Finite Group Databasebeta

← #29597025

Subgroups — #29597025

Order 15616.

492 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455539C211yesmaximal, characteristic
2#21454414C211yesmaximal, characteristic, complemented
3#21447556C211yesmaximal, characteristic, complemented
4C2×#16839573C211yesmaximal, characteristic, complemented
5#21443934C211yesmaximal, characteristic, complemented
6#21443918C211yesmaximal, characteristic, complemented
7C61×#2117C211yesmaximal, characteristic, complemented
8#16840560—21no—
9C2×#14519799C2211yescharacteristic
10#16840245—21no—
11#16840037C412yescomplemented
12#16840032C412yescomplemented
13#16840025—22no—
14C2×#14519713C2211yescharacteristic
15#16839804—22no—
16#16839804—21no—
17#16839800—21no—
18#16839800C2211yescharacteristic
19C2×#14519709—21no—
20C2×#14519709C2211yescharacteristic
21#16839744—21no—
22#16839573—22no—
23#16839573—21no—
24#16839573—21no—
25C2×#131×C61C2211yescharacteristic
26C2×M32×C61C2211yescharacteristic
27C2×M32×C61C2211yescharacteristic
28C61×#350—22no—
29C2×#11758222—21no—
30#14519808—41no—
31#14519807—22no—
32#14519803—41no—
33#14519802—22no—
34#14519799—21no—
35#14519799—21no—
36#14519799—21no—
37#14519799—21no—
38#14519797—42no—
39C2×Dic244C2×C411yescharacteristic, complemented
40C2×#11758190—21no—
41C2×D488C2×C411yescharacteristic, complemented
42C2×#11758194—21no—
43Q8×Dic61—41no—
44D4×Dic61—42no—
45D4×Dic61—41no—
46#14519736—22no—
47#14519736—21no—
48C2×C4⋊Dic61—22no—
49C2×C4⋊Dic61—21no—
50#14519713—21no—
51#14519713C2×C412yescomplemented
52#14519711—22no—
53#14519711—21no—
54#14519709—22no—
55#14519709—21no—
56#14519709D411yescharacteristic
57#14519709D412yes—
58#14519709D411yescharacteristic
59#14519695—22no—
60#14519693—22no—
61C2×#41×C61D411yescharacteristic
62#131×C61—21no—
63#131×C61—21no—
64#131×C61C2×C412yes—
65#131×C61D412yes—
66C2×M16×C61—21no—
67C2×M16×C61D411yescharacteristic
68C22×C8×C61—21no—
69C22×C8×C61C2311yescharacteristic
70M32×C61—21no—
71M32×C61—21no—
72M32×C61—21no—
73M32×C61—21no—
74C2×C16×C61—21no—
75C2×C16×C61—21no—
76#41×C61—21no—
77#41×C61—21no—
78#41×C61C22⋊C412yes—
79#41×C61SD1612yes—
80C2×Q8×C61D811yescharacteristic
81C2×D4×C61—21no—
82C2×D4×C61D811yescharacteristic
83C22×C4×C61—21no—
84C22×C4×C61C2×D411yescharacteristic
85C2×C22⋊D61—41no—
86C22×Dic61—42no—
87C22×Dic61—41no—
88#11758222—41no—
89#11758222—41no—
90#11758222—41no—
91#11758222—41no—
92C2×D244—21no—
93C2×C4×D61—41no—
94C2×Dic122—21no—
95M16×C61—21no—
96M16×C61—22no—
97M16×C61—22no—
98M16×C61C22⋊C412yes—
99C2×C8×C61—21no—
100C2×C8×C61—22no—
101C2×C8×C61—22no—
102C2×C8×C61C22×C411yescharacteristic
103C2×C8×C61C2×D411yescharacteristic
104C2×C8×C61C2×D411yescharacteristic
105C2×C8×C61C2×D411yescharacteristic
106C2×C8×C61#4111yescharacteristic
107C2×C8×C61#4111yescharacteristic
108C22⋊Dic61—42no—
109C22⋊Dic61—41no—
110#11758202—42no—
111#11758201—42no—
112#11758200—42no—
113#11758199—42no—
114C4⋊Dic61—21no—
115C4⋊Dic61—22no—
116C4⋊Dic61—21no—
117C4⋊Dic61—42no—
118C4⋊Dic61—41no—
119C4×Dic61—42no—
120C4×Dic61—41no—
121#11758194—22no—
122#11758194—41no—
123C2×#92418—41no—
124Dic244—21no—
125Dic244—21no—
126D488—21no—
127D488—21no—
128#11758190—41no—
129#11758190—41no—
130C16×C61—41no—
131C16×C61—41no—
132C23×C61—21no—
133C23×C61AGL(1,ℤ/8)11yescharacteristic
134C22×D61—41no—
135Q8×C61—21no—
136Q8×C61#10212yes—
137D4×C61—22no—
138D4×C61—22no—
139D4×C61—21no—
140D4×C61#10212yes—
141C2×C4×C61—21no—
142C2×C4×C61—22no—
143C2×C4×C61—22no—
144C2×C4×C61C2×C22⋊C411yescharacteristic
145C2×C4×C61#12011yescharacteristic
146C2×C4×C61#12311yescharacteristic
147C2×C4×C61C2×D811yescharacteristic
148C2×C4×C61C2×SD1611yescharacteristic
149C2×C4×C61#13711yescharacteristic
150C22⋊D61—81no—
151C22⋊D61—81no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—42no—
155C2×Dic61—41no—
156C2×Dic61—161no—
157C2×Dic61—162no—
158D244—41no—
159D244—41no—
160C4×D61—81no—
161C4×D61—81no—
162Dic122—41no—
163Dic122—41no—
164C8×C61—22no—
165C8×C61C4×D411yescharacteristic
166C8×C61C4×D411yescharacteristic
167C8×C61#12111yescharacteristic
168C8×C61#12111yescharacteristic
169#92418—42no—
170#13005—611nomaximal, Sylow
171C22×C61—22no—
172C22×C61—41no—
173C22×C61—42no—
174C22×C61#41711yescharacteristic
175C22×C61#44711yescharacteristic
176C22×C61#48111yescharacteristic
177C2×D61—81no—
178C2×D61—161no—
179C4×C61—22no—
180C4×C61#38611yescharacteristic
181C4×C61C2×#10211yescharacteristic
182C4×C61#44811yescharacteristic
183C4×C61#48011yescharacteristic
184Dic61—81no—
185Dic61—82no—
186Dic61—81no—
187Dic61—81no—
188#3067—611no—
189#2912—611no—
190#2145—611no—
191C2×#360—611no—
192#2117D6111yescharacteristic, complemented
193#1389—611no—
194#1383—611no—
195C2×C61—21no—
196C2×C61—21no—
197C2×C61—21no—
198C2×C61—42no—
199C2×C61#187411yescharacteristic
200D61—161no—
201#575—1221no—
202C2×#135—611no—
203C2×#131C2×D6111yescharacteristic
204#520—1221no—
205C2×M32C2×D6111yescharacteristic
206C2×M32C2×D6111yescharacteristic
207#473—612no—
208#472—612no—
209#471—1222no—
210C2×#108—611no—
211#428—1222no—
212#428—1221no—
213#427—611no—
214#427—1221no—
215C2×#106—611no—
216C2×#106—1221no—
217#413—1221no—
218#360—1222no—
219#360—1221no—
220#360—1221no—
221#350—22no—
222C61#1300511yescharacteristic, complemented, Sylow
223C2×#41—1221no—
224C2×#41D24411yescharacteristic
225#137—1222no—
226#137—2441no—
227AGL(1,ℤ/8)—1222no—
228AGL(1,ℤ/8)—2441no—
229#135—1221no—
230#135—1221no—
231#135—1221no—
232#135—1221no—
233#135—2442no—
234C2×Q16—611no—
235C2×SD16—1221no—
236C2×D8—611no—
237#131—21no—
238#131—21no—
239#131C4×D6112yes—
240#131C22⋊D6112yes—
241C2×M16—21no—
242C2×M16—1221no—
243C2×M16D24411yescharacteristic
244C22×C8—21no—
245C22×C8C22×D6111yescharacteristic
246C4×Q8—2441no—
247C4×D4—2442no—
248C4×D4—2441no—
249#117—1222no—
250#117—1221no—
251C2×C4⋊C4—1222no—
252C2×C4⋊C4—1221no—
253M32—21no—
254M32—21no—
255M32—21no—
256M32—21no—
257C2×C16—21no—
258C2×C16—21no—
259#108—612no—
260#108—1221no—
261#107—1222no—
262#107—1221no—
263#106—612no—
264#106—611no—
265#106—611no—
266#106—1222no—
267#106—1221no—
268#101—1222no—
269#100—1222no—
270#41—21no—
271#41—21no—
272#41—2441no—
273#41—2441no—
274#41—2441no—
275#41—2441no—
276#41#1175819012yes—
277#41#1175819812yes—
278C2×Q8—1221no—
279C2×Q8D48811yescharacteristic
280C2×D4—21no—
281C2×D4—1221no—
282C2×D4—2441no—
283C2×D4D48811yescharacteristic
284C22×C4—21no—
285C22×C4—2442no—
286C22×C4—2441no—
287C22×C4—2441no—
288C22×C4C2×D24411yescharacteristic
289Q16—1221no—
290Q16—1221no—
291Q16—2442no—
292SD16—2442no—
293SD16—2442no—
294SD16—2441no—
295SD16—2441no—
296D8—1221no—
297D8—1221no—
298D8—2442no—
299M16—21no—
300M16—22no—
301M16—22no—
302M16—1222no—
303M16—2441no—
304M16#1175819812yes—
305C2×C8—21no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—2441no—
309C2×C8C2×C4×D6111yescharacteristic
310C2×C8D4×D6111yescharacteristic
311C2×C8D4×D6111yescharacteristic
312C2×C8#1175822411yescharacteristic
313C2×C8#1175822411yescharacteristic
314C2×C8C2×C22⋊D6111yescharacteristic
315C4⋊C4—1222no—
316C4⋊C4—1221no—
317C4⋊C4—1221no—
318C4⋊C4—2442no—
319C4⋊C4—2441no—
320C22⋊C4—2442no—
321C22⋊C4—2441no—
322C42—2442no—
323C42—2441no—
324C16—41no—
325C16—41no—
326C23—21no—
327C23—2441no—
328C23#1451980511yescharacteristic
329Q8—21no—
330Q8—2441no—
331Q8—2441no—
332Q8#1451969912yes—
333D4—22no—
334D4—22no—
335D4—21no—
336D4—4881no—
337D4—4881no—
338D4—2441no—
339D4—2441no—
340D4#1451969912yes—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—9761no—
345C2×C4—9762no—
346C2×C4—4881no—
347C2×C4—4881no—
348C2×C4—2441no—
349C2×C4—2441no—
350C2×C4—2442no—
351C2×C4—2441no—
352C2×C4C2×#1175819811yescharacteristic
353C2×C4#1451974711yescharacteristic
354C2×C4#1451976111yescharacteristic
355C2×C4C2×D48811yescharacteristic
356C2×C4C2×#1175819011yescharacteristic
357C2×C4#1451981011yescharacteristic
358C8—22no—
359C8—2442no—
360C8#1451973711yescharacteristic
361C8#1451973711yescharacteristic
362C8#1451974911yescharacteristic
363C8#1451974911yescharacteristic
364C22—22no—
365C22—41no—
366C22—42no—
367C22—9761no—
368C22—4881no—
369C22#1683976411yescharacteristic
370C22#1683989811yescharacteristic
371C22#1684008111yescharacteristic
372C4—22no—
373C4—4882no—
374C4—4881no—
375C4—4881no—
376C4—4881no—
377C4#1683963911yescharacteristic
378C4C2×#1451969911yescharacteristic
379C4#1683990311yescharacteristic
380C4#1684007611yescharacteristic
381C2—21no—
382C2—21no—
383C2—21no—
384C2—42no—
385C2—9761no—
386C2#2144613411yescharacteristic
subgroupG/H
centerC2#21446134
commutatorC2×C8×C61C22×C4
FrattiniC22×C8C22×D61
FittingC61×#2117C2
socleC2×C61#1874
prime radical#21455539C2
cosocleC22×C8×C61C23
hypercenter#2117D61
hypocenterC61#13005
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #2117

Complements

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

NH ≅ G/Ntype
2. #21454414C2semidirect
3. #21447556C2semidirect
4. C2×#16839573C2semidirect
5. #21443934C2semidirect
6. #21443918C2semidirect
7. C61×#2117C2semidirect
11. #16840037C4semidirect
12. #16840032C4semidirect
39. C2×Dic244C2×C4semidirect
41. C2×D488C2×C4semidirect
51. #14519713C2×C4semidirect
192. #2117D61semidirect
222. C61#13005semidirect

Maximal subgroups

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

subgroupindex
#214555392
#214544142
#214475562
C2×#168395732
#214439342
#214439182
C61×#21172
#1300561