Finite Group Databasebeta

← #29834554

Subgroups — #29834554

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×#16840197C211yesmaximal, characteristic, complemented
2C2×#16840178C211yesmaximal, characteristic, complemented
3#21455485C211yesmaximal, characteristic, complemented
4#21455468C211yesmaximal, characteristic, complemented
5#21447935C212yesmaximal, complemented
6#21447921C212yesmaximal, complemented
7#21447918C212yesmaximal, complemented
8#21447904C212yesmaximal, complemented
9#21447608C211yesmaximal, characteristic, complemented
10C2×#16839564C211yesmaximal, characteristic, complemented
11C61×#2138C211yesmaximal, characteristic, complemented
12C22×D488C2211yescharacteristic
13C2×#14519783C2211yescharacteristic
14#16840340—21no—
15#16840331—21no—
16#16840197—24no—
17C2×#14519720C2212yes—
18C2×#14519718C2212yes—
19#16840178—24no—
20C2×#14519717C2211yescharacteristic
21C2×#14519716C2212yescomplemented
22C2×C16×D61C2212yescomplemented
23#16840016—22no—
24#16840005—22no—
25#16840000C2212yescomplemented
26#16839999C2212yes—
27#16839999C2212yes—
28#16839995C2212yescomplemented
29C2×#14519711C2211yescharacteristic
30#16839766—21no—
31#16839763—21no—
32#16839589C2212yescomplemented
33#16839587C2212yescomplemented
34#16839564C2212yes—
35#16839564C2212yes—
36C2×#135×C61C2211yescharacteristic
37C22×C16×C61C2211yescharacteristic
38C61×#427C2211yescharacteristic
39C61×#358C2211yescharacteristic, complemented
40C61×#358C2211yescharacteristic, complemented
41C61×#357C2211yescharacteristic, complemented
42C61×#357C2211yescharacteristic, complemented
43C2×#11758222D411yescharacteristic
44C22×D244—21no—
45C2×D488—22no—
46C2×D488—22no—
47C2×D488—21no—
48C2×D488C2312yes—
49#14519783—22no—
50#14519783—22no—
51C2×#11758194D411yescharacteristic
52C2×#11758189D412yes—
53C2×C8×D61C2312yes—
54#14519774—41no—
55#14519765—41no—
56#14519756—41no—
57#14519753—22no—
58#14519752—22no—
59#14519752—22no—
60#14519751—41no—
61#14519749—42no—
62#14519748—22no—
63#14519747—42no—
64C2×C4⋊Dic61—21no—
65#14519720—24no—
66#14519718—24no—
67#14519717D414yescomplemented
68#14519716—24no—
69C2×#11758185C2312yes—
70C16×D61—24no—
71#14519711—21no—
72#14519711C2312yes—
73#14519701—22no—
74#14519698—22no—
75#14519697—22no—
76#14519697—22no—
77C2×#41×C61—21no—
78#135×C61—22no—
79#135×C61—22no—
80C2×Q16×C61C2311yescharacteristic
81C2×SD16×C61—21no—
82C2×D8×C61C2311yescharacteristic
83C22×C8×C61C2311yescharacteristic
84#117×C61—21no—
85C2×C16×C61—22no—
86C2×C16×C61C2311yescharacteristic
87C2×C16×C61C2311yescharacteristic
88#107×C61C2311yescharacteristic
89#107×C61C2311yescharacteristic
90#106×C61—21no—
91C23×D61—41no—
92#41×C61—42no—
93#41×C61—42no—
94C2×Q8×C61—21no—
95C2×D4×C61—21no—
96C2×D4×C61—41no—
97C22×C4×C61—41no—
98C22×C4×C61C2×D411yescharacteristic
99C2×C22⋊D61—42no—
100C2×C22⋊D61D812yes—
101C22×Dic61—41no—
102#11758222—22no—
103#11758222—22no—
104C2×D244—22no—
105C2×D244—42no—
106C2×D244—42no—
107C2×D244—41no—
108C2×D244—42no—
109C2×D244—42no—
110C2×D244D811yescharacteristic
111C2×C4×D61C2×D412yes—
112C2×Dic122D811yescharacteristic
113Q16×C61—21no—
114Q16×C61—21no—
115SD16×C61—42no—
116D8×C61—21no—
117D8×C61—21no—
118C2×C8×C61C2×D411yescharacteristic
119C2×C8×C61C2×D412yes—
120C2×C8×C61#4112yes—
121C2×C8×C61C2411yescharacteristic
122C4⋊C4×C61—21no—
123C4⋊C4×C61—21no—
124C22⋊C4×C61—41no—
125C42×C61—41no—
126C22⋊Dic61—42no—
127#11758198—42no—
128#11758198—42no—
129#11758198—42no—
130C4⋊Dic61—22no—
131C4⋊Dic61—41no—
132#11758196—42no—
133#11758194—22no—
134C2×#92418C2×D412yes—
135D488—22no—
136D488—22no—
137D488—42no—
138#11758189—24no—
139C8×D61—22no—
140C8×D61—22no—
141C16×C61—21no—
142C16×C61—21no—
143#11758185C2×D414yes—
144C23×C61—41no—
145C23×C61C2×D811yescharacteristic
146C22×D61—42no—
147C22×D61—82no—
148C22×D61—161no—
149C22×D61—161no—
150C22×D61C2×D812yes—
151Q8×C61—41no—
152Q8×C61—41no—
153D4×C61—41no—
154D4×C61—41no—
155D4×C61—82no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—82no—
160C2×C4×C61#12112yes—
161C2×C4×C61C2×D811yescharacteristic
162C2×C4×C61#13512yes—
163C2×C4×C61C22×D411yescharacteristic
164C22⋊D61—24no—
165C22⋊D61—162no—
166C2×Dic61—42no—
167C2×Dic61—161no—
168C2×Dic61C2×D812yes—
169D244—22no—
170D244—42no—
171D244—42no—
172D244—82no—
173D244—162no—
174D244—162no—
175C4×D61—22no—
176C4×D61—22no—
177Dic122—22no—
178C8×C61#12112yes—
179C8×C61C22×D411yescharacteristic
180C8×C61C2×#4111yescharacteristic
181#92418—22no—
182#33612—611nomaximal, Sylow
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—81no—
186C22×C61—81no—
187C22×C61—81no—
188C22×C61#46612yes—
189C22×C61C22×D811yescharacteristic
190C2×D61—22no—
191C2×D61—22no—
192C2×D61—22no—
193C2×D61—82no—
194C2×D61—82no—
195C2×D61—82no—
196C2×D61—164no—
197C4×C61—81no—
198C4×C61—81no—
199C4×C61—81no—
200C4×C61#46612yes—
201C4×C61C2×#12111yescharacteristic
202C4×C61C2×#13511yescharacteristic
203Dic61—22no—
204Dic61—82no—
205C2×#509—611no—
206C2×#504—611no—
207#3061—611no—
208#3058—611no—
209#2220—612no—
210#2218—612no—
211#2217—612no—
212#2215—612no—
213#2159—611no—
214#2138D6111yescharacteristic, complemented
215C2×#357—611no—
216C2×C61—22no—
217C2×C61—161no—
218C2×C61C2×#46611yescharacteristic
219C2×C61#341511yescharacteristic
220C2×C61#341611yescharacteristic
221D61—42no—
222D61—162no—
223C2×#135C2×D6111yescharacteristic
224C22×D8—611no—
225C2×#131—611no—
226#537—1221no—
227#535—1221no—
228#509—1224no—
229C2×SD32—612no—
230C2×D16—612no—
231#504—1224no—
232C2×M32—611no—
233C2×M32—612no—
234C22×C16—612no—
235C22×C16C2×D6111yescharacteristic
236#469—1222no—
237#468—1222no—
238#467—612no—
239#466—612no—
240#466—612no—
241#466—612no—
242#427C2×D6111yescharacteristic
243C2×#107—611no—
244#418—1221no—
245#417—1221no—
246#367—612no—
247#366—612no—
248#358C2×D6111yescharacteristic, complemented
249#358C2×D6111yescharacteristic, complemented
250#357—612no—
251#357—612no—
252#357C2×D6111yescharacteristic, complemented
253#357C2×D6111yescharacteristic, complemented
254C61#3361211yescharacteristic, complemented, Sylow
255C2×#41—21no—
256C2×#41—611no—
257C22×D4—1221no—
258#135—22no—
259#135—22no—
260C2×Q16C22×D6111yescharacteristic
261C2×SD16—21no—
262C2×D8—612no—
263C2×D8—1222no—
264C2×D8—1222no—
265C2×D8—1221no—
266C2×D8C22×D6111yescharacteristic
267#131—1222no—
268#131—1222no—
269C2×M16—611no—
270C2×M16—612no—
271C22×C8—612no—
272C22×C8C22×D6111yescharacteristic
273#127—2441no—
274#124—2441no—
275C22⋊Q8—1222no—
276C22⋊Q8—2441no—
277#121—1222no—
278#121—1222no—
279#121—1222no—
280#121—2441no—
281#121—2442no—
282#120—2442no—
283#117—21no—
284C2×C4⋊C4—1221no—
285SD32—1224no—
286D16—1224no—
287M32—614no—
288M32—1224no—
289C2×C16—22no—
290C2×C16—612no—
291C2×C16—1224no—
292C2×C16C22×D6111yescharacteristic
293C2×C16C22×D6111yescharacteristic
294#107—612no—
295#107—1221no—
296#107C22×D6111yescharacteristic
297#107C22×D6111yescharacteristic
298#106—21no—
299#103—1222no—
300#102—1222no—
301#102—1222no—
302#102—1222no—
303C24—2441no—
304#41—42no—
305#41—42no—
306#41—1222no—
307#41—1222no—
308C2×Q8—21no—
309C2×Q8—611no—
310C2×D4—21no—
311C2×D4—41no—
312C2×D4—612no—
313C2×D4—611no—
314C2×D4—1222no—
315C2×D4—2442no—
316C2×D4—2442no—
317C2×D4—2442no—
318C2×D4—2442no—
319C2×D4—2441no—
320C2×D4—2442no—
321C22×C4—41no—
322C22×C4—612no—
323C22×C4—2441no—
324C22×C4D4×D6111yescharacteristic
325Q16—21no—
326Q16—21no—
327SD16—42no—
328D8—21no—
329D8—21no—
330D8—1222no—
331D8—1222no—
332D8—2442no—
333M16—1222no—
334M16—1224no—
335C2×C8—612no—
336C2×C8—1222no—
337C2×C8—1222no—
338C2×C8#1175822212yes—
339C2×C8D4×D6112yes—
340C2×C8D4×D6111yescharacteristic
341C2×C8C23×D6111yescharacteristic
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—1222no—
345C4⋊C4—2442no—
346C4⋊C4—2441no—
347C22⋊C4—41no—
348C22⋊C4—2442no—
349C22⋊C4—2442no—
350C22⋊C4—2442no—
351C22⋊C4—2442no—
352C42—41no—
353C16—21no—
354C16—21no—
355C16—614no—
356C23—41no—
357C23—612no—
358C23—9761no—
359C23—9761no—
360C23—4882no—
361C23—2442no—
362C23D8×D6111yescharacteristic
363Q8—41no—
364Q8—41no—
365Q8—1222no—
366D4—41no—
367D4—41no—
368D4—82no—
369D4—1224no—
370D4—1222no—
371D4—9762no—
372D4—9762no—
373D4—9762no—
374D4—4882no—
375D4—2442no—
376D4—2442no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—82no—
381C2×C4—612no—
382C2×C4—1222no—
383C2×C4—1222no—
384C2×C4—9761no—
385C2×C4—2442no—
386C2×C4#1451975012yes—
387C2×C4D8×D6111yescharacteristic
388C2×C4#1451979812yes—
389C2×C4C2×D4×D6111yescharacteristic
390C8—1222no—
391C8#1451975012yes—
392C8C2×D4×D6111yescharacteristic
393C8C2×#1175822211yescharacteristic
394C22—21no—
395C22—21no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—1222no—
400C22—1222no—
401C22—1222no—
402C22—9764no—
403C22—4882no—
404C22—4882no—
405C22—4882no—
406C22#1683999712yes—
407C22C2×D8×D6111yescharacteristic
408C4—81no—
409C4—81no—
410C4—81no—
411C4—1222no—
412C4—4882no—
413C4#1683999712yes—
414C4C2×#1451975011yescharacteristic
415C4C2×#1451979811yescharacteristic
416C2—22no—
417C2—161no—
418C2—9762no—
419C2—2442no—
420C2C2×#1683999711yescharacteristic
421C2#2145895711yescharacteristic
422C2#2145896211yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2138C2
socleC22×C61C22×D8
hypercenter#2138D61
hypocenterC61#33612
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
1. C2×#16840197C2semidirect
2. C2×#16840178C2semidirect
3. #21455485C2semidirect
4. #21455468C2semidirect
5. #21447935C2semidirect
6. #21447921C2semidirect
7. #21447918C2semidirect
8. #21447904C2semidirect
9. #21447608C2semidirect
10. C2×#16839564C2semidirect
11. C61×#2138C2semidirect
21. C2×#14519716C22semidirect
22. C2×C16×D61C22semidirect
25. #16840000C22semidirect
28. #16839995C22semidirect
32. #16839589C22semidirect
33. #16839587C22semidirect
39. C61×#358C22semidirect
40. C61×#358C22semidirect
41. C61×#357C22semidirect
42. C61×#357C22semidirect
67. #14519717D4semidirect
214. #2138D61semidirect
248. #358C2×D61semidirect
249. #358C2×D61semidirect
252. #357C2×D61semidirect
253. #357C2×D61semidirect
254. C61#33612semidirect

Maximal subgroups

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

subgroupindex
C2×#168401972
C2×#168401782
#214554852
#214554682
#214479352
#214479212
#214479182
#214479042
#214476082
C2×#168395642
C61×#21382
#3361261