Finite Group Databasebeta

← #29598721

Subgroups — #29598721

Order 15616.

566 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519723C211yesmaximal, characteristic, complemented
2C2×#16840004C211yesmaximal, characteristic
3C2×#16840000C211yesmaximal, characteristic
4C2×#16839567C211yesmaximal, characteristic, complemented
5C2×#16839541C211yesmaximal, characteristic
6#21443910C211yesmaximal, characteristic, complemented
7C2×C61×#358C211yesmaximal, characteristic, complemented
8C22×Dic244C2211yescharacteristic
9C22×C8×D61C2211yescharacteristic
10C2×#14519757—21no—
11C2×#14519753—21no—
12C2×#14519723—22no—
13C2×#14519723—22no—
14C2×#14519723—22no—
15C22×#11758185C2211yescharacteristic
16#16840004—22no—
17#16840004—22no—
18#16840000—22no—
19#16840000—22no—
20C2×#14519711C2211yescharacteristic
21C2×#14519701—21no—
22C2×#14519700—21no—
23#16839567—22no—
24#16839541—22no—
25C22×Q16×C61C2211yescharacteristic
26C22×C16×C61C2211yescharacteristic
27C2×#107×C61C2211yescharacteristic
28C61×#358—22no—
29C22×Dic122—21no—
30C22×C4×D61D411yescharacteristic
31C2×Dic244—21no—
32C2×Dic244—21no—
33C2×Dic244—21no—
34C2×Dic244—21no—
35C2×Dic244—21no—
36C2×Dic244D412yes—
37C22×#92418D411yescharacteristic
38C2×C8×D61—21no—
39C2×C8×D61—21no—
40C2×C8×D61—21no—
41C2×C8×D61—21no—
42C2×C8×D61—21no—
43C2×C8×D61—21no—
44#14519757—24no—
45#14519757—42no—
46#14519753—24no—
47#14519753—42no—
48C2×C4⋊Dic61—21no—
49C2×C4⋊Dic61—41no—
50C2×#11758196—41no—
51C2×#11758198—41no—
52C2×#11758198—41no—
53#14519723—28no—
54C2×#11758185—21no—
55C2×#11758185—21no—
56C2×#11758185D412yes—
57#14519711—21no—
58#14519711D412yes—
59#14519701—24no—
60#14519700—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×C61D412yes—
68C22×C8×C61C2311yescharacteristic
69C2×C4⋊C4×C61—21no—
70C2×C16×C61—21no—
71C2×C16×C61—21no—
72C2×C16×C61—21no—
73#107×C61—21no—
74#107×C61D412yes—
75C23×D61D811yescharacteristic
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×C61—41no—
84C22×C4×C61C2×D411yescharacteristic
85C22×Dic61—41no—
86C22×Dic61—41no—
87C22×Dic61D811yescharacteristic
88C2×C4×D61—21no—
89C2×C4×D61—21no—
90C2×C4×D61—21no—
91C2×C4×D61—21no—
92C2×C4×D61—21no—
93C2×C4×D61—21no—
94C2×Dic122—22no—
95C2×Dic122—41no—
96C2×Dic122—41no—
97C2×Dic122—41no—
98C2×Dic122—41no—
99C2×Dic122—41no—
100Q16×C61—22no—
101Q16×C61—22no—
102Q16×C61—41no—
103Q16×C61—41no—
104C2×C8×C61C2×D411yescharacteristic
105C2×C8×C61C2×D411yescharacteristic
106C2×C8×C61C2×D411yescharacteristic
107C2×C8×C61C2×D411yescharacteristic
108C2×C8×C61C2×D411yescharacteristic
109C2×C8×C61#4111yescharacteristic
110C4⋊C4×C61—22no—
111C4⋊C4×C61—41no—
112#11758198—44no—
113#11758198—44no—
114C4⋊Dic61—22no—
115C4⋊Dic61—44no—
116C4⋊Dic61—41no—
117#11758196—44no—
118C2×#92418—21no—
119C2×#92418—21no—
120C2×#92418—21no—
121Dic244—22no—
122Dic244—22no—
123Dic244—41no—
124Dic244—41no—
125C8×D61—41no—
126C8×D61—41no—
127C8×D61—41no—
128C8×D61—41no—
129C16×C61—41no—
130#11758185—22no—
131C23×C61C2×D811yescharacteristic
132C22×D61—21no—
133C22×D61—21no—
134C22×D61—21no—
135C22×D61—21no—
136C22×D61—21no—
137C22×D61—21no—
138C22×D61Q3212yescomplemented
139Q8×C61—42no—
140Q8×C61—42no—
141Q8×C61—81no—
142Q8×C61—81no—
143C2×C4×C61—42no—
144C2×C4×C61—42no—
145C2×C4×C61—81no—
146C2×C4×C61—81no—
147C2×C4×C61—81no—
148C2×C4×C61—81no—
149C2×C4×C61#12011yescharacteristic
150C2×C4×C61#12011yescharacteristic
151C2×C4×C61#12111yescharacteristic
152C2×C4×C61#13511yescharacteristic
153C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
155C2×Dic61—21no—
156C2×Dic61—21no—
157C2×Dic61—21no—
158C2×Dic61—42no—
159C2×Dic61—42no—
160C2×Dic61—81no—
161C2×Dic61—81no—
162C2×Dic61—81no—
163C2×Dic61—81no—
164C4×D61—41no—
165C4×D61—41no—
166C4×D61—41no—
167C4×D61—41no—
168Dic122—42no—
169Dic122—42no—
170Dic122—81no—
171Dic122—81no—
172C8×C61#12011yescharacteristic
173C8×C61#12111yescharacteristic
174C8×C61#12111yescharacteristic
175C8×C61#12711yescharacteristic
176#92418—41no—
177#13306—611nomaximal, Sylow
178C22×C61#44711yescharacteristic
179C22×C61#44711yescharacteristic
180C22×C61#46611yescharacteristic
181C22×C61C2×Q3211yescharacteristic
182C22×C61#50811yescharacteristic
183C22×C61#51011yescharacteristic
184C22×C61#51011yescharacteristic
185C2×D61—22no—
186C2×D61—22no—
187C2×D61—22no—
188C2×D61—41no—
189C2×D61—41no—
190C2×D61—41no—
191C2×D61—41no—
192C4×C61—82no—
193C4×C61—82no—
194C4×C61#39211yescharacteristic
195C4×C61#44911yescharacteristic
196C4×C61#44911yescharacteristic
197C4×C61#46911yescharacteristic
198Dic61—41no—
199Dic61—82no—
200Dic61—82no—
201C22×Q32—611no—
202C2×#467—611no—
203C2×#467—611no—
204C2×#358—611no—
205C2×#358D6111yescharacteristic, complemented
206C2×C22⋊C16—611no—
207#1380—611no—
208C2×C61#199911yescharacteristic
209C2×C61#218811yescharacteristic
210C2×C61#218811yescharacteristic
211C2×C61#218911yescharacteristic
212C2×C61#218911yescharacteristic
213C2×C61#221611yescharacteristic
214C2×C61#221911yescharacteristic
215D61—42no—
216C22×Q16—611no—
217C22×Q16C2×D6111yescharacteristic
218C23×C8—611no—
219C2×C22⋊Q8—1221no—
220C2×C22⋊Q8—1221no—
221C2×Q32—1222no—
222C2×Q32—1222no—
223C2×Q32—1222no—
224C22×C16—611no—
225C22×C16C2×D6111yescharacteristic
226#467—1222no—
227#467—1222no—
228#467—1222no—
229#467—1222no—
230C2×#107—611no—
231C2×#107C2×D6111yescharacteristic
232C2×#103—1221no—
233C2×#103—1221no—
234#358—22no—
235#358—1222no—
236C22⋊C16—1222no—
237C61#1330611yescharacteristic, complemented, Sylow
238C22×Q8—21no—
239C22×Q8—1221no—
240C23×C4—611no—
241C2×Q16—21no—
242C2×Q16—21no—
243C2×Q16—21no—
244C2×Q16—21no—
245C2×Q16—21no—
246C2×Q16—612no—
247C2×Q16—1221no—
248C2×Q16—1221no—
249C2×Q16—1221no—
250C2×Q16—1221no—
251C2×Q16—1221no—
252C2×Q16D24412yes—
253C22×C8—611no—
254C22×C8—1221no—
255C22×C8—1221no—
256C22×C8—1221no—
257C22×C8—1221no—
258C22×C8—1221no—
259C22×C8—1221no—
260C22×C8C22×D6111yescharacteristic
261C22⋊Q8—1224no—
262C22⋊Q8—1224no—
263C22⋊Q8—2442no—
264C22⋊Q8—2442no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—1221no—
267C2×C4⋊C4—2441no—
268C2×C4⋊C4—2441no—
269C2×C22⋊C4—2441no—
270C2×C22⋊C4—2441no—
271Q32—1228no—
272C2×C16—21no—
273C2×C16—21no—
274C2×C16—21no—
275C2×C16—612no—
276C2×C16—1221no—
277C2×C16—1221no—
278#107—21no—
279#107—612no—
280#107—1221no—
281#107C22⋊D6112yes—
282#103—1224no—
283#103—1224no—
284C24—611no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—1222no—
292C2×Q8—2441no—
293C2×Q8—2441no—
294C2×Q8—2441no—
295C2×Q8—2441no—
296C2×Q8—2441no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—611no—
300C22×C4—1221no—
301C22×C4—1221no—
302C22×C4—1221no—
303C22×C4—1221no—
304C22×C4—1221no—
305C22×C4—1221no—
306C22×C4—2441no—
307C22×C4—2441no—
308C22×C4D4×D6111yescharacteristic
309Q16—22no—
310Q16—22no—
311Q16—41no—
312Q16—41no—
313Q16—1222no—
314Q16—1222no—
315Q16—2441no—
316Q16—2441no—
317C2×C8—1221no—
318C2×C8—1221no—
319C2×C8—1221no—
320C2×C8—2441no—
321C2×C8—2441no—
322C2×C8—2441no—
323C2×C8—2441no—
324C2×C8C2×D24411yescharacteristic
325C2×C8#1175822211yescharacteristic
326C2×C8D4×D6111yescharacteristic
327C2×C8D4×D6111yescharacteristic
328C2×C8D4×D6111yescharacteristic
329C2×C8C2×C22⋊D6111yescharacteristic
330C4⋊C4—22no—
331C4⋊C4—41no—
332C4⋊C4—1222no—
333C4⋊C4—2444no—
334C4⋊C4—2444no—
335C4⋊C4—2441no—
336C22⋊C4—2444no—
337C22⋊C4—2444no—
338C16—41no—
339C16—1222no—
340C23—612no—
341C23—1221no—
342C23—1221no—
343C23—1221no—
344C23—1221no—
345C23—1221no—
346C23—1221no—
347C23D8×D6111yescharacteristic
348Q8—42no—
349Q8—42no—
350Q8—81no—
351Q8—81no—
352Q8—4881no—
353Q8—4881no—
354Q8—2442no—
355Q8—2442no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—1221no—
363C2×C4—1221no—
364C2×C4—1221no—
365C2×C4—4881no—
366C2×C4—4881no—
367C2×C4—4881no—
368C2×C4—4881no—
369C2×C4—2442no—
370C2×C4—2442no—
371C2×C4—2441no—
372C2×C4—2441no—
373C2×C4—2441no—
374C2×C4—2441no—
375C2×C4#1451974511yescharacteristic
376C2×C4#1451974711yescharacteristic
377C2×C4#1451975011yescharacteristic
378C2×C4#1451979811yescharacteristic
379C2×C4#1451980111yescharacteristic
380C2×C4#1451980411yescharacteristic
381C8—2441no—
382C8#1451974711yescharacteristic
383C8#1451975011yescharacteristic
384C8#1451975111yescharacteristic
385C8#1451977311yescharacteristic
386C22—1222no—
387C22—1222no—
388C22—1222no—
389C22—2441no—
390C22—2441no—
391C22—2441no—
392C22—2441no—
393C22#1683989411yescharacteristic
394C22#1683989711yescharacteristic
395C22#1683999711yescharacteristic
396C22Q32×D6111yescharacteristic
397C22#1684019311yescharacteristic
398C22#1684020111yescharacteristic
399C22#1684020411yescharacteristic
400C4—82no—
401C4—82no—
402C4—4882no—
403C4—4882no—
404C4—2441no—
405C4#1683967111yescharacteristic
406C4#1683990611yescharacteristic
407C4#1683990811yescharacteristic
408C4#1684001411yescharacteristic
409C2—2442no—
410C2#2144681611yescharacteristic
411C2#2144775111yescharacteristic
412C2#2144775411yescharacteristic
413C2#2144775611yescharacteristic
414C2#2144776211yescharacteristic
415C2#2144790911yescharacteristic
416C2#2144792611yescharacteristic
subgroupG/H
centerC23D8×D61
commutator, cosocleC22×C8×C61C23
FrattiniC22×C8C22×D61
FittingC2×C61×#358C2
socleC23×C61C2×D8
prime radicalC23×D61D8
hypercenterC2×#358D61
hypocenterC61#13306
LCSG ▷ C22×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
1. C22×#14519723C2semidirect
4. C2×#16839567C2semidirect
6. #21443910C2semidirect
7. C2×C61×#358C2semidirect
138. C22×D61Q32semidirect
205. C2×#358D61semidirect
237. C61#13306semidirect

Maximal subgroups

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

subgroupindex
C22×#145197232
C2×#168400042
C2×#168400002
C2×#168395672
C2×#168395412
#214439102
C2×C61×#3582
#1330661