Finite Group Databasebeta

← #29725140

Subgroups — #29725140

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×#16840032C211yesmaximal, characteristic
2#21454670C211yesmaximal, characteristic, complemented
3#21447278C212yesmaximal, complemented
4#21447093C212yesmaximal, complemented
5#21446727C211yesmaximal, characteristic, complemented
6#21446610C211yesmaximal, characteristic, complemented
7#21446303C212yesmaximal, complemented
8#21445942C211yesmaximal, characteristic, complemented
9#21445929C212yesmaximal, complemented
10#21445619C211yesmaximal, characteristic, complemented
11C61×#1814C211yesmaximal, characteristic, complemented
12C2×#14519802C2212yes—
13C22×D488C2211yescharacteristic
14#16840236C2212yes—
15#16840032—22no—
16#16840032—22no—
17#16839873—22no—
18#16839844—22no—
19#16839811C2212yes—
20C2×#14519713C2211yescharacteristic
21#16839768C2212yes—
22#16839766C2212yes—
23#16839746C2211yescharacteristic
24C2×#14519693C2212yes—
25C2×#14519693C2212yes—
26#16839736C2211yescharacteristic
27#16839717C2211yescharacteristic
28#16839717C2211yescharacteristic
29#16839534C2212yes—
30#16839533C2211yescharacteristic
31#16839533C2211yescharacteristic
32#16839532C2212yes—
33#16839518C2212yes—
34C2×#131×C61C2211yescharacteristic
35C2×#108×C61C2211yescharacteristic
36C61×#427C2211yescharacteristic
37C61×#421C2212yes—
38C61×#344C2212yes—
39C22×D244C2312yes—
40#14519802—24no—
41#14519802—42no—
42C2×#11758201—22no—
43C2×D488—21no—
44C2×D488—21no—
45C2×D488—21no—
46C2×D488—21no—
47C2×D488D412yes—
48C2×D488D412yes—
49C2×#11758199—22no—
50C2×#11758194C2311yescharacteristic
51C2×#11758194C2311yescharacteristic
52C2×#11758194C2312yes—
53C4×D244—42no—
54#14519739—24no—
55#14519736C2311yescharacteristic
56#14519736C2311yescharacteristic
57C22⋊C4×D61—24no—
58#14519713—21no—
59#14519713—21no—
60#14519713—42no—
61#14519699—42no—
62#14519697—24no—
63#14519696—21no—
64#14519694—21no—
65#14519693—24no—
66#14519693—42no—
67#14519692—22no—
68#14519687—21no—
69#14519687—21no—
70C8×Dic61—21no—
71C8×Dic61—21no—
72C2×#41×C61C2311yescharacteristic
73#131×C61—21no—
74#131×C61—21no—
75#131×C61—21no—
76#131×C61—21no—
77C2×M16×C61—21no—
78C2×M16×C61C2311yescharacteristic
79C2×M16×C61C2312yes—
80C22×C8×C61—21no—
81C22×C8×C61C2311yescharacteristic
82#117×C61C2312yes—
83#108×C61—21no—
84#108×C61—21no—
85#107×C61—21no—
86#106×C61—21no—
87#104×C61—42no—
88C23×D61—22no—
89#41×C61—21no—
90#41×C61—21no—
91#41×C61—21no—
92#41×C61—21no—
93C2×Q8×C61#4111yescharacteristic
94C2×D4×C61—21no—
95C2×D4×C61#4111yescharacteristic
96C22×C4×C61—21no—
97C22×C4×C61C2411yescharacteristic
98C2×D244—22no—
99C2×D244—22no—
100C2×D244—22no—
101C2×D244—22no—
102C2×D244—22no—
103C2×D244C2×D414yes—
104C2×C4×D61—44no—
105M16×C61—21no—
106M16×C61—22no—
107M16×C61—21no—
108M16×C61—22no—
109M16×C61—41no—
110M16×C61—41no—
111C2×C8×C61—21no—
112C2×C8×C61—22no—
113C2×C8×C61—41no—
114C2×C8×C61—41no—
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—22no—
122C22⋊C4×C61—22no—
123C42×C61—22no—
124C22⋊Dic61—21no—
125C22⋊Dic61#4112yes—
126#11758201—44no—
127#11758199—44no—
128#11758198—44no—
129C4⋊Dic61—21no—
130C4⋊Dic61—21no—
131C4×Dic61—21no—
132C4×Dic61—21no—
133#11758194—22no—
134#11758194—21no—
135#11758194—21no—
136#11758194—21no—
137#11758194—21no—
138#11758194—22no—
139C2×#92418—21no—
140C2×#92418—21no—
141C2×#92418—22no—
142D488—22no—
143D488—22no—
144D488—22no—
145D488—22no—
146C23×C61—21no—
147C23×C61C2×#4111yescharacteristic
148C22×D61—24no—
149C22×D61—42no—
150C22×D61—42no—
151C22×D61—162no—
152Q8×C61—21no—
153Q8×C61—21no—
154D4×C61—21no—
155D4×C61—21no—
156D4×C61—41no—
157D4×C61—41no—
158C2×C4×C61—21no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—42no—
162C2×C4×C61C22×D411yescharacteristic
163C2×C4×C61C2×#4111yescharacteristic
164C2×C4×C61C2×#4111yescharacteristic
165C2×C4×C61C2×#4111yescharacteristic
166C2×C4×C61H(5,𝔽2)11yescharacteristic
167C2×C4×C61H(5,𝔽2)11yescharacteristic
168C2×Dic61—22no—
169C2×Dic61—41no—
170D244—24no—
171D244—24no—
172D244—42no—
173D244—42no—
174C4×D61—164no—
175C8×C61—41no—
176C8×C61—42no—
177C8×C61#12311yescharacteristic
178C8×C61#12311yescharacteristic
179C8×C61#12311yescharacteristic
180C8×C61#12311yescharacteristic
181#92418—41no—
182#92418—42no—
183#92418—41no—
184#24988—611nomaximal, Sylow
185C22×C61—41no—
186C22×C61—41no—
187C22×C61—81no—
188C22×C61#52911yescharacteristic
189C22×C61#54611yescharacteristic
190C22×C61#55511yescharacteristic
191C2×D61—44no—
192C2×D61—84no—
193C2×D61—162no—
194C2×D61—162no—
195C4×C61—41no—
196C4×C61—82no—
197C4×C61C2×#12311yescharacteristic
198C4×C61#54011yescharacteristic
199C4×C61#54611yescharacteristic
200C4×C61#55311yescharacteristic
201Dic61—42no—
202Dic61—81no—
203C2×#472—611no—
204#2948—611no—
205#2081—612no—
206#2048—612no—
207#1978—611no—
208#1953—611no—
209#1904—612no—
210#1833—611no—
211#1829—612no—
212#1814D6111yescharacteristic, complemented
213#1757—611no—
214C2×C61—21no—
215C2×C61—21no—
216C2×C61—21no—
217C2×C61—81no—
218C2×C61#242011yescharacteristic
219D61—84no—
220C2×AGL(1,ℤ/8)—612no—
221C22×D8—611no—
222C2×#131C2×D6111yescharacteristic
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#427C2×D6111yescharacteristic
232#421C2×D6112yes—
233#418—612no—
234#418—612no—
235#413—611no—
236C2×#100—612no—
237C2×#100—612no—
238#410—611no—
239#405—611no—
240#405—611no—
241#344—611no—
242#344—611no—
243#344—612no—
244#344—612no—
245#344C2×D6112yes—
246H(3,ℤ/4)—612no—
247C61#2498811yescharacteristic, complemented, Sylow
248C2×#41C22×D6111yescharacteristic
249C22×D4—612no—
250AGL(1,ℤ/8)—1224no—
251AGL(1,ℤ/8)—2442no—
252C2×SD16—1222no—
253C2×D8—612no—
254C2×D8—612no—
255C2×D8—1222no—
256C2×D8—1221no—
257C2×D8—1221no—
258C2×D8—1221no—
259C2×D8—1221no—
260#131—21no—
261#131—21no—
262#131—21no—
263#131—21no—
264C2×M16—21no—
265C2×M16—612no—
266C2×M16—611no—
267C2×M16—611no—
268C2×M16C22×D6111yescharacteristic
269C2×M16C22×D6112yes—
270C22×C8—21no—
271C22×C8C22×D6111yescharacteristic
272C4×D4—1224no—
273C4×D4—2442no—
274#117—611no—
275#117—611no—
276#117C22×D6112yes—
277C2×C22⋊C4—1224no—
278#108—21no—
279#108—21no—
280#108—1221no—
281#108—1221no—
282#108—2442no—
283#107—21no—
284#106—21no—
285#104—42no—
286#102—1224no—
287#102—2442no—
288#101—1221no—
289#100—1224no—
290#100—1221no—
291#100—2442no—
292#99—1222no—
293#97—1221no—
294#97—1221no—
295C4×C8—1221no—
296C4×C8—1221no—
297C24—1222no—
298#41—21no—
299#41—21no—
300#41—21no—
301#41—21no—
302C2×Q8#1175822611yescharacteristic
303C2×D4—21no—
304C2×D4—614no—
305C2×D4—1222no—
306C2×D4—1222no—
307C2×D4—1222no—
308C2×D4—1222no—
309C2×D4—1222no—
310C2×D4#1175822611yescharacteristic
311C22×C4—21no—
312C22×C4—2444no—
313C22×C4C23×D6111yescharacteristic
314SD16—2444no—
315D8—1222no—
316D8—1222no—
317D8—1222no—
318D8—1222no—
319D8—2444no—
320M16—21no—
321M16—22no—
322M16—21no—
323M16—22no—
324M16—41no—
325M16—41no—
326M16—1222no—
327M16—1221no—
328M16—1221no—
329M16—1221no—
330M16—1221no—
331M16—1222no—
332C2×C8—21no—
333C2×C8—22no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—1221no—
337C2×C8—1221no—
338C2×C8—1222no—
339C2×C8#1175822211yescharacteristic
340C2×C8#1175822211yescharacteristic
341C2×C8D4×D6111yescharacteristic
342C2×C8D4×D6111yescharacteristic
343C2×C8#1175822411yescharacteristic
344C2×C8#1175822411yescharacteristic
345C4⋊C4—22no—
346C4⋊C4—1221no—
347C4⋊C4—1221no—
348C22⋊C4—22no—
349C22⋊C4—612no—
350C22⋊C4—1221no—
351C22⋊C4—2444no—
352C42—22no—
353C42—1221no—
354C42—1221no—
355C23—21no—
356C23—1224no—
357C23—9762no—
358C23—2442no—
359C23—2442no—
360C23#41×D6111yescharacteristic
361Q8—21no—
362Q8—21no—
363D4—21no—
364D4—21no—
365D4—41no—
366D4—41no—
367D4—1224no—
368D4—1224no—
369D4—2442no—
370D4—2442no—
371C2×C4—21no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—1222no—
376C2×C4—9764no—
377C2×C4—2441no—
378C2×C4C2×D4×D6111yescharacteristic
379C2×C4C2×#1175822411yescharacteristic
380C2×C4C2×#1175822211yescharacteristic
381C2×C4C2×#1175822611yescharacteristic
382C2×C4#1451982211yescharacteristic
383C2×C4#1451982311yescharacteristic
384C8—41no—
385C8—42no—
386C8—2441no—
387C8—2441no—
388C8—2442no—
389C8#1451975911yescharacteristic
390C8#1451975911yescharacteristic
391C8#1451975911yescharacteristic
392C8#1451975911yescharacteristic
393C22—41no—
394C22—41no—
395C22—81no—
396C22—9762no—
397C22—9762no—
398C22—4884no—
399C22—2444no—
400C22#1684029811yescharacteristic
401C22#1684040911yescharacteristic
402C22#1684047511yescharacteristic
403C4—41no—
404C4—82no—
405C4—4881no—
406C4—2442no—
407C4C2×#1451975911yescharacteristic
408C4#1684036511yescharacteristic
409C4#1684040311yescharacteristic
410C4#1684046011yescharacteristic
411C2—21no—
412C2—21no—
413C2—21no—
414C2—81no—
415C2—4884no—
416C2#2144931311yescharacteristic
subgroupG/H
centerC2#21449313
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1814C2
socleC2×C61#2420
prime radicalC2×#16840032C2
hypercenter#1814D61
hypocenterC61#24988
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1814

Complements

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

NH ≅ G/Ntype
2. #21454670C2semidirect
3. #21447278C2semidirect
4. #21447093C2semidirect
5. #21446727C2semidirect
6. #21446610C2semidirect
7. #21446303C2semidirect
8. #21445942C2semidirect
9. #21445929C2semidirect
10. #21445619C2semidirect
11. C61×#1814C2semidirect
212. #1814D61semidirect
247. C61#24988semidirect

Maximal subgroups

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

subgroupindex
C2×#168400322
#214546702
#214472782
#214470932
#214467272
#214466102
#214463032
#214459422
#214459292
#214456192
C61×#18142
#2498861