Finite Group Databasebeta

← #29596032

Subgroups — #29596032

Order 15616.

420 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 89 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#21454525C211yesmaximal, characteristic
2#21447461C211yesmaximal, characteristic
3#21447458C211yesmaximal, characteristic
4#21445867C211yesmaximal, characteristic, complemented
5#21445863C211yesmaximal, characteristic, complemented
6#21443948C211yesmaximal, characteristic, complemented
7C61×#1395C211yesmaximal, characteristic, complemented
8C2×#14519784C2211yescharacteristic
9C22×#11758189C411yescharacteristic, complemented
10#16840269C2211yescharacteristic
11#16839837—21no—
12#16839837—21no—
13#16839830—21no—
14#16839830—21no—
15C2×#14519712—21no—
16#16839798—21no—
17#16839788C411yescharacteristic, complemented
18#16839746C2211yescharacteristic
19#16839736C2211yescharacteristic
20#16839725C411yescharacteristic, complemented
21#16839720C411yescharacteristic, complemented
22#16839553—21no—
23#16839551—21no—
24#16839550—21no—
25#16839548—21no—
26#16839520—21no—
27#16839518—21no—
28C61×#413C2211yescharacteristic
29C61×#410C2211yescharacteristic
30C61×#408C2211yescharacteristic
31C22×C4×D61C2×C411yescharacteristic
32#14519784—41no—
33#14519784—41no—
34C2×#11758194—21no—
35C2×#11758194C2×C411yescharacteristic
36C2×#11758194D411yescharacteristic
37C2×#11758194D411yescharacteristic
38C2×#11758189—21no—
39C2×#11758189—21no—
40C2×#11758189—41no—
41C2×#11758189—41no—
42C22×#92418C2×C411yescharacteristic
43C22×#92418D411yescharacteristic
44C22×#92418D411yescharacteristic
45#14519758D411yescharacteristic
46#14519758Q811yescharacteristic
47#14519753—21no—
48#14519748—21no—
49#14519745D411yescharacteristic
50#14519745Q811yescharacteristic
51C4×C22⋊D61—21no—
52C4×C22⋊D61—21no—
53#14519736—21no—
54#14519736C2×C411yescharacteristic
55#14519712—41no—
56#14519712—41no—
57#14519710—41no—
58#14519708—41no—
59#14519706—21no—
60#14519706—21no—
61#14519696—21no—
62#14519694—21no—
63#14519692—21no—
64#14519692—21no—
65#14519689—21no—
66#14519689—21no—
67#14519687—21no—
68C8×Dic61—21no—
69C2×#41×C61C2311yescharacteristic
70C2×M16×C61—21no—
71C2×M16×C61C2×C411yescharacteristic
72C22×C8×C61C2×C411yescharacteristic
73#117×C61—21no—
74#101×C61—21no—
75#100×C61—21no—
76#99×C61—21no—
77#99×C61—21no—
78C22⋊C8×C61—21no—
79C22⋊C8×C61—21no—
80C23×D61C4⋊C411yescharacteristic
81#41×C61—41no—
82#41×C61—41no—
83C2×Q8×C61#4111yescharacteristic
84C2×D4×C61C2×D411yescharacteristic
85C2×D4×C61C2×Q811yescharacteristic
86C2×D4×C61#4111yescharacteristic
87C22×C4×C61C22×C411yescharacteristic
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C2×C22⋊D61—21no—
91C2×C22⋊D61—21no—
92C22×Dic61C4⋊C411yescharacteristic
93C2×C4×D61—21no—
94C2×C4×D61—21no—
95C2×C4×D61—41no—
96C2×C4×D61—41no—
97M16×C61—41no—
98M16×C61—41no—
99M16×C61—41no—
100C2×C8×C61—21no—
101C2×C8×C61—21no—
102C2×C8×C61—41no—
103C2×C8×C61—41no—
104C4⋊C4×C61—41no—
105C22⋊C4×C61—41no—
106C42×C61—41no—
107C22⋊Dic61—21no—
108C22⋊Dic61—21no—
109C22⋊Dic61—41no—
110C22⋊Dic61C4⋊C411yescharacteristic
111C22⋊Dic61C4⋊C411yescharacteristic
112#11758198—21no—
113#11758198—21no—
114C4⋊Dic61—21no—
115C4⋊Dic61—41no—
116#11758196—21no—
117#11758196—21no—
118C4×Dic61—21no—
119C4×Dic61—41no—
120#11758194—41no—
121#11758194—41no—
122#11758194—41no—
123#11758194—41no—
124#11758194—41no—
125C2×#92418—21no—
126C2×#92418—21no—
127C2×#92418—21no—
128C2×#92418—21no—
129C2×#92418—21no—
130C2×#92418—21no—
131C2×#92418—41no—
132C2×#92418C22⋊C411yescharacteristic
133C2×#92418C22⋊C411yescharacteristic
134C2×#92418C22⋊C411yescharacteristic
135C2×#92418C22⋊C411yescharacteristic
136#11758189—42no—
137#11758189—41no—
138#11758189—41no—
139C23×C61C2×C4⋊C411yescharacteristic
140C23×C61#12111yescharacteristic
141C23×C61C22⋊Q811yescharacteristic
142C22×D61—21no—
143C22×D61—41no—
144C22×D61—81no—
145Q8×C61—41no—
146D4×C61—41no—
147D4×C61—41no—
148D4×C61—41no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—81no—
156C2×C4×C61C2×C22⋊C411yescharacteristic
157C2×C4×C61#11711yescharacteristic
158C2×C4×C61#12111yescharacteristic
159C2×C4×C61C22⋊Q811yescharacteristic
160C22⋊D61—81no—
161C22⋊D61—81no—
162C2×Dic61—21no—
163C2×Dic61—21no—
164C2×Dic61—21no—
165C2×Dic61—41no—
166C2×Dic61—81no—
167C4×D61—42no—
168C4×D61—41no—
169C4×D61—41no—
170C8×C61—41no—
171C8×C61—41no—
172C8×C61—81no—
173#92418—21no—
174#92418—21no—
175#92418—21no—
176#92418—81no—
177#92418#9911yescharacteristic
178#92418#9911yescharacteristic
179#12806—611nomaximal, Sylow
180C22×C61—21no—
181C22×C61—21no—
182C22×C61—21no—
183C22×C61—41no—
184C22×C61—41no—
185C22×C61—41no—
186C22×C61#38011yescharacteristic
187C2×D61—41no—
188C2×D61—41no—
189C2×D61—41no—
190C2×D61—82no—
191C4×C61—21no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—161no—
195C4×C61C2×#9911yescharacteristic
196C4×C61#41011yescharacteristic
197Dic61—41no—
198Dic61—41no—
199Dic61—41no—
200Dic61—161no—
201#2930—611no—
202#2120—611no—
203#2120—611no—
204#1813—611no—
205#1812—611no—
206#1395—611no—
207#1395D6111yescharacteristic, complemented
208C2×C61—21no—
209C2×C61—41no—
210C2×C61—41no—
211C2×C61—41no—
212C2×C61#179911yescharacteristic
213D61—81no—
214C2×#131—611no—
215C22×M16—611no—
216#525—611no—
217#436—1221no—
218#436—1221no—
219#435—1221no—
220#435—1221no—
221C2×#108—1221no—
222#427—1221no—
223#424—611no—
224#413—611no—
225#413C2×D6111yescharacteristic
226#410—611no—
227#410C2×D6111yescharacteristic
228#408C2×D6111yescharacteristic
229#407—611no—
230#405—611no—
231#352—1221no—
232#352—1221no—
233AΣL(1,𝔽16)—1221no—
234AΣL(1,𝔽16)—1221no—
235#338—1221no—
236H(3,ℤ/4)—1221no—
237C61#1280611yescharacteristic, complemented, Sylow
238C2×#41C22×D6111yescharacteristic
239C23×C4—611no—
240#131—2441no—
241#131—2441no—
242C2×M16—21no—
243C2×M16—611no—
244C2×M16—611no—
245C2×M16—611no—
246C2×M16—1221no—
247C2×M16—1221no—
248C2×M16—1221no—
249C2×M16—2441no—
250C2×M16—2441no—
251C2×M16C4×D6111yescharacteristic
252C22×C8—611no—
253C22×C8—611no—
254C22×C8—611no—
255C22×C8C4×D6111yescharacteristic
256#123—611no—
257#123—611no—
258C22⋊Q8—1221no—
259#121—1221no—
260#120—611no—
261#120—611no—
262C4×D4—1221no—
263C4×D4—1221no—
264#117—21no—
265#117—611no—
266#117—1221no—
267#108—2441no—
268#108—2441no—
269#107—2441no—
270#106—2441no—
271C4⋊C8—1221no—
272C4⋊C8—1221no—
273#101—21no—
274#101—1221no—
275#100—21no—
276#100—1221no—
277#99—21no—
278#99—21no—
279#99—1221no—
280#99—1221no—
281C22⋊C8—21no—
282C22⋊C8—21no—
283C22⋊C8—1221no—
284C22⋊C8—1221no—
285#97—1221no—
286C4×C8—1221no—
287C24—611no—
288#41—41no—
289#41—41no—
290C2×Q8#1175822211yescharacteristic
291C2×D4—1221no—
292C2×D4—1221no—
293C2×D4#1175822211yescharacteristic
294C2×D4D4×D6111yescharacteristic
295C2×D4Q8×D6111yescharacteristic
296C22×C4—611no—
297C22×C4—1221no—
298C22×C4—1221no—
299C22×C4—2441no—
300C22×C4—2441no—
301C22×C4C2×C4×D6111yescharacteristic
302C22×C4D4×D6111yescharacteristic
303C22×C4D4×D6111yescharacteristic
304M16—41no—
305M16—41no—
306M16—41no—
307M16—2441no—
308M16—2441no—
309M16—2442no—
310M16—2441no—
311M16—2441no—
312M16—2441no—
313M16—2441no—
314M16—2441no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—611no—
320C2×C8—611no—
321C2×C8—611no—
322C2×C8—611no—
323C2×C8—1221no—
324C2×C8—1221no—
325C2×C8—1221no—
326C2×C8—1221no—
327C2×C8—1221no—
328C2×C8—1221no—
329C2×C8—2441no—
330C4⋊C4—41no—
331C4⋊C4—1221no—
332C4⋊C4—1221no—
333C4⋊C4—1221no—
334C4⋊C4—2441no—
335C22⋊C4—41no—
336C22⋊C4—611no—
337C22⋊C4—611no—
338C22⋊C4—1221no—
339C22⋊C4—1221no—
340C22⋊C4—1221no—
341C22⋊C4—1221no—
342C22⋊C4—2441no—
343C42—41no—
344C42—1221no—
345C42—2441no—
346C23—1221no—
347C23—4881no—
348C23—2441no—
349C23C4⋊C4×D6111yescharacteristic
350C23#1451975011yescharacteristic
351C23#1451975511yescharacteristic
352Q8—41no—
353D4—41no—
354D4—41no—
355D4—41no—
356D4—4881no—
357D4—4881no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—81no—
365C2×C4—1221no—
366C2×C4—1221no—
367C2×C4—1221no—
368C2×C4—4881no—
369C2×C4—2442no—
370C2×C4—2441no—
371C2×C4—2441no—
372C2×C4—2441no—
373C2×C4C22⋊C4×D6111yescharacteristic
374C2×C4#1451973411yescharacteristic
375C2×C4#1451975011yescharacteristic
376C2×C4#1451975511yescharacteristic
377C8—41no—
378C8—41no—
379C8—81no—
380C8—611no—
381C8—611no—
382C8—1221no—
383C8—1221no—
384C8—1221no—
385C8—4881no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—4882no—
393C22—2441no—
394C22—2441no—
395C22—2441no—
396C22#1683962111yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—161no—
401C4—9761no—
402C4—2441no—
403C4—2441no—
404C4—2441no—
405C4#99×D6111yescharacteristic
406C4#1683973411yescharacteristic
407C2—21no—
408C2—41no—
409C2—41no—
410C2—41no—
411C2—4881no—
412C2#2144580611yescharacteristic
subgroupG/H
centerC4#16839734
commutatorC22×C4×C61C22×C4
FrattiniC2×#41C22×D61
FittingC61×#1395C2
socleC2×C61#1799
prime radical#16840269C22
cosocleC2×#41×C61C23
hypercenter#1395D61
hypocenterC61#12806
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #1395

Complements

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

NH ≅ G/Ntype
4. #21445867C2semidirect
5. #21445863C2semidirect
6. #21443948C2semidirect
7. C61×#1395C2semidirect
9. C22×#11758189C4semidirect
17. #16839788C4semidirect
20. #16839725C4semidirect
21. #16839720C4semidirect
207. #1395D61semidirect
237. C61#12806semidirect

Maximal subgroups

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

subgroupindex
#214545252
#214474612
#214474582
#214458672
#214458632
#214439482
C61×#13952
#1280661