Finite Group Databasebeta

← #29574193

Subgroups — #29574193

Order 15616.

546 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448766C211yesmaximal, characteristic
2#21445978C211yesmaximal, characteristic, complemented
3#21445816C211yesmaximal, characteristic
4C2×#16839489C211yesmaximal, characteristic
5#21443703C211yesmaximal, characteristic, complemented
6#21443660C211yesmaximal, characteristic, complemented
7C61×#1948C211yesmaximal, characteristic, complemented
8C2×C4×D244C2211yescharacteristic
9C2×#14519735—21no—
10C2×#14519710—22no—
11C2×C4⋊#92418C2211yescharacteristic
12C2×#14519697—21no—
13C2×C4×#92418C2211yescharacteristic
14#16839664—21no—
15#16839654—22no—
16#16839632C2211yescharacteristic
17#16839619—21no—
18C4×C4⋊Dic61—21no—
19#16839524—21no—
20#16839489C414yescomplemented
21C2×#106×C61—21no—
22C2×C4⋊C8×C61C2211yescharacteristic
23C61×#384C2211yescharacteristic
24C4×C4⋊C4×C61C2211yescharacteristic
25C61×#340—21no—
26C22×D244D411yescharacteristic
27C22×C4×D61—21no—
28C22×#92418—21no—
29C22×#92418—21no—
30C4×D244—21no—
31C4×D244—21no—
32C4×D244C2×C412yes—
33C4×D244Q812yes—
34#14519735—42no—
35#14519735—42no—
36C2×C4⋊Dic61—21no—
37C2×C4⋊Dic61—21no—
38C2×C4⋊Dic61D411yescharacteristic
39C2×C4⋊Dic61D412yes—
40C2×#11758198—21no—
41C2×#11758198—22no—
42C2×#11758198—22no—
43C2×C4×Dic61—22no—
44#14519710—44no—
45C4⋊#92418—21no—
46C4⋊#92418—21no—
47#14519697—42no—
48C4×#92418—21no—
49C4×#92418C2×C412yes—
50#14519684—22no—
51C22×C8×C61—21no—
52C2×C4⋊C4×C61—21no—
53C2×C4⋊C4×C61—21no—
54C2×C4⋊C4×C61—21no—
55C2×C4⋊C4×C61D411yescharacteristic
56C2×C4⋊C4×C61D411yescharacteristic
57C2×C42×C61—21no—
58C2×C42×C61C2311yescharacteristic
59#106×C61—42no—
60C4⋊C8×C61—21no—
61C4⋊C8×C61C2×C412yes—
62#95×C61—21no—
63C23×D61—21no—
64C22×C4×C61—21no—
65C22×C4×C61—21no—
66C22×C4×C61—41no—
67C22×C4×C61C2×D411yescharacteristic
68C22×C4×C61C2×D411yescharacteristic
69C22×C4×C61C2×D411yescharacteristic
70C22×Dic61—21no—
71C22×Dic61—22no—
72C2×D244—21no—
73C2×D244—21no—
74C2×D244—21no—
75C2×D244—21no—
76C2×D244—21no—
77C2×D244C4⋊C412yes—
78C2×C4×D61—22no—
79C2×C4×D61—22no—
80C2×C4×D61—41no—
81C2×C4×D61—41no—
82C2×C4×D61—41no—
83C2×C4×D61—41no—
84C2×C8×C61—22no—
85C2×C8×C61—41no—
86C2×C8×C61—41no—
87C4⋊C4×C61—21no—
88C4⋊C4×C61—21no—
89C4⋊C4×C61—21no—
90C4⋊C4×C61—42no—
91C4⋊C4×C61—41no—
92C4⋊C4×C61—41no—
93C4⋊C4×C61—41no—
94C4⋊C4×C61—41no—
95C4⋊C4×C61D812yes—
96C42×C61—41no—
97C42×C61—41no—
98C42×C61C22×C411yescharacteristic
99C42×C61C2×Q811yescharacteristic
100C42×C61#4111yescharacteristic
101C42×C61#4111yescharacteristic
102#11758198—22no—
103#11758198—22no—
104#11758198—44no—
105#11758198—82no—
106C4⋊Dic61—21no—
107C4⋊Dic61—41no—
108C4⋊Dic61—41no—
109C4⋊Dic61—41no—
110C4⋊Dic61—41no—
111C4⋊Dic61—42no—
112C4⋊Dic61C4⋊C412yes—
113C4×Dic61—44no—
114C2×#92418—22no—
115C2×#92418—22no—
116C2×#92418—22no—
117C2×#92418—41no—
118C2×#92418—41no—
119C2×#92418—41no—
120C23×C61#12011yescharacteristic
121C22×D61—22no—
122C22×D61—41no—
123C22×D61—41no—
124C22×D61—41no—
125C22×D61—41no—
126C22×D61—41no—
127C22×D61—41no—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
130C2×C4×C61—22no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—81no—
137C2×C4×C61—81no—
138C2×C4×C61—81no—
139C2×C4×C61C2×C4⋊C411yescharacteristic
140C2×C4×C61C4×D411yescharacteristic
141C2×C4×C61C4×D411yescharacteristic
142C2×C4×C61C22⋊Q811yescharacteristic
143C2×C4×C61C22⋊Q811yescharacteristic
144C2×C4×C61#12311yescharacteristic
145C2×C4×C61C2×D811yescharacteristic
146C2×C4×C61#13511yescharacteristic
147C2×C4×C61#13511yescharacteristic
148C2×C4×C61#13511yescharacteristic
149C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C61#13711yescharacteristic
152C2×C4×C61#13711yescharacteristic
153C2×Dic61—22no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—42no—
157C2×Dic61—42no—
158C2×Dic61—42no—
159D244—22no—
160D244—22no—
161D244—41no—
162D244—41no—
163C4×D61—42no—
164C4×D61—42no—
165C4×D61—42no—
166C4×D61—42no—
167C8×C61—42no—
168#92418—24no—
169#92418—81no—
170#8885—611nomaximal, Sylow
171C22×C61#38511yescharacteristic
172C22×C61#44711yescharacteristic
173C22×C61#44911yescharacteristic
174C22×C61#45211yescharacteristic
175C22×C61#45211yescharacteristic
176C22×C61#45311yescharacteristic
177C22×C61#45611yescharacteristic
178C2×D61—42no—
179C2×D61—42no—
180C2×D61—42no—
181C2×D61—81no—
182C2×D61—81no—
183C2×D61—81no—
184C2×D61—81no—
185C4×C61—21no—
186C4×C61—21no—
187C4×C61—42no—
188C4×C61—81no—
189C4×C61—161no—
190C4×C61#42711yescharacteristic
191C4×C61#42811yescharacteristic
192C4×C61C4×D811yescharacteristic
193C4×C61#44011yescharacteristic
194C4×C61#47411yescharacteristic
195C4×C61#47811yescharacteristic
196C4×C61#48211yescharacteristic
197C4×C61#48511yescharacteristic
198Dic61—42no—
199Dic61—82no—
200#2354—611no—
201#1948D6111yescharacteristic, complemented
202#1845—611no—
203#1802—611no—
204C2×D4⋊C8—611no—
205#1296—611no—
206#1279—611no—
207C2×C61#162111yescharacteristic
208C2×C61#163411yescharacteristic
209C2×C61#164711yescharacteristic
210C2×C61#165011yescharacteristic
211C2×C61#185411yescharacteristic
212C2×C61#185911yescharacteristic
213C2×C61#186211yescharacteristic
214D61—82no—
215C2×C4×D4—611no—
216C2×#117—1221no—
217C2×#107—1222no—
218C2×#106—21no—
219C2×C4⋊C8—611no—
220C2×C4⋊C8C2×D6111yescharacteristic
221C2×#102—1221no—
222C2×C4×C8—611no—
223#390—1221no—
224#388—1222no—
225#384—611no—
226#384C2×D6111yescharacteristic
227#380—1221no—
228C4×C4⋊C4—1221no—
229C4×C4⋊C4C2×D6111yescharacteristic
230#340—21no—
231#340—1221no—
232D4⋊C8—614no—
233C61#888511yescharacteristic, complemented, Sylow
234C22×D4—611no—
235C23×C4—1221no—
236C22×C8—21no—
237C22×C8—1221no—
238C22×C8—1221no—
239C4×D4—612no—
240C4×D4—612no—
241C4×D4—1221no—
242C4×D4—1221no—
243#117—2442no—
244#117—2442no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—611no—
249C2×C4⋊C4—612no—
250C2×C4⋊C4—1221no—
251C2×C4⋊C4—1221no—
252C2×C4⋊C4C22⋊D6111yescharacteristic
253C2×C4⋊C4C22⋊D6111yescharacteristic
254C2×C22⋊C4—1221no—
255C2×C22⋊C4—1222no—
256C2×C22⋊C4—1222no—
257C2×C42—21no—
258C2×C42—1222no—
259C2×C42C22×D6111yescharacteristic
260#107—2444no—
261#106—42no—
262C4⋊C8—21no—
263C4⋊C8—1221no—
264C4⋊C8—1221no—
265C4⋊C8C4×D6112yes—
266#102—2442no—
267C4×C8—612no—
268C4×C8—1221no—
269#95—21no—
270#95—1222no—
271C24—1221no—
272C2×D4—612no—
273C2×D4—1221no—
274C2×D4—1221no—
275C2×D4—1221no—
276C2×D4—1221no—
277C2×D4—1221no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—41no—
281C22×C4—1221no—
282C22×C4—1222no—
283C22×C4—1222no—
284C22×C4—1222no—
285C22×C4—2441no—
286C22×C4—2441no—
287C22×C4—2441no—
288C22×C4—2441no—
289C22×C4D4×D6111yescharacteristic
290C22×C4D4×D6111yescharacteristic
291C22×C4C2×C22⋊D6111yescharacteristic
292C2×C8—22no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—1222no—
296C2×C8—1222no—
297C2×C8—1222no—
298C2×C8—2441no—
299C2×C8—2441no—
300C2×C8—2441no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—612no—
310C4⋊C4—1221no—
311C4⋊C4—2442no—
312C4⋊C4—2441no—
313C4⋊C4—2441no—
314C4⋊C4—2441no—
315C4⋊C4—2441no—
316C4⋊C4#1175819912yes—
317C22⋊C4—1222no—
318C22⋊C4—1222no—
319C22⋊C4—4882no—
320C22⋊C4—2444no—
321C42—41no—
322C42—41no—
323C42—2444no—
324C42C2×C4×D6111yescharacteristic
325C42#1175822211yescharacteristic
326C42Q8×D6111yescharacteristic
327C42#1175822611yescharacteristic
328C23—1222no—
329C23—2441no—
330C23—2441no—
331C23—2441no—
332C23—2441no—
333C23—2441no—
334C23—2441no—
335C23#1451974511yescharacteristic
336D4—1222no—
337D4—1222no—
338D4—2441no—
339D4—2441no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—1222no—
352C2×C4—2442no—
353C2×C4—2442no—
354C2×C4—2442no—
355C2×C4—2441no—
356C2×C4—2441no—
357C2×C4—2442no—
358C2×C4—2442no—
359C2×C4—2442no—
360C2×C4—2442no—
361C2×C4C4⋊C4×D6111yescharacteristic
362C2×C4C4×C22⋊D6111yescharacteristic
363C2×C4#1451973911yescharacteristic
364C2×C4#1451975311yescharacteristic
365C2×C4#1451975511yescharacteristic
366C2×C4#1451976211yescharacteristic
367C2×C4C2×#1175819911yescharacteristic
368C2×C4#1451979711yescharacteristic
369C2×C4#1451979811yescharacteristic
370C2×C4#1451979811yescharacteristic
371C2×C4#1451980211yescharacteristic
372C2×C4#1451980411yescharacteristic
373C2×C4#1451980611yescharacteristic
374C2×C4#1451980811yescharacteristic
375C8—42no—
376C8—1224no—
377C8—4881no—
378C22—4881no—
379C22—4881no—
380C22—4881no—
381C22—4881no—
382C22—2442no—
383C22—2442no—
384C22—2442no—
385C22#1683963311yescharacteristic
386C22#1683989511yescharacteristic
387C22#1683990611yescharacteristic
388C22#1683992111yescharacteristic
389C22#1683992211yescharacteristic
390C22#1683992811yescharacteristic
391C22#1683994311yescharacteristic
392C4—21no—
393C4—21no—
394C4—42no—
395C4—81no—
396C4—161no—
397C4—4882no—
398C4—2442no—
399C4#1683979711yescharacteristic
400C4#1683980111yescharacteristic
401C4C4×#1175819911yescharacteristic
402C4#1683986011yescharacteristic
403C4#1684003911yescharacteristic
404C4#1684005911yescharacteristic
405C4#1684008811yescharacteristic
406C4#1684010611yescharacteristic
407C2—4882no—
408C2#2144488211yescharacteristic
409C2#2144496311yescharacteristic
410C2#2144505211yescharacteristic
411C2#2144507411yescharacteristic
412C2#2144601711yescharacteristic
413C2#2144604211yescharacteristic
414C2#2144606511yescharacteristic
subgroupG/H
centerC23#14519745
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1948C2
socleC23×C61#120
prime radicalC22×D244D4
cosocleC2×C42×C61C23
hypercenter#1948D61
hypocenterC61#8885
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
2. #21445978C2semidirect
5. #21443703C2semidirect
6. #21443660C2semidirect
7. C61×#1948C2semidirect
20. #16839489C4semidirect
201. #1948D61semidirect
233. C61#8885semidirect

Maximal subgroups

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

subgroupindex
#214487662
#214459782
#214458162
C2×#168394892
#214437032
#214436602
C61×#19482
#888561