Finite Group Databasebeta

← #29580820

Subgroups — #29580820

Order 15616.

534 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 85 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#21457009C211yesmaximal, characteristic
2#21455941C211yesmaximal, characteristic
3#21446737C211yesmaximal, characteristic
4#21444421C211yesmaximal, characteristic, complemented
5#21444266C211yesmaximal, characteristic, complemented
6#21443675C211yesmaximal, characteristic, complemented
7C61×#2584C211yesmaximal, characteristic, complemented
8#16840312C2211yescharacteristic
9#16840172—21no—
10#16840128—21no—
11#16840080—21no—
12#16840075—21no—
13#16840075—21no—
14#16840059—22no—
15#16840057—21no—
16#16839962—21no—
17#16839901—21no—
18#16839860—22no—
19#16839819C2211yescharacteristic
20#16839819C2211yescharacteristic
21C2×#14519705—22no—
22C4×#11758194C2211yescharacteristic
23#16839504—22no—
24Q8⋊#92418—22no—
25C2×#125×C61C2211yescharacteristic
26C2×C4×Q8×C61C2211yescharacteristic
27C61×#398—21no—
28C61×#395—21no—
29C61×#389—21no—
30C61×#389—21no—
31C61×#389—22no—
32C4×C4⋊C4×C61C2211yescharacteristic
33C2×#11758222—21no—
34C2×#11758201—22no—
35C2×#11758194—21no—
36#14519777D411yescharacteristic
37#14519774D411yescharacteristic
38#14519765—21no—
39#14519751—21no—
40#14519751—21no—
41C4×D244—21no—
42C4×D244—21no—
43#14519710—22no—
44#14519710—22no—
45#14519708—22no—
46C4⋊#92418—21no—
47C4⋊#92418D412yes—
48#14519705—44no—
49#14519701—22no—
50#14519697—22no—
51#14519697—22no—
52#14519697—22no—
53C22⋊#92418—21no—
54C22⋊#92418—21no—
55#14519688—21no—
56#14519688—21no—
57#14519688D412yes—
58C4×#92418—21no—
59C22×Q8×C61D411yescharacteristic
60#125×C61—21no—
61#125×C61—21no—
62#125×C61—21no—
63#125×C61—21no—
64C4×Q8×C61—21no—
65C4×Q8×C61—21no—
66C4×Q8×C61—21no—
67C4×Q8×C61—21no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61—22no—
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61—21no—
72C2×C4⋊C4×C61—21no—
73C2×C4⋊C4×C61—22no—
74C2×C4⋊C4×C61D411yescharacteristic
75C2×C4⋊C4×C61D411yescharacteristic
76C2×C4⋊C4×C61D411yescharacteristic
77C2×C42×C61—21no—
78C2×C42×C61D412yes—
79C2×C42×C61C2311yescharacteristic
80#95×C61—22no—
81#95×C61—22no—
82#95×C61—21no—
83C2×Q8×C61—21no—
84C2×Q8×C61—21no—
85C2×Q8×C61—42no—
86C22×C4×C61—22no—
87C22×C4×C61—21no—
88C22×C4×C61—21no—
89C22×C4×C61—21no—
90C22×C4×C61—21no—
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C2×C22⋊D61—41no—
95#11758222—42no—
96#11758222—42no—
97C2×D244—21no—
98C2×D244—21no—
99C2×D244—21no—
100C2×C4×D61—41no—
101C2×Dic122—21no—
102C4⋊C4×C61—24no—
103C4⋊C4×C61—21no—
104C4⋊C4×C61—21no—
105C4⋊C4×C61—21no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—44no—
110C4⋊C4×C61—41no—
111C4⋊C4×C61—41no—
112C4⋊C4×C61—41no—
113C4⋊C4×C61—41no—
114C4⋊C4×C61—41no—
115C4⋊C4×C61—41no—
116C4⋊C4×C61—82no—
117C42×C61—42no—
118C42×C61—41no—
119C42×C61—41no—
120C42×C61C2×D411yescharacteristic
121C42×C61C2×D411yescharacteristic
122C42×C61C2×D411yescharacteristic
123C42×C61#4111yescharacteristic
124#11758201—44no—
125#11758198—41no—
126#11758198—41no—
127C4⋊Dic61—21no—
128C4⋊Dic61—21no—
129#11758194—22no—
130#11758194—41no—
131C2×#92418—22no—
132C2×#92418—22no—
133C2×#92418—41no—
134C23×C61#12011yescharacteristic
135C22×D61—41no—
136Q8×C61—22no—
137Q8×C61—22no—
138Q8×C61—41no—
139Q8×C61—41no—
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—22no—
143C2×C4×C61—24no—
144C2×C4×C61—22no—
145C2×C4×C61—41no—
146C2×C4×C61—42no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—42no—
150C2×C4×C61—42no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61#12011yescharacteristic
159C2×C4×C61#12011yescharacteristic
160C2×C4×C61#12111yescharacteristic
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C2×C4×C61#12711yescharacteristic
164C22⋊D61—81no—
165C22⋊D61—81no—
166C2×Dic61—41no—
167D244—42no—
168D244—42no—
169D244—42no—
170C4×D61—82no—
171Dic122—42no—
172#92418—42no—
173#92418—42no—
174#92418—42no—
175#10007—611nomaximal, Sylow
176C22×C61—21no—
177C22×C61—21no—
178C22×C61#39211yescharacteristic
179C22×C61#45311yescharacteristic
180C22×C61#45411yescharacteristic
181C2×D61—81no—
182C2×D61—81no—
183C2×D61—81no—
184C4×C61—21no—
185C4×C61—21no—
186C4×C61—21no—
187C4×C61—21no—
188C4×C61—21no—
189C4×C61—44no—
190C4×C61—42no—
191C4×C61—81no—
192C4×C61—81no—
193C4×C61—81no—
194C4×C61#45612yes—
195Dic61—81no—
196#3234—611no—
197#3112—611no—
198#2584D6111yescharacteristic, complemented
199#1980—611no—
200#1528—611no—
201#1491—611no—
202#1285—611no—
203C2×C61—21no—
204C2×C61—21no—
205C2×C61#200511yescharacteristic
206C2×C61#200611yescharacteristic
207C2×C61#201011yescharacteristic
208D61—161no—
209#532—611no—
210C2×#125C2×D6111yescharacteristic
211C2×C4×Q8C2×D6111yescharacteristic
212#501—1221no—
213#490—1221no—
214#481—1221no—
215#480—1221no—
216#480—1221no—
217#478—1222no—
218#477—1221no—
219#460—1221no—
220#448—1221no—
221#440—1222no—
222#432—611no—
223#432—611no—
224C2×#104—1222no—
225C4×M16—611no—
226#398—21no—
227#395—21no—
228#389—21no—
229#389—21no—
230#389—22no—
231C4×C4⋊C4C2×D6111yescharacteristic
232#330—1222no—
233Q8⋊C8—1222no—
234C61#1000711yescharacteristic, complemented, Sylow
235C2×#41—1221no—
236C22×Q8C22⋊D6111yescharacteristic
237C2×SD16—1222no—
238C2×M16—1221no—
239C4⋊Q8—611no—
240#127—611no—
241#125—21no—
242#125—21no—
243#125—21no—
244#125—21no—
245#124—1221no—
246#121—1221no—
247#121—1221no—
248C4×Q8—21no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×D4—1221no—
253C4×D4—1221no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4C22⋊D6111yescharacteristic
261C2×C4⋊C4C22⋊D6111yescharacteristic
262C2×C4⋊C4C22⋊D6111yescharacteristic
263C2×C42—21no—
264C2×C42C22⋊D6112yes—
265C2×C42C22×D6111yescharacteristic
266#107—1222no—
267#107—1222no—
268#106—1222no—
269C4⋊C8—612no—
270C4⋊C8—1221no—
271#104—2444no—
272#103—1222no—
273#102—1222no—
274#102—1222no—
275#102—1222no—
276C22⋊C8—1221no—
277C22⋊C8—1221no—
278#97—612no—
279#97—1221no—
280#97—1221no—
281C4×C8—1221no—
282#95—22no—
283#95—22no—
284#95—21no—
285#41—2442no—
286#41—2442no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—42no—
290C2×Q8—1221no—
291C2×D4—1221no—
292C2×D4—1221no—
293C2×D4—1221no—
294C2×D4—2441no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—2441no—
301C22×C4C2×C22⋊D6111yescharacteristic
302C22×C4C2×C22⋊D6111yescharacteristic
303C22×C4C2×C22⋊D6111yescharacteristic
304SD16—2444no—
305M16—1222no—
306M16—2441no—
307C2×C8—1222no—
308C2×C8—1222no—
309C2×C8—2441no—
310C4⋊C4—24no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—44no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—82no—
325C4⋊C4—1221no—
326C4⋊C4—1221no—
327C22⋊C4—2441no—
328C22⋊C4—2441no—
329C42—42no—
330C42—41no—
331C42—41no—
332C42D4×D6111yescharacteristic
333C42D4×D6111yescharacteristic
334C42D4×D6111yescharacteristic
335C42#1175822411yescharacteristic
336C23—2441no—
337C23#1451974611yescharacteristic
338Q8—22no—
339Q8—22no—
340Q8—41no—
341Q8—41no—
342Q8—2442no—
343D4—4881no—
344D4—4881no—
345D4—2442no—
346D4—2442no—
347D4—2442no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—4882no—
367C2×C4—2441no—
368C2×C4#1451974511yescharacteristic
369C2×C4#1451974511yescharacteristic
370C2×C4#1451974811yescharacteristic
371C2×C4#1451974911yescharacteristic
372C2×C4#1451974911yescharacteristic
373C2×C4#1451977311yescharacteristic
374C8—2442no—
375C8—2442no—
376C8—2442no—
377C22—21no—
378C22—21no—
379C22—4881no—
380C22—4881no—
381C22—4881no—
382C22#1683967011yescharacteristic
383C22#1683993011yescharacteristic
384C22#1683993511yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—44no—
391C4—42no—
392C4—81no—
393C4—81no—
394C4—81no—
395C4—4881no—
396C4#1683994412yes—
397C2—21no—
398C2—21no—
399C2—9761no—
400C2#2144685511yescharacteristic
401C2#2144686111yescharacteristic
402C2#2144688411yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#2584C2
socleC22×C61#392
prime radical#16840312C22
hypercenter#2584D61
hypocenterC61#10007
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2584

Complements

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

NH ≅ G/Ntype
4. #21444421C2semidirect
5. #21444266C2semidirect
6. #21443675C2semidirect
7. C61×#2584C2semidirect
198. #2584D61semidirect
234. C61#10007semidirect

Maximal subgroups

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

subgroupindex
#214570092
#214559412
#214467372
#214444212
#214442662
#214436752
C61×#25842
#1000761