Finite Group Databasebeta

← #29725143

Subgroups — #29725143

Order 15616.

620 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840034C211yesmaximal, characteristic
2#21447287C212yesmaximal, complemented
3#21447097C212yesmaximal
4#21446735C211yesmaximal, characteristic, complemented
5#21446609C211yesmaximal, characteristic, complemented
6#21446313C212yesmaximal
7#21445942C211yesmaximal, characteristic, complemented
8#21445933C212yesmaximal, complemented
9#21445870C211yesmaximal, characteristic
10#21445620C211yesmaximal, characteristic
11C61×#2950C211yesmaximal, characteristic, complemented
12C2×#14519808C2212yes—
13C2×#14519783C2211yescharacteristic
14#16840034—22no—
15#16840034—22no—
16#16839810C2212yes—
17C2×#14519713C2211yescharacteristic
18C2×#14519713C2211yescharacteristic
19#16839780C2212yes—
20#16839771C2212yes—
21#16839744C2211yescharacteristic
22C2×#14519696C2212yes—
23C2×#14519696C2212yes—
24#16839735C2211yescharacteristic
25#16839717C2211yescharacteristic
26#16839534C2211yescharacteristic
27#16839532C2211yescharacteristic
28#16839532C2212yes—
29#16839532C2212yes—
30#16839532C2212yes—
31#16839518C2212yes—
32C22×Q16×C61C2211yescharacteristic
33C61×#519C2212yes—
34C61×#441—22no—
35C4×Q16×C61—22no—
36C61×#427C2211yescharacteristic
37C61×#419C2212yes—
38C61×#405C2211yescharacteristic
39C2×#11758222C2311yescharacteristic
40#14519808—24no—
41#14519808—42no—
42C2×#11758202—22no—
43C2×#11758201—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#14519701—42no—
62#14519696—24no—
63#14519696—42no—
64#14519695—21no—
65#14519693—21no—
66#14519691—22no—
67#14519687—21no—
68C8×Dic61—21no—
69C22×Q8×C61C2312yes—
70C2×Q16×C61—21no—
71C2×Q16×C61—21no—
72C2×Q16×C61—21no—
73C2×Q16×C61—21no—
74C2×Q16×C61D412yes—
75C2×Q16×C61D412yes—
76C2×M16×C61C2311yescharacteristic
77C22×C8×C61C2311yescharacteristic
78C4×Q8×C61—24no—
79C4×Q8×C61—42no—
80#117×C61—24no—
81#117×C61C2312yes—
82#117×C61C2311yescharacteristic
83#107×C61—21no—
84#106×C61—21no—
85#103×C61—24no—
86#97×C61—21no—
87C4×C8×C61—21no—
88C2×Q8×C61—22no—
89C2×Q8×C61—22no—
90C2×Q8×C61—22no—
91C2×Q8×C61—22no—
92C2×Q8×C61—22no—
93C2×Q8×C61C2×D414yes—
94C22×C4×C61—22no—
95C22×C4×C61C2411yescharacteristic
96C2×C22⋊D61—21no—
97#11758222—21no—
98#11758222—21no—
99#11758222—21no—
100#11758222—21no—
101C2×D244#4111yescharacteristic
102C2×C4×D61—21no—
103C2×Dic122#4111yescharacteristic
104Q16×C61—22no—
105Q16×C61—22no—
106Q16×C61—22no—
107Q16×C61—22no—
108M16×C61—21no—
109M16×C61—21no—
110C2×C8×C61—21no—
111C2×C8×C61C2×D411yescharacteristic
112C2×C8×C61C2×D411yescharacteristic
113C2×C8×C61#4111yescharacteristic
114C2×C8×C61#4111yescharacteristic
115C2×C8×C61#4111yescharacteristic
116C2×C8×C61#4111yescharacteristic
117C4⋊C4×C61—21no—
118C4⋊C4×C61—22no—
119C4⋊C4×C61—44no—
120C22⋊C4×C61—22no—
121C22⋊C4×C61#4112yes—
122C42×C61—21no—
123C42×C61—22no—
124C42×C61—44no—
125C22⋊Dic61—21no—
126#11758202—44no—
127#11758201—44no—
128C4⋊Dic61—21no—
129C4×Dic61—21no—
130#11758194—22no—
131#11758194—21no—
132#11758194—21no—
133#11758194—22no—
134#11758194—21no—
135#11758194—22no—
136#11758194—21no—
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×C61C2×#4111yescharacteristic
147C22×D61—21no—
148Q8×C61—24no—
149Q8×C61—24no—
150Q8×C61—42no—
151Q8×C61—42no—
152C2×C4×C61—22no—
153C2×C4×C61—24no—
154C2×C4×C61—42no—
155C2×C4×C61—42no—
156C2×C4×C61—42no—
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—41no—
167D244—21no—
168D244—21no—
169C4×D61—41no—
170C4×D61—41no—
171Dic122—21no—
172Dic122—21no—
173C8×C61—41no—
174C8×C61#12311yescharacteristic
175C8×C61#12311yescharacteristic
176C8×C61#12311yescharacteristic
177C8×C61#12311yescharacteristic
178#92418—41no—
179#92418—42no—
180#92418—42no—
181#92418—41no—
182#24989—611nomaximal, Sylow
183C22×C61—41no—
184C22×C61#52911yescharacteristic
185C22×C61#54611yescharacteristic
186C22×C61#55511yescharacteristic
187C2×D61—41no—
188C2×D61—81no—
189C4×C61—42no—
190C4×C61—44no—
191C4×C61—82no—
192C4×C61C2×#12311yescharacteristic
193C4×C61#54011yescharacteristic
194C4×C61#54611yescharacteristic
195C4×C61#55311yescharacteristic
196Dic61—41no—
197Dic61—81no—
198C2×#473—611no—
199#2950D6111yescharacteristic, complemented
200#2082—612no—
201#2049—612no—
202#1979—611no—
203#1953—611no—
204#1905—612no—
205#1833—611no—
206#1830—612no—
207#1814—611no—
208#1757—611no—
209C2×C61—21no—
210C2×C61—21no—
211C2×C61—21no—
212C2×C61#242011yescharacteristic
213D61—81no—
214C2×#137—612no—
215C22×Q16C2×D6111yescharacteristic
216C2×#131—611no—
217#519C2×D6112yes—
218#473—1222no—
219#473—1222no—
220#441—22no—
221C4×Q16—22no—
222#430—612no—
223C2×#108—611no—
224C2×#108—611no—
225#427C2×D6111yescharacteristic
226#421—612no—
227#419—612no—
228#419C2×D6112yes—
229#413—611no—
230C2×#101—612no—
231C2×#101—612no—
232#410—611no—
233#405—611no—
234#405C2×D6111yescharacteristic
235#344—611no—
236#344—612no—
237#344—612no—
238#344—611no—
239#344—612no—
240H(3,ℤ/4)—612no—
241C61#2498911yescharacteristic, complemented, Sylow
242C2×#41—611no—
243C22×Q8C22×D6112yes—
244#137—1224no—
245#137—2442no—
246C2×Q16—21no—
247C2×Q16—21no—
248C2×Q16—21no—
249C2×Q16—21no—
250C2×Q16—1222no—
251C2×Q16C22⋊D6112yes—
252C2×Q16C22⋊D6112yes—
253C2×SD16—1222no—
254#131—1221no—
255#131—1221no—
256#131—1221no—
257#131—1221no—
258C2×M16—611no—
259C2×M16—612no—
260C2×M16—611no—
261C2×M16—612no—
262C2×M16—1221no—
263C2×M16C22×D6111yescharacteristic
264C22×C8—1221no—
265C22×C8C22×D6111yescharacteristic
266C4×Q8—24no—
267C4×Q8—42no—
268#117—24no—
269#117—611no—
270#117C22×D6112yes—
271#117C22×D6111yescharacteristic
272#108—1221no—
273#108—1221no—
274#108—1221no—
275#108—1221no—
276#108—2442no—
277#107—21no—
278#106—21no—
279#104—2442no—
280#103—24no—
281#103—2442no—
282#101—1221no—
283#101—1224no—
284#101—2442no—
285#100—1221no—
286#99—1222no—
287#97—21no—
288#97—1221no—
289C4×C8—21no—
290C4×C8—1221no—
291#41—1221no—
292#41—1221no—
293#41—1221no—
294#41—1221no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—611no—
301C2×Q8C2×C22⋊D6114yes—
302C2×D4—611no—
303C2×D4—1221no—
304C22×C4—22no—
305C22×C4—1221no—
306C22×C4C23×D6111yescharacteristic
307Q16—22no—
308Q16—22no—
309Q16—22no—
310Q16—22no—
311Q16—2444no—
312SD16—2444no—
313M16—21no—
314M16—21no—
315M16—1221no—
316M16—1222no—
317M16—1221no—
318M16—1222no—
319M16—1221no—
320M16—1221no—
321M16—1222no—
322M16—1222no—
323M16—2441no—
324M16—2441no—
325C2×C8—21no—
326C2×C8—1222no—
327C2×C8—1221no—
328C2×C8—1221no—
329C2×C8—1222no—
330C2×C8—2441no—
331C2×C8—2441no—
332C2×C8#1175822211yescharacteristic
333C2×C8#1175822211yescharacteristic
334C2×C8#1175822211yescharacteristic
335C2×C8#1175822211yescharacteristic
336C2×C8C2×C22⋊D6111yescharacteristic
337C2×C8C2×C22⋊D6111yescharacteristic
338C4⋊C4—21no—
339C4⋊C4—22no—
340C4⋊C4—44no—
341C4⋊C4—1221no—
342C22⋊C4—22no—
343C22⋊C4—1221no—
344C22⋊C4#1175822412yes—
345C42—21no—
346C42—22no—
347C42—44no—
348C42—1221no—
349C23—1221no—
350C23C2×#1175822411yescharacteristic
351Q8—24no—
352Q8—24no—
353Q8—42no—
354Q8—42no—
355Q8—1221no—
356Q8—1221no—
357D4—1221no—
358D4—1221no—
359D4—2441no—
360D4—2441no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—1221no—
367C2×C4—2441no—
368C2×C4—2441no—
369C2×C4—2441no—
370C2×C4C22×C22⋊D6111yescharacteristic
371C2×C4#41×D6111yescharacteristic
372C2×C4C2×#1175822211yescharacteristic
373C2×C4C2×#1175822211yescharacteristic
374C2×C4#1451982211yescharacteristic
375C2×C4#1451982311yescharacteristic
376C8—41no—
377C8—2441no—
378C8—2441no—
379C8—2442no—
380C8—2442no—
381C8#1451976011yescharacteristic
382C8#1451976011yescharacteristic
383C8#1451976011yescharacteristic
384C8#1451976011yescharacteristic
385C22—41no—
386C22—4881no—
387C22—2441no—
388C22#1684029411yescharacteristic
389C22#1684040611yescharacteristic
390C22#1684047711yescharacteristic
391C4—42no—
392C4—44no—
393C4—82no—
394C4—4881no—
395C4—2441no—
396C4C2×#1451976011yescharacteristic
397C4#1684036311yescharacteristic
398C4#1684040011yescharacteristic
399C4#1684046211yescharacteristic
400C2—21no—
401C2—21no—
402C2—21no—
403C2—4881no—
404C2#2144930511yescharacteristic
subgroupG/H
centerC2#21449305
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2950C2
socleC2×C61#2420
prime radicalC2×#11758222C23
hypercenter#2950D61
hypocenterC61#24989
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #2950

Complements

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

NH ≅ G/Ntype
2. #21447287C2semidirect
4. #21446735C2semidirect
5. #21446609C2semidirect
7. #21445942C2semidirect
8. #21445933C2semidirect
11. C61×#2950C2semidirect
199. #2950D61semidirect
241. C61#24989semidirect

Maximal subgroups

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

subgroupindex
C2×#168400342
#214472872
#214470972
#214467352
#214466092
#214463132
#214459422
#214459332
#214458702
#214456202
C61×#29502
#2498961