Finite Group Databasebeta

← #29663163

Subgroups — #29663163

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#21456746C211yesmaximal, characteristic
3C2×#16840113C211yesmaximal, characteristic, complemented
4#21454532C211yesmaximal, characteristic
5#21445367C214yesmaximal, complemented
6#21445243C214yesmaximal, complemented
7#21444503C211yesmaximal, characteristic, complemented
8#21444490C211yesmaximal, characteristic, complemented
9C61×#1564C211yesmaximal, characteristic, complemented
10C2×#14519783—21no—
11#16840312C2211yescharacteristic
12C2×#14519777C2211yescharacteristic
13C2×#14519777C2211yescharacteristic
14#16840167—21no—
15#16840135—21no—
16#16840113—21no—
17#16840113—21no—
18#16840113—21no—
19#16840113C2212yes—
20#16840113C2212yes—
21#16840109—21no—
22#16840076—22no—
23#16840043C2214yes—
24#16839968C2214yes—
25#16839903—22no—
26#16839841—21no—
27#16839841C2212yes—
28C8×D244—21no—
29C8×D244C2212yes—
30#16839786C2211yescharacteristic
31C2×#14519702—21no—
32C2×#14519702—21no—
33#16839728—21no—
34#16839509C2214yes—
35#16839507C2214yes—
36C61×#431C2211yescharacteristic
37C61×#431C2211yescharacteristic
38C2×C4×C8×C61C2211yescharacteristic
39C61×#335C2212yes—
40C61×#334C2212yes—
41C2×#11758222—21no—
42C22×Dic122—21no—
43C22×Dic122—21no—
44C2×#11758190—24no—
45#14519783—44no—
46C2×#11758194—21no—
47C2×#11758189—22no—
48C2×C8×D61—22no—
49#14519777—21no—
50#14519777—21no—
51#14519777—21no—
52#14519777—21no—
53#14519777—21no—
54#14519777—21no—
55#14519777D411yescharacteristic
56#14519777C2312yes—
57#14519777C2312yes—
58#14519774D411yescharacteristic
59#14519765—21no—
60#14519751—21no—
61#14519751D412yes—
62C4×D244—21no—
63C4×D244C2312yes—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68#14519711—24no—
69C4⋊#92418—21no—
70C4⋊#92418C2312yes—
71#14519702—22no—
72#14519702—22no—
73#14519702—22no—
74#14519702—22no—
75#14519702—24no—
76#14519699—24no—
77#14519689—22no—
78#14519689—22no—
79C22×C8×C61D411yescharacteristic
80C22×C8×C61D411yescharacteristic
81C2×C42×C61C2311yescharacteristic
82C4⋊C8×C61C2312yes—
83C4⋊C8×C61C2312yes—
84C22⋊C8×C61—21no—
85C22⋊C8×C61—21no—
86C4×C8×C61—21no—
87C4×C8×C61—21no—
88C4×C8×C61D411yescharacteristic
89C4×C8×C61D411yescharacteristic
90C4×C8×C61C2311yescharacteristic
91C4×C8×C61C2311yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61#4111yescharacteristic
95C2×C22⋊D61—22no—
96C22×Dic61—41no—
97C22×Dic61—41no—
98#11758222—42no—
99#11758222—42no—
100C2×D244—21no—
101C2×D244—21no—
102C2×D244C2×D412yes—
103C2×C4×D61—22no—
104C2×Dic122—21no—
105C2×Dic122—22no—
106C2×Dic122—22no—
107C2×Dic122—42no—
108C2×Dic122—42no—
109C2×Dic122—41no—
110C2×Dic122—41no—
111C2×Dic122—82no—
112C2×C8×C61—22no—
113C2×C8×C61—22no—
114C2×C8×C61—22no—
115C2×C8×C61—22no—
116C2×C8×C61C2×D411yescharacteristic
117C2×C8×C61C2×D411yescharacteristic
118C2×C8×C61C2×D411yescharacteristic
119C2×C8×C61C2×D411yescharacteristic
120C42×C61C2×D411yescharacteristic
121C42×C61C2×D411yescharacteristic
122C42×C61#4111yescharacteristic
123C42×C61C2411yescharacteristic
124#11758198—22no—
125#11758198—22no—
126C4⋊Dic61—21no—
127C4⋊Dic61—22no—
128C4⋊Dic61—22no—
129C4⋊Dic61—22no—
130C4⋊Dic61—22no—
131C4⋊Dic61—41no—
132C4⋊Dic61—41no—
133C4⋊Dic61—41no—
134C4⋊Dic61—41no—
135C4⋊Dic61C2×D412yes—
136#11758194—42no—
137C2×#92418—22no—
138#11758190—84no—
139#11758189—44no—
140C8×D61—44no—
141C23×C61#12111yescharacteristic
142C22×D61—22no—
143C2×C4×C61—22no—
144C2×C4×C61—22no—
145C2×C4×C61#12111yescharacteristic
146C2×C4×C61#12111yescharacteristic
147C2×C4×C61#12111yescharacteristic
148C2×C4×C61#13512yes—
149C2×C4×C61AGL(1,ℤ/8)12yes—
150C2×C4×C61C22×D411yescharacteristic
151C2×C4×C61C22×D411yescharacteristic
152C2×C4×C61C2×#4111yescharacteristic
153C22⋊D61—44no—
154C2×Dic61—22no—
155C2×Dic61—42no—
156C2×Dic61—42no—
157C2×Dic61—82no—
158C2×Dic61—82no—
159D244—24no—
160D244—42no—
161D244—42no—
162C4×D61—42no—
163C4×D61—42no—
164Dic122—42no—
165Dic122—42no—
166Dic122—42no—
167Dic122—44no—
168Dic122—82no—
169Dic122—82no—
170C8×C61—21no—
171C8×C61—21no—
172C8×C61—21no—
173C8×C61—21no—
174C8×C61—42no—
175C8×C61—42no—
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#46312yes—
188C4×C61#47312yes—
189C4×C61C2×#13511yescharacteristic
190C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
191C4×C61#57712yes—
192Dic61—42no—
193Dic61—82no—
194Dic61—82no—
195#3226—611no—
196#3206—611no—
197C2×#487—611no—
198#2931—611no—
199#1699—614no—
200#1679—614no—
201#1564D6111yescharacteristic, complemented
202#1549—611no—
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⋊Q8—611no—
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#431C2×D6111yescharacteristic
230#431C2×D6111yescharacteristic
231#423—611no—
232C2×#103—1221no—
233C2×#103—1221no—
234#408—1221no—
235C2×C4×C8C2×D6111yescharacteristic
236#335C2×D6112yes—
237#334C2×D6112yes—
238#332—614no—
239#332—614no—
240C61#1919711yescharacteristic, complemented, Sylow
241C2×#41—1221no—
242C22×Q8—1221no—
243C22×Q8—1221no—
244C2×SD16—1224no—
245#131—2444no—
246C2×M16—1221no—
247C2×M16—1222no—
248C22×C8—1222no—
249C22×C8D24411yescharacteristic
250C22×C8D24411yescharacteristic
251C4⋊Q8—611no—
252C4⋊Q8—612no—
253C4⋊Q8—612no—
254C4⋊Q8—1221no—
255C4⋊Q8—1221no—
256C4⋊Q8—1221no—
257C4⋊Q8—1221no—
258C4⋊Q8—1221no—
259C4⋊Q8—1221no—
260#127—611no—
261#124—1221no—
262#121—612no—
263#121—1221no—
264C4×D4—612no—
265C4×D4—1221no—
266C2×C4⋊C4—1221no—
267C2×C4⋊C4—1221no—
268C2×C4⋊C4—1221no—
269C2×C4⋊C4—1221no—
270C2×C42C22×D6111yescharacteristic
271#107—1224no—
272C4⋊C8—612no—
273C4⋊C8—1221no—
274C4⋊C8C22×D6112yes—
275C4⋊C8C22×D6112yes—
276#103—1224no—
277#103—1222no—
278#103—1222no—
279#103—1222no—
280#103—1222no—
281#102—1224no—
282C22⋊C8—21no—
283C22⋊C8—21no—
284C22⋊C8—1222no—
285C22⋊C8—1222no—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8D24411yescharacteristic
289C4×C8D24411yescharacteristic
290C4×C8C22×D6111yescharacteristic
291C4×C8C22×D6111yescharacteristic
292#41—2442no—
293#41—2442no—
294C2×Q8—1221no—
295C2×Q8—1222no—
296C2×Q8—1222no—
297C2×Q8—4882no—
298C2×Q8—2442no—
299C2×Q8—2442no—
300C2×Q8—2441no—
301C2×Q8—2441no—
302C2×D4—612no—
303C2×D4—1222no—
304C2×D4—1221no—
305C2×D4—1221no—
306C22×C4—1222no—
307C22×C4—2441no—
308C22×C4—2441no—
309C22×C4C2×D24411yescharacteristic
310C22×C4D4×D6111yescharacteristic
311C22×C4#1175822611yescharacteristic
312SD16—4884no—
313M16—2444no—
314M16—2442no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—1222no—
320C2×C8—2444no—
321C2×C8C2×D24411yescharacteristic
322C2×C8C2×D24411yescharacteristic
323C2×C8C2×D24411yescharacteristic
324C2×C8C2×D24411yescharacteristic
325C4⋊C4—612no—
326C4⋊C4—1221no—
327C4⋊C4—1222no—
328C4⋊C4—1222no—
329C4⋊C4—1222no—
330C4⋊C4—1222no—
331C4⋊C4—2441no—
332C4⋊C4—2441no—
333C4⋊C4—2441no—
334C4⋊C4—2441no—
335C22⋊C4—1222no—
336C22⋊C4—1222no—
337C42C2×D24411yescharacteristic
338C42D4×D6111yescharacteristic
339C42#1175822611yescharacteristic
340C42C23×D6111yescharacteristic
341C23—1222no—
342C23#1451975211yescharacteristic
343Q8—4882no—
344Q8—4882no—
345Q8—2442no—
346Q8—2442no—
347Q8—2444no—
348Q8—2442no—
349D4—1224no—
350D4—2444no—
351D4—2442no—
352D4—2442no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—1222no—
356C2×C4—4882no—
357C2×C4—4882no—
358C2×C4—2442no—
359C2×C4—2442no—
360C2×C4—2442no—
361C2×C4—2442no—
362C2×C4#1451975211yescharacteristic
363C2×C4#1451975211yescharacteristic
364C2×C4#1451975211yescharacteristic
365C2×C4#1451979912yes—
366C2×C4#1451980512yes—
367C2×C4C2×D4×D6111yescharacteristic
368C2×C4C22×D24411yescharacteristic
369C2×C4C2×#1175822611yescharacteristic
370C8—21no—
371C8—21no—
372C8—21no—
373C8—21no—
374C8—42no—
375C8—42no—
376C8—2442no—
377C22—21no—
378C22—21no—
379C22—2442no—
380C22—2442no—
381C22—2442no—
382C22#1683997812yes—
383C22C2×#1451975211yescharacteristic
384C4—21no—
385C4—21no—
386C4—4882no—
387C4—4882no—
388C4—2442no—
389C4#1683997812yes—
390C4#1684003812yes—
391C4C2×#1451979911yescharacteristic
392C4C2×#1451980511yescharacteristic
393C4#1684056912yes—
394C2—22no—
395C2—4882no—
396C2C2×#1683997811yescharacteristic
397C2#2145545911yescharacteristic
398C2C2×#1684003811yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1564C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenter#1564D61
hypocenterC61#19197
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1564

Complements

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

NH ≅ G/Ntype
3. C2×#16840113C2semidirect
5. #21445367C2semidirect
6. #21445243C2semidirect
7. #21444503C2semidirect
8. #21444490C2semidirect
9. C61×#1564C2semidirect
201. #1564D61semidirect
240. C61#19197semidirect

Maximal subgroups

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

subgroupindex
#214569422
#214567462
C2×#168401132
#214545322
#214453672
#214452432
#214445032
#214444902
C61×#15642
#1919761