Finite Group Databasebeta

← #29725132

Subgroups — #29725132

Order 15616.

656 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
1C2×#16840029C211yesmaximal, characteristic
2#21447280C212yesmaximal, complemented
3#21447090C212yesmaximal, complemented
4#21446730C211yesmaximal, characteristic, complemented
5#21446609C211yesmaximal, characteristic, complemented
6#21446306C212yesmaximal, complemented
7#21445942C211yesmaximal, characteristic, complemented
8#21445930C212yesmaximal, complemented
9#21445870C211yesmaximal, characteristic, complemented
10#21445620C211yesmaximal, characteristic, complemented
11C61×#2948C211yesmaximal, characteristic, complemented
12C2×#14519803C2212yes—
13C2×#14519783C2211yescharacteristic
14#16840029—22no—
15#16840029—22no—
16#16839810C2212yes—
17C2×#14519713C2211yescharacteristic
18C2×#14519713C2211yescharacteristic
19#16839780C2212yes—
20#16839767C2212yes—
21#16839744C2211yescharacteristic
22C2×#14519694C2212yes—
23C2×#14519694C2212yes—
24#16839735C2211yescharacteristic
25#16839717C2211yescharacteristic
26#16839534C2211yescharacteristic
27#16839532C2211yescharacteristic
28#16839532C2212yes—
29#16839532C2212yes—
30#16839532C2212yes—
31#16839518C2212yes—
32C22×D8×C61C2211yescharacteristic
33C61×#518C2212yes—
34C61×#442—22no—
35C4×D8×C61—22no—
36C61×#427C2211yescharacteristic
37C61×#418C2212yes—
38C61×#405C2211yescharacteristic
39C2×#11758222C2311yescharacteristic
40#14519803—24no—
41#14519803—42no—
42C2×#11758200—22no—
43C2×#11758199—22no—
44#14519783—21no—
45#14519783—21no—
46#14519783—21no—
47#14519783—21no—
48C2×#11758194C2311yescharacteristic
49C2×#11758194C2311yescharacteristic
50C2×#11758194C2312yes—
51C2×#11758194C2312yes—
52C2×#11758189—21no—
53C2×C8×D61—21no—
54#14519736C2311yescharacteristic
55#14519713—21no—
56#14519713—21no—
57#14519713—21no—
58#14519713—21no—
59#14519712—42no—
60#14519705—42no—
61#14519698—42no—
62#14519695—21no—
63#14519694—24no—
64#14519694—42no—
65#14519693—21no—
66#14519691—22no—
67#14519687—21no—
68C8×Dic61—21no—
69C22×D4×C61C2312yes—
70C2×D8×C61—21no—
71C2×D8×C61—21no—
72C2×D8×C61—21no—
73C2×D8×C61—21no—
74C2×D8×C61D412yes—
75C2×D8×C61D412yes—
76C2×M16×C61C2311yescharacteristic
77C22×C8×C61C2311yescharacteristic
78C4×D4×C61—24no—
79C4×D4×C61—42no—
80#117×C61C2312yes—
81#117×C61C2311yescharacteristic
82C2×C22⋊C4×C61—24no—
83#107×C61—21no—
84#106×C61—21no—
85#102×C61—24no—
86#97×C61—21no—
87C4×C8×C61—21no—
88C24×C61—22no—
89C2×D4×C61—22no—
90C2×D4×C61—22no—
91C2×D4×C61—22no—
92C2×D4×C61—22no—
93C2×D4×C61—22no—
94C2×D4×C61C2×D414yes—
95C22×C4×C61—44no—
96C22×C4×C61C2411yescharacteristic
97C2×C22⋊D61—21no—
98#11758222—21no—
99#11758222—21no—
100#11758222—21no—
101#11758222—21no—
102C2×D244#4111yescharacteristic
103C2×C4×D61—21no—
104C2×Dic122#4111yescharacteristic
105D8×C61—22no—
106D8×C61—22no—
107D8×C61—22no—
108D8×C61—22no—
109M16×C61—21no—
110M16×C61—21no—
111C2×C8×C61—21no—
112C2×C8×C61C2×D411yescharacteristic
113C2×C8×C61C2×D411yescharacteristic
114C2×C8×C61#4111yescharacteristic
115C2×C8×C61#4111yescharacteristic
116C2×C8×C61#4111yescharacteristic
117C2×C8×C61#4111yescharacteristic
118C4⋊C4×C61—21no—
119C4⋊C4×C61—22no—
120C22⋊C4×C61—22no—
121C22⋊C4×C61—44no—
122C22⋊C4×C61#4112yes—
123C42×C61—21no—
124C42×C61—22no—
125C22⋊Dic61—21no—
126#11758200—44no—
127#11758199—44no—
128C4⋊Dic61—21no—
129C4×Dic61—21no—
130#11758194—21no—
131#11758194—22no—
132#11758194—21no—
133#11758194—22no—
134#11758194—21no—
135#11758194—21no—
136#11758194—22no—
137#11758194—22no—
138C2×#92418—21no—
139C2×#92418—21no—
140C2×#92418—22no—
141C2×#92418—22no—
142#11758189—41no—
143#11758189—41no—
144C8×D61—41no—
145C8×D61—41no—
146C23×C61—24no—
147C23×C61—42no—
148C23×C61—42no—
149C23×C61—162no—
150C23×C61C2×#4111yescharacteristic
151C22×D61—21no—
152D4×C61—24no—
153D4×C61—24no—
154D4×C61—42no—
155D4×C61—42no—
156C2×C4×C61—22no—
157C2×C4×C61—42no—
158C2×C4×C61—164no—
159C2×C4×C61C22×D411yescharacteristic
160C2×C4×C61C2×#4111yescharacteristic
161C2×C4×C61C2×#4111yescharacteristic
162C2×C4×C61C2×#4111yescharacteristic
163C2×C4×C61H(5,𝔽2)11yescharacteristic
164C2×C4×C61H(5,𝔽2)11yescharacteristic
165C22⋊D61—41no—
166C22⋊D61—41no—
167C2×Dic61—21no—
168C2×Dic61—41no—
169D244—21no—
170D244—21no—
171C4×D61—41no—
172C4×D61—41no—
173Dic122—21no—
174Dic122—21no—
175C8×C61—41no—
176C8×C61#12311yescharacteristic
177C8×C61#12311yescharacteristic
178C8×C61#12311yescharacteristic
179C8×C61#12311yescharacteristic
180#92418—41no—
181#92418—42no—
182#92418—42no—
183#92418—41no—
184#24988—611nomaximal, Sylow
185C22×C61—41no—
186C22×C61—44no—
187C22×C61—84no—
188C22×C61—162no—
189C22×C61—162no—
190C22×C61#52911yescharacteristic
191C22×C61#54611yescharacteristic
192C22×C61#55511yescharacteristic
193C2×D61—41no—
194C2×D61—81no—
195C4×C61—42no—
196C4×C61—82no—
197C4×C61C2×#12311yescharacteristic
198C4×C61#54011yescharacteristic
199C4×C61#54611yescharacteristic
200C4×C61#55311yescharacteristic
201Dic61—41no—
202Dic61—81no—
203C2×#472—611no—
204#2948D6111yescharacteristic, complemented
205#2081—612no—
206#2048—612no—
207#1978—611no—
208#1953—611no—
209#1904—612no—
210#1833—611no—
211#1829—612no—
212#1814—611no—
213#1757—611no—
214C2×C61—21no—
215C2×C61—21no—
216C2×C61—21no—
217C2×C61—84no—
218C2×C61#242011yescharacteristic
219D61—81no—
220C2×AGL(1,ℤ/8)—612no—
221C22×D8C2×D6111yescharacteristic
222C2×#131—611no—
223#518C2×D6112yes—
224#472—1222no—
225#472—1222no—
226#442—22no—
227C4×D8—22no—
228#430—612no—
229C2×#108—611no—
230C2×#108—611no—
231#427C2×D6111yescharacteristic
232#421—612no—
233#418—612no—
234#418C2×D6112yes—
235#413—611no—
236C2×#100—612no—
237C2×#100—612no—
238#410—611no—
239#405—611no—
240#405C2×D6111yescharacteristic
241#344—611no—
242#344—612no—
243#344—612no—
244#344—611no—
245#344—612no—
246H(3,ℤ/4)—612no—
247C61#2498811yescharacteristic, complemented, Sylow
248C2×#41—611no—
249C22×D4C22×D6112yes—
250AGL(1,ℤ/8)—1224no—
251AGL(1,ℤ/8)—2442no—
252C2×SD16—1222no—
253C2×D8—21no—
254C2×D8—21no—
255C2×D8—21no—
256C2×D8—21no—
257C2×D8—1222no—
258C2×D8C22⋊D6112yes—
259C2×D8C22⋊D6112yes—
260#131—1221no—
261#131—1221no—
262#131—1221no—
263#131—1221no—
264C2×M16—611no—
265C2×M16—612no—
266C2×M16—611no—
267C2×M16—612no—
268C2×M16—1221no—
269C2×M16C22×D6111yescharacteristic
270C22×C8—1221no—
271C22×C8C22×D6111yescharacteristic
272C4×D4—24no—
273C4×D4—42no—
274#117—611no—
275#117C22×D6112yes—
276#117C22×D6111yescharacteristic
277C2×C22⋊C4—24no—
278#108—1221no—
279#108—1221no—
280#108—1221no—
281#108—1221no—
282#108—2442no—
283#107—21no—
284#106—21no—
285#104—2442no—
286#102—24no—
287#102—2442no—
288#101—1221no—
289#100—1221no—
290#100—1224no—
291#100—2442no—
292#99—1222no—
293#97—21no—
294#97—1221no—
295C4×C8—21no—
296C4×C8—1221no—
297C24—22no—
298#41—1221no—
299#41—1221no—
300#41—1221no—
301#41—1221no—
302C2×Q8—611no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—611no—
309C2×D4—1221no—
310C2×D4C2×C22⋊D6114yes—
311C22×C4—44no—
312C22×C4—1221no—
313C22×C4C23×D6111yescharacteristic
314SD16—2444no—
315D8—22no—
316D8—22no—
317D8—22no—
318D8—22no—
319D8—2444no—
320M16—21no—
321M16—21no—
322M16—1221no—
323M16—1222no—
324M16—1221no—
325M16—1222no—
326M16—1221no—
327M16—1221no—
328M16—1222no—
329M16—1222no—
330M16—2441no—
331M16—2441no—
332C2×C8—21no—
333C2×C8—1222no—
334C2×C8—1221no—
335C2×C8—1221no—
336C2×C8—1222no—
337C2×C8—2441no—
338C2×C8—2441no—
339C2×C8#1175822211yescharacteristic
340C2×C8#1175822211yescharacteristic
341C2×C8#1175822211yescharacteristic
342C2×C8#1175822211yescharacteristic
343C2×C8C2×C22⋊D6111yescharacteristic
344C2×C8C2×C22⋊D6111yescharacteristic
345C4⋊C4—21no—
346C4⋊C4—22no—
347C4⋊C4—1221no—
348C22⋊C4—22no—
349C22⋊C4—44no—
350C22⋊C4—1221no—
351C22⋊C4#1175822412yes—
352C42—21no—
353C42—22no—
354C42—1221no—
355C23—24no—
356C23—42no—
357C23—42no—
358C23—162no—
359C23—1221no—
360C23C2×#1175822411yescharacteristic
361Q8—1221no—
362Q8—1221no—
363D4—24no—
364D4—24no—
365D4—42no—
366D4—42no—
367D4—1221no—
368D4—1221no—
369D4—2441no—
370D4—2441no—
371C2×C4—22no—
372C2×C4—42no—
373C2×C4—164no—
374C2×C4—1221no—
375C2×C4—2441no—
376C2×C4—2441no—
377C2×C4—2441no—
378C2×C4C22×C22⋊D6111yescharacteristic
379C2×C4#41×D6111yescharacteristic
380C2×C4C2×#1175822211yescharacteristic
381C2×C4C2×#1175822211yescharacteristic
382C2×C4#1451982211yescharacteristic
383C2×C4#1451982311yescharacteristic
384C8—41no—
385C8—2441no—
386C8—2441no—
387C8—2442no—
388C8—2442no—
389C8#1451976011yescharacteristic
390C8#1451976011yescharacteristic
391C8#1451976011yescharacteristic
392C8#1451976011yescharacteristic
393C22—41no—
394C22—44no—
395C22—84no—
396C22—162no—
397C22—162no—
398C22—4881no—
399C22—2441no—
400C22#1684029411yescharacteristic
401C22#1684040611yescharacteristic
402C22#1684047711yescharacteristic
403C4—42no—
404C4—82no—
405C4—4881no—
406C4—2441no—
407C4C2×#1451976011yescharacteristic
408C4#1684036311yescharacteristic
409C4#1684040011yescharacteristic
410C4#1684046211yescharacteristic
411C2—21no—
412C2—21no—
413C2—21no—
414C2—84no—
415C2—4881no—
416C2#2144930511yescharacteristic
subgroupG/H
centerC2#21449305
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2948C2
socleC2×C61#2420
prime radicalC2×#16840029C2
hypercenter#2948D61
hypocenterC61#24988
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #2948

Complements

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

NH ≅ G/Ntype
2. #21447280C2semidirect
3. #21447090C2semidirect
4. #21446730C2semidirect
5. #21446609C2semidirect
6. #21446306C2semidirect
7. #21445942C2semidirect
8. #21445930C2semidirect
9. #21445870C2semidirect
10. #21445620C2semidirect
11. C61×#2948C2semidirect
204. #2948D61semidirect
247. C61#24988semidirect

Maximal subgroups

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

subgroupindex
C2×#168400292
#214472802
#214470902
#214467302
#214466092
#214463062
#214459422
#214459302
#214458702
#214456202
C61×#29482
#2498861