Finite Group Databasebeta

← #29834866

Subgroups — #29834866

Order 15616.

556 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455470C211yesmaximal, characteristic
2#21454416C211yesmaximal, characteristic
3#21448026C212yesmaximal, complemented
4#21448025C212yesmaximal, complemented
5#21448020C212yesmaximal, complemented
6#21448015C212yesmaximal, complemented
7#21447607C211yesmaximal, characteristic, complemented
8#21447601C211yesmaximal, characteristic, complemented
9C2×#16839566C211yesmaximal, characteristic, complemented
10#21447434C211yesmaximal, characteristic
11C61×#2141C211yesmaximal, characteristic, complemented
12C2×#14519783C2211yescharacteristic
13#16840336—21no—
14#16840242—21no—
15C22×#11758185C2211yescharacteristic
16#16840017—22no—
17#16840000C2212yes—
18#16840000C2212yes—
19#16839805—21no—
20#16839801—22no—
21#16839800C2211yescharacteristic
22#16839799C2212yes—
23C2×#14519711C2211yescharacteristic
24#107×D61C2212yes—
25#16839771—21no—
26#16839590C2212yes—
27#16839589C2211yescharacteristic
28#16839589C2211yescharacteristic
29#16839588C2212yes—
30#16839587C2211yescharacteristic
31#16839587C2211yescharacteristic
32#16839566C2212yes—
33#16839566C2212yes—
34#16839541C2212yes—
35#16839541C2212yes—
36C22×Q16×C61C2211yescharacteristic
37C2×M32×C61C2211yescharacteristic
38C61×#427C2211yescharacteristic
39C61×#358C2212yes—
40C61×#358C2212yes—
41C2×#11758222D411yescharacteristic
42#14519783—22no—
43#14519783—22no—
44C2×#11758194D411yescharacteristic
45C2×#11758189D412yes—
46C2×C8×D61C2312yes—
47#14519775—41no—
48#14519766—41no—
49#14519760—42no—
50#14519753—22no—
51#14519753—22no—
52C4×Dic122—41no—
53C4×D244—41no—
54#14519737—42no—
55#14519736—21no—
56#14519735—22no—
57C2×C4⋊Dic61—21no—
58C4⋊C4×D61—22no—
59C2×#11758185—22no—
60C2×#11758185C2311yescharacteristic
61C2×#11758185C2311yescharacteristic
62#14519711—21no—
63#14519711C2311yescharacteristic
64#14519711C2311yescharacteristic
65#14519711C2312yes—
66#14519710—22no—
67#14519709—21no—
68#14519708—22no—
69#14519701—22no—
70#14519701—22no—
71C22×Q8×C61—21no—
72C2×Q16×C61—21no—
73C2×Q16×C61—22no—
74C2×Q16×C61—22no—
75C2×Q16×C61C2312yes—
76C22×C8×C61C2311yescharacteristic
77#117×C61—21no—
78M32×C61—22no—
79C2×C16×C61C2312yes—
80#107×C61C2311yescharacteristic
81#107×C61C2311yescharacteristic
82#106×C61—21no—
83C2×Q8×C61—22no—
84C2×Q8×C61—41no—
85C2×Q8×C61—42no—
86C2×Q8×C61—42no—
87C22×C4×C61—41no—
88C22×C4×C61C2×D411yescharacteristic
89C2×C22⋊D61D812yes—
90C22×Dic61—41no—
91#11758222—22no—
92#11758222—22no—
93C2×D244D811yescharacteristic
94C2×C4×D61—42no—
95C2×C4×D61C2×D412yes—
96C2×Dic122D811yescharacteristic
97Q16×C61—22no—
98Q16×C61—22no—
99Q16×C61—42no—
100C2×C8×C61C2×D411yescharacteristic
101C2×C8×C61#4112yes—
102C2×C8×C61#4112yes—
103C2×C8×C61C2411yescharacteristic
104C4⋊C4×C61—21no—
105C4⋊C4×C61—21no—
106C22⋊C4×C61—41no—
107C42×C61—41no—
108C22⋊Dic61—41no—
109#11758198—42no—
110#11758198—42no—
111#11758198—42no—
112C4⋊Dic61—21no—
113C4⋊Dic61—21no—
114C4⋊Dic61—22no—
115C4⋊Dic61—41no—
116#11758196—42no—
117#11758196—42no—
118#11758196—42no—
119C4×Dic61—41no—
120C4×Dic61—42no—
121#11758194—22no—
122C2×#92418C2×D412yes—
123#11758189—24no—
124C8×D61—22no—
125C8×D61—22no—
126C16×C61—22no—
127#11758185—21no—
128#11758185—21no—
129C23×C61C2×D811yescharacteristic
130C22×D61C2×D812yes—
131Q8×C61—42no—
132Q8×C61—42no—
133Q8×C61—82no—
134C2×C4×C61—41no—
135C2×C4×C61—42no—
136C2×C4×C61—41no—
137C2×C4×C61—82no—
138C2×C4×C61#12312yes—
139C2×C4×C61C2×D811yescharacteristic
140C2×C4×C61#13712yes—
141C2×C4×C61C22×D411yescharacteristic
142C22⋊D61—24no—
143C2×Dic61—41no—
144C2×Dic61—41no—
145C2×Dic61—42no—
146C2×Dic61—161no—
147C2×Dic61C2×D812yes—
148D244—22no—
149C4×D61—22no—
150C4×D61—22no—
151C4×D61—162no—
152Dic122—22no—
153C8×C61#12312yes—
154C8×C61C2×#4111yescharacteristic
155C8×C61C2×#4111yescharacteristic
156#92418—22no—
157#33651—611nomaximal, Sylow
158C22×C61—21no—
159C22×C61—21no—
160C22×C61#48012yes—
161C22×C61C22×D811yescharacteristic
162C2×D61—22no—
163C2×D61—22no—
164C2×D61—22no—
165C4×C61—82no—
166C4×C61—81no—
167C4×C61—81no—
168C4×C61#48012yes—
169C4×C61C2×#12311yescharacteristic
170C4×C61C2×#13711yescharacteristic
171Dic61—22no—
172Dic61—82no—
173Dic61—81no—
174Dic61—81no—
175#3059—611no—
176#2913—611no—
177#2237—612no—
178#2237—612no—
179#2236—612no—
180#2235—612no—
181#2159—611no—
182#2158—611no—
183#2141D6111yescharacteristic, complemented
184C2×#358—611no—
185#2113—611no—
186C2×C61—22no—
187C2×C61C2×#48011yescharacteristic
188C2×C61#341611yescharacteristic
189C2×C61#341711yescharacteristic
190D61—42no—
191C22×Q16C2×D6111yescharacteristic
192C2×#131—611no—
193#536—1221no—
194#520—1221no—
195C2×M32C2×D6111yescharacteristic
196C22×C16—611no—
197#470—1222no—
198#467—612no—
199#467—612no—
200#428—1222no—
201#428—1221no—
202#427—611no—
203#427—612no—
204#427C2×D6111yescharacteristic
205C2×#107—611no—
206C2×#107—612no—
207#419—1221no—
208#367—612no—
209#367—611no—
210#367—611no—
211#366—612no—
212#366—611no—
213#366—611no—
214#358—612no—
215#358—612no—
216#358C2×D6112yes—
217#358C2×D6112yes—
218C22⋊C16—612no—
219C22⋊C16—612no—
220C61#3365111yescharacteristic, complemented, Sylow
221C2×#41—611no—
222C22×Q8—21no—
223C2×Q16—22no—
224C2×Q16—21no—
225C2×Q16—22no—
226C2×Q16C22×D6112yes—
227#131—1222no—
228#131—1222no—
229C2×M16—611no—
230C2×M16—612no—
231C22×C8—612no—
232C22×C8C22×D6111yescharacteristic
233C4⋊Q8—2441no—
234#124—2441no—
235#123—2442no—
236C22⋊Q8—1222no—
237C22⋊Q8—1222no—
238C4×Q8—2441no—
239C4×D4—2441no—
240C4×D4—2442no—
241#117—21no—
242#117—1222no—
243#117—1221no—
244C2×C4⋊C4—1222no—
245C2×C4⋊C4—1221no—
246M32—22no—
247C2×C16—611no—
248C2×C16—611no—
249C2×C16—1222no—
250C2×C16C22×D6112yes—
251#107—612no—
252#107—611no—
253#107—611no—
254#107—1222no—
255#107—1221no—
256#107C22×D6111yescharacteristic
257#107C22×D6111yescharacteristic
258#106—21no—
259#106—1222no—
260#106—1221no—
261#103—1222no—
262#103—1222no—
263#41—1222no—
264#41—1222no—
265C2×Q8—22no—
266C2×Q8—41no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—611no—
270C2×D4—612no—
271C2×D4—611no—
272C22×C4—41no—
273C22×C4—612no—
274C22×C4—2442no—
275C22×C4—2441no—
276C22×C4D4×D6111yescharacteristic
277Q16—22no—
278Q16—22no—
279Q16—42no—
280M16—1222no—
281M16—1224no—
282C2×C8—612no—
283C2×C8—1222no—
284C2×C8—1222no—
285C2×C8#1175822212yes—
286C2×C8D4×D6111yescharacteristic
287C2×C8#1175822412yes—
288C2×C8C23×D6111yescharacteristic
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—1221no—
292C4⋊C4—1221no—
293C4⋊C4—1222no—
294C4⋊C4—2442no—
295C4⋊C4—2442no—
296C4⋊C4—2442no—
297C4⋊C4—2441no—
298C22⋊C4—41no—
299C22⋊C4—2442no—
300C22⋊C4—2442no—
301C22⋊C4—2442no—
302C22⋊C4—2441no—
303C42—41no—
304C42—2442no—
305C42—2441no—
306C16—22no—
307C16—1221no—
308C16—1221no—
309C23—612no—
310C23D8×D6111yescharacteristic
311Q8—42no—
312Q8—42no—
313Q8—82no—
314Q8—1222no—
315D4—1224no—
316D4—1222no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—82no—
321C2×C4—612no—
322C2×C4—1222no—
323C2×C4—1222no—
324C2×C4—9762no—
325C2×C4—9761no—
326C2×C4—2441no—
327C2×C4—2441no—
328C2×C4—2442no—
329C2×C4#1451975912yes—
330C2×C4D8×D6111yescharacteristic
331C2×C4#1451980612yes—
332C2×C4C2×D4×D6111yescharacteristic
333C8—1222no—
334C8#1451975912yes—
335C8C2×#1175822411yescharacteristic
336C8C2×#1175822211yescharacteristic
337C22—21no—
338C22—21no—
339C22—1222no—
340C22—1222no—
341C22—1222no—
342C22#1684007412yes—
343C22C2×D8×D6111yescharacteristic
344C4—81no—
345C4—81no—
346C4—82no—
347C4—1222no—
348C4—4882no—
349C4—4881no—
350C4—4881no—
351C4#1684007412yes—
352C4C2×#1451975911yescharacteristic
353C4C2×#1451980611yescharacteristic
354C2—22no—
355C2—2442no—
356C2C2×#1684007411yescharacteristic
357C2#2145896411yescharacteristic
358C2#2145897111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2141C2
socleC22×C61C22×D8
prime radicalC2×#11758222D4
hypercenter#2141D61
hypocenterC61#33651
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2141

Complements

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

NH ≅ G/Ntype
3. #21448026C2semidirect
4. #21448025C2semidirect
5. #21448020C2semidirect
6. #21448015C2semidirect
7. #21447607C2semidirect
8. #21447601C2semidirect
9. C2×#16839566C2semidirect
11. C61×#2141C2semidirect
183. #2141D61semidirect
220. C61#33651semidirect

Maximal subgroups

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

subgroupindex
#214554702
#214544162
#214480262
#214480252
#214480202
#214480152
#214476072
#214476012
C2×#168395662
#214474342
C61×#21412
#3365161