Finite Group Databasebeta

← #29834540

Subgroups — #29834540

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840192C211yesmaximal, characteristic, complemented
2C2×#16840180C211yesmaximal, characteristic, complemented
3#21455491C211yesmaximal, characteristic, complemented
4#21455485C211yesmaximal, characteristic, complemented
5#21447926C212yesmaximal, complemented
6#21447921C212yesmaximal, complemented
7#21447918C212yesmaximal, complemented
8#21447914C212yesmaximal, complemented
9C2×#16839589C211yesmaximal, characteristic, complemented
10#21447532C211yesmaximal, characteristic, complemented
11C61×#2139C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13C2×#14519783C2211yescharacteristic
14#16840343—21no—
15#16840340—21no—
16#16840192—24no—
17C2×#14519723C2211yescharacteristic
18C2×#14519721C2211yescharacteristic
19C2×#14519720C2211yescharacteristic
20C2×#14519718C2211yescharacteristic
21#16840180—24no—
22C2×#14519716C2212yes—
23C22×#11758185C2211yescharacteristic
24C2×C16×D61C2212yes—
25#16840009—22no—
26#16840005—22no—
27#16840004C2212yes—
28#16840000C2212yes—
29#16839999C2212yes—
30#16839995C2212yes—
31C2×#14519711C2211yescharacteristic
32#16839763—21no—
33#16839761—21no—
34#16839589C2212yes—
35#16839589C2212yes—
36#16839567C2212yes—
37#16839564C2212yes—
38C2×#135×C61C2211yescharacteristic
39C2×M32×C61C2211yescharacteristic
40C2×#107×C61C2211yescharacteristic
41C61×#358C2212yes—
42C61×#357C2212yes—
43C2×#11758222—21no—
44C2×#11758222D411yescharacteristic
45#14519799—22no—
46#14519799—22no—
47C2×Dic244C2311yescharacteristic
48C2×#11758190—21no—
49C2×D488C2311yescharacteristic
50#14519783—22no—
51#14519783—22no—
52C2×#11758194D411yescharacteristic
53C2×#11758189D412yes—
54C2×C8×D61C2312yes—
55#14519757—22no—
56#14519757—41no—
57#14519756—41no—
58#14519753—22no—
59#14519752—22no—
60#14519752—41no—
61#14519751—42no—
62#14519751—41no—
63#14519749—42no—
64#14519748—22no—
65C2×C4⋊Dic61—21no—
66#14519723—22no—
67#14519721—22no—
68#14519720—22no—
69#14519718—22no—
70#14519716—24no—
71C2×#11758185D414yescomplemented
72C2×#11758185C2311yescharacteristic
73C2×#11758185C2311yescharacteristic
74C16×D61—24no—
75#14519711—21no—
76#14519711C2312yes—
77#14519701—22no—
78#14519700—22no—
79#14519698—22no—
80#14519697—22no—
81C2×#41×C61—21no—
82#135×C61—22no—
83#135×C61—22no—
84C2×Q16×C61C2311yescharacteristic
85C2×SD16×C61—21no—
86C2×D8×C61C2311yescharacteristic
87C22×C8×C61C2311yescharacteristic
88C2×C4⋊C4×C61—21no—
89M32×C61—22no—
90C2×C16×C61C2312yes—
91#107×C61—21no—
92#107×C61C2312yes—
93#41×C61—42no—
94#41×C61—42no—
95C2×Q8×C61—21no—
96C2×D4×C61—21no—
97C2×D4×C61—41no—
98C22×C4×C61—41no—
99C22×C4×C61—41no—
100C22×C4×C61C2×D411yescharacteristic
101C2×C22⋊D61—41no—
102C2×C22⋊D61—42no—
103C2×C22⋊D61D812yes—
104C22×Dic61—41no—
105#11758222—22no—
106#11758222—22no—
107#11758222—42no—
108#11758222—42no—
109C2×D244—21no—
110C2×D244—42no—
111C2×D244D811yescharacteristic
112C2×C4×D61—41no—
113C2×C4×D61C2×D412yes—
114C2×Dic122—21no—
115C2×Dic122D811yescharacteristic
116Q16×C61—21no—
117Q16×C61—21no—
118SD16×C61—42no—
119D8×C61—21no—
120D8×C61—21no—
121C2×C8×C61C2×D411yescharacteristic
122C2×C8×C61C2×D412yes—
123C2×C8×C61#4112yes—
124C2×C8×C61C2411yescharacteristic
125C4⋊C4×C61—22no—
126C4⋊C4×C61—41no—
127C22⋊Dic61—42no—
128#11758198—42no—
129#11758198—42no—
130#11758198—42no—
131C4⋊Dic61—22no—
132C4⋊Dic61—41no—
133C4⋊Dic61—42no—
134#11758196—42no—
135#11758194—22no—
136C2×#92418C2×D412yes—
137Dic244—21no—
138Dic244—21no—
139D488—21no—
140D488—21no—
141#11758190—42no—
142#11758189—24no—
143C8×D61—22no—
144C8×D61—22no—
145C16×C61—22no—
146#11758185C2×D412yes—
147#11758185C2×D412yes—
148C23×C61—41no—
149C23×C61C2×D811yescharacteristic
150C22×D61—41no—
151C22×D61C2×D812yes—
152Q8×C61—41no—
153Q8×C61—41no—
154D4×C61—41no—
155D4×C61—41no—
156D4×C61—82no—
157C2×C4×C61—41no—
158C2×C4×C61—42no—
159C2×C4×C61—82no—
160C2×C4×C61—161no—
161C2×C4×C61#12112yes—
162C2×C4×C61C2×D811yescharacteristic
163C2×C4×C61#13512yes—
164C2×C4×C61C22×D411yescharacteristic
165C22⋊D61—24no—
166C22⋊D61—82no—
167C22⋊D61—162no—
168C2×Dic61—42no—
169C2×Dic61—41no—
170C2×Dic61—161no—
171C2×Dic61C2×D812yes—
172D244—22no—
173D244—41no—
174D244—41no—
175D244—162no—
176C4×D61—22no—
177C4×D61—22no—
178C4×D61—82no—
179Dic122—22no—
180Dic122—41no—
181Dic122—41no—
182C8×C61#12112yes—
183C8×C61C22×D411yescharacteristic
184C8×C61C2×#4111yescharacteristic
185#92418—22no—
186#33611—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—81no—
190C22×C61—81no—
191C22×C61—81no—
192C22×C61#46612yes—
193C22×C61C22×D811yescharacteristic
194C2×D61—22no—
195C2×D61—22no—
196C2×D61—22no—
197C2×D61—81no—
198C2×D61—81no—
199C2×D61—81no—
200C4×C61—82no—
201C4×C61—81no—
202C4×C61#46612yes—
203C4×C61C2×#12111yescharacteristic
204C4×C61C2×#13511yescharacteristic
205Dic61—22no—
206Dic61—82no—
207Dic61—81no—
208C2×#508—611no—
209C2×#504—611no—
210#3061—611no—
211#3061—611no—
212#2219—612no—
213#2218—612no—
214#2217—612no—
215#2217—612no—
216C2×#367—611no—
217#2139—611no—
218#2139D6111yescharacteristic, complemented
219C2×C61—22no—
220C2×C61—161no—
221C2×C61C2×#46611yescharacteristic
222C2×C61#341611yescharacteristic
223C2×C61#341611yescharacteristic
224D61—42no—
225D61—161no—
226C2×#135—611no—
227C2×#135C2×D6111yescharacteristic
228C2×#131—611no—
229#537—1221no—
230#537—1221no—
231#508—1224no—
232C2×Q32—611no—
233C2×SD32—611no—
234C2×SD32—611no—
235C2×D16—611no—
236#504—1224no—
237C2×M32—612no—
238C2×M32C2×D6111yescharacteristic
239C22×C16—611no—
240C22×C16—612no—
241#468—1222no—
242#468—1222no—
243#467—612no—
244#467—612no—
245#466—612no—
246#466—612no—
247C2×#107—611no—
248C2×#107C2×D6111yescharacteristic
249#417—1221no—
250#417—1221no—
251#367—612no—
252#367—612no—
253#358—612no—
254#358C2×D6112yes—
255#357—612no—
256#357C2×D6112yes—
257C61#3361111yescharacteristic, complemented, Sylow
258C2×#41—21no—
259C2×#41—611no—
260C2×#41—1221no—
261#135—22no—
262#135—22no—
263#135—1222no—
264#135—1222no—
265C2×Q16—611no—
266C2×Q16C22×D6111yescharacteristic
267C2×SD16—21no—
268C2×SD16—1221no—
269C2×D8—611no—
270C2×D8C22×D6111yescharacteristic
271#131—1222no—
272#131—1222no—
273C2×M16—611no—
274C2×M16—612no—
275C22×C8—612no—
276C22×C8C22×D6111yescharacteristic
277C22⋊Q8—1222no—
278C22⋊Q8—1222no—
279C22⋊Q8—2441no—
280C22⋊Q8—2441no—
281#121—1222no—
282#121—1222no—
283#121—2441no—
284#121—2442no—
285#121—2441no—
286#121—2442no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—1221no—
289Q32—1222no—
290SD32—1222no—
291SD32—1222no—
292D16—1222no—
293M32—22no—
294M32—1224no—
295C2×C16—614no—
296C2×C16—611no—
297C2×C16—611no—
298C2×C16—1224no—
299C2×C16C22×D6112yes—
300#107—21no—
301#107—612no—
302#107—1221no—
303#107C22×D6112yes—
304#103—1222no—
305#103—1222no—
306#102—1222no—
307#102—1222no—
308#41—42no—
309#41—42no—
310#41—1222no—
311#41—1222no—
312#41—2442no—
313#41—2442no—
314C2×Q8—21no—
315C2×Q8—611no—
316C2×Q8—1221no—
317C2×D4—21no—
318C2×D4—41no—
319C2×D4—612no—
320C2×D4—611no—
321C2×D4—1221no—
322C2×D4—2442no—
323C2×D4—2442no—
324C2×D4—2441no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—612no—
328C22×C4—2441no—
329C22×C4—2441no—
330C22×C4D4×D6111yescharacteristic
331Q16—21no—
332Q16—21no—
333Q16—1221no—
334Q16—1221no—
335SD16—42no—
336SD16—2442no—
337D8—21no—
338D8—21no—
339D8—1221no—
340D8—1221no—
341M16—1222no—
342M16—1224no—
343C2×C8—612no—
344C2×C8—1222no—
345C2×C8—1222no—
346C2×C8#1175822212yes—
347C2×C8D4×D6112yes—
348C2×C8D4×D6111yescharacteristic
349C2×C8C23×D6111yescharacteristic
350C4⋊C4—22no—
351C4⋊C4—41no—
352C4⋊C4—1222no—
353C4⋊C4—2442no—
354C4⋊C4—2442no—
355C4⋊C4—2441no—
356C22⋊C4—2442no—
357C22⋊C4—2442no—
358C22⋊C4—2442no—
359C22⋊C4—2442no—
360C16—22no—
361C16—612no—
362C16—612no—
363C23—41no—
364C23—612no—
365C23—2441no—
366C23D8×D6111yescharacteristic
367Q8—41no—
368Q8—41no—
369Q8—1222no—
370Q8—2441no—
371Q8—2441no—
372D4—41no—
373D4—41no—
374D4—82no—
375D4—1224no—
376D4—1222no—
377D4—9762no—
378D4—9762no—
379D4—4882no—
380D4—2441no—
381D4—2441no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—82no—
385C2×C4—161no—
386C2×C4—612no—
387C2×C4—1222no—
388C2×C4—1222no—
389C2×C4—9761no—
390C2×C4—4882no—
391C2×C4—2442no—
392C2×C4—2441no—
393C2×C4#1451975012yes—
394C2×C4D8×D6111yescharacteristic
395C2×C4#1451979812yes—
396C2×C4C2×D4×D6111yescharacteristic
397C8—1222no—
398C8#1451975012yes—
399C8C2×D4×D6111yescharacteristic
400C8C2×#1175822211yescharacteristic
401C22—21no—
402C22—21no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—1222no—
407C22—1222no—
408C22—1222no—
409C22—4881no—
410C22—4881no—
411C22—4881no—
412C22#1683999712yes—
413C22C2×D8×D6111yescharacteristic
414C4—82no—
415C4—81no—
416C4—1222no—
417C4—4882no—
418C4—4881no—
419C4#1683999712yes—
420C4C2×#1451975011yescharacteristic
421C4C2×#1451979811yescharacteristic
422C2—22no—
423C2—161no—
424C2—9761no—
425C2—2442no—
426C2C2×#1683999711yescharacteristic
427C2#2145896411yescharacteristic
428C2#2145896511yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2139C2
socleC22×C61C22×D8
hypercenter#2139D61
hypocenterC61#33611
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2139

Complements

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

NH ≅ G/Ntype
1. C2×#16840192C2semidirect
2. C2×#16840180C2semidirect
3. #21455491C2semidirect
4. #21455485C2semidirect
5. #21447926C2semidirect
6. #21447921C2semidirect
7. #21447918C2semidirect
8. #21447914C2semidirect
9. C2×#16839589C2semidirect
10. #21447532C2semidirect
11. C61×#2139C2semidirect
71. C2×#11758185D4semidirect
218. #2139D61semidirect
257. C61#33611semidirect

Maximal subgroups

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

subgroupindex
C2×#168401922
C2×#168401802
#214554912
#214554852
#214479262
#214479212
#214479182
#214479142
C2×#168395892
#214475322
C61×#21392
#3361161