Finite Group Databasebeta

← #29834560

Subgroups — #29834560

Order 15616.

662 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840203C211yesmaximal, characteristic, complemented
2C2×#16840178C211yesmaximal, characteristic, complemented
3#21455491C211yesmaximal, characteristic
4#21455470C211yesmaximal, characteristic, complemented
5#21447935C212yesmaximal, complemented
6#21447926C212yesmaximal, complemented
7#21447918C212yesmaximal, complemented
8#21447909C212yesmaximal, complemented
9#21447605C211yesmaximal, characteristic, complemented
10#21447527C211yesmaximal, characteristic, complemented
11C2×C61×#358C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13C2×#14519783C2211yescharacteristic
14#16840343—21no—
15#16840336—21no—
16#16840203—24no—
17C2×#14519723C2212yes—
18C2×#14519720C2212yes—
19#16840178—24no—
20C2×#14519717C2211yescharacteristic
21C2×#14519716C2212yes—
22C2×C16×D61C2212yes—
23#16840017—22no—
24#16840009—22no—
25#16840004C2212yes—
26#16840000C2212yes—
27#16840000C2212yes—
28#16839999C2212yes—
29#16839800C2211yescharacteristic
30#16839771—21no—
31#16839761—21no—
32#16839589C2212yes—
33#16839587C2212yes—
34#16839567C2211yescharacteristic
35#16839567C2211yescharacteristic
36#16839564C2211yescharacteristic
37#16839564C2211yescharacteristic
38C22×Q16×C61C2211yescharacteristic
39C22×C16×C61C2211yescharacteristic
40C2×#107×C61C2211yescharacteristic
41C61×#358C2212yes—
42C61×#358C2212yes—
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#14519775—41no—
56#14519766—41no—
57#14519760—42no—
58#14519757—22no—
59#14519757—41no—
60#14519753—22no—
61#14519753—22no—
62#14519752—22no—
63#14519752—41no—
64#14519751—42no—
65#14519736—21no—
66#14519723—24no—
67#14519720—24no—
68#14519717D414yescomplemented
69#14519716—24no—
70C2×#11758185C2312yes—
71C16×D61—24no—
72#14519711C2311yescharacteristic
73#14519711C2311yescharacteristic
74#14519709—21no—
75#14519701—22no—
76#14519701—22no—
77#14519700—22no—
78#14519697—22no—
79C22×Q8×C61—21no—
80C2×Q16×C61—22no—
81C2×Q16×C61—21no—
82C2×Q16×C61—22no—
83C2×Q16×C61C2312yes—
84C22×C8×C61C2311yescharacteristic
85C2×C4⋊C4×C61—21no—
86C2×C16×C61—22no—
87C2×C16×C61C2311yescharacteristic
88C2×C16×C61C2311yescharacteristic
89#107×C61—21no—
90#107×C61C2312yes—
91C2×Q8×C61—22no—
92C2×Q8×C61—41no—
93C2×Q8×C61—42no—
94C2×Q8×C61—42no—
95C22×C4×C61—41no—
96C22×C4×C61—41no—
97C22×C4×C61C2×D411yescharacteristic
98C2×C22⋊D61—41no—
99C2×C22⋊D61D812yes—
100#11758222—22no—
101#11758222—22no—
102#11758222—42no—
103#11758222—42no—
104C2×D244—21no—
105C2×D244—42no—
106C2×D244D811yescharacteristic
107C2×C4×D61—41no—
108C2×C4×D61C2×D412yes—
109C2×Dic122—21no—
110C2×Dic122D811yescharacteristic
111Q16×C61—22no—
112Q16×C61—22no—
113Q16×C61—42no—
114C2×C8×C61C2×D411yescharacteristic
115C2×C8×C61C2×D412yes—
116C2×C8×C61#4112yes—
117C2×C8×C61C2411yescharacteristic
118C4⋊C4×C61—22no—
119C4⋊C4×C61—41no—
120C22⋊Dic61—41no—
121#11758198—42no—
122#11758198—42no—
123#11758198—42no—
124#11758198—42no—
125C4⋊Dic61—21no—
126C4⋊Dic61—21no—
127C4⋊Dic61—42no—
128#11758196—42no—
129#11758196—42no—
130C4×Dic61—41no—
131#11758194—22no—
132C2×#92418C2×D412yes—
133Dic244—21no—
134Dic244—21no—
135D488—21no—
136D488—21no—
137#11758190—42no—
138#11758189—24no—
139C8×D61—22no—
140C8×D61—22no—
141C16×C61—21no—
142C16×C61—21no—
143#11758185C2×D414yes—
144C23×C61C2×D811yescharacteristic
145C22×D61—41no—
146C22×D61C2×D812yes—
147Q8×C61—42no—
148Q8×C61—42no—
149Q8×C61—82no—
150C2×C4×C61—42no—
151C2×C4×C61—42no—
152C2×C4×C61—82no—
153C2×C4×C61—161no—
154C2×C4×C61#12112yes—
155C2×C4×C61C2×D811yescharacteristic
156C2×C4×C61#13512yes—
157C2×C4×C61C22×D411yescharacteristic
158C22⋊D61—24no—
159C22⋊D61—82no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61C2×D812yes—
164D244—22no—
165D244—41no—
166D244—41no—
167D244—162no—
168C4×D61—22no—
169C4×D61—22no—
170C4×D61—82no—
171Dic122—22no—
172Dic122—41no—
173Dic122—41no—
174C8×C61#12112yes—
175C8×C61C22×D411yescharacteristic
176C8×C61C2×#4111yescharacteristic
177#92418—22no—
178#33613—611nomaximal, Sylow
179C22×C61—21no—
180C22×C61—21no—
181C22×C61#46612yes—
182C22×C61C22×D811yescharacteristic
183C2×D61—22no—
184C2×D61—22no—
185C2×D61—22no—
186C2×D61—81no—
187C2×D61—81no—
188C2×D61—81no—
189C4×C61—82no—
190C4×C61—82no—
191C4×C61#46612yes—
192C4×C61C2×#12111yescharacteristic
193C4×C61C2×#13511yescharacteristic
194Dic61—22no—
195Dic61—81no—
196Dic61—81no—
197Dic61—81no—
198C2×#510—611no—
199C2×#504—611no—
200#3061—611no—
201#3059—611no—
202#2220—612no—
203#2219—612no—
204#2217—612no—
205#2216—612no—
206#2159—611no—
207#2138—611no—
208C2×#358D6111yescharacteristic, complemented
209C2×C61—22no—
210C2×C61C2×#46611yescharacteristic
211C2×C61#341611yescharacteristic
212C2×C61#341711yescharacteristic
213D61—42no—
214D61—161no—
215C2×#135—611no—
216C22×Q16C2×D6111yescharacteristic
217C2×#131—611no—
218#537—1221no—
219#536—1221no—
220#510—1224no—
221C2×Q32—612no—
222C2×SD32—612no—
223#504—1224no—
224C2×M32—611no—
225C2×M32—612no—
226C22×C16—612no—
227C22×C16C2×D6111yescharacteristic
228#470—1222no—
229#468—1222no—
230#467—612no—
231#467—612no—
232#467—612no—
233#466—612no—
234#427—611no—
235C2×#107C2×D6111yescharacteristic
236#419—1221no—
237#417—1221no—
238#367—612no—
239#366—612no—
240#358—611no—
241#358—611no—
242#358C2×D6112yes—
243#358C2×D6112yes—
244#357—611no—
245#357—611no—
246C61#3361311yescharacteristic, complemented, Sylow
247C2×#41—611no—
248C2×#41—1221no—
249C22×Q8—21no—
250#135—1222no—
251#135—1222no—
252C2×Q16—22no—
253C2×Q16—22no—
254C2×Q16—21no—
255C2×Q16—611no—
256C2×Q16C22×D6112yes—
257C2×SD16—1221no—
258C2×D8—611no—
259#131—1222no—
260#131—1222no—
261C2×M16—611no—
262C2×M16—612no—
263C22×C8—612no—
264C22×C8C22×D6111yescharacteristic
265C4⋊Q8—2441no—
266#124—2441no—
267#123—2442no—
268C22⋊Q8—1222no—
269C22⋊Q8—1222no—
270C22⋊Q8—1222no—
271C22⋊Q8—2441no—
272#121—1222no—
273#121—2441no—
274#121—2442no—
275#117—1221no—
276C2×C4⋊C4—21no—
277Q32—1224no—
278SD32—1224no—
279M32—614no—
280M32—1224no—
281C2×C16—22no—
282C2×C16—612no—
283C2×C16—1224no—
284C2×C16C22×D6111yescharacteristic
285C2×C16C22×D6111yescharacteristic
286#107—21no—
287#107—611no—
288#107—611no—
289#107C22×D6112yes—
290#106—1221no—
291#103—1222no—
292#103—1222no—
293#103—1222no—
294#102—1222no—
295#41—1222no—
296#41—1222no—
297#41—2442no—
298#41—2442no—
299C2×Q8—22no—
300C2×Q8—41no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—611no—
304C2×Q8—1221no—
305C2×D4—612no—
306C2×D4—611no—
307C2×D4—1221no—
308C2×D4—2442no—
309C2×D4—2441no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—612no—
313C22×C4—2441no—
314C22×C4D4×D6111yescharacteristic
315Q16—22no—
316Q16—22no—
317Q16—42no—
318Q16—1221no—
319Q16—1221no—
320SD16—2442no—
321D8—1221no—
322D8—1221no—
323M16—1222no—
324M16—1224no—
325C2×C8—612no—
326C2×C8—1222no—
327C2×C8—1222no—
328C2×C8#1175822212yes—
329C2×C8D4×D6112yes—
330C2×C8D4×D6111yescharacteristic
331C2×C8C23×D6111yescharacteristic
332C4⋊C4—22no—
333C4⋊C4—41no—
334C4⋊C4—1221no—
335C4⋊C4—1221no—
336C4⋊C4—2442no—
337C4⋊C4—2442no—
338C4⋊C4—2442no—
339C22⋊C4—2442no—
340C22⋊C4—2442no—
341C22⋊C4—2442no—
342C22⋊C4—2442no—
343C22⋊C4—2441no—
344C42—2441no—
345C16—21no—
346C16—21no—
347C16—614no—
348C23—612no—
349C23—2441no—
350C23D8×D6111yescharacteristic
351Q8—42no—
352Q8—42no—
353Q8—82no—
354Q8—1222no—
355Q8—2441no—
356Q8—2441no—
357D4—1224no—
358D4—1222no—
359D4—9762no—
360D4—4882no—
361D4—2441no—
362D4—2441no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—82no—
366C2×C4—161no—
367C2×C4—612no—
368C2×C4—1222no—
369C2×C4—1222no—
370C2×C4—4882no—
371C2×C4—2441no—
372C2×C4—2441no—
373C2×C4—2441no—
374C2×C4#1451975012yes—
375C2×C4D8×D6111yescharacteristic
376C2×C4#1451979812yes—
377C2×C4C2×D4×D6111yescharacteristic
378C8—1222no—
379C8#1451975012yes—
380C8C2×D4×D6111yescharacteristic
381C8C2×#1175822211yescharacteristic
382C22—21no—
383C22—21no—
384C22—1222no—
385C22—1222no—
386C22—1222no—
387C22—4881no—
388C22—4881no—
389C22—4881no—
390C22#1683999712yes—
391C22C2×D8×D6111yescharacteristic
392C4—82no—
393C4—82no—
394C4—1222no—
395C4—4881no—
396C4—4881no—
397C4—4881no—
398C4#1683999712yes—
399C4C2×#1451975011yescharacteristic
400C4C2×#1451979811yescharacteristic
401C2—22no—
402C2—9761no—
403C2—2442no—
404C2C2×#1683999711yescharacteristic
405C2#2145896111yescharacteristic
406C2#2145896811yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#358C2
socleC22×C61C22×D8
prime radical#21455491C2
hypercenterC2×#358D61
hypocenterC61#33613
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
1. C2×#16840203C2semidirect
2. C2×#16840178C2semidirect
4. #21455470C2semidirect
5. #21447935C2semidirect
6. #21447926C2semidirect
7. #21447918C2semidirect
8. #21447909C2semidirect
9. #21447605C2semidirect
10. #21447527C2semidirect
11. C2×C61×#358C2semidirect
68. #14519717D4semidirect
208. C2×#358D61semidirect
246. C61#33613semidirect

Maximal subgroups

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

subgroupindex
C2×#168402032
C2×#168401782
#214554912
#214554702
#214479352
#214479262
#214479182
#214479092
#214476052
#214475272
C2×C61×#3582
#3361361