2×C61—42no—
220C2×C61C2×#46611yescharacteristic
221D61—161no—
222#575—1221no—
223C2×#137—21no—
224C2×#135—611no—
225C22×Q16C2×D6111yescharacteristic
226C2×#131C2×D6111yescharacteristic
227#510—1222no—
228#510—1222no—
229C2×Q32—612no—
230C2×SD32—612no—
231#504—614no—
232#504—1222no—
233C2×M32—611no—
234C2×M32—611no—
235C2×M32—1222no—
236#473—22no—
237#473—612no—
238#473C2×D6114yes—
239#472—612no—
240#471—1222no—
241C2×#108—611no—
242C2×#108C2×D6111yescharacteristic
243#413—1221no—
244C2×#101—21no—
245#368—614no—
246#362—614no—
247#362—612no—
248#361—612no—
249C61#3361911yescharacteristic, complemented, Sylow
250C2×#41—1221no—
251C2×#41C22⋊D6111yescharacteristic
252C22×Q8—21no—
253#137—24no—
254#137—42no—
255#137—1222no—
256#137—2441no—
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—41no—
270C2×Q16—611no—
271C2×Q16C22×D6112yes—
272C2×SD16—41no—
273C2×SD16—1221no—
274C2×D8—611no—
275#131—21no—
276#131—21no—
277#131C22⋊D6112yes—
278#131C22×D6112yes—
279C2×M16—21no—
280C2×M16—21no—
281C2×M16—1221no—
282C2×M16C22⋊D6111yescharacteristic
283C22×C8—21no—
284C22×C8C22×D6111yescharacteristic
285Q32—1224no—
286SD32—1224no—
287M32—612no—
288M32—612no—
289M32—612no—
290M32—1224no—
291M32—1221no—
292C2×C16—612no—
293C2×C16—1224no—
294C2×C16—1221no—
295#108—21no—
296#108—612no—
297#108—1221no—
298#108C22×D6112yes—
299#101—24no—
300#101—1222no—
301#100—1222no—
302#41—21no—
303#41—21no—
304#41—2441no—
305#41—2441no—
306#41—2441no—
307#41—2441no—
308#41#1175819912yes—
309#41C2×C22⋊D6112yes—
310C2×Q8—22no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—1221no—
317C2×Q8#1175819911yescharacteristic
318C2×D4—21no—
319C2×D4—1221no—
320C2×D4—2441no—
321C2×D4#1175819911yescharacteristic
322C22×C4—21no—
323C22×C4—41no—
324C22×C4—2441no—
325C22×C4C2×C22⋊D6111yescharacteristic
326Q16—22no—
327Q16—22no—
328Q16—44no—
329Q16—41no—
330Q16—41no—
331Q16—1221no—
332Q16—1221no—
333Q16—2442no—
334SD16—44no—
335SD16—2442no—
336SD16—2442no—
337SD16—2441no—
338SD16—2441no—
339D8—1221no—
340D8—1221no—
341D8—2442no—
342M16—21no—
343M16—22no—
344M16—22no—
345M16—22no—
346M16—41no—
347M16—1222no—
348M16—2441no—
349M16C2×C22⋊D6112yes—
350C2×C8—21no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—41no—
354C2×C8—2441no—
355C2×C8D4×D6111yescharacteristic
356C2×C8D4×D6111yescharacteristic
357C2×C8#1175822411yescharacteristic
358C2×C8#1175822411yescharacteristic
359C2×C8C2×C22⋊D6111yescharacteristic
360C2×C8C23×D6111yescharacteristic
361C16—614no—
362C16—1222no—
363C23—21no—
364C23—2441no—
365C23#1451979711yescharacteristic
366Q8—21no—
367Q8—42no—
368Q8—42no—
369Q8—81no—
370Q8—81no—
371Q8—2441no—
372Q8—2441no—
373Q8C2×#1175819912yes—
374D4—22no—
375D4—22no—
376D4—21no—
377D4—4881no—
378D4—4881no—
379D4—2441no—
380D4—2441no—
381D4C2×#1175819912yes—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—42no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—4881no—
389C2×C4—4881no—
390C2×C4—2441no—
391C2×C4#1451974911yescharacteristic
392C2×C4#1451974911yescharacteristic
393C2×C4C2×#1175819911yescharacteristic
394C2×C4C2×#1175819911yescharacteristic
395C2×C4#1451979711yescharacteristic
396C2×C4C22×C22⋊D6111yescharacteristic
397C8—22no—
398C8—42no—
399C8—2442no—
400C8#1451974911yescharacteristic
401C8#1451974911yescharacteristic
402C8C2×D4×D6111yescharacteristic
403C8C2×#1175822411yescharacteristic
404C22—22no—
405C22—41no—
406C22—81no—
407C22—9761no—
408C22—4881no—
409C22#1683999611yescharacteristic
410C22#1683999611yescharacteristic
411C22C2×#1451979711yescharacteristic
412C4—22no—
413C4—82no—
414C4—4881no—
415C4#1683999611yescharacteristic
416C4#1683999611yescharacteristic
417C4C2×#1451974911yescharacteristic
418C4C22×#1175819911yescharacteristic
419C2—21no—
420C2—21no—
421C2—21no—
422C2—42no—
423C2—9761no—
424C2C2×#1683999611yescharacteristic
subgroupG/H
centerC2C2×#16839996
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#473C2
socleC2×C61C2×#466
prime radical#21455539C2
hypercenterC2×#473D61
hypocenterC61#33619
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#473

Complements

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

NH ≅ G/Ntype
1. C2×#16840203C2semidirect
2. #21458922C2semidirect
4. #21447951C2semidirect
5. #21447942C2semidirect
6. #21447613C2semidirect
7. #21447566C2semidirect
8. C2×#16839578C2semidirect
9. C2×C61×#473C2semidirect
14. C2×#14519723C22semidirect
15. C2×#14519720C22semidirect
26. #16839591C22semidirect
27. #16839578C22semidirect
28. #16839577C22semidirect
29. #16839574C22semidirect
210. C2×#473D61semidirect
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
#214589222
#214555392
#214479512
#214479422
#214476132
#214475662
C2×#168395782
C2×C61×#4732
#3361961