Finite Group Databasebeta

← #29597443

Subgroups — #29597443

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
1C2×#16840179C211yesmaximal, characteristic, complemented
2#21447555C211yesmaximal, characteristic, complemented
3C2×#16839569C211yesmaximal, characteristic, complemented
4#21445873C211yesmaximal, characteristic, complemented
5#21443932C211yesmaximal, characteristic, complemented
6#21443916C211yesmaximal, characteristic, complemented
7C61×#3067C211yesmaximal, characteristic
8#16840179—22no—
9#16840179—22no—
10C2×#14519717—21no—
11C2×#14519717C2211yescharacteristic
12C22×#11758185—21no—
13C22×#11758185C2211yescharacteristic
14C2×#14519713C2211yescharacteristic
15#16839800C2211yescharacteristic
16#16839779—21no—
17#16839572—22no—
18#16839569—21no—
19#16839569—21no—
20#16839534—21no—
21C61×#575—21no—
22C2×#135×C61C2211yescharacteristic
23C2×#131×C61C2211yescharacteristic
24C61×#473C412yescomplemented
25C61×#472C412yescomplemented
26C61×#471—22no—
27C2×#108×C61C2211yescharacteristic
28C61×#413—21no—
29C2×#11758194—21no—
30#14519736—21no—
31#14519717—24no—
32#14519717—21no—
33#14519717D412yes—
34C2×#11758185—24no—
35C2×#11758185—21no—
36C2×#11758185—21no—
37C2×#11758185—21no—
38C2×#11758185D411yescharacteristic
39C2×#11758185D411yescharacteristic
40#14519713—21no—
41#14519713—21no—
42#14519711—21no—
43#14519709—21no—
44#14519704—42no—
45C2×#41×C61—21no—
46C2×#41×C61D411yescharacteristic
47#137×C61—22no—
48#137×C61—41no—
49AGL(1,ℤ/8)×C61—22no—
50AGL(1,ℤ/8)×C61—41no—
51#135×C61—21no—
52#135×C61—21no—
53#135×C61—21no—
54#135×C61—21no—
55#135×C61—42no—
56C2×Q16×C61C2×C411yescharacteristic
57C2×SD16×C61—21no—
58C2×D8×C61C2×C411yescharacteristic
59#131×C61—21no—
60#131×C61—21no—
61#131×C61C2×C412yes—
62#131×C61D412yes—
63C2×M16×C61—21no—
64C2×M16×C61—21no—
65C2×M16×C61D411yescharacteristic
66C22×C8×C61—21no—
67C22×C8×C61C2311yescharacteristic
68#108×C61—21no—
69#108×C61C2×C412yes—
70#101×C61—22no—
71#100×C61—22no—
72#41×C61—21no—
73#41×C61—21no—
74#41×C61—41no—
75#41×C61—41no—
76#41×C61—41no—
77#41×C61—41no—
78#41×C61C22⋊C412yes—
79#41×C61D812yes—
80C2×Q8×C61—21no—
81C2×Q8×C61SD1611yescharacteristic
82C2×D4×C61—21no—
83C2×D4×C61—21no—
84C2×D4×C61—41no—
85C2×D4×C61SD1611yescharacteristic
86C22×C4×C61—21no—
87C22×C4×C61—41no—
88C22×C4×C61C2×D411yescharacteristic
89Q16×C61—21no—
90Q16×C61—21no—
91Q16×C61—42no—
92SD16×C61—42no—
93SD16×C61—42no—
94SD16×C61—41no—
95SD16×C61—41no—
96D8×C61—21no—
97D8×C61—21no—
98D8×C61—42no—
99M16×C61—21no—
100M16×C61—22no—
101M16×C61—22no—
102M16×C61—22no—
103M16×C61—41no—
104M16×C61C22⋊C412yes—
105C2×C8×C61—21no—
106C2×C8×C61—22no—
107C2×C8×C61—22no—
108C2×C8×C61—41no—
109C2×C8×C61C22×C411yescharacteristic
110C2×C8×C61C2×D411yescharacteristic
111C2×C8×C61C2×D411yescharacteristic
112C2×C8×C61C2×D411yescharacteristic
113C2×C8×C61#4111yescharacteristic
114C2×C8×C61#4111yescharacteristic
115C22⋊Dic61—41no—
116C4⋊Dic61—41no—
117C4×Dic61—41no—
118#11758194—41no—
119#11758194—41no—
120C2×#92418—41no—
121#11758185—21no—
122#11758185—22no—
123#11758185—21no—
124C23×C61—21no—
125C23×C61—41no—
126C23×C61#13511yescharacteristic
127Q8×C61—21no—
128Q8×C61—41no—
129Q8×C61—41no—
130Q8×C61#10212yes—
131D4×C61—22no—
132D4×C61—22no—
133D4×C61—21no—
134D4×C61—41no—
135D4×C61—41no—
136D4×C61—81no—
137D4×C61—81no—
138D4×C61#10212yes—
139C2×C4×C61—21no—
140C2×C4×C61—22no—
141C2×C4×C61—22no—
142C2×C4×C61—41no—
143C2×C4×C61—81no—
144C2×C4×C61—81no—
145C2×C4×C61C2×C22⋊C411yescharacteristic
146C2×C4×C61#12111yescharacteristic
147C2×C4×C61#12111yescharacteristic
148C2×C4×C61C2×D811yescharacteristic
149C2×C4×C61C2×SD1611yescharacteristic
150C2×C4×C61#13511yescharacteristic
151C2×Dic61—81no—
152C8×C61—22no—
153C8×C61—42no—
154C8×C61C4×D411yescharacteristic
155C8×C61C4×D411yescharacteristic
156C8×C61#12011yescharacteristic
157C8×C61#12311yescharacteristic
158#92418—81no—
159#13071—611nomaximal, Sylow
160C22×C61—22no—
161C22×C61—41no—
162C22×C61—42no—
163C22×C61—81no—
164C22×C61—161no—
165C22×C61#41611yescharacteristic
166C22×C61#46511yescharacteristic
167C22×C61#46611yescharacteristic
168C4×C61—22no—
169C4×C61—81no—
170C4×C61#38611yescharacteristic
171C4×C61C2×#10211yescharacteristic
172C4×C61#46511yescharacteristic
173C4×C61#46611yescharacteristic
174Dic61—161no—
175C2×#504—611no—
176#3067D6111yescharacteristic
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—161no—
187C2×C61#189311yescharacteristic
188#575—21no—
189C2×#135C2×D6111yescharacteristic
190C2×#131C2×D6111yescharacteristic
191#504—1222no—
192#504—1222no—
193C2×M32—611no—
194C2×M32—1221no—
195C22×C16—611no—
196C22×C16—1221no—
197#473Dic6112yescomplemented
198#472Dic6112yescomplemented
199#471—22no—
200C2×#108—611no—
201C2×#108C2×D6111yescharacteristic
202#427—611no—
203#421—1221no—
204#413—21no—
205#360—1222no—
206#359—1221no—
207#359—1221no—
208#344—1221no—
209C61#1307111yescharacteristic, complemented, Sylow
210C2×#41—21no—
211C2×#41C22⋊D6111yescharacteristic
212#137—22no—
213#137—41no—
214AGL(1,ℤ/8)—22no—
215AGL(1,ℤ/8)—41no—
216#135—21no—
217#135—21no—
218#135—21no—
219#135—21no—
220#135—42no—
221C2×Q16C2×Dic6111yescharacteristic
222C2×SD16—21no—
223C2×D8C2×Dic6111yescharacteristic
224#131—21no—
225#131—21no—
226#131C2×Dic6112yes—
227#131C22⋊D6112yes—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—1221no—
231C2×M16C22⋊D6111yescharacteristic
232C22×C8—21no—
233C22×C8C22×D6111yescharacteristic
234#117—1221no—
235M32—612no—
236M32—1224no—
237M32—1221no—
238C2×C16—611no—
239C2×C16—611no—
240C2×C16—1224no—
241C2×C16—1221no—
242C2×C16—1221no—
243C2×C16—1221no—
244#108—21no—
245#108—1221no—
246#108—1221no—
247#108C2×Dic6112yes—
248#107—1221no—
249#106—1221no—
250#104—2442no—
251#101—22no—
252#100—22no—
253#41—21no—
254#41—21no—
255#41—41no—
256#41—41no—
257#41—41no—
258#41—41no—
259#41#1175819912yes—
260#41C22⋊Dic6112yes—
261C2×Q8—21no—
262C2×Q8#1175820011yescharacteristic
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—41no—
266C2×D4#1175820011yescharacteristic
267C22×C4—21no—
268C22×C4—41no—
269C22×C4C2×C22⋊D6111yescharacteristic
270Q16—21no—
271Q16—21no—
272Q16—42no—
273SD16—42no—
274SD16—42no—
275SD16—41no—
276SD16—41no—
277D8—21no—
278D8—21no—
279D8—42no—
280M16—21no—
281M16—22no—
282M16—22no—
283M16—22no—
284M16—41no—
285M16—2441no—
286M16—2441no—
287M16C22⋊Dic6112yes—
288C2×C8—21no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—41no—
292C2×C8—2441no—
293C2×C8D4×D6111yescharacteristic
294C2×C8D4×D6111yescharacteristic
295C2×C8#1175822411yescharacteristic
296C2×C8#1175822411yescharacteristic
297C2×C8C22×Dic6111yescharacteristic
298C2×C8C2×C22⋊D6111yescharacteristic
299C4⋊C4—2441no—
300C22⋊C4—2441no—
301C42—2441no—
302C16—1221no—
303C16—1221no—
304C16—1222no—
305C23—21no—
306C23—41no—
307C23#1451979711yescharacteristic
308Q8—21no—
309Q8—41no—
310Q8—41no—
311Q8#1451969812yes—
312D4—22no—
313D4—22no—
314D4—21no—
315D4—41no—
316D4—41no—
317D4—81no—
318D4—81no—
319D4#1451969812yes—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—41no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—4881no—
327C2×C4C2×C22⋊Dic6111yescharacteristic
328C2×C4#1451974911yescharacteristic
329C2×C4#1451974911yescharacteristic
330C2×C4C2×#1175819911yescharacteristic
331C2×C4C2×#1175820011yescharacteristic
332C2×C4#1451979711yescharacteristic
333C8—22no—
334C8—42no—
335C8—4881no—
336C8D4×Dic6111yescharacteristic
337C8D4×Dic6111yescharacteristic
338C8#1451974511yescharacteristic
339C8#1451975811yescharacteristic
340C22—22no—
341C22—41no—
342C22—42no—
343C22—81no—
344C22—161no—
345C22#1683975811yescharacteristic
346C22#1683998911yescharacteristic
347C22#1683999611yescharacteristic
348C4—22no—
349C4—81no—
350C4—9761no—
351C4#1683964211yescharacteristic
352C4C2×#1451969811yescharacteristic
353C4#1683998911yescharacteristic
354C4#1683999611yescharacteristic
355C2—21no—
356C2—21no—
357C2—21no—
358C2—42no—
359C2—161no—
360C2#2144624811yescharacteristic
subgroupG/H
centerC4#16839996
commutatorC2×C8×C61C22×C4
FrattiniC22×C8C22×D61
Fitting, prime radicalC61×#3067C2
socleC2×C61#1893
cosocleC22×C8×C61C23
hypercenter#3067D61
hypocenterC61#13071
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ #3067

Complements

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

NH ≅ G/Ntype
1. C2×#16840179C2semidirect
2. #21447555C2semidirect
3. C2×#16839569C2semidirect
4. #21445873C2semidirect
5. #21443932C2semidirect
6. #21443916C2semidirect
24. C61×#473C4semidirect
25. C61×#472C4semidirect
197. #473Dic61semidirect
198. #472Dic61semidirect
209. C61#13071semidirect

Maximal subgroups

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

subgroupindex
C2×#168401792
#214475552
C2×#168395692
#214458732
#214439322
#214439162
C61×#30672
#1307161