Finite Group Databasebeta

← #29596027

Subgroups — #29596027

Order 15616.

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

normal subgroups only

This group has 434 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#21454516C211yesmaximal, 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×C8×D61C411yescharacteristic, complemented
10#16840269C2211yescharacteristic
11#16839836—21no—
12#16839836—21no—
13#16839825—21no—
14#16839825—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
38C22×#92418C2×C411yescharacteristic
39C22×#92418D411yescharacteristic
40C22×#92418D411yescharacteristic
41C2×C8×D61—21no—
42C2×C8×D61—21no—
43C2×C8×D61—41no—
44C2×C8×D61—41no—
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
136C8×D61—42no—
137C8×D61—41no—
138C8×D61—41no—
139C23×C61C2×C4⋊C411yescharacteristic
140C23×C61#12111yescharacteristic
141C23×C61C22⋊Q811yescharacteristic
142C22×D61—21no—
143C22×D61—42no—
144C22×D61—41no—
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—42no—
161C22⋊D61—42no—
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#12805—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—22no—
188C2×D61—41no—
189C2×D61—42no—
190C2×D61—41no—
191C2×D61—42no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—21no—
195C4×C61—161no—
196C4×C61C2×#9911yescharacteristic
197C4×C61#41011yescharacteristic
198Dic61—41no—
199Dic61—41no—
200Dic61—41no—
201Dic61—161no—
202#2929—611no—
203#2120—611no—
204#2120—611no—
205#1813—611no—
206#1812—611no—
207#1395—611no—
208#1395D6111yescharacteristic, complemented
209C2×C61—21no—
210C2×C61—41no—
211C2×C61—41no—
212C2×C61—41no—
213C2×C61#179911yescharacteristic
214D61—42no—
215C2×#131—611no—
216C23×C8—611no—
217#525—611no—
218#435—1221no—
219#435—1221no—
220C8×D4—1221no—
221C8×D4—1221no—
222C2×#108—1221no—
223#427—1221no—
224#424—611no—
225#413—611no—
226#413C2×D6111yescharacteristic
227#410—611no—
228#410C2×D6111yescharacteristic
229#408C2×D6111yescharacteristic
230#407—611no—
231#405—611no—
232#352—1221no—
233#352—1221no—
234AΣL(1,𝔽16)—1221no—
235AΣL(1,𝔽16)—1221no—
236#338—1221no—
237H(3,ℤ/4)—1221no—
238C61#1280511yescharacteristic, complemented, Sylow
239C2×#41C22×D6111yescharacteristic
240C23×C4—611no—
241#131—2441no—
242#131—2441no—
243C2×M16—21no—
244C2×M16—611no—
245C2×M16—611no—
246C2×M16—611no—
247C2×M16—1221no—
248C2×M16C4×D6111yescharacteristic
249C22×C8—611no—
250C22×C8—611no—
251C22×C8—611no—
252C22×C8—1221no—
253C22×C8—1221no—
254C22×C8—2441no—
255C22×C8—2441no—
256C22×C8C4×D6111yescharacteristic
257#123—611no—
258#123—611no—
259C22⋊Q8—1221no—
260#121—1221no—
261#120—611no—
262#120—611no—
263C4×D4—1221no—
264C4×D4—1221no—
265#117—21no—
266#117—611no—
267#117—1221no—
268#108—2441no—
269#108—2441no—
270#107—2441no—
271#106—2441no—
272C4⋊C8—1221no—
273C4⋊C8—1221no—
274#101—21no—
275#101—1221no—
276#100—21no—
277#100—1221no—
278#99—21no—
279#99—21no—
280#99—1221no—
281#99—1221no—
282C22⋊C8—21no—
283C22⋊C8—21no—
284C22⋊C8—1221no—
285C22⋊C8—1221no—
286#97—1221no—
287C4×C8—1221no—
288C24—611no—
289#41—41no—
290#41—41no—
291C2×Q8#1175822211yescharacteristic
292C2×D4—1221no—
293C2×D4—1221no—
294C2×D4#1175822211yescharacteristic
295C2×D4D4×D6111yescharacteristic
296C2×D4Q8×D6111yescharacteristic
297C22×C4—611no—
298C22×C4—1221no—
299C22×C4—1221no—
300C22×C4—2441no—
301C22×C4—2441no—
302C22×C4C2×C4×D6111yescharacteristic
303C22×C4D4×D6111yescharacteristic
304C22×C4D4×D6111yescharacteristic
305M16—41no—
306M16—41no—
307M16—41no—
308M16—2441no—
309M16—2441no—
310M16—2441no—
311M16—2441no—
312M16—2441no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—611no—
318C2×C8—611no—
319C2×C8—611no—
320C2×C8—611no—
321C2×C8—1221no—
322C2×C8—1221no—
323C2×C8—1221no—
324C2×C8—1221no—
325C2×C8—1221no—
326C2×C8—1221no—
327C2×C8—2441no—
328C2×C8—2441no—
329C2×C8—2442no—
330C2×C8—2441no—
331C4⋊C4—41no—
332C4⋊C4—1221no—
333C4⋊C4—1221no—
334C4⋊C4—1221no—
335C4⋊C4—2441no—
336C22⋊C4—41no—
337C22⋊C4—611no—
338C22⋊C4—611no—
339C22⋊C4—1221no—
340C22⋊C4—1221no—
341C22⋊C4—1221no—
342C22⋊C4—1221no—
343C22⋊C4—2441no—
344C42—41no—
345C42—1221no—
346C42—2441no—
347C23—1221no—
348C23—2441no—
349C23—2442no—
350C23C4⋊C4×D6111yescharacteristic
351C23#1451975011yescharacteristic
352C23#1451975511yescharacteristic
353Q8—41no—
354D4—41no—
355D4—41no—
356D4—41no—
357D4—2442no—
358D4—2442no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—81no—
366C2×C4—1221no—
367C2×C4—1221no—
368C2×C4—1221no—
369C2×C4—4881no—
370C2×C4—2442no—
371C2×C4—2441no—
372C2×C4—2441no—
373C2×C4—2441no—
374C2×C4C22⋊C4×D6111yescharacteristic
375C2×C4#1451973411yescharacteristic
376C2×C4#1451975011yescharacteristic
377C2×C4#1451975511yescharacteristic
378C8—41no—
379C8—41no—
380C8—81no—
381C8—611no—
382C8—611no—
383C8—1221no—
384C8—1221no—
385C8—1221no—
386C8—4881no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—1222no—
394C22—2441no—
395C22—2442no—
396C22—2442no—
397C22—2441no—
398C22#1683962111yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—161no—
403C4—9761no—
404C4—2441no—
405C4—2441no—
406C4—2441no—
407C4#99×D6111yescharacteristic
408C4#1683973411yescharacteristic
409C2—21no—
410C2—41no—
411C2—41no—
412C2—41no—
413C2—2442no—
414C2#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#12805
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×C8×D61C4semidirect
17. #16839788C4semidirect
20. #16839725C4semidirect
21. #16839720C4semidirect
208. #1395D61semidirect
238. C61#12805semidirect

Maximal subgroups

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

subgroupindex
#214545162
#214474612
#214474582
#214458672
#214458632
#214439482
C61×#13952
#1280561