Finite Group Databasebeta

← #29574880

Subgroups — #29574880

Order 15616.

500 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455851C211yesmaximal, characteristic
2#21454770C211yesmaximal, characteristic, complemented
3#21446737C211yesmaximal, characteristic
4C4×#14519700C211yesmaximal, characteristic, complemented
5#21444269C211yesmaximal, characteristic
6#21443655C211yesmaximal, characteristic, complemented
7C61×#1303C211yesmaximal, characteristic, complemented
8#16840312C2211yescharacteristic
9C2×C4×Dic122C2211yescharacteristic
10#16840081—21no—
11#16840048C411yescharacteristic, complemented
12#16840048C411yescharacteristic, complemented
13#16839963C411yescharacteristic, complemented
14#16839963C411yescharacteristic, complemented
15#16839903—21no—
16#16839891—22no—
17#16839889—21no—
18#16839886—21no—
19C4×#11758190—21no—
20C4×#11758190—21no—
21#16839785—21no—
22C2×#14519705—21no—
23C2×#14519705—21no—
24C2×#14519700—21no—
25C4×#11758194C2211yescharacteristic
26M16×Dic61—21no—
27C2×C4×#92418C2211yescharacteristic
28#16839524—21no—
29#16839489—22no—
30C61×#431C2211yescharacteristic
31C61×#423C2211yescharacteristic
32C4×C4⋊C4×C61C2211yescharacteristic
33C2×#11758222—21no—
34C22×Dic122D411yescharacteristic
35C2×#11758190—21no—
36C2×#11758190—21no—
37C2×#11758194D411yescharacteristic
38C2×#11758194D411yescharacteristic
39C22×#92418—21no—
40#14519777C2×C411yescharacteristic
41#14519774C2×C411yescharacteristic
42#14519765—21no—
43#14519751—21no—
44#14519751—21no—
45C4×Dic122—21no—
46C4×Dic122—21no—
47C4×Dic122C2×C411yescharacteristic
48C4×Dic122C2×C411yescharacteristic
49C4×Dic122D411yescharacteristic
50C4×Dic122D411yescharacteristic
51C4×D244—21no—
52C4×D244—21no—
53C2×C4⋊Dic61D411yescharacteristic
54C2×#11758196—21no—
55C2×C4×Dic61—21no—
56#14519709—21no—
57#14519709—21no—
58#14519706—22no—
59#14519706—22no—
60#14519705—42no—
61#14519705—42no—
62#14519702—21no—
63#14519702—21no—
64#14519700—24no—
65#14519699—21no—
66#14519699—21no—
67#14519688—21no—
68#14519687—22no—
69C4×#92418—21no—
70C4×#92418—21no—
71C4×#92418—21no—
72C8×Dic61—22no—
73C2×M16×C61—21no—
74C2×C4⋊C4×C61—21no—
75C2×C42×C61—21no—
76C2×C42×C61C2311yescharacteristic
77C4⋊C8×C61—21no—
78C4⋊C8×C61C2×C411yescharacteristic
79C4⋊C8×C61C2×C411yescharacteristic
80C4⋊C8×C61D412yes—
81C22⋊C8×C61—21no—
82C4×C8×C61—21no—
83#95×C61—21no—
84C22×C4×C61—41no—
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61#4111yescharacteristic
88C2×C22⋊D61—41no—
89C22×Dic61—21no—
90C22×Dic61—21no—
91#11758222—41no—
92#11758222—41no—
93#11758222—41no—
94#11758222—41no—
95C2×D244—21no—
96C2×D244—21no—
97C2×D244—21no—
98C2×C4×D61—41no—
99C2×Dic122—22no—
100C2×Dic122—21no—
101C2×Dic122—21no—
102C2×Dic122—41no—
103C2×Dic122C22⋊C411yescharacteristic
104C2×Dic122C22⋊C411yescharacteristic
105M16×C61—41no—
106M16×C61—41no—
107C2×C8×C61—21no—
108C2×C8×C61—21no—
109C2×C8×C61—22no—
110C4⋊C4×C61—22no—
111C4⋊C4×C61—41no—
112C42×C61—41no—
113C42×C61—41no—
114C42×C61C22×C411yescharacteristic
115C42×C61C2×D411yescharacteristic
116C42×C61C2×D411yescharacteristic
117C42×C61#4111yescharacteristic
118#11758198—41no—
119#11758198—41no—
120C4⋊Dic61—21no—
121C4⋊Dic61—21no—
122C4⋊Dic61—21no—
123C4⋊Dic61C22⋊C411yescharacteristic
124C4⋊Dic61C22⋊C411yescharacteristic
125#11758196—21no—
126#11758196—21no—
127#11758196—21no—
128#11758196—21no—
129C4×Dic61—21no—
130C4×Dic61—21no—
131C4×Dic61—21no—
132C4×Dic61—21no—
133#11758194—21no—
134#11758194—21no—
135#11758194—21no—
136#11758194SD1612yescomplemented
137C2×#92418—21no—
138C2×#92418—22no—
139C2×#92418—21no—
140C2×#92418—41no—
141C2×#92418—41no—
142#11758190—42no—
143#11758190—42no—
144C23×C61#12111yescharacteristic
145C22×D61—41no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—41no—
149C2×C4×C61—42no—
150C2×C4×C61—81no—
151C2×C4×C61—81no—
152C2×C4×C61C2×C22⋊C411yescharacteristic
153C2×C4×C61C4×D411yescharacteristic
154C2×C4×C61C4×D411yescharacteristic
155C2×C4×C61#12011yescharacteristic
156C2×C4×C61#12111yescharacteristic
157C2×C4×C61#12311yescharacteristic
158C2×C4×C61C2×SD1611yescharacteristic
159C2×C4×C61#13511yescharacteristic
160C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C61#13711yescharacteristic
162C22⋊D61—81no—
163C22⋊D61—81no—
164C2×Dic61—21no—
165C2×Dic61—21no—
166C2×Dic61—21no—
167C2×Dic61—21no—
168C2×Dic61—42no—
169C2×Dic61—41no—
170C2×Dic61—81no—
171D244—42no—
172D244—41no—
173D244—41no—
174D244—41no—
175D244—41no—
176C4×D61—81no—
177C4×D61—81no—
178Dic122—21no—
179Dic122—21no—
180Dic122—21no—
181Dic122—42no—
182Dic122—41no—
183Dic122—41no—
184Dic122#10412yescomplemented
185C8×C61—41no—
186C8×C61—41no—
187C8×C61—42no—
188#92418—22no—
189#92418—42no—
190#92418—41no—
191#8992—611nomaximal, Sylow
192C22×C61—21no—
193C22×C61—21no—
194C22×C61#38611yescharacteristic
195C22×C61#46011yescharacteristic
196C22×C61#46411yescharacteristic
197C2×D61—81no—
198C2×D61—81no—
199C2×D61—81no—
200C4×C61—21no—
201C4×C61—21no—
202C4×C61—21no—
203C4×C61—82no—
204C4×C61C2×#10411yescharacteristic
205C4×C61#42111yescharacteristic
206C4×C61C4×SD1611yescharacteristic
207C4×C61#44011yescharacteristic
208C4×C61#44811yescharacteristic
209C4×C61#44911yescharacteristic
210Dic61—22no—
211Dic61—41no—
212Dic61—41no—
213Dic61—41no—
214Dic61—81no—
215#3103—611no—
216Q8×M16—611no—
217#1980—611no—
218C4×#103—611no—
219#1492—611no—
220#1303D6111yescharacteristic, complemented
221#1278—611no—
222C2×C61—21no—
223C2×C61—21no—
224C2×C61#187111yescharacteristic
225C2×C61#188011yescharacteristic
226C2×C61#188111yescharacteristic
227D61—161no—
228#532—611no—
229C2×C4×Q8—611no—
230#481—1221no—
231#475—611no—
232#475—611no—
233#460—611no—
234#460—611no—
235#448—1221no—
236#446—1221no—
237#446—1222no—
238C8×Q8—1221no—
239C4×SD16—1221no—
240C4×SD16—1221no—
241#431C2×D6111yescharacteristic
242#423—1221no—
243#423C2×D6111yescharacteristic
244C2×#104—1221no—
245C2×#104—1221no—
246C2×#103—1221no—
247C4×M16—611no—
248C4×M16—1221no—
249C2×C4×C8—611no—
250C4×C4⋊C4C2×D6111yescharacteristic
251#340—1221no—
252D4⋊C8—1222no—
253C61#899211yescharacteristic, complemented, Sylow
254C2×#41—1221no—
255C22×Q8—611no—
256C2×SD16—1221no—
257C2×SD16—1221no—
258C2×M16—21no—
259C2×M16—611no—
260C2×M16—611no—
261C22×C8—1221no—
262C4⋊Q8—611no—
263#127—611no—
264#124—1221no—
265#121—1221no—
266#121—1221no—
267C4×Q8—611no—
268C4×Q8—611no—
269C4×Q8—611no—
270C4×Q8—611no—
271C4×Q8—1221no—
272C4×Q8—1221no—
273C4×D4—1221no—
274C4×D4—1221no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—611no—
277C2×C4⋊C4—1221no—
278C2×C42—21no—
279C2×C42—1221no—
280C2×C42C22×D6111yescharacteristic
281#106—1221no—
282#106—1221no—
283C4⋊C8—21no—
284C4⋊C8—1222no—
285C4⋊C8—1222no—
286C4⋊C8C4×D6111yescharacteristic
287C4⋊C8C4×D6111yescharacteristic
288C4⋊C8D24412yes—
289#104—2442no—
290#104—2442no—
291#103—1224no—
292#103—1221no—
293#103—1221no—
294#102—1221no—
295#102—1221no—
296C22⋊C8—21no—
297#97—1222no—
298#97—1221no—
299C4×C8—21no—
300C4×C8—1222no—
301C4×C8—1221no—
302C4×C8—1221no—
303C4×C8—1221no—
304#95—21no—
305#41—2441no—
306#41—2441no—
307#41—2441no—
308#41—2441no—
309C2×Q8—611no—
310C2×Q8—611no—
311C2×Q8—1222no—
312C2×Q8—1221no—
313C2×Q8—1221no—
314C2×Q8—2441no—
315C2×D4—1221no—
316C2×D4—1221no—
317C2×D4—1221no—
318C2×D4—2441no—
319C22×C4—41no—
320C22×C4—1221no—
321C22×C4—1221no—
322C22×C4—2441no—
323C22×C4#1175822211yescharacteristic
324C22×C4D4×D6111yescharacteristic
325C22×C4D4×D6111yescharacteristic
326SD16—2442no—
327SD16—2442no—
328M16—41no—
329M16—41no—
330M16—612no—
331M16—1221no—
332M16—1221no—
333M16—1221no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—22no—
337C2×C8—1221no—
338C2×C8—1222no—
339C2×C8—1221no—
340C2×C8—2441no—
341C2×C8—2441no—
342C4⋊C4—22no—
343C4⋊C4—41no—
344C4⋊C4—611no—
345C4⋊C4—611no—
346C4⋊C4—1221no—
347C4⋊C4—1221no—
348C4⋊C4—1221no—
349C4⋊C4—1221no—
350C4⋊C4—1221no—
351C4⋊C4—1221no—
352C4⋊C4—1221no—
353C22⋊C4—2441no—
354C22⋊C4—2441no—
355C42—41no—
356C42—41no—
357C42—1221no—
358C42—1221no—
359C42—1221no—
360C42—1221no—
361C42C2×C4×D6111yescharacteristic
362C42C2×D24411yescharacteristic
363C42D4×D6111yescharacteristic
364C42#1175822611yescharacteristic
365C23—2441no—
366C23#1451975011yescharacteristic
367Q8—612no—
368Q8—1221no—
369Q8—1221no—
370Q8—1221no—
371Q8—2442no—
372Q8—2441no—
373Q8—2441no—
374D4—4881no—
375D4—4881no—
376D4—2442no—
377D4—2441no—
378D4—2441no—
379D4—2441no—
380D4—2441no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—1221no—
388C2×C4—1221no—
389C2×C4—1221no—
390C2×C4—1221no—
391C2×C4—4881no—
392C2×C4—4881no—
393C2×C4—4881no—
394C2×C4—2442no—
395C2×C4—2441no—
396C2×C4C22⋊C4×D6111yescharacteristic
397C2×C4#1451973911yescharacteristic
398C2×C4C4×D24411yescharacteristic
399C2×C4#1451974711yescharacteristic
400C2×C4#1451975211yescharacteristic
401C2×C4#1451976211yescharacteristic
402C2×C4SD16×D6111yescharacteristic
403C2×C4#1451979811yescharacteristic
404C2×C4#1451980111yescharacteristic
405C2×C4#1451980911yescharacteristic
406C8—41no—
407C8—41no—
408C8—42no—
409C8—1222no—
410C8—2442no—
411C8—2441no—
412C22—21no—
413C22—21no—
414C22—4881no—
415C22—4881no—
416C22—4881no—
417C22#1683964011yescharacteristic
418C22#1683996111yescharacteristic
419C22#1683998711yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—82no—
424C4—1222no—
425C4—4881no—
426C4—2441no—
427C4—2441no—
428C4—2441no—
429C4#104×D6111yescharacteristic
430C4#1683977711yescharacteristic
431C4#1683984911yescharacteristic
432C4#1683986111yescharacteristic
433C4#1683990211yescharacteristic
434C4#1683990811yescharacteristic
435C2—21no—
436C2—21no—
437C2—9761no—
438C2#2144612011yescharacteristic
439C2#2144617511yescharacteristic
440C2#2144618011yescharacteristic
subgroupG/H
centerC2×C4#14519747
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1303C2
socleC22×C61#386
prime radical#16840312C22
cosocleC2×C42×C61C23
hypercenter#1303D61
hypocenterC61#8992
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
2. #21454770C2semidirect
4. C4×#14519700C2semidirect
6. #21443655C2semidirect
7. C61×#1303C2semidirect
11. #16840048C4semidirect
12. #16840048C4semidirect
13. #16839963C4semidirect
14. #16839963C4semidirect
136. #11758194SD16semidirect
184. Dic122#104semidirect
220. #1303D61semidirect
253. C61#8992semidirect

Maximal subgroups

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

subgroupindex
#214558512
#214547702
#214467372
C4×#145197002
#214442692
#214436552
C61×#13032
#899261