Finite Group Databasebeta

← #29598816

Subgroups — #29598816

Order 15616.

552 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C22×#14519720C211yesmaximal, characteristic, complemented
2C2×#16840034C211yesmaximal, characteristic, complemented
3#21455468C211yesmaximal, characteristic
4C2×#16839576C211yesmaximal, characteristic, complemented
5C2×#16839547C211yesmaximal, characteristic, complemented
6#21443914C211yesmaximal, characteristic, complemented
7C61×#2141C211yesmaximal, characteristic, complemented
8C2×#14519808—21no—
9C22×D488C2211yescharacteristic
10C2×#14519783C2211yescharacteristic
11#16840331—21no—
12C2×#14519720—22no—
13C2×#14519720—22no—
14C2×#14519720—22no—
15C22×#11758185C2211yescharacteristic
16#16840034—22no—
17#16840034—22no—
18#16840016—22no—
19#16839999—22no—
20C2×#14519713C2211yescharacteristic
21#16839766—21no—
22C2×#14519696—21no—
23#16839576—22no—
24#16839547—22no—
25C22×Q16×C61C2211yescharacteristic
26C2×M32×C61C2211yescharacteristic
27C61×#427C2211yescharacteristic
28C61×#358—22no—
29C2×#11758222D411yescharacteristic
30C22×D244—21no—
31#14519808—24no—
32#14519808—42no—
33C2×#11758202—41no—
34C2×#11758201—41no—
35C2×D488—21no—
36C2×D488—21no—
37C2×D488—21no—
38C2×D488—21no—
39C2×D488—21no—
40C2×D488D412yes—
41#14519783—21no—
42#14519783—21no—
43#14519783—21no—
44#14519783—21no—
45C2×#11758194—21no—
46C2×#11758194D411yescharacteristic
47C2×#11758189—21no—
48C2×C8×D61—21no—
49#14519774—41no—
50#14519765—41no—
51#14519752—24no—
52#14519747—42no—
53#14519720—28no—
54C2×#11758185—21no—
55C2×#11758185—21no—
56C2×#11758185D412yescomplemented
57#14519713—21no—
58#14519713D412yescomplemented
59#14519697—24no—
60#14519696—24no—
61C22×Q8×C61—21no—
62C2×Q16×C61—21no—
63C2×Q16×C61—21no—
64C2×Q16×C61—21no—
65C2×Q16×C61—21no—
66C2×Q16×C61—21no—
67C2×Q16×C61D412yescomplemented
68C22×C8×C61C2311yescharacteristic
69#117×C61—21no—
70M32×C61—21no—
71M32×C61—21no—
72C2×C16×C61—21no—
73#107×C61D412yes—
74#106×C61—21no—
75C23×D61—41no—
76C2×Q8×C61—22no—
77C2×Q8×C61—41no—
78C2×Q8×C61—41no—
79C2×Q8×C61—41no—
80C2×Q8×C61—41no—
81C2×Q8×C61—41no—
82C22×C4×C61—41no—
83C22×C4×C61C2×D411yescharacteristic
84C2×C22⋊D61—21no—
85#11758222—21no—
86#11758222—21no—
87#11758222—21no—
88#11758222—21no—
89C2×D244—22no—
90C2×D244—41no—
91C2×D244—41no—
92C2×D244—41no—
93C2×D244—41no—
94C2×D244—41no—
95C2×D244—44no—
96C2×D244D811yescharacteristic
97C2×C4×D61—21no—
98C2×Dic122D811yescharacteristic
99Q16×C61—22no—
100Q16×C61—22no—
101Q16×C61—41no—
102Q16×C61—41no—
103C2×C8×C61C2×D411yescharacteristic
104C2×C8×C61C2×D411yescharacteristic
105C2×C8×C61C2×D411yescharacteristic
106C2×C8×C61C2×D411yescharacteristic
107C2×C8×C61C2×D411yescharacteristic
108C2×C8×C61#4111yescharacteristic
109C4⋊C4×C61—22no—
110C22⋊C4×C61—41no—
111C42×C61—41no—
112#11758202—44no—
113#11758201—44no—
114#11758198—44no—
115#11758194—22no—
116#11758194—21no—
117#11758194—21no—
118#11758194—41no—
119C2×#92418—21no—
120C2×#92418—41no—
121D488—22no—
122D488—22no—
123D488—41no—
124D488—41no—
125#11758189—41no—
126#11758189—41no—
127C8×D61—41no—
128C8×D61—41no—
129C16×C61—41no—
130#11758185—22no—
131C23×C61AGL(1,ℤ/8)11yescharacteristic
132C22×D61—21no—
133C22×D61—42no—
134C22×D61—81no—
135C22×D61—81no—
136C22×D61—162no—
137Q8×C61—42no—
138Q8×C61—42no—
139Q8×C61—81no—
140Q8×C61—81no—
141C2×C4×C61—42no—
142C2×C4×C61—42no—
143C2×C4×C61—81no—
144C2×C4×C61—81no—
145C2×C4×C61#12011yescharacteristic
146C2×C4×C61#12011yescharacteristic
147C2×C4×C61#12111yescharacteristic
148C2×C4×C61C2×D811yescharacteristic
149C2×C4×C61#13511yescharacteristic
150C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
151C22⋊D61—41no—
152C22⋊D61—41no—
153C2×Dic61—21no—
154D244—21no—
155D244—21no—
156D244—42no—
157D244—42no—
158D244—81no—
159D244—81no—
160D244—164no—
161C4×D61—41no—
162C4×D61—41no—
163Dic122—21no—
164Dic122—21no—
165C8×C61#12011yescharacteristic
166C8×C61#12111yescharacteristic
167C8×C61#12111yescharacteristic
168C8×C61#12711yescharacteristic
169#92418—41no—
170#92418—42no—
171#13321—611nomaximal, Sylow
172C22×C61—22no—
173C22×C61#44711yescharacteristic
174C22×C61#44911yescharacteristic
175C22×C61#46911yescharacteristic
176C2×D61—42no—
177C2×D61—41no—
178C2×D61—84no—
179C2×D61—82no—
180C2×D61—162no—
181C2×D61—162no—
182C4×C61—82no—
183C4×C61—82no—
184C4×C61#39211yescharacteristic
185C4×C61#44711yescharacteristic
186C4×C61#44911yescharacteristic
187C4×C61#46611yescharacteristic
188Dic61—41no—
189C22×SD32—611no—
190C2×#473—611no—
191#3058—611no—
192C2×#361—611no—
193#2141D6111yescharacteristic, complemented
194C2×#350—611no—
195#1382—611no—
196C2×C61—21no—
197C2×C61—21no—
198C2×C61#199911yescharacteristic
199C2×C61#219412yes—
200D61—81no—
201D61—162no—
202C2×#137—1221no—
203C22×Q16C2×D6111yescharacteristic
204C22×D8—611no—
205C2×#131—611no—
206#535—1221no—
207C2×SD32—1222no—
208C2×SD32—1222no—
209C2×SD32—1222no—
210C2×M32C2×D6111yescharacteristic
211C22×C16—611no—
212#473—1222no—
213#473—1222no—
214#469—1222no—
215#466—1222no—
216C2×#108—611no—
217#427C2×D6111yescharacteristic
218#418—1221no—
219C2×#101—1221no—
220#361—1222no—
221#358—22no—
222#350—1222no—
223C61#1332111yescharacteristic, complemented, Sylow
224C2×#41—611no—
225C22×Q8—21no—
226C22×D4—1221no—
227#137—1224no—
228#137—2442no—
229C2×Q16—21no—
230C2×Q16—21no—
231C2×Q16—21no—
232C2×Q16—21no—
233C2×Q16—21no—
234C2×Q16—2441no—
235C2×Q16D24412yescomplemented
236C2×SD16—2441no—
237C2×D8—612no—
238C2×D8—1221no—
239C2×D8—1221no—
240C2×D8—1221no—
241C2×D8—1221no—
242C2×D8—1221no—
243#131—1221no—
244#131—1221no—
245#131—1221no—
246#131—1221no—
247C2×M16—611no—
248C2×M16—1221no—
249C2×M16—1221no—
250C22×C8—1221no—
251C22×C8C22×D6111yescharacteristic
252#127—2441no—
253#124—2441no—
254#121—1224no—
255#120—2442no—
256#117—21no—
257SD32—1228no—
258M32—21no—
259M32—21no—
260C2×C16—21no—
261C2×C16—612no—
262C2×C16—1221no—
263C2×C16—1221no—
264#108—612no—
265#108—1221no—
266#107C22⋊D6112yes—
267#106—21no—
268#102—1224no—
269#101—1224no—
270C24—2441no—
271#41—1221no—
272#41—1221no—
273#41—1221no—
274#41—1221no—
275C2×Q8—22no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—611no—
282C2×D4—611no—
283C2×D4—1221no—
284C2×D4—1222no—
285C2×D4—2444no—
286C2×D4—2441no—
287C2×D4—2441no—
288C2×D4—2441no—
289C2×D4—2441no—
290C2×D4—2441no—
291C22×C4—41no—
292C22×C4—1221no—
293C22×C4D4×D6111yescharacteristic
294Q16—22no—
295Q16—22no—
296Q16—41no—
297Q16—41no—
298Q16—2444no—
299SD16—2444no—
300D8—1222no—
301D8—1222no—
302D8—2441no—
303D8—2441no—
304M16—1221no—
305M16—1222no—
306M16—1221no—
307M16—2441no—
308M16—2441no—
309M16—2441no—
310C2×C8—1221no—
311C2×C8—2441no—
312C2×C8—2441no—
313C2×C8—2441no—
314C2×C8C2×D24411yescharacteristic
315C2×C8#1175822211yescharacteristic
316C2×C8D4×D6111yescharacteristic
317C2×C8D4×D6111yescharacteristic
318C2×C8D4×D6111yescharacteristic
319C2×C8C2×C22⋊D6111yescharacteristic
320C4⋊C4—22no—
321C22⋊C4—41no—
322C22⋊C4—2444no—
323C42—41no—
324C16—41no—
325C16—1222no—
326C23—1221no—
327C23—9762no—
328C23—4881no—
329C23—4881no—
330C23—2442no—
331C23#1451980111yescharacteristic
332Q8—42no—
333Q8—42no—
334Q8—81no—
335Q8—81no—
336Q8—1221no—
337Q8—1221no—
338D4—1221no—
339D4—1221no—
340D4—9764no—
341D4—4881no—
342D4—4881no—
343D4—2441no—
344D4—2441no—
345D4—2442no—
346D4—2442no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—1221no—
352C2×C4—2441no—
353C2×C4—2441no—
354C2×C4#1451974511yescharacteristic
355C2×C4#1451974711yescharacteristic
356C2×C4#1451975011yescharacteristic
357C2×C4D8×D6111yescharacteristic
358C2×C4#1451979811yescharacteristic
359C2×C4#1451980411yescharacteristic
360C8—2442no—
361C8—2441no—
362C8#1451974711yescharacteristic
363C8#1451975011yescharacteristic
364C8#1451975111yescharacteristic
365C8#1451977311yescharacteristic
366C22—22no—
367C22—9762no—
368C22—9762no—
369C22—4884no—
370C22—4882no—
371C22—2442no—
372C22—2441no—
373C22#1683989411yescharacteristic
374C22#1683990811yescharacteristic
375C22#1684001411yescharacteristic
376C4—82no—
377C4—82no—
378C4—2441no—
379C4#1683967111yescharacteristic
380C4#1683989711yescharacteristic
381C4#1683990611yescharacteristic
382C4#1683999711yescharacteristic
383C2—21no—
384C2—21no—
385C2—9762no—
386C2—4881no—
387C2#2144681611yescharacteristic
388C2#2144778712yes—
subgroupG/H
centerC22#16839894
commutator, cosocleC22×C8×C61C23
FrattiniC22×C8C22×D61
FittingC61×#2141C2
socleC22×C61#447
prime radical#21455468C2
hypercenter#2141D61
hypocenterC61#13321
LCSG ▷ C22×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2141

Complements

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

NH ≅ G/Ntype
1. C22×#14519720C2semidirect
2. C2×#16840034C2semidirect
4. C2×#16839576C2semidirect
5. C2×#16839547C2semidirect
6. #21443914C2semidirect
7. C61×#2141C2semidirect
56. C2×#11758185D4semidirect
58. #14519713D4semidirect
67. C2×Q16×C61D4semidirect
193. #2141D61semidirect
223. C61#13321semidirect
235. C2×Q16D244semidirect

Maximal subgroups

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

subgroupindex
C22×#145197202
C2×#168400342
#214554682
C2×#168395762
C2×#168395472
#214439142
C61×#21412
#1332161