Finite Group Databasebeta

← #29834684

Subgroups — #29834684

Order 15616.

574 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840166C211yesmaximal, characteristic, complemented
2#21454590C211yesmaximal, characteristic
3#21447970C212yesmaximal, complemented
4#21447965C212yesmaximal, complemented
5#21447960C212yesmaximal, complemented
6#21447955C212yesmaximal, complemented
7#21447608C211yesmaximal, characteristic, complemented
8#21447574C211yesmaximal, characteristic, complemented
9#21447532C211yesmaximal, characteristic
10#21447529C211yesmaximal, characteristic
11C61×#2159C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13C2×#14519776—21no—
14#16840230—21no—
15C2×#14519717C2211yescharacteristic
16#16840166—21no—
17#16840166—21no—
18#16840166C2212yes—
19#16840166C2212yes—
20#16839856C2211yescharacteristic
21#16839856C2211yescharacteristic
22#16839849—21no—
23#16839849—21no—
24#16839844C2211yescharacteristic
25#16839844C2211yescharacteristic
26C2×#14519711C2211yescharacteristic
27C2×#14519710—21no—
28#16839756—21no—
29C2×C8×Dic61C2211yescharacteristic
30#16839589C2212yes—
31#16839587C2212yes—
32#16839580C2212yes—
33#16839580C2212yes—
34#16839567C2212yes—
35#16839565C2212yes—
36#16839564C2212yes—
37#16839562C2212yes—
38C2×M32×C61C2211yescharacteristic
39C61×#427C2211yescharacteristic
40C2×#107×C61C2211yescharacteristic
41C61×#367C2212yes—
42C61×#366C2212yes—
43C2×#11758222—21no—
44C22×Dic122—41no—
45#14519799—22no—
46#14519799Q814yescomplemented
47C2×Dic244C2311yescharacteristic
48C2×#11758190—21no—
49C2×D488C2311yescharacteristic
50C22×#92418D411yescharacteristic
51#14519776—22no—
52#14519776—22no—
53#14519776—41no—
54#14519776—41no—
55#14519742—21no—
56#14519742—21no—
57#14519739—21no—
58#14519739—21no—
59#14519737—41no—
60#14519737—41no—
61#14519734—41no—
62C2×C4⋊Dic61—21no—
63C2×#11758196—41no—
64C2×C4×Dic61D411yescharacteristic
65C42×D61—41no—
66#14519717—22no—
67C2×#11758185C2312yes—
68#14519711—21no—
69#14519711C2312yes—
70#14519710—22no—
71#14519710—22no—
72#14519700—21no—
73#14519700—21no—
74#14519697—21no—
75#14519697—21no—
76C8×Dic61D411yescharacteristic
77C8×Dic61D411yescharacteristic
78C8×Dic61C2311yescharacteristic
79C8×Dic61C2311yescharacteristic
80C22×C8×C61C2311yescharacteristic
81#117×C61—21no—
82C2×C4⋊C4×C61—21no—
83M32×C61—22no—
84C2×C16×C61C2312yes—
85#107×C61—21no—
86#107×C61C2311yescharacteristic
87#107×C61C2312yes—
88#107×C61C2311yescharacteristic
89#106×C61—21no—
90C22×C4×C61—41no—
91C22×C4×C61C2×D411yescharacteristic
92C2×C22⋊D61—41no—
93C22×Dic61—41no—
94C22×Dic61D811yescharacteristic
95C22×Dic61D811yescharacteristic
96#11758222—24no—
97#11758222—42no—
98C2×D244—21no—
99C2×C4×D61—41no—
100C2×C4×D61—41no—
101C2×C4×D61—41no—
102C2×Dic122—21no—
103C2×Dic122—42no—
104C2×Dic122—42no—
105C2×Dic122—84no—
106C2×C8×C61C2×D411yescharacteristic
107C2×C8×C61C2×Q812yes—
108C2×C8×C61#4112yes—
109C2×C8×C61C2411yescharacteristic
110C4⋊C4×C61—21no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—22no—
113C4⋊C4×C61—41no—
114C22⋊C4×C61—41no—
115C42×C61—41no—
116#11758198—41no—
117#11758198—41no—
118C4⋊Dic61—22no—
119C4⋊Dic61—41no—
120#11758196—42no—
121#11758196—42no—
122#11758196—41no—
123#11758196—41no—
124C4×Dic61—41no—
125C4×Dic61—41no—
126C4×Dic61D811yescharacteristic
127C4×Dic61D811yescharacteristic
128C4×Dic61C2×D411yescharacteristic
129C4×Dic61C2×D411yescharacteristic
130C2×#92418—22no—
131C2×#92418C2×D411yescharacteristic
132C2×#92418C2×D411yescharacteristic
133Dic244—21no—
134Dic244C2×Q812yescomplemented
135D488—21no—
136D488C2×Q812yescomplemented
137#11758190—24no—
138C16×C61—22no—
139#11758185—22no—
140C23×C61C2×D811yescharacteristic
141C22×D61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—42no—
145C2×C4×C61—82no—
146C2×C4×C61C22⋊Q812yes—
147C2×C4×C61C2×D811yescharacteristic
148C2×C4×C61#13712yes—
149C2×C4×C61C22×D411yescharacteristic
150C22⋊D61—44no—
151C2×Dic61—22no—
152C2×Dic61—22no—
153C2×Dic61—41no—
154C2×Dic61—42no—
155C2×Dic61—82no—
156C2×Dic61C2×D811yescharacteristic
157C2×Dic61C2×D811yescharacteristic
158C2×Dic61C2×D811yescharacteristic
159C2×Dic61C2×D811yescharacteristic
160D244—22no—
161D244—41no—
162C4×D61—44no—
163C4×D61—82no—
164C4×D61—82no—
165Dic122—22no—
166Dic122—41no—
167Dic122—84no—
168Dic122—84no—
169C8×C61C22⋊Q812yes—
170C8×C61C22×Q811yescharacteristic
171C8×C61C2×#4111yescharacteristic
172#92418—21no—
173#92418—21no—
174#33629—611nomaximal, Sylow
175C22×C61—21no—
176C22×C61—21no—
177C22×C61#47412yes—
178C22×C61C22×D811yescharacteristic
179C2×D61—42no—
180C2×D61—81no—
181C2×D61—81no—
182C4×C61—82no—
183C4×C61—81no—
184C4×C61—81no—
185C4×C61#47412yes—
186C4×C61C2×C22⋊Q811yescharacteristic
187C4×C61C2×#13711yescharacteristic
188Dic61—21no—
189Dic61—21no—
190Dic61—42no—
191Dic61—82no—
192Dic61#50912yes—
193Dic61#51012yes—
194C2×#500—611no—
195C4×#135—611no—
196#2227—612no—
197#2226—612no—
198#2225—612no—
199#2224—612no—
200#2159—611no—
201#2159D6111yescharacteristic, complemented
202#2150—611no—
203#2139—611no—
204#2139—611no—
205C2×C61—22no—
206C2×C61C2×#47411yescharacteristic
207C2×C61C2×#50911yescharacteristic
208C2×C61C2×#51011yescharacteristic
209D61—82no—
210C2×#135—611no—
211C2×C4⋊Q8—1221no—
212C4×#41—1221no—
213C2×M32—611no—
214C2×M32C2×D6111yescharacteristic
215#500—612no—
216#500—612no—
217#500—1221no—
218#500—1221no—
219C4×Q16—611no—
220C4×Q16—611no—
221C4×SD16—1221no—
222C4×SD16—1221no—
223C4×D8—611no—
224C4×D8—611no—
225#427C2×D6111yescharacteristic
226C2×#107—611no—
227C2×#107—1221no—
228C2×#107C2×D6111yescharacteristic
229#416—1221no—
230C2×C4×C8—611no—
231#367—612no—
232#367C2×D6112yes—
233#366—612no—
234#366C2×D6112yes—
235C4⋊C16—612no—
236C4⋊C16—612no—
237#358—612no—
238#358—612no—
239#357—612no—
240#357—612no—
241C61#3362911yescharacteristic, complemented, Sylow
242C2×#41—1221no—
243C22×Q8—2441no—
244#135—614no—
245#135—1222no—
246C2×Q16—611no—
247C2×SD16—1221no—
248C2×D8—611no—
249C22×C8—611no—
250C22×C8C22×D6111yescharacteristic
251C4⋊Q8—1222no—
252C4⋊Q8—1222no—
253C4⋊Q8—2441no—
254C4⋊Q8—2441no—
255C4×Q8—1221no—
256C4×Q8—1221no—
257C4×D4—1221no—
258C4×D4—1221no—
259C4×D4—2441no—
260C4×D4—2441no—
261#117—21no—
262#117—2441no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—1221no—
265C2×C4⋊C4—2441no—
266C2×C42—611no—
267C2×C42—2441no—
268M32—22no—
269M32—1222no—
270C2×C16—612no—
271C2×C16C22×D6112yes—
272#107—21no—
273#107—612no—
274#107—1222no—
275#107—1222no—
276#107—1221no—
277#107C22×D6112yes—
278#107C22×D6111yescharacteristic
279#107C22×D6111yescharacteristic
280#106—21no—
281#103—1221no—
282#103—1221no—
283#102—1221no—
284#102—1221no—
285C4×C8—611no—
286C4×C8—611no—
287C4×C8—611no—
288C4×C8—611no—
289#41—1224no—
290#41—2442no—
291C2×Q8—1221no—
292C2×Q8—4884no—
293C2×Q8—2442no—
294C2×Q8—2442no—
295C2×D4—1221no—
296C2×D4—2441no—
297C22×C4—41no—
298C22×C4—611no—
299C22×C4—611no—
300C22×C4—2441no—
301C22×C4—2441no—
302C22×C4—2441no—
303C22×C4—2441no—
304C22×C4D4×D6111yescharacteristic
305Q16—612no—
306Q16—1221no—
307SD16—1224no—
308D8—612no—
309D8—1221no—
310C2×C8—611no—
311C2×C8—611no—
312C2×C8—1222no—
313C2×C8#1175822212yes—
314C2×C8D4×D6111yescharacteristic
315C2×C8Q8×D6112yes—
316C2×C8C23×D6111yescharacteristic
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—41no—
321C4⋊C4—1222no—
322C4⋊C4—2442no—
323C4⋊C4—2442no—
324C4⋊C4—2441no—
325C4⋊C4—2441no—
326C4⋊C4—2441no—
327C22⋊C4—41no—
328C22⋊C4—2441no—
329C22⋊C4—2441no—
330C42—41no—
331C42—611no—
332C42—611no—
333C42—611no—
334C42—611no—
335C42—2441no—
336C42—2441no—
337C16—22no—
338C16—1222no—
339C23—2441no—
340C23D8×D6111yescharacteristic
341Q8—1222no—
342Q8—4884no—
343Q8—4884no—
344Q8—2441no—
345D4—1222no—
346D4—2444no—
347D4—2441no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—82no—
352C2×C4—611no—
353C2×C4—611no—
354C2×C4—611no—
355C2×C4—611no—
356C2×C4—1222no—
357C2×C4—1222no—
358C2×C4—4882no—
359C2×C4—4882no—
360C2×C4—4882no—
361C2×C4—2442no—
362C2×C4—2441no—
363C2×C4—2444no—
364C2×C4#1451975512yes—
365C2×C4D8×D6111yescharacteristic
366C2×C4#1451980912yes—
367C2×C4C2×D4×D6111yescharacteristic
368C8—1221no—
369C8—1221no—
370C8#1451975512yes—
371C8C2×Q8×D6111yescharacteristic
372C8C2×#1175822211yescharacteristic
373C22—21no—
374C22—21no—
375C22—4881no—
376C22—4881no—
377C22—2442no—
378C22#1684004112yes—
379C22C2×D8×D6111yescharacteristic
380C4—81no—
381C4—82no—
382C4—81no—
383C4—612no—
384C4—612no—
385C4—1221no—
386C4—1221no—
387C4—4882no—
388C4—2442no—
389C4#1684004112yes—
390C4C2×#1451975511yescharacteristic
391C4C2×#1451980911yescharacteristic
392C2—22no—
393C2—4882no—
394C2C2×#1684004111yescharacteristic
395C2#509×D6111yescharacteristic
396C2#510×D6111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2159C2
socleC22×C61C22×D8
prime radicalC2×#14519799C22
hypercenter#2159D61
hypocenterC61#33629
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2159

Complements

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

NH ≅ G/Ntype
1. C2×#16840166C2semidirect
3. #21447970C2semidirect
4. #21447965C2semidirect
5. #21447960C2semidirect
6. #21447955C2semidirect
7. #21447608C2semidirect
8. #21447574C2semidirect
11. C61×#2159C2semidirect
46. #14519799Q8semidirect
134. Dic244C2×Q8semidirect
136. D488C2×Q8semidirect
201. #2159D61semidirect
241. C61#33629semidirect

Maximal subgroups

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

subgroupindex
C2×#168401662
#214545902
#214479702
#214479652
#214479602
#214479552
#214476082
#214475742
#214475322
#214475292
C61×#21592
#3362961