Finite Group Databasebeta

← #29834882

Subgroups — #29834882

Order 15616.

568 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455491C211yesmaximal, characteristic
2#21454415C211yesmaximal, characteristic, complemented
3#21448029C212yesmaximal, complemented
4#21448023C212yesmaximal, complemented
5#21448013C212yesmaximal, complemented
6#21448001C212yesmaximal, complemented
7#21447605C211yesmaximal, characteristic, complemented
8#21447529C211yesmaximal, characteristic, complemented
9#21447527C211yesmaximal, characteristic, complemented
10#21447432C211yesmaximal, characteristic, complemented
11C2×C61×#366C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13C2×#14519783C2211yescharacteristic
14#16840343—21no—
15#16840243—21no—
16C2×#14519717C2211yescharacteristic
17#16840009—22no—
18#16840004C2212yes—
19#16839999C2212yes—
20#16839803—22no—
21#16839802—21no—
22#16839800C2211yescharacteristic
23#16839799C2212yes—
24#107×D61C2212yes—
25#16839761—21no—
26#16839589C2212yes—
27#16839587C2212yes—
28#16839567C2211yescharacteristic
29#16839567C2211yescharacteristic
30#16839565C2212yes—
31#16839564C2211yescharacteristic
32#16839564C2211yescharacteristic
33#16839562C2212yes—
34#16839541C2212yes—
35#16839541C2212yes—
36C22×C16×C61C2211yescharacteristic
37C2×#107×C61C2211yescharacteristic
38C2×#107×C61C2211yescharacteristic
39C61×#366C2212yes—
40C61×#366C2212yes—
41C2×#11758222—21no—
42C2×#11758222D411yescharacteristic
43#14519799—22no—
44#14519799—22no—
45C2×Dic244C2311yescharacteristic
46C2×#11758190—21no—
47C2×D488C2311yescharacteristic
48#14519783—22no—
49#14519783—22no—
50C2×#11758194D411yescharacteristic
51C2×#11758189D412yes—
52C2×C8×D61C2312yes—
53#14519757—22no—
54#14519757—41no—
55#14519752—22no—
56#14519752—41no—
57#14519751—42no—
58#14519742—41no—
59#14519739—41no—
60C4×C22⋊D61—42no—
61#14519736—21no—
62#14519735—22no—
63C4⋊C4×D61—22no—
64#14519717—22no—
65C2×#11758185C2312yes—
66#14519711C2311yescharacteristic
67#14519711C2311yescharacteristic
68#14519710—22no—
69#14519709—21no—
70#14519708—22no—
71#14519700—22no—
72#14519697—22no—
73C22×C8×C61C2311yescharacteristic
74C2×C4⋊C4×C61—21no—
75C2×C4⋊C4×C61—21no—
76C2×C16×C61—22no—
77C2×C16×C61C2311yescharacteristic
78C2×C16×C61C2311yescharacteristic
79#107×C61—21no—
80#107×C61—21no—
81#107×C61C2312yes—
82#107×C61C2312yes—
83C22×C4×C61—41no—
84C22×C4×C61—41no—
85C22×C4×C61C2×D411yescharacteristic
86C2×C22⋊D61—41no—
87C2×C22⋊D61D812yes—
88#11758222—22no—
89#11758222—22no—
90#11758222—42no—
91#11758222—42no—
92C2×D244—21no—
93C2×D244—42no—
94C2×D244D811yescharacteristic
95C2×C4×D61—41no—
96C2×C4×D61—42no—
97C2×C4×D61C2×D412yes—
98C2×Dic122—21no—
99C2×Dic122D811yescharacteristic
100C2×C8×C61C2×D411yescharacteristic
101C2×C8×C61#4112yes—
102C2×C8×C61#4112yes—
103C2×C8×C61C2411yescharacteristic
104C4⋊C4×C61—22no—
105C4⋊C4×C61—22no—
106C4⋊C4×C61—41no—
107C4⋊C4×C61—41no—
108C22⋊Dic61—41no—
109#11758198—42no—
110#11758198—42no—
111#11758198—42no—
112C4⋊Dic61—21no—
113C4⋊Dic61—21no—
114C4⋊Dic61—42no—
115#11758196—42no—
116C4×Dic61—41no—
117C4×Dic61—42no—
118#11758194—22no—
119C2×#92418C2×D412yes—
120Dic244—21no—
121Dic244—21no—
122D488—21no—
123D488—21no—
124#11758190—42no—
125#11758189—24no—
126C8×D61—22no—
127C8×D61—22no—
128C16×C61—21no—
129C16×C61—21no—
130#11758185—22no—
131C23×C61C2×D811yescharacteristic
132C22×D61—41no—
133C22×D61C2×D812yes—
134C2×C4×C61—42no—
135C2×C4×C61—42no—
136C2×C4×C61—161no—
137C2×C4×C61—161no—
138C2×C4×C61#12312yes—
139C2×C4×C61C2×D811yescharacteristic
140C2×C4×C61#13712yes—
141C2×C4×C61C22×D411yescharacteristic
142C22⋊D61—24no—
143C22⋊D61—82no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61C2×D812yes—
148D244—22no—
149D244—41no—
150D244—41no—
151D244—162no—
152C4×D61—22no—
153C4×D61—22no—
154C4×D61—82no—
155C4×D61—162no—
156Dic122—22no—
157Dic122—41no—
158Dic122—41no—
159C8×C61#12312yes—
160C8×C61C2×#4111yescharacteristic
161C8×C61C2×#4111yescharacteristic
162#92418—22no—
163#33652—611nomaximal, Sylow
164C22×C61—21no—
165C22×C61—21no—
166C22×C61#48012yes—
167C22×C61C22×D811yescharacteristic
168C2×D61—22no—
169C2×D61—22no—
170C2×D61—22no—
171C2×D61—81no—
172C2×D61—81no—
173C2×D61—81no—
174C4×C61—82no—
175C4×C61—82no—
176C4×C61#48012yes—
177C4×C61C2×#12311yescharacteristic
178C4×C61C2×#13711yescharacteristic
179Dic61—22no—
180Dic61—81no—
181Dic61—81no—
182Dic61—81no—
183#3061—611no—
184#2913—611no—
185#2237—612no—
186#2236—612no—
187#2234—612no—
188#2232—612no—
189#2159—611no—
190C2×#366D6111yescharacteristic, complemented
191#2139—611no—
192#2138—611no—
193#2113—611no—
194C2×C61—22no—
195C2×C61C2×#48011yescharacteristic
196C2×C61#341511yescharacteristic
197C2×C61#341711yescharacteristic
198D61—42no—
199D61—161no—
200C2×#135—611no—
201C2×#131—611no—
202#537—1221no—
203#520—1221no—
204C2×M32—611no—
205C22×C16C2×D6111yescharacteristic
206#468—1222no—
207#467—612no—
208#466—612no—
209#428—1222no—
210#428—1221no—
211#427—611no—
212#427—612no—
213C2×#107—612no—
214C2×#107C2×D6111yescharacteristic
215C2×#107C2×D6111yescharacteristic
216#417—1221no—
217#367—612no—
218#366—612no—
219#366C2×D6112yes—
220#366C2×D6112yes—
221#358—611no—
222#358—611no—
223#358—612no—
224#357—611no—
225#357—611no—
226#357—612no—
227C22⋊C16—612no—
228C22⋊C16—612no—
229C61#3365211yescharacteristic, complemented, Sylow
230C2×#41—611no—
231C2×#41—1221no—
232#135—1222no—
233#135—1222no—
234C2×Q16—611no—
235C2×SD16—1221no—
236C2×D8—611no—
237#131—1222no—
238#131—1222no—
239C2×M16—611no—
240C2×M16—612no—
241C22×C8—612no—
242C22×C8C22×D6111yescharacteristic
243C22⋊Q8—1222no—
244C22⋊Q8—2441no—
245#121—1222no—
246#121—2441no—
247#121—2442no—
248C4×Q8—2441no—
249C4×D4—2441no—
250C4×D4—2442no—
251#117—1222no—
252#117—1221no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—1222no—
256M32—1222no—
257C2×C16—22no—
258C2×C16—612no—
259C2×C16C22×D6111yescharacteristic
260C2×C16C22×D6111yescharacteristic
261#107—21no—
262#107—21no—
263#107—611no—
264#107—611no—
265#107—1222no—
266#107C22×D6112yes—
267#107C22×D6112yes—
268#106—1222no—
269#106—1221no—
270#103—1222no—
271#102—1222no—
272#41—1222no—
273#41—1222no—
274#41—2442no—
275#41—2442no—
276C2×Q8—611no—
277C2×Q8—1221no—
278C2×D4—612no—
279C2×D4—611no—
280C2×D4—1221no—
281C2×D4—2442no—
282C2×D4—2441no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—612no—
286C22×C4—2442no—
287C22×C4—2441no—
288C22×C4D4×D6111yescharacteristic
289Q16—1221no—
290Q16—1221no—
291SD16—2442no—
292D8—1221no—
293D8—1221no—
294M16—1222no—
295M16—1224no—
296C2×C8—612no—
297C2×C8—1222no—
298C2×C8—1222no—
299C2×C8#1175822212yes—
300C2×C8D4×D6111yescharacteristic
301C2×C8#1175822612yes—
302C2×C8C23×D6111yescharacteristic
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—1221no—
308C4⋊C4—1221no—
309C4⋊C4—2442no—
310C4⋊C4—2442no—
311C22⋊C4—2442no—
312C22⋊C4—2442no—
313C22⋊C4—2442no—
314C22⋊C4—2441no—
315C42—2442no—
316C42—2441no—
317C16—21no—
318C16—21no—
319C16—1222no—
320C23—612no—
321C23—2441no—
322C23D8×D6111yescharacteristic
323Q8—1222no—
324Q8—2441no—
325Q8—2441no—
326D4—1224no—
327D4—1222no—
328D4—9762no—
329D4—4882no—
330D4—2441no—
331D4—2441no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—161no—
335C2×C4—161no—
336C2×C4—612no—
337C2×C4—1222no—
338C2×C4—1222no—
339C2×C4—9762no—
340C2×C4—4882no—
341C2×C4—2441no—
342C2×C4—2441no—
343C2×C4—2441no—
344C2×C4#1451976212yes—
345C2×C4D8×D6111yescharacteristic
346C2×C4#1451980912yes—
347C2×C4C2×D4×D6111yescharacteristic
348C8—1222no—
349C8#1451976212yes—
350C8C2×#1175822211yescharacteristic
351C8C2×#1175822611yescharacteristic
352C22—21no—
353C22—21no—
354C22—1222no—
355C22—1222no—
356C22—1222no—
357C22—4881no—
358C22—4881no—
359C22—4881no—
360C22#1684007712yes—
361C22C2×D8×D6111yescharacteristic
362C4—82no—
363C4—82no—
364C4—1222no—
365C4—4881no—
366C4—4881no—
367C4—4881no—
368C4#1684007712yes—
369C4C2×#1451976211yescharacteristic
370C4C2×#1451980911yescharacteristic
371C2—22no—
372C2—9761no—
373C2—2442no—
374C2C2×#1684007711yescharacteristic
375C2#2145895611yescharacteristic
376C2#2145896811yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#366C2
socleC22×C61C22×D8
prime radical#21455491C2
hypercenterC2×#366D61
hypocenterC61#33652
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#366

Complements

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

NH ≅ G/Ntype
2. #21454415C2semidirect
3. #21448029C2semidirect
4. #21448023C2semidirect
5. #21448013C2semidirect
6. #21448001C2semidirect
7. #21447605C2semidirect
8. #21447529C2semidirect
9. #21447527C2semidirect
10. #21447432C2semidirect
11. C2×C61×#366C2semidirect
190. C2×#366D61semidirect
229. C61#33652semidirect

Maximal subgroups

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

subgroupindex
#214554912
#214544152
#214480292
#214480232
#214480132
#214480012
#214476052
#214475292
#214475272
#214474322
C2×C61×#3662
#3365261