Finite Group Databasebeta

← #29834812

Subgroups — #29834812

Order 15616.

590 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455400C211yesmaximal, characteristic
2#21448027C212yesmaximal, complemented
3#21448021C211yesmaximal, characteristic, complemented
4#21448016C211yesmaximal, characteristic, complemented
5#21448010C212yesmaximal, complemented
6#21448004C211yesmaximal, characteristic, complemented
7#21447999C211yesmaximal, characteristic, complemented
8#21447601C212yesmaximal, complemented
9#21447524C212yesmaximal, complemented
10C2×C22⋊#11758185C211yesmaximal, characteristic, complemented
11C2×C61×#427C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13#16840273—21no—
14C22×#11758185C2212yes—
15#16840003C2211yescharacteristic
16#16840003C2211yescharacteristic
17#16839998C2211yescharacteristic
18#16839998C2211yescharacteristic
19#16839991—21no—
20#16839991—21no—
21#16839800C2211yescharacteristic
22#16839759—21no—
23#16839589C2212yescomplemented
24#16839589C2212yescomplemented
25#16839587C2212yescomplemented
26#16839587C2212yescomplemented
27#16839565C2212yescomplemented
28#16839565C2212yescomplemented
29#16839562C2212yescomplemented
30#16839562C2212yescomplemented
31C22⋊#11758185C2211yescharacteristic
32C22⋊#11758185C2211yescharacteristic
33C22⋊#11758185C2212yes—
34C23×C8×C61C2211yescharacteristic
35C2×#117×C61—21no—
36C61×#427—21no—
37C61×#427—21no—
38C61×#427C2212yes—
39C61×#427C2212yes—
40C2×#107×C61C2211yescharacteristic
41C2×#107×C61C2211yescharacteristic
42C2×#106×C61—21no—
43C2×#11758222—21no—
44#14519799—24no—
45C2×Dic244C2311yescharacteristic
46C2×#11758190—21no—
47C2×D488C2311yescharacteristic
48#14519760—41no—
49#14519756—21no—
50#14519756—21no—
51#14519751—21no—
52#14519751—21no—
53#14519746—41no—
54C4×C22⋊D61—41no—
55C4×C22⋊D61—41no—
56#14519736—21no—
57C2×#11758185—24no—
58C2×#11758185C2312yes—
59C2×#11758185C2312yes—
60#14519711C2311yescharacteristic
61#14519711C2311yescharacteristic
62#14519709—21no—
63#14519702—21no—
64#14519702—21no—
65#14519699—21no—
66#14519699—21no—
67C23×C4×C61D411yescharacteristic
68C22×C8×C61—24no—
69C22×C8×C61D411yescharacteristic
70C22×C8×C61D411yescharacteristic
71C22×C8×C61D411yescharacteristic
72C22×C8×C61C2311yescharacteristic
73C22×C8×C61C2311yescharacteristic
74C22×C8×C61C2311yescharacteristic
75#117×C61—22no—
76#117×C61—22no—
77#117×C61—41no—
78#117×C61—41no—
79C2×C4⋊C4×C61—21no—
80C2×C4⋊C4×C61—21no—
81C2×C22⋊C4×C61—41no—
82C2×C42×C61—41no—
83#107×C61—21no—
84#107×C61—21no—
85#107×C61C2312yes—
86#107×C61C2312yes—
87#106×C61—22no—
88#106×C61—22no—
89C24×C61D811yescharacteristic
90C22×C4×C61—22no—
91C22×C4×C61—22no—
92C22×C4×C61—41no—
93C22×C4×C61—41no—
94C22×C4×C61D811yescharacteristic
95C22×C4×C61D811yescharacteristic
96C22×C4×C61D811yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C2×C22⋊D61—41no—
101C2×C22⋊D61—41no—
102C2×C22⋊D61—41no—
103#11758222—42no—
104#11758222—42no—
105C2×D244—21no—
106C2×C4×D61—41no—
107C2×Dic122—21no—
108C2×C8×C61—22no—
109C2×C8×C61—22no—
110C2×C8×C61—22no—
111C2×C8×C61—22no—
112C2×C8×C61—22no—
113C2×C8×C61C2×D411yescharacteristic
114C2×C8×C61C2×D411yescharacteristic
115C2×C8×C61C2×D411yescharacteristic
116C2×C8×C61#4114yes—
117C2×C8×C61C2411yescharacteristic
118C4⋊C4×C61—22no—
119C4⋊C4×C61—22no—
120C4⋊C4×C61—41no—
121C4⋊C4×C61—41no—
122C22⋊C4×C61—42no—
123C22⋊C4×C61—42no—
124C42×C61—42no—
125C42×C61—42no—
126C22⋊Dic61—41no—
127C22⋊Dic61—41no—
128C22⋊Dic61—41no—
129#11758198—41no—
130#11758198—41no—
131C4⋊Dic61—21no—
132C4⋊Dic61—21no—
133#11758196—41no—
134#11758196—41no—
135C4×Dic61—41no—
136Dic244—22no—
137D488—22no—
138#11758190—42no—
139#11758185—22no—
140#11758185—22no—
141C23×C61—22no—
142C23×C61—22no—
143C23×C61—41no—
144C23×C61C2×D811yescharacteristic
145C23×C61C2×D811yescharacteristic
146C23×C61C2×D811yescharacteristic
147C22×D61—41no—
148C2×C4×C61—22no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—22no—
154C2×C4×C61—22no—
155C2×C4×C61—42no—
156C2×C4×C61—42no—
157C2×C4×C61—82no—
158C2×C4×C61—82no—
159C2×C4×C61#12312yes—
160C2×C4×C61C2×D811yescharacteristic
161C2×C4×C61C2×D811yescharacteristic
162C2×C4×C61C2×D811yescharacteristic
163C2×C4×C61#13712yes—
164C2×C4×C61C22×D411yescharacteristic
165C22⋊D61—82no—
166C22⋊D61—82no—
167C22⋊D61—82no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—41no—
171D244—41no—
172D244—41no—
173C4×D61—82no—
174Dic122—41no—
175Dic122—41no—
176C8×C61—21no—
177C8×C61—21no—
178C8×C61#12312yes—
179C8×C61C2×#4112yes—
180#33645—611nomaximal, Sylow
181C22×C61—21no—
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—44no—
189C22×C61#48012yes—
190C22×C61#50812yes—
191C22×C61C22×D811yescharacteristic
192C2×D61—81no—
193C2×D61—81no—
194C2×D61—81no—
195C4×C61—21no—
196C4×C61—21no—
197C4×C61—82no—
198C4×C61—82no—
199C4×C61#48012yes—
200C4×C61C2×#12311yescharacteristic
201C4×C61C2×#13711yescharacteristic
202Dic61—81no—
203Dic61—81no—
204Dic61—81no—
205#3050—611no—
206C2×#427D6111yescharacteristic, complemented
207#2237—612no—
208#2236—611no—
209#2235—611no—
210#2234—612no—
211#2233—611no—
212#2232—611no—
213#2158—612no—
214#2138—612no—
215C2×C22⋊C16—611no—
216C2×C61—22no—
217C2×C61—22no—
218C2×C61—41no—
219C2×C61C2×#48011yescharacteristic
220C2×C61C2×#50811yescharacteristic
221C2×C61#341411yescharacteristic
222D61—161no—
223C2×#135—611no—
224C23×C8C2×D6111yescharacteristic
225#525—1221no—
226C2×#117—21no—
227C22×C16—612no—
228#467—611no—
229#467—611no—
230#466—611no—
231#466—611no—
232#465—1221no—
233#465—1221no—
234#427—21no—
235#427—21no—
236#427—611no—
237#427C2×D6112yes—
238#427C2×D6112yes—
239C2×#107C2×D6111yescharacteristic
240C2×#107C2×D6111yescharacteristic
241C2×#106—21no—
242#416—1221no—
243#367—612no—
244#367—612no—
245#366—612no—
246#366—612no—
247#358—612no—
248#358—612no—
249#357—612no—
250#357—612no—
251C22⋊C16—611no—
252C22⋊C16—611no—
253C22⋊C16—612no—
254C61#3364511yescharacteristic, complemented, Sylow
255C2×#41—1221no—
256C23×C4C22⋊D6111yescharacteristic
257#135—1224no—
258C2×Q16—611no—
259C2×SD16—1221no—
260C2×D8—611no—
261C22×C8—24no—
262C22×C8C22⋊D6111yescharacteristic
263C22×C8C22⋊D6111yescharacteristic
264C22×C8C22⋊D6111yescharacteristic
265C22×C8C22×D6111yescharacteristic
266C22×C8C22×D6111yescharacteristic
267C22×C8C22×D6111yescharacteristic
268#123—2441no—
269C22⋊Q8—1221no—
270C22⋊Q8—1221no—
271#121—1221no—
272#121—1221no—
273#120—2441no—
274C4×D4—2441no—
275C4×D4—2441no—
276#117—22no—
277#117—22no—
278#117—41no—
279#117—41no—
280#117—1221no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C22⋊C4—41no—
284C2×C42—41no—
285C2×C16—612no—
286C2×C16—612no—
287C2×C16—1224no—
288#107—21no—
289#107—21no—
290#107—611no—
291#107—611no—
292#107C22×D6112yes—
293#107C22×D6112yes—
294#106—22no—
295#106—22no—
296#106—1221no—
297#103—1221no—
298#103—1221no—
299#102—1221no—
300#102—1221no—
301C24#1175819911yescharacteristic
302#41—2442no—
303#41—2442no—
304C2×Q8—1221no—
305C2×D4—1221no—
306C2×D4—2441no—
307C2×D4—2441no—
308C2×D4—2441no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—2441no—
314C22×C4#1175819911yescharacteristic
315C22×C4#1175819911yescharacteristic
316C22×C4#1175819911yescharacteristic
317C22×C4C2×C22⋊D6111yescharacteristic
318C22×C4C2×C22⋊D6111yescharacteristic
319C22×C4C2×C22⋊D6111yescharacteristic
320Q16—1222no—
321SD16—2442no—
322D8—1222no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8#1175822214yes—
329C2×C8C2×C22⋊D6111yescharacteristic
330C2×C8C2×C22⋊D6111yescharacteristic
331C2×C8C2×C22⋊D6111yescharacteristic
332C2×C8C23×D6111yescharacteristic
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—1221no—
338C4⋊C4—1221no—
339C4⋊C4—2441no—
340C4⋊C4—2441no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—2441no—
344C22⋊C4—2441no—
345C22⋊C4—2441no—
346C22⋊C4—2441no—
347C22⋊C4—2441no—
348C42—42no—
349C42—42no—
350C42—2441no—
351C16—1222no—
352C16—1222no—
353C23—22no—
354C23—22no—
355C23—41no—
356C23—2441no—
357C23C2×#1175819911yescharacteristic
358C23C2×#1175819911yescharacteristic
359C23C2×#1175819911yescharacteristic
360Q8—2441no—
361Q8—2441no—
362D4—4882no—
363D4—4882no—
364D4—4882no—
365D4—2441no—
366D4—2441no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—4882no—
379C2×C4—2441no—
380C2×C4—2441no—
381C2×C4—2441no—
382C2×C4#1451976012yes—
383C2×C4C2×#1175819911yescharacteristic
384C2×C4C2×#1175819911yescharacteristic
385C2×C4C2×#1175819911yescharacteristic
386C2×C4#1451980812yes—
387C2×C4C22×C22⋊D6111yescharacteristic
388C8—21no—
389C8—21no—
390C8#1451976012yes—
391C8C2×#1175822212yes—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—44no—
400C22—4881no—
401C22—4881no—
402C22—4881no—
403C22#1684007512yes—
404C22#1684019212yes—
405C22C22×#1175819911yescharacteristic
406C4—21no—
407C4—21no—
408C4—82no—
409C4—82no—
410C4—4881no—
411C4—4881no—
412C4—4881no—
413C4#1684007512yes—
414C4C2×#1451976011yescharacteristic
415C4C2×#1451980811yescharacteristic
416C2—22no—
417C2—22no—
418C2—41no—
419C2—9761no—
420C2C2×#1684007511yescharacteristic
421C2C2×#1684019211yescharacteristic
422C2#2145895311yescharacteristic
subgroupG/H
centerC2×C4C2×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#427C2
socleC22×C61C22×D8
prime radical#21455400C2
hypercenterC2×#427D61
hypocenterC61#33645
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ C2×#427

Complements

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

NH ≅ G/Ntype
2. #21448027C2semidirect
3. #21448021C2semidirect
4. #21448016C2semidirect
5. #21448010C2semidirect
6. #21448004C2semidirect
7. #21447999C2semidirect
8. #21447601C2semidirect
9. #21447524C2semidirect
10. C2×C22⋊#11758185C2semidirect
11. C2×C61×#427C2semidirect
23. #16839589C22semidirect
24. #16839589C22semidirect
25. #16839587C22semidirect
26. #16839587C22semidirect
27. #16839565C22semidirect
28. #16839565C22semidirect
29. #16839562C22semidirect
30. #16839562C22semidirect
206. C2×#427D61semidirect
254. C61#33645semidirect

Maximal subgroups

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

subgroupindex
#214554002
#214480272
#214480212
#214480162
#214480102
#214480042
#214479992
#214476012
#214475242
C2×C22⋊#117581852
C2×C61×#4272
#3364561