Finite Group Databasebeta

← #29595906

Subgroups — #29595906

Order 15616.

860 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 860 conjugacy classes of subgroups — too many to draw. Only its 127 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#21447491C212yesmaximal, complemented
2#21447484C211yesmaximal, characteristic, complemented
3#21446347C212yesmaximal, complemented
4#21446203C211yesmaximal, characteristic, complemented
5C61×#3549C211yesmaximal, characteristic
6#16840529C2211yescharacteristic, complemented
7#16839779—21no—
8#16839767—22no—
9#16839736C2211yescharacteristic, complemented
10#16839736C2212yescomplemented
11#16839557—21no—
12#16839555—21no—
13#16839555—24no—
14#16839522—21no—
15#16839518—22no—
16C61×H(3,𝔽4)—21no—
17C61×H(3,𝔽4)C412yescomplemented
18C61×#555—22no—
19C61×#555C412yescomplemented
20D42×C61—22no—
21D42×C61—21no—
22C61×#535—21no—
23C61×#535—24no—
24C61×#535C2212yes—
25C61×#532—21no—
26C61×#532C2211yescharacteristic
27#14519784—42no—
28C2×#11758194—22no—
29C2×#11758194—21no—
30#14519736—22no—
31#14519736—21no—
32#14519704—42no—
33#14519698—44no—
34#14519692—22no—
35#14519692—22no—
36#14519692—22no—
37C2×#41×C61—22no—
38C2×#41×C61D411yescharacteristic
39C2×#41×C61C2311yescharacteristic
40C22×D4×C61—24no—
41C22×D4×C61—21no—
42C22×D4×C61D412yes—
43C4⋊Q8×C61—21no—
44C4⋊Q8×C61C2×C411yescharacteristic
45#127×C61—21no—
46#127×C61—22no—
47#127×C61D411yescharacteristic
48#127×C61D414yes—
49#124×C61—24no—
50#124×C61—22no—
51#124×C61—22no—
52#124×C61—24no—
53#124×C61C2×C411yescharacteristic
54#124×C61C2×C414yes—
55#124×C61D411yescharacteristic
56#121×C61—22no—
57#121×C61—22no—
58#121×C61—24no—
59#121×C61—24no—
60#121×C61—22no—
61#121×C61—22no—
62#121×C61—22no—
63#120×C61—22no—
64#120×C61—24no—
65#120×C61—22no—
66#120×C61—24no—
67#120×C61—22no—
68#120×C61—22no—
69C4×D4×C61—22no—
70C4×D4×C61—22no—
71C4×D4×C61—22no—
72#117×C61—21no—
73#117×C61—24no—
74#117×C61D412yes—
75C2×C42×C61D411yescharacteristic
76C24×C61—22no—
77C24×C61—22no—
78#41×C61—41no—
79#41×C61—41no—
80#41×C61—41no—
81#41×C61—41no—
82#41×C61—84no—
83C2×Q8×C61—22no—
84C2×Q8×C61C22⋊C411yescharacteristic
85C2×Q8×C61C22×C411yescharacteristic
86C2×D4×C61—22no—
87C2×D4×C61—22no—
88C2×D4×C61—22no—
89C2×D4×C61—22no—
90C2×D4×C61—24no—
91C2×D4×C61—24no—
92C2×D4×C61—21no—
93C2×D4×C61—24no—
94C2×D4×C61—22no—
95C2×D4×C61—42no—
96C2×D4×C61—42no—
97C2×D4×C61—88no—
98C2×D4×C61—81no—
99C2×D4×C61—81no—
100C2×D4×C61C22⋊C411yescharacteristic
101C2×D4×C61C22⋊C414yes—
102C2×D4×C61C2×D411yescharacteristic
103C2×D4×C61C2×D412yes—
104C22×C4×C61—24no—
105C22×C4×C61—21no—
106C22×C4×C61—21no—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D412yes—
109C4⋊C4×C61—22no—
110C4⋊C4×C61—22no—
111C4⋊C4×C61—22no—
112C22⋊C4×C61—24no—
113C22⋊C4×C61—22no—
114C22⋊C4×C61—24no—
115C22⋊C4×C61—22no—
116C22⋊C4×C61—22no—
117C22⋊C4×C61—24no—
118C22⋊C4×C61—22no—
119C22⋊C4×C61—24no—
120C42×C61—22no—
121C42×C61—21no—
122C42×C61C22⋊C411yescharacteristic
123C42×C61C22⋊C411yescharacteristic
124C42×C61C22⋊C414yes—
125C22⋊Dic61—42no—
126C22⋊Dic61—41no—
127C4⋊Dic61—42no—
128C4⋊Dic61—41no—
129C4×Dic61—41no—
130C4×Dic61—42no—
131#11758194—22no—
132#11758194—42no—
133#11758194—41no—
134C2×#92418—24no—
135C2×#92418—41no—
136C23×C61—24no—
137C23×C61—21no—
138C23×C61—82no—
139C23×C61—84no—
140C23×C61—84no—
141C23×C61—82no—
142C23×C61#12012yes—
143C23×C61#12011yescharacteristic
144Q8×C61—41no—
145Q8×C61—41no—
146Q8×C61—82no—
147D4×C61—42no—
148D4×C61—41no—
149D4×C61—44no—
150D4×C61—41no—
151D4×C61—44no—
152D4×C61—44no—
153D4×C61—42no—
154D4×C61—42no—
155D4×C61—81no—
156D4×C61—84no—
157D4×C61—84no—
158D4×C61—82no—
159D4×C61—84no—
160D4×C61—82no—
161C2×C4×C61—21no—
162C2×C4×C61—22no—
163C2×C4×C61—21no—
164C2×C4×C61—22no—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—21no—
168C2×C4×C61—24no—
169C2×C4×C61—48no—
170C2×C4×C61—42no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61C2×C22⋊C411yescharacteristic
174C2×C4×C61C2×C22⋊C412yes—
175C2×C4×C61#12011yescharacteristic
176C2×Dic61—81no—
177C2×Dic61—82no—
178#92418—42no—
179#12778—611nomaximal, Sylow
180C22×C61—21no—
181C22×C61—22no—
182C22×C61—41no—
183C22×C61—42no—
184C22×C61—44no—
185C22×C61—44no—
186C22×C61—42no—
187C22×C61—41no—
188C22×C61—44no—
189C22×C61—164no—
190C22×C61—164no—
191C22×C61#37911yescharacteristic
192C4×C61—22no—
193C4×C61—22no—
194C4×C61—21no—
195C4×C61—44no—
196C4×C61—41no—
197C4×C61—41no—
198C4×C61B(4,𝔽2)11yescharacteristic
199C4×C61#45811yescharacteristic
200Dic61—161no—
201Dic61—162no—
202#3549D6111yescharacteristic
203#2128—612no—
204#2126—611no—
205#1910—612no—
206#1884—611no—
207C2×C61—21no—
208C2×C61—41no—
209C2×C61—42no—
210C2×C61—84no—
211C2×C61—81no—
212C2×C61#179011yescharacteristic
213#568—611no—
214H(3,𝔽4)—21no—
215H(3,𝔽4)Dic6112yescomplemented
216#555—22no—
217#555Dic6112yescomplemented
218D42—22no—
219D42—21no—
220#535—21no—
221#535—24no—
222#535C2×D6112yes—
223#532—21no—
224#532C2×D6111yescharacteristic
225#421—1221no—
226#418—1222no—
227#410—612no—
228#410—611no—
229#354—1221no—
230#353—1221no—
231#353—1224no—
232#339—1221no—
233H(3,ℤ/4)—1222no—
234C61#1277811yescharacteristic, complemented, Sylow
235C2×#41—22no—
236C2×#41C22⋊D6111yescharacteristic
237C2×#41C22×D6111yescharacteristic
238C22×D4—24no—
239C22×D4—21no—
240C22×D4C22⋊D6112yes—
241#131—2442no—
242C2×M16—1222no—
243C2×M16—1221no—
244C4⋊Q8—21no—
245C4⋊Q8C2×Dic6111yescharacteristic
246#127—21no—
247#127—22no—
248#127C22⋊D6111yescharacteristic
249#127C22⋊D6114yes—
250#124—24no—
251#124—22no—
252#124—22no—
253#124—24no—
254#124C2×Dic6111yescharacteristic
255#124C2×Dic6114yes—
256#124C22⋊D6111yescharacteristic
257#121—22no—
258#121—22no—
259#121—24no—
260#121—24no—
261#121—22no—
262#121—22no—
263#121—22no—
264#120—22no—
265#120—24no—
266#120—22no—
267#120—24no—
268#120—22no—
269#120—22no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—22no—
273#117—21no—
274#117—24no—
275#117—1222no—
276#117—1221no—
277#117C22⋊D6112yes—
278C2×C42C22⋊D6111yescharacteristic
279#104—2442no—
280#102—2444no—
281#99—1222no—
282#99—1222no—
283#99—1222no—
284C24—22no—
285C24—22no—
286#41—41no—
287#41—41no—
288#41—41no—
289#41—41no—
290#41—84no—
291C2×Q8—22no—
292C2×Q8C22⋊Dic6111yescharacteristic
293C2×Q8C22×Dic6111yescharacteristic
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—21no—
301C2×D4—24no—
302C2×D4—22no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—88no—
306C2×D4—81no—
307C2×D4—81no—
308C2×D4C22⋊Dic6111yescharacteristic
309C2×D4C22⋊Dic6114yes—
310C2×D4C2×C22⋊D6111yescharacteristic
311C2×D4C2×C22⋊D6112yes—
312C22×C4—24no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4C2×C22⋊D6111yescharacteristic
316C22×C4C2×C22⋊D6112yes—
317M16—1222no—
318M16—2442no—
319M16—2441no—
320C2×C8—1224no—
321C2×C8—2441no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—2442no—
326C4⋊C4—2441no—
327C22⋊C4—24no—
328C22⋊C4—22no—
329C22⋊C4—24no—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—24no—
333C22⋊C4—22no—
334C22⋊C4—24no—
335C22⋊C4—2442no—
336C22⋊C4—2441no—
337C42—22no—
338C42—21no—
339C42—2441no—
340C42—2442no—
341C42C22⋊Dic6111yescharacteristic
342C42C22⋊Dic6111yescharacteristic
343C42C22⋊Dic6114yes—
344C23—24no—
345C23—21no—
346C23—82no—
347C23—84no—
348C23—84no—
349C23—82no—
350C23#1451974612yes—
351C23#1451974611yescharacteristic
352Q8—41no—
353Q8—41no—
354Q8—82no—
355D4—42no—
356D4—41no—
357D4—44no—
358D4—41no—
359D4—44no—
360D4—44no—
361D4—42no—
362D4—42no—
363D4—81no—
364D4—84no—
365D4—84no—
366D4—82no—
367D4—84no—
368D4—82no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—24no—
377C2×C4—48no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—4881no—
382C2×C4—4882no—
383C2×C4C2×C22⋊Dic6111yescharacteristic
384C2×C4C2×C22⋊Dic6112yes—
385C2×C4#1451974611yescharacteristic
386C8—2442no—
387C22—21no—
388C22—22no—
389C22—41no—
390C22—42no—
391C22—44no—
392C22—44no—
393C22—42no—
394C22—41no—
395C22—44no—
396C22—164no—
397C22—164no—
398C22#1683961811yescharacteristic
399C4—22no—
400C4—22no—
401C4—21no—
402C4—44no—
403C4—41no—
404C4—41no—
405C4—9762no—
406C4—9761no—
407C4#1683994911yescharacteristic
408C4#1683995211yescharacteristic
409C2—21no—
410C2—42no—
411C2—41no—
412C2—84no—
413C2—81no—
414C2#2144576711yescharacteristic
subgroupG/H
centerC2#21445767
commutatorC2×Q8×C61C22×C4
FrattiniC2×#41C22×D61
Fitting, prime radicalC61×#3549C2
socleC2×C61#1790
cosocleC2×#41×C61C23
hypercenter#3549D61
hypocenterC61#12778
LCSG ▷ C2×Q8×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #3549

Complements

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

NH ≅ G/Ntype
1. #21447491C2semidirect
2. #21447484C2semidirect
3. #21446347C2semidirect
4. #21446203C2semidirect
6. #16840529C22semidirect
9. #16839736C22semidirect
10. #16839736C22semidirect
17. C61×H(3,𝔽4)C4semidirect
19. C61×#555C4semidirect
215. H(3,𝔽4)Dic61semidirect
217. #555Dic61semidirect
234. C61#12778semidirect

Maximal subgroups

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

subgroupindex
#214474912
#214474842
#214463472
#214462032
C61×#35492
#1277861