Finite Group Databasebeta

← #29657842

Subgroups — #29657842

Order 15616.

1082 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 1082 conjugacy classes of subgroups — too many to draw. Only its 179 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457197C212yesmaximal, complemented
2#21456245C211yesmaximal, characteristic, complemented
3#21444964C212yesmaximal, complemented
4#21444935C212yesmaximal, complemented
5#21444902C212yesmaximal, complemented
6#21444873C212yesmaximal, complemented
7#21444554C211yesmaximal, characteristic, complemented
8#21444270C212yesmaximal, complemented
9C61×#3549C211yesmaximal, characteristic, complemented
10C2×#14519797—22no—
11#16840312C2211yescharacteristic, complemented
12#16840144—22no—
13#16840141C2211yescharacteristic, complemented
14#16840138C2211yescharacteristic
15#16840136C2211yescharacteristic, complemented
16#16840134C2211yescharacteristic
17#16840052C2214yescomplemented
18#16840049—22no—
19#16840013—24no—
20#16839957C2214yescomplemented
21#16839954—22no—
22#16839786C2212yescomplemented
23C2×C4×#92418C2211yescharacteristic, complemented
24#16839511C2212yescomplemented
25#16839506C2212yes—
26D4⋊#92418C2214yescomplemented
27D4⋊#92418C2214yescomplemented
28C61×H(3,𝔽4)—22no—
29C61×#555—22no—
30C61×#555C2212yes—
31D42×C61—22no—
32D42×C61C2212yes—
33C61×#535—22no—
34C61×#535—24no—
35C61×#532C2211yescharacteristic
36C61×#532C2212yes—
37C2×#11758222—21no—
38#14519797—48no—
39C2×#11758202—22no—
40C2×#11758200—22no—
41C2×#11758201—22no—
42C2×#11758199—22no—
43C2×#11758199—24no—
44C2×#11758194—22no—
45C22×#92418—21no—
46#14519777C2311yescharacteristic
47#14519774C2311yescharacteristic
48#14519765—21no—
49#14519751—22no—
50C4×D244—22no—
51#14519708—24no—
52C4⋊#92418—22no—
53C4⋊#92418C2314yes—
54#14519698—24no—
55#14519697—24no—
56C4×#92418—21no—
57C4×#92418C2311yescharacteristic
58C4×#92418C2311yescharacteristic
59C2×#41×C61—22no—
60C2×#41×C61D412yes—
61C22×D4×C61—22no—
62C22×D4×C61—24no—
63C4⋊Q8×C61D412yes—
64C4⋊Q8×C61C2311yescharacteristic
65#127×C61—24no—
66#127×C61D412yes—
67#127×C61C2311yescharacteristic
68#124×C61—28no—
69#124×C61—24no—
70#124×C61—22no—
71#124×C61D412yes—
72#121×C61—24no—
73#121×C61—24no—
74#121×C61—22no—
75#121×C61—22no—
76#121×C61—24no—
77#121×C61D414yes—
78#120×C61—24no—
79#120×C61—28no—
80#120×C61—24no—
81C4×D4×C61—22no—
82C4×D4×C61—22no—
83C4×D4×C61C2314yes—
84#117×C61—22no—
85#117×C61—24no—
86C2×C42×C61C2311yescharacteristic
87C24×C61—24no—
88#41×C61—44no—
89#41×C61—82no—
90#41×C61—82no—
91C2×Q8×C61—22no—
92C2×Q8×C61C2×D412yes—
93C2×D4×C61—24no—
94C2×D4×C61—28no—
95C2×D4×C61—22no—
96C2×D4×C61—24no—
97C2×D4×C61—22no—
98C2×D4×C61—22no—
99C2×D4×C61—22no—
100C2×D4×C61—84no—
101C2×D4×C61—84no—
102C2×D4×C61—84no—
103C2×D4×C61C2×D414yes—
104C2×D4×C61C2×D412yes—
105C22×C4×C61—22no—
106C22×C4×C61—24no—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D412yes—
109C2×C22⋊D61—41no—
110#11758222—44no—
111C2×D244—21no—
112C2×D244—22no—
113C2×C4×D61—41no—
114C2×Dic122—21no—
115C4⋊C4×C61—22no—
116C4⋊C4×C61—22no—
117C4⋊C4×C61C2×D414yes—
118C22⋊C4×C61—28no—
119C22⋊C4×C61—24no—
120C22⋊C4×C61—24no—
121C22⋊C4×C61—24no—
122C22⋊C4×C61—24no—
123C42×C61—24no—
124C42×C61C2×D412yes—
125C42×C61C2×D411yescharacteristic
126C42×C61C2411yescharacteristic
127#11758202—44no—
128#11758201—44no—
129#11758200—44no—
130#11758199—44no—
131#11758199—84no—
132#11758198—42no—
133C4⋊Dic61—22no—
134#11758194—82no—
135C2×#92418—24no—
136C2×#92418—21no—
137C2×#92418—21no—
138C2×#92418—24no—
139C23×C61—24no—
140C23×C61—22no—
141C23×C61—88no—
142C23×C61—84no—
143C23×C61#12011yescharacteristic
144C22×D61—41no—
145Q8×C61—42no—
146Q8×C61—82no—
147D4×C61—24no—
148D4×C61—24no—
149D4×C61—42no—
150D4×C61—42no—
151D4×C61—42no—
152D4×C61—44no—
153D4×C61—82no—
154D4×C61—82no—
155D4×C61—84no—
156D4×C61—88no—
157D4×C61—84no—
158C2×C4×C61—22no—
159C2×C4×C61—22no—
160C2×C4×C61—22no—
161C2×C4×C61—22no—
162C2×C4×C61—24no—
163C2×C4×C61—44no—
164C2×C4×C61—44no—
165C2×C4×C61—44no—
166C2×C4×C61#12012yes—
167C2×C4×C61#12011yescharacteristic
168C2×C4×C61C22×D411yescharacteristic
169C2×C4×C61C22×D412yes—
170C22⋊D61—82no—
171C2×Dic61—41no—
172D244—42no—
173D244—42no—
174D244—42no—
175C4×D61—82no—
176Dic122—42no—
177#92418—22no—
178#92418—22no—
179#92418—44no—
180#18686—611nomaximal, Sylow
181C22×C61—22no—
182C22×C61—21no—
183C22×C61—44no—
184C22×C61—44no—
185C22×C61—44no—
186C22×C61—42no—
187C22×C61—44no—
188C22×C61—168no—
189C22×C61C2×#12011yescharacteristic
190C2×D61—82no—
191C2×D61—81no—
192C4×C61—22no—
193C4×C61—21no—
194C4×C61—44no—
195C4×C61—42no—
196C4×C61#45312yes—
197C4×C61#57612yes—
198C4×C61#57712yes—
199Dic61—81no—
200#3549D6111yescharacteristic, complemented
201#3253—612no—
202#3147—611no—
203#1634—612no—
204#1629—612no—
205#1624—612no—
206#1619—612no—
207#1560—611no—
208#1492—612no—
209C2×C61—41no—
210C2×C61—84no—
211C2×C61—82no—
212C2×C61#301212yes—
213C2×C61C2×#45311yescharacteristic
214D61—161no—
215C2×#135—1222no—
216H(3,𝔽4)—22no—
217#555—22no—
218#555C2×D6112yes—
219D42—22no—
220D42C2×D6112yes—
221#535—22no—
222#535—24no—
223#532—611no—
224#532C2×D6111yescharacteristic
225#532C2×D6112yes—
226#495—1222no—
227#494—611no—
228#493—611no—
229#492—611no—
230#492—611no—
231#476—614no—
232#476—1222no—
233#469—1224no—
234#459—614no—
235#459—1222no—
236#423—612no—
237C2×C4×C8—611no—
238#333—612no—
239#331—612no—
240D4⋊C8—614no—
241D4⋊C8—614no—
242C61#1868611yescharacteristic, complemented, Sylow
243C2×#41—22no—
244C2×#41—1221no—
245C2×#41C22⋊D6112yes—
246C22×D4—22no—
247C22×D4—24no—
248#135—2448no—
249C2×Q16—1222no—
250C2×SD16—1222no—
251C2×SD16—1222no—
252C2×D8—1224no—
253C2×D8—1222no—
254C2×M16—1222no—
255C22×C8—1221no—
256C4⋊Q8—611no—
257C4⋊Q8C22⋊D6112yes—
258C4⋊Q8C22×D6111yescharacteristic
259#127—24no—
260#127—611no—
261#127C22⋊D6112yes—
262#127C22×D6111yescharacteristic
263#124—28no—
264#124—24no—
265#124—22no—
266#124—1221no—
267#124C22⋊D6112yes—
268#121—24no—
269#121—24no—
270#121—22no—
271#121—22no—
272#121—24no—
273#121—1222no—
274#121C22⋊D6114yes—
275#120—24no—
276#120—28no—
277#120—24no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—1222no—
281C4×D4C22×D6114yes—
282#117—22no—
283#117—24no—
284C2×C42C22×D6111yescharacteristic
285#106—1224no—
286C4⋊C8—614no—
287C4⋊C8—1222no—
288#102—1224no—
289#102—1224no—
290C4×C8—611no—
291C4×C8—611no—
292C4×C8—1221no—
293C24—24no—
294#41—44no—
295#41—82no—
296#41—82no—
297#41—2444no—
298C2×Q8—22no—
299C2×Q8—1221no—
300C2×Q8C2×C22⋊D6112yes—
301C2×D4—24no—
302C2×D4—28no—
303C2×D4—22no—
304C2×D4—24no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—84no—
309C2×D4—84no—
310C2×D4—84no—
311C2×D4—1222no—
312C2×D4—1221no—
313C2×D4—2441no—
314C2×D4C2×C22⋊D6114yes—
315C2×D4C2×C22⋊D6112yes—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—2441no—
319C22×C4C2×C22⋊D6111yescharacteristic
320C22×C4C2×C22⋊D6112yes—
321Q16—2444no—
322SD16—2444no—
323SD16—2444no—
324D8—4884no—
325D8—2444no—
326M16—4882no—
327C2×C8—1224no—
328C2×C8—1221no—
329C2×C8—1221no—
330C2×C8—1224no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—1222no—
334C4⋊C4C2×C22⋊D6114yes—
335C22⋊C4—28no—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4—24no—
340C22⋊C4—2442no—
341C42—24no—
342C42C2×C22⋊D6111yescharacteristic
343C42C2×C22⋊D6112yes—
344C42C23×D6111yescharacteristic
345C23—24no—
346C23—22no—
347C23—88no—
348C23—84no—
349C23—2441no—
350C23#1451974611yescharacteristic
351Q8—42no—
352Q8—82no—
353Q8—2442no—
354D4—24no—
355D4—24no—
356D4—42no—
357D4—42no—
358D4—42no—
359D4—44no—
360D4—82no—
361D4—82no—
362D4—84no—
363D4—88no—
364D4—84no—
365D4—4882no—
366D4—2442no—
367D4—2442no—
368D4—2442no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—4882no—
378C2×C4—2441no—
379C2×C4#1451974612yes—
380C2×C4#1451974611yescharacteristic
381C2×C4C22×C22⋊D6111yescharacteristic
382C2×C4C22×C22⋊D6112yes—
383C8—1222no—
384C8—1222no—
385C8—2444no—
386C22—22no—
387C22—21no—
388C22—44no—
389C22—44no—
390C22—44no—
391C22—42no—
392C22—44no—
393C22—168no—
394C22—4882no—
395C22—4881no—
396C22C2×#1451974611yescharacteristic
397C4—22no—
398C4—21no—
399C4—44no—
400C4—42no—
401C4—4881no—
402C4#1683993012yes—
403C4#1684056512yes—
404C4#1684057312yes—
405C2—41no—
406C2—84no—
407C2—82no—
408C2—9761no—
409C2#2145512312yes—
410C2C2×#1683993011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3549C2
socleC22×C61C2×#120
hypercenter#3549D61
hypocenterC61#18686
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3549

Complements

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

NH ≅ G/Ntype
1. #21457197C2semidirect
2. #21456245C2semidirect
3. #21444964C2semidirect
4. #21444935C2semidirect
5. #21444902C2semidirect
6. #21444873C2semidirect
7. #21444554C2semidirect
8. #21444270C2semidirect
9. C61×#3549C2semidirect
11. #16840312C22semidirect
13. #16840141C22semidirect
15. #16840136C22semidirect
17. #16840052C22semidirect
20. #16839957C22semidirect
22. #16839786C22semidirect
23. C2×C4×#92418C22semidirect
24. #16839511C22semidirect
26. D4⋊#92418C22semidirect
27. D4⋊#92418C22semidirect
200. #3549D61semidirect
242. C61#18686semidirect

Maximal subgroups

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

subgroupindex
#214571972
#214562452
#214449642
#214449352
#214449022
#214448732
#214445542
#214442702
C61×#35492
#1868661