Finite Group Databasebeta

← #29597441

Subgroups — #29597441

Order 15616.

456 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455539C211yesmaximal, characteristic
2#21447556C211yesmaximal, characteristic, complemented
3C2×#16839570C211yesmaximal, characteristic, complemented
4#21445873C211yesmaximal, characteristic, complemented
5#21443931C211yesmaximal, characteristic, complemented
6#21443915C211yesmaximal, characteristic, complemented
7C2×C61×#504C211yesmaximal, characteristic, complemented
8#16840560—21no—
9C2×#14519799C2211yescharacteristic
10#16840037C412yescomplemented
11#16840032C412yescomplemented
12#16840025—22no—
13C2×#14519713C2211yescharacteristic
14C2×#14519713C2211yescharacteristic
15#16839800C2211yescharacteristic
16#16839779—21no—
17#16839744—21no—
18#16839573—22no—
19#16839570—21no—
20#16839570—21no—
21#16839534—21no—
22C2×#131×C61C2211yescharacteristic
23C61×#504—22no—
24C61×#504—22no—
25C2×M32×C61—21no—
26C2×M32×C61C2211yescharacteristic
27C22×C16×C61—21no—
28C22×C16×C61C2211yescharacteristic
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
40C2×#11758190—21no—
41C2×D488C2×C411yescharacteristic
42C2×#11758194—21no—
43C2×#11758194—21no—
44#14519736—21no—
45#14519713—21no—
46#14519713—21no—
47#14519713—21no—
48#14519713C2×C412yes—
49#14519711—21no—
50#14519709—21no—
51#14519704—42no—
52#14519695—22no—
53#14519693—22no—
54C2×#41×C61D411yescharacteristic
55#131×C61—21no—
56#131×C61—21no—
57#131×C61C2×C412yes—
58#131×C61D412yes—
59C2×M16×C61—21no—
60C2×M16×C61D411yescharacteristic
61C22×C8×C61—21no—
62C22×C8×C61C2311yescharacteristic
63M32×C61—24no—
64M32×C61—21no—
65M32×C61D412yes—
66C2×C16×C61—24no—
67C2×C16×C61—21no—
68C2×C16×C61—21no—
69C2×C16×C61—21no—
70C2×C16×C61D411yescharacteristic
71C2×C16×C61D411yescharacteristic
72#41×C61—21no—
73#41×C61—21no—
74#41×C61C22⋊C412yes—
75#41×C61D812yes—
76C2×Q8×C61SD1611yescharacteristic
77C2×D4×C61—21no—
78C2×D4×C61SD1611yescharacteristic
79C22×C4×C61—21no—
80C22×C4×C61C2×D411yescharacteristic
81C2×C22⋊D61—41no—
82#11758222—41no—
83#11758222—41no—
84#11758222—41no—
85#11758222—41no—
86C2×D244—21no—
87C2×C4×D61—41no—
88C2×Dic122—21no—
89M16×C61—21no—
90M16×C61—22no—
91M16×C61—22no—
92M16×C61C22⋊C412yes—
93C2×C8×C61—21no—
94C2×C8×C61—22no—
95C2×C8×C61—22no—
96C2×C8×C61C22×C411yescharacteristic
97C2×C8×C61C2×D411yescharacteristic
98C2×C8×C61C2×D411yescharacteristic
99C2×C8×C61C2×D411yescharacteristic
100C2×C8×C61#4111yescharacteristic
101C2×C8×C61#4111yescharacteristic
102C22⋊Dic61—41no—
103#11758202—42no—
104#11758201—42no—
105#11758200—42no—
106#11758199—42no—
107C4⋊Dic61—41no—
108C4×Dic61—41no—
109#11758194—22no—
110#11758194—41no—
111#11758194—41no—
112#11758194—41no—
113C2×#92418—41no—
114C2×#92418—41no—
115Dic244—21no—
116Dic244—21no—
117D488—21no—
118D488—21no—
119#11758190—41no—
120#11758190—41no—
121C16×C61—21no—
122C16×C61—22no—
123C16×C61—21no—
124C23×C61—21no—
125C23×C61#13511yescharacteristic
126C22×D61—41no—
127Q8×C61—21no—
128Q8×C61#10212yes—
129D4×C61—22no—
130D4×C61—22no—
131D4×C61—21no—
132D4×C61#10212yes—
133C2×C4×C61—21no—
134C2×C4×C61—22no—
135C2×C4×C61—22no—
136C2×C4×C61C2×C22⋊C411yescharacteristic
137C2×C4×C61#12111yescharacteristic
138C2×C4×C61#12111yescharacteristic
139C2×C4×C61C2×D811yescharacteristic
140C2×C4×C61C2×SD1611yescharacteristic
141C2×C4×C61#13511yescharacteristic
142C22⋊D61—81no—
143C22⋊D61—81no—
144C2×Dic61—41no—
145C2×Dic61—81no—
146D244—41no—
147D244—41no—
148C4×D61—81no—
149C4×D61—81no—
150Dic122—41no—
151Dic122—41no—
152C8×C61—22no—
153C8×C61C4×D411yescharacteristic
154C8×C61C4×D411yescharacteristic
155C8×C61#12011yescharacteristic
156C8×C61#12311yescharacteristic
157#92418—42no—
158#92418—81no—
159#13071—611nomaximal, Sylow
160C22×C61—22no—
161C22×C61—41no—
162C22×C61—42no—
163C22×C61#41611yescharacteristic
164C22×C61#46511yescharacteristic
165C22×C61#46611yescharacteristic
166C2×D61—81no—
167C2×D61—161no—
168C4×C61—22no—
169C4×C61#38611yescharacteristic
170C4×C61C2×#10211yescharacteristic
171C4×C61#46511yescharacteristic
172C4×C61#46611yescharacteristic
173Dic61—81no—
174Dic61—161no—
175C2×#504D6111yescharacteristic, complemented
176#3067—611no—
177#2145—611no—
178C2×#359—611no—
179#1814—611no—
180#1388—611no—
181#1382—611no—
182C2×C61—21no—
183C2×C61—21no—
184C2×C61—21no—
185C2×C61—42no—
186C2×C61#189311yescharacteristic
187D61—161no—
188#575—1221no—
189C2×#135—611no—
190C2×#131C2×D6111yescharacteristic
191#504—22no—
192#504—22no—
193C2×M32—21no—
194C2×M32C2×D6111yescharacteristic
195C22×C16—21no—
196C22×C16C2×D6111yescharacteristic
197#473—612no—
198#472—612no—
199#471—1222no—
200C2×#108—611no—
201C2×#108—611no—
202#427—611no—
203#421—1221no—
204#413—1221no—
205#360—1222no—
206#359—1221no—
207#359—1221no—
208#344—1221no—
209C61#1307111yescharacteristic, complemented, Sylow
210C2×#41—1221no—
211C2×#41D24411yescharacteristic
212#137—1222no—
213#137—2441no—
214AGL(1,ℤ/8)—1222no—
215AGL(1,ℤ/8)—2441no—
216#135—1221no—
217#135—1221no—
218#135—1221no—
219#135—1221no—
220#135—2442no—
221C2×Q16—611no—
222C2×SD16—1221no—
223C2×D8—611no—
224#131—21no—
225#131—21no—
226#131C4×D6112yes—
227#131C22⋊D6112yes—
228C2×M16—21no—
229C2×M16—1221no—
230C2×M16—1221no—
231C2×M16D24411yescharacteristic
232C22×C8—21no—
233C22×C8C22×D6111yescharacteristic
234#117—1221no—
235M32—24no—
236M32—21no—
237M32C22⋊D6112yes—
238C2×C16—24no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16C22⋊D6111yescharacteristic
243C2×C16C22⋊D6111yescharacteristic
244#108—612no—
245#108—1221no—
246#108—1221no—
247#108—1221no—
248#107—1221no—
249#106—1221no—
250#104—2442no—
251#101—1222no—
252#100—1222no—
253#41—21no—
254#41—21no—
255#41—2441no—
256#41—2441no—
257#41—2441no—
258#41—2441no—
259#41D48812yes—
260#41#1175819812yes—
261C2×Q8—1221no—
262C2×Q8#1175819011yescharacteristic
263C2×D4—21no—
264C2×D4—1221no—
265C2×D4—2441no—
266C2×D4#1175819011yescharacteristic
267C22×C4—21no—
268C22×C4—2441no—
269C22×C4C2×D24411yescharacteristic
270Q16—1221no—
271Q16—1221no—
272Q16—2442no—
273SD16—2442no—
274SD16—2442no—
275SD16—2441no—
276SD16—2441no—
277D8—1221no—
278D8—1221no—
279D8—2442no—
280M16—21no—
281M16—22no—
282M16—22no—
283M16—1222no—
284M16—2441no—
285M16—2441no—
286M16—2441no—
287M16#1175819812yes—
288C2×C8—21no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—2441no—
292C2×C8—2441no—
293C2×C8C2×C4×D6111yescharacteristic
294C2×C8#1175822211yescharacteristic
295C2×C8#1175822211yescharacteristic
296C2×C8C2×C22⋊D6111yescharacteristic
297C2×C8C2×C22⋊D6111yescharacteristic
298C2×C8C2×C22⋊D6111yescharacteristic
299C4⋊C4—2441no—
300C22⋊C4—2441no—
301C42—2441no—
302C16—21no—
303C16—22no—
304C16—21no—
305C23—21no—
306C23—2441no—
307C23#1451979911yescharacteristic
308Q8—21no—
309Q8—2441no—
310Q8—2441no—
311Q8#1451969912yes—
312D4—22no—
313D4—22no—
314D4—21no—
315D4—4881no—
316D4—4881no—
317D4—2441no—
318D4—2441no—
319D4#1451969912yes—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—4881no—
324C2×C4—4881no—
325C2×C4—4881no—
326C2×C4—2441no—
327C2×C4C2×#1175819811yescharacteristic
328C2×C4#1451975111yescharacteristic
329C2×C4#1451975111yescharacteristic
330C2×C4C2×D48811yescharacteristic
331C2×C4C2×#1175819011yescharacteristic
332C2×C4#1451979911yescharacteristic
333C8—22no—
334C8—4881no—
335C8—2442no—
336C8C4×C22⋊D6111yescharacteristic
337C8C4×C22⋊D6111yescharacteristic
338C8#1451974611yescharacteristic
339C8#1451976011yescharacteristic
340C22—22no—
341C22—41no—
342C22—42no—
343C22—9761no—
344C22—4881no—
345C22#1683975911yescharacteristic
346C22#1683999111yescharacteristic
347C22#1683999811yescharacteristic
348C4—22no—
349C4—9761no—
350C4—4881no—
351C4#1683964111yescharacteristic
352C4C2×#1451969911yescharacteristic
353C4#1683999111yescharacteristic
354C4#1683999811yescharacteristic
355C2—21no—
356C2—21no—
357C2—21no—
358C2—42no—
359C2—9761no—
360C2#2144624711yescharacteristic
subgroupG/H
centerC4#16839998
commutatorC2×C8×C61C22×C4
FrattiniC22×C8C22×D61
FittingC2×C61×#504C2
socleC2×C61#1893
prime radical#21455539C2
cosocleC22×C8×C61C23
hypercenterC2×#504D61
hypocenterC61#13071
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504

Complements

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

NH ≅ G/Ntype
2. #21447556C2semidirect
3. C2×#16839570C2semidirect
4. #21445873C2semidirect
5. #21443931C2semidirect
6. #21443915C2semidirect
7. C2×C61×#504C2semidirect
10. #16840037C4semidirect
11. #16840032C4semidirect
175. C2×#504D61semidirect
209. C61#13071semidirect

Maximal subgroups

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

subgroupindex
#214555392
#214475562
C2×#168395702
#214458732
#214439312
#214439152
C2×C61×#5042
#1307161