Finite Group Databasebeta

← #29663165

Subgroups — #29663165

Order 15616.

662 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456942C211yesmaximal, characteristic
2#21456744C211yesmaximal, characteristic
3C2×#16840114C211yesmaximal, characteristic, complemented
4#21454530C211yesmaximal, characteristic
5#21445369C214yesmaximal, complemented
6#21445245C214yesmaximal, complemented
7#21444565C211yesmaximal, characteristic, complemented
8#21444492C211yesmaximal, characteristic, complemented
9C61×#1549C211yesmaximal, characteristic, complemented
10C2×#14519785—21no—
11#16840312C2211yescharacteristic
12C2×#14519777C2211yescharacteristic
13#16840167—21no—
14#16840136—21no—
15#16840114—21no—
16#16840114—21no—
17#16840114C2212yes—
18#16840114C2212yes—
19#16840112—21no—
20#16840109—21no—
21#16840076—22no—
22#16840043C2214yes—
23#16839964C2214yes—
24#16839901—22no—
25#16839839—21no—
26#16839839C2212yes—
27#16839827—21no—
28#16839827C2212yes—
29#16839815C2211yescharacteristic
30C2×#14519700—21no—
31C2×#14519700—21no—
32#16839730—21no—
33C2×C4×#92418C2211yescharacteristic
34#16839514C2212yes—
35#16839512C2212yes—
36#16839509C2214yes—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C61×#431C2211yescharacteristic
39C61×#423C2211yescharacteristic
40C61×#332C2214yes—
41C2×#11758222—21no—
42C22×Dic122—21no—
43C2×#11758201—24no—
44#14519785—44no—
45C2×#11758189—22no—
46C22×#92418D411yescharacteristic
47C22×#92418D411yescharacteristic
48C2×C8×D61—22no—
49#14519777—21no—
50#14519777—21no—
51#14519777—21no—
52#14519777D411yescharacteristic
53#14519777C2312yes—
54#14519774D411yescharacteristic
55#14519765—21no—
56#14519751—21no—
57#14519751D412yes—
58C4×D244—21no—
59C4×D244C2312yes—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61—21no—
62#14519711—24no—
63C4⋊#92418C2312yes—
64#14519701—24no—
65#14519700—22no—
66#14519700—22no—
67#14519700—22no—
68#14519700—22no—
69#14519699—24no—
70C22⋊#92418—21no—
71#14519689—22no—
72#14519689—22no—
73C4×#92418—21no—
74C4×#92418D411yescharacteristic
75C4×#92418D411yescharacteristic
76C4×#92418C2311yescharacteristic
77C4×#92418C2311yescharacteristic
78C22×Q8×C61—21no—
79C2×M16×C61—21no—
80C4⋊Q8×C61—21no—
81C4⋊Q8×C61—21no—
82C4⋊Q8×C61—21no—
83C4⋊Q8×C61C2312yes—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61—21no—
86C2×C42×C61C2311yescharacteristic
87C4⋊C8×C61—21no—
88C4⋊C8×C61C2312yes—
89C4⋊C8×C61C2312yes—
90C22⋊C8×C61—21no—
91C4×C8×C61—21no—
92C2×Q8×C61—22no—
93C2×Q8×C61—41no—
94C2×Q8×C61—82no—
95C22×C4×C61—41no—
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61#4111yescharacteristic
99C2×C22⋊D61—22no—
100C22×Dic61—41no—
101#11758222—42no—
102#11758222—42no—
103C2×D244—21no—
104C2×D244—21no—
105C2×D244C2×D412yes—
106C2×C4×D61—22no—
107C2×Dic122—21no—
108C2×Dic122—22no—
109C2×Dic122—42no—
110C2×Dic122—42no—
111C2×Dic122—41no—
112M16×C61—42no—
113C2×C8×C61—22no—
114C2×C8×C61—22no—
115C4⋊C4×C61—22no—
116C4⋊C4×C61—22no—
117C4⋊C4×C61—41no—
118C4⋊C4×C61—41no—
119C42×C61C2×D411yescharacteristic
120C42×C61C2×D411yescharacteristic
121C42×C61#4111yescharacteristic
122C42×C61C2411yescharacteristic
123#11758201—84no—
124#11758198—22no—
125#11758198—22no—
126C4⋊Dic61—21no—
127C4⋊Dic61—22no—
128C4⋊Dic61—22no—
129C4⋊Dic61—41no—
130C4⋊Dic61—41no—
131C4⋊Dic61C2×D412yes—
132C2×#92418—22no—
133C2×#92418—22no—
134C2×#92418—22no—
135C2×#92418C2×D411yescharacteristic
136C2×#92418C2×D411yescharacteristic
137C2×#92418C2×D411yescharacteristic
138C2×#92418C2×D411yescharacteristic
139#11758189—44no—
140C8×D61—44no—
141C23×C61#12111yescharacteristic
142C22×D61—22no—
143Q8×C61—44no—
144Q8×C61—82no—
145C2×C4×C61—22no—
146C2×C4×C61—22no—
147C2×C4×C61—42no—
148C2×C4×C61—82no—
149C2×C4×C61#12111yescharacteristic
150C2×C4×C61#12111yescharacteristic
151C2×C4×C61#12111yescharacteristic
152C2×C4×C61#13512yes—
153C2×C4×C61AGL(1,ℤ/8)12yes—
154C2×C4×C61C22×D411yescharacteristic
155C2×C4×C61C22×D411yescharacteristic
156C2×C4×C61C2×#4111yescharacteristic
157C22⋊D61—44no—
158C2×Dic61—22no—
159C2×Dic61—42no—
160C2×Dic61—82no—
161D244—24no—
162D244—42no—
163D244—42no—
164C4×D61—42no—
165C4×D61—42no—
166Dic122—42no—
167Dic122—42no—
168Dic122—42no—
169Dic122—82no—
170C8×C61—42no—
171C8×C61—42no—
172#92418—21no—
173#92418—21no—
174#92418—21no—
175#92418—21no—
176#92418—42no—
177#19197—611nomaximal, Sylow
178C22×C61—21no—
179C22×C61—21no—
180C22×C61#46312yes—
181C22×C61C2×#12111yescharacteristic
182C2×D61—42no—
183C2×D61—42no—
184C2×D61—42no—
185C4×C61—21no—
186C4×C61—21no—
187C4×C61—82no—
188C4×C61#46312yes—
189C4×C61#47312yes—
190C4×C61C2×#13511yescharacteristic
191C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
192C4×C61#57712yes—
193Dic61—42no—
194Dic61—82no—
195#3226—611no—
196#3206—611no—
197C2×#487—611no—
198#2931—611no—
199#1699—614no—
200#1679—614no—
201#1564—611no—
202#1549D6111yescharacteristic, complemented
203#1546—611no—
204C2×C61—22no—
205C2×C61C2×#46311yescharacteristic
206C2×C61#305711yescharacteristic
207C2×C61C2×#47311yescharacteristic
208D61—82no—
209C2×#131—1221no—
210#532—611no—
211C2×C4⋊Q8—611no—
212C2×C4⋊Q8C2×D6111yescharacteristic
213#500—1221no—
214#492—1221no—
215#487—612no—
216#487—612no—
217#487—1221no—
218#487—1221no—
219#487—1221no—
220#486—1221no—
221#480—1222no—
222#474—614no—
223#461—614no—
224#448—1222no—
225#436—612no—
226#436—1221no—
227C8×D4—612no—
228C8×D4—1221no—
229#431—611no—
230#431C2×D6111yescharacteristic
231#423C2×D6111yescharacteristic
232C2×#103—1221no—
233C2×#103—1221no—
234#408—1221no—
235C2×C4×C8—611no—
236#335—612no—
237#334—612no—
238#332—614no—
239#332C2×D6114yes—
240C61#1919711yescharacteristic, complemented, Sylow
241C2×#41—1221no—
242C22×Q8—21no—
243C22×Q8—1221no—
244C2×SD16—1224no—
245#131—2444no—
246C2×M16—21no—
247C2×M16—1222no—
248C22×C8—611no—
249C22×C8—611no—
250C22×C8—1222no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—611no—
255C4⋊Q8—612no—
256C4⋊Q8—1221no—
257C4⋊Q8—1221no—
258C4⋊Q8—1221no—
259C4⋊Q8C22×D6112yes—
260#127—611no—
261#124—1221no—
262#121—612no—
263#121—1221no—
264C4×D4—612no—
265C4×D4—1221no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—1221no—
269C2×C4⋊C4—1221no—
270C2×C42C22×D6111yescharacteristic
271#107—1224no—
272C4⋊C8—21no—
273C4⋊C8—612no—
274C4⋊C8C22×D6112yes—
275C4⋊C8C22×D6112yes—
276#103—1222no—
277#103—1222no—
278#103—1222no—
279#103—1222no—
280#103—1224no—
281#102—1224no—
282C22⋊C8—21no—
283C22⋊C8—1222no—
284C22⋊C8—1222no—
285C22⋊C8—1221no—
286C4×C8—21no—
287C4×C8—611no—
288C4×C8—611no—
289C4×C8—611no—
290C4×C8—611no—
291C4×C8—1221no—
292#41—2442no—
293#41—2442no—
294C2×Q8—22no—
295C2×Q8—41no—
296C2×Q8—82no—
297C2×Q8—1221no—
298C2×Q8—1222no—
299C2×Q8—2442no—
300C2×Q8—2442no—
301C2×Q8—2441no—
302C2×D4—612no—
303C2×D4—1222no—
304C2×D4—1221no—
305C2×D4—1221no—
306C22×C4—41no—
307C22×C4—1222no—
308C22×C4—2441no—
309C22×C4#1175822211yescharacteristic
310C22×C4D4×D6111yescharacteristic
311C22×C4D4×D6111yescharacteristic
312SD16—4884no—
313M16—42no—
314M16—2444no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—611no—
318C2×C8—611no—
319C2×C8—611no—
320C2×C8—611no—
321C2×C8—1222no—
322C2×C8—1222no—
323C2×C8—1222no—
324C2×C8—2444no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—612no—
330C4⋊C4—1221no—
331C4⋊C4—1222no—
332C4⋊C4—1222no—
333C4⋊C4—2441no—
334C4⋊C4—2441no—
335C22⋊C4—1222no—
336C22⋊C4—1222no—
337C42#1175822211yescharacteristic
338C42D4×D6111yescharacteristic
339C42D4×D6111yescharacteristic
340C42C23×D6111yescharacteristic
341C23—1222no—
342C23#1451975011yescharacteristic
343Q8—44no—
344Q8—82no—
345Q8—4882no—
346Q8—2442no—
347Q8—2442no—
348Q8—2442no—
349D4—1224no—
350D4—2444no—
351D4—2442no—
352D4—2442no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—42no—
356C2×C4—82no—
357C2×C4—1222no—
358C2×C4—4882no—
359C2×C4—2442no—
360C2×C4—2442no—
361C2×C4—2442no—
362C2×C4#1451975011yescharacteristic
363C2×C4#1451975011yescharacteristic
364C2×C4#1451975011yescharacteristic
365C2×C4#1451980012yes—
366C2×C4#1451980412yes—
367C2×C4C2×D4×D6111yescharacteristic
368C2×C4C2×D4×D6111yescharacteristic
369C2×C4C2×#1175822211yescharacteristic
370C8—42no—
371C8—42no—
372C8—1221no—
373C8—1221no—
374C8—1221no—
375C8—1221no—
376C8—2442no—
377C22—21no—
378C22—21no—
379C22—2442no—
380C22—2442no—
381C22—2442no—
382C22#1683998012yes—
383C22C2×#1451975011yescharacteristic
384C4—21no—
385C4—21no—
386C4—82no—
387C4—4882no—
388C4—2442no—
389C4#1683998012yes—
390C4#1684003612yes—
391C4C2×#1451980011yescharacteristic
392C4C2×#1451980411yescharacteristic
393C4#1684057212yes—
394C2—22no—
395C2—4882no—
396C2C2×#1683998011yescharacteristic
397C2#2145545711yescharacteristic
398C2C2×#1684003611yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1549C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenter#1549D61
hypocenterC61#19197
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
3. C2×#16840114C2semidirect
5. #21445369C2semidirect
6. #21445245C2semidirect
7. #21444565C2semidirect
8. #21444492C2semidirect
9. C61×#1549C2semidirect
202. #1549D61semidirect
240. C61#19197semidirect

Maximal subgroups

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

subgroupindex
#214569422
#214567442
C2×#168401142
#214545302
#214453692
#214452452
#214445652
#214444922
C61×#15492
#1919761