Finite Group Databasebeta

← #29725136

Subgroups — #29725136

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#21454671C211yesmaximal, characteristic, complemented
3#21447276C212yesmaximal, complemented
4#21447091C212yesmaximal, complemented
5#21446610C211yesmaximal, characteristic, complemented
6#21446304C212yesmaximal, complemented
7#21445943C211yesmaximal, characteristic, complemented
8#21445928C212yesmaximal, complemented
9#21445874C211yesmaximal, characteristic, complemented
10#21445620C211yesmaximal, characteristic, complemented
11C61×#1978C211yesmaximal, characteristic, complemented
12C2×#14519803C2212yes—
13C2×#14519783C2211yescharacteristic
14#16840237C2212yes—
15#16840029—22no—
16#16840029—22no—
17#16839874—22no—
18D8×Dic61—22no—
19#16839811C2212yes—
20C2×#14519713C2211yescharacteristic
21#16839800C2211yescharacteristic
22#16839778C2212yes—
23#16839767C2212yes—
24#16839744C2211yescharacteristic
25C2×#14519694C2212yes—
26#16839735C2211yescharacteristic
27#16839717C2211yescharacteristic
28#16839534C2211yescharacteristic
29#16839534C2212yes—
30#16839533C2212yes—
31#16839533C2212yes—
32#16839532C2211yescharacteristic
33#16839518C2212yes—
34C22×D8×C61C2211yescharacteristic
35C2×#108×C61C2211yescharacteristic
36C61×#418C2212yes—
37C2×#100×C61C2212yes—
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×#11758189—21no—
52C2×C8×D61—21no—
53D4×Dic61—24no—
54D4×Dic61—42no—
55#14519736C2312yes—
56#14519736C2311yescharacteristic
57C2×C22⋊Dic61—24no—
58#14519713—21no—
59#14519713—21no—
60#14519713—42no—
61#14519711—21no—
62#14519709—21no—
63#14519703—42no—
64#14519698—24no—
65#14519695—21no—
66#14519694—24no—
67#14519693—21no—
68#14519691—22no—
69#14519687—21no—
70C8×Dic61—21no—
71C22×D4×C61C2312yes—
72C2×D8×C61—21no—
73C2×D8×C61—21no—
74C2×D8×C61—21no—
75C2×D8×C61—21no—
76C2×D8×C61D412yes—
77C2×D8×C61D412yes—
78C2×M16×C61C2311yescharacteristic
79C2×M16×C61C2312yes—
80C22×C8×C61C2311yescharacteristic
81#117×C61C2311yescharacteristic
82#108×C61—21no—
83#108×C61—21no—
84#102×C61—42no—
85#100×C61—42no—
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×C61C2411yescharacteristic
96C2×C22⋊D61—21no—
97C22×Dic61—44no—
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—22no—
110M16×C61—21no—
111M16×C61—21no—
112M16×C61—22no—
113C2×C8×C61—21no—
114C2×C8×C61—22no—
115C2×C8×C61C2×D411yescharacteristic
116C2×C8×C61C2×D411yescharacteristic
117C2×C8×C61#4111yescharacteristic
118C2×C8×C61#4111yescharacteristic
119C2×C8×C61#4111yescharacteristic
120C2×C8×C61#4111yescharacteristic
121C4⋊C4×C61—21no—
122C22⋊C4×C61—21no—
123C42×C61—21no—
124C22⋊Dic61—22no—
125C22⋊Dic61—44no—
126C22⋊Dic61#4112yes—
127#11758200—44no—
128#11758199—44no—
129C4⋊Dic61—21no—
130C4⋊Dic61—22no—
131C4×Dic61—21no—
132C4×Dic61—22no—
133#11758194—21no—
134#11758194—22no—
135#11758194—21no—
136#11758194—21no—
137#11758194—21no—
138#11758194—22no—
139C2×#92418—21no—
140C2×#92418—21no—
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—41no—
157C2×C4×C61C22×D411yescharacteristic
158C2×C4×C61C2×#4111yescharacteristic
159C2×C4×C61C2×#4111yescharacteristic
160C2×C4×C61C2×#4111yescharacteristic
161C2×C4×C61H(5,𝔽2)11yescharacteristic
162C2×C4×C61H(5,𝔽2)11yescharacteristic
163C22⋊D61—41no—
164C22⋊D61—41no—
165C2×Dic61—21no—
166C2×Dic61—22no—
167C2×Dic61—42no—
168C2×Dic61—164no—
169D244—21no—
170D244—21no—
171C4×D61—41no—
172C4×D61—41no—
173Dic122—21no—
174Dic122—21no—
175C8×C61—42no—
176C8×C61—41no—
177C8×C61#12311yescharacteristic
178C8×C61#12311yescharacteristic
179C8×C61#12311yescharacteristic
180C8×C61#12311yescharacteristic
181#92418—41no—
182#92418—41no—
183#92418—42no—
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—81no—
196C4×C61C2×#12311yescharacteristic
197C4×C61#54011yescharacteristic
198C4×C61#54611yescharacteristic
199C4×C61#55311yescharacteristic
200Dic61—42no—
201Dic61—41no—
202Dic61—82no—
203C2×#472—611no—
204#2948—611no—
205#2081—612no—
206#2048—612no—
207#1978D6111yescharacteristic, complemented
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#518—612no—
224#472—1222no—
225#472—1222no—
226#442—1222no—
227C4×D8—1222no—
228#430—612no—
229C2×#108—611no—
230C2×#108C2×D6111yescharacteristic
231#427—611no—
232#421—612no—
233#418—612no—
234#418C2×D6112yes—
235#413—611no—
236C2×#100—612no—
237C2×#100C2×D6112yes—
238#410—611no—
239#405—611no—
240#405C2×D6111yescharacteristic
241#344—612no—
242#344—612no—
243#344—611no—
244#344—612no—
245#344—611no—
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×D8D24412yes—
259C2×D8D24412yes—
260#131—1221no—
261#131—1221no—
262#131—1221no—
263#131—1221no—
264C2×M16—611no—
265C2×M16—612no—
266C2×M16—611no—
267C2×M16—1221no—
268C2×M16C22×D6111yescharacteristic
269C2×M16C22×D6112yes—
270C22×C8—1221no—
271C22×C8C22×D6111yescharacteristic
272C4×D4—1224no—
273C4×D4—2442no—
274#117—611no—
275#117—612no—
276#117C22×D6111yescharacteristic
277C2×C22⋊C4—1224no—
278#108—21no—
279#108—21no—
280#108—1221no—
281#108—1221no—
282#108—2442no—
283#107—1221no—
284#106—1221no—
285#104—2442no—
286#102—42no—
287#102—1224no—
288#101—1221no—
289#100—42no—
290#100—1221no—
291#100—1224no—
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×D24414yes—
311C22×C4—1221no—
312C22×C4—2444no—
313C22×C4C23×D6111yescharacteristic
314SD16—2444no—
315D8—22no—
316D8—22no—
317D8—22no—
318D8—22no—
319D8—2444no—
320M16—21no—
321M16—22no—
322M16—21no—
323M16—22no—
324M16—1221no—
325M16—1222no—
326M16—1221no—
327M16—1221no—
328M16—1221no—
329M16—1222no—
330M16—2441no—
331M16—2441no—
332C2×C8—21no—
333C2×C8—22no—
334C2×C8—1221no—
335C2×C8—1222no—
336C2×C8—1221no—
337C2×C8—2441no—
338C2×C8—2441no—
339C2×C8C2×D24411yescharacteristic
340C2×C8C2×D24411yescharacteristic
341C2×C8#1175822411yescharacteristic
342C2×C8#1175822411yescharacteristic
343C2×C8#1175822411yescharacteristic
344C2×C8#1175822411yescharacteristic
345C4⋊C4—21no—
346C4⋊C4—1221no—
347C4⋊C4—1222no—
348C22⋊C4—21no—
349C22⋊C4—612no—
350C22⋊C4—1222no—
351C22⋊C4—2444no—
352C42—21no—
353C42—1221no—
354C42—1222no—
355C23—24no—
356C23—42no—
357C23—42no—
358C23—162no—
359C23—1221no—
360C23#41×D6111yescharacteristic
361Q8—1221no—
362Q8—1221no—
363D4—24no—
364D4—24no—
365D4—42no—
366D4—42no—
367D4—1221no—
368D4—1221no—
369D4—2441no—
370D4—2441no—
371C2×C4—41no—
372C2×C4—1221no—
373C2×C4—1222no—
374C2×C4—9764no—
375C2×C4—2442no—
376C2×C4—2441no—
377C2×C4—2441no—
378C2×C4C22×D24411yescharacteristic
379C2×C4#41×D6111yescharacteristic
380C2×C4C2×#1175822411yescharacteristic
381C2×C4C2×#1175822411yescharacteristic
382C2×C4#1451982211yescharacteristic
383C2×C4#1451982211yescharacteristic
384C8—41no—
385C8—42no—
386C8—2441no—
387C8—2441no—
388C8—2442no—
389C8#1451976111yescharacteristic
390C8#1451976111yescharacteristic
391C8#1451976111yescharacteristic
392C8#1451976111yescharacteristic
393C22—41no—
394C22—44no—
395C22—84no—
396C22—162no—
397C22—162no—
398C22—4881no—
399C22—2441no—
400C22#1684029111yescharacteristic
401C22#1684040711yescharacteristic
402C22#1684047611yescharacteristic
403C4—81no—
404C4—4882no—
405C4—2442no—
406C4—2441no—
407C4C2×#1451976111yescharacteristic
408C4#1684036611yescharacteristic
409C4#1684040711yescharacteristic
410C4#1684045911yescharacteristic
411C2—21no—
412C2—21no—
413C2—21no—
414C2—84no—
415C2—4881no—
416C2#2144930911yescharacteristic
subgroupG/H
centerC2#21449309
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1978C2
socleC2×C61#2420
prime radicalC2×#16840029C2
hypercenter#1978D61
hypocenterC61#24988
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1978

Complements

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

NH ≅ G/Ntype
2. #21454671C2semidirect
3. #21447276C2semidirect
4. #21447091C2semidirect
5. #21446610C2semidirect
6. #21446304C2semidirect
7. #21445943C2semidirect
8. #21445928C2semidirect
9. #21445874C2semidirect
10. #21445620C2semidirect
11. C61×#1978C2semidirect
207. #1978D61semidirect
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
#214546712
#214472762
#214470912
#214466102
#214463042
#214459432
#214459282
#214458742
#214456202
C61×#19782
#2498861