Finite Group Databasebeta

← #29596931

Subgroups — #29596931

Order 15616.

456 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840179C211yesmaximal, characteristic, complemented
2#21447447C211yesmaximal, characteristic, complemented
3C2×#16839546C211yesmaximal, characteristic, complemented
4#21447440C211yesmaximal, characteristic, complemented
5#21443908C211yesmaximal, characteristic, complemented
6#21443905C211yesmaximal, characteristic, complemented
7C61×#2955C211yesmaximal, characteristic
8#16840179—22no—
9#16840179—22no—
10C2×#14519717—21no—
11C2×#14519717C2211yescharacteristic
12C2×#14519717C2211yescharacteristic
13C2×#14519717C2211yescharacteristic
14C22×#11758185—21no—
15C22×#11758185C2211yescharacteristic
16#16839546—21no—
17#16839546—21no—
18#16839546—22no—
19#16839544—22no—
20C61×#575C411yescharacteristic
21C2×#131×C61C2211yescharacteristic
22C2×#131×C61C2211yescharacteristic
23C61×#444—22no—
24C61×#443—22no—
25C61×#430C411yescharacteristic
26C61×#421C411yescharacteristic
27C61×#421C411yescharacteristic
28C61×#405C2211yescharacteristic
29#14519717—24no—
30#14519717—21no—
31#14519717—21no—
32#14519717—21no—
33#14519717—21no—
34#14519717—21no—
35#14519717D412yes—
36C2×#11758185—24no—
37C2×#11758185—21no—
38C2×#11758185—21no—
39C2×#11758185—21no—
40C2×#11758185—21no—
41C2×#11758185—21no—
42C2×#11758185D411yescharacteristic
43C2×#11758185D411yescharacteristic
44C2×#41×C61C2×C411yescharacteristic
45C2×#41×C61C2×C411yescharacteristic
46#137×C61—21no—
47#137×C61C812yes—
48AGL(1,ℤ/8)×C61—21no—
49AGL(1,ℤ/8)×C61C812yes—
50#135×C61—22no—
51#131×C61—21no—
52#131×C61—21no—
53#131×C61—21no—
54#131×C61—21no—
55#131×C61—21no—
56#131×C61—21no—
57#131×C61D412yes—
58#131×C61D412yes—
59C2×M16×C61—21no—
60C2×M16×C61—21no—
61C2×M16×C61C2×C411yescharacteristic
62C2×M16×C61C2×C411yescharacteristic
63C2×M16×C61C2×C411yescharacteristic
64C22×C8×C61—21no—
65C22×C8×C61—21no—
66C22×C8×C61C2311yescharacteristic
67#117×C61C2×C411yescharacteristic
68#108×C61—22no—
69#104×C61—22no—
70#104×C61—22no—
71#97×C61—21no—
72C4×C8×C61—21no—
73#41×C61—21no—
74#41×C61—21no—
75#41×C61—21no—
76#41×C61—21no—
77#41×C61—21no—
78#41×C61—21no—
79#41×C61C22⋊C412yes—
80#41×C61C2×C812yes—
81C2×Q8×C61C2×C811yescharacteristic
82C2×Q8×C61M1611yescharacteristic
83C2×D4×C61—21no—
84C2×D4×C61—21no—
85C2×D4×C61C2×C811yescharacteristic
86C2×D4×C61M1611yescharacteristic
87C22×C4×C61—21no—
88C22×C4×C61—21no—
89C22×C4×C61C22×C411yescharacteristic
90Q16×C61—22no—
91SD16×C61—22no—
92SD16×C61—22no—
93D8×C61—22no—
94M16×C61—21no—
95M16×C61—21no—
96M16×C61—22no—
97M16×C61—22no—
98M16×C61—21no—
99M16×C61—21no—
100M16×C61—41no—
101M16×C61—41no—
102M16×C61C22⋊C412yes—
103M16×C61C2×C812yes—
104C2×C8×C61—21no—
105C2×C8×C61—21no—
106C2×C8×C61—22no—
107C2×C8×C61—22no—
108C2×C8×C61—21no—
109C2×C8×C61—41no—
110C2×C8×C61—41no—
111C2×C8×C61C2×D411yescharacteristic
112C2×C8×C61C2×D411yescharacteristic
113C2×C8×C61C2×D411yescharacteristic
114C2×C8×C61C2×D411yescharacteristic
115C2×C8×C61#4111yescharacteristic
116C2×C8×C61#4111yescharacteristic
117C4⋊C4×C61—21no—
118C22⋊C4×C61—21no—
119C42×C61—21no—
120#11758185—21no—
121#11758185—22no—
122#11758185—21no—
123#11758185—41no—
124#11758185—41no—
125C23×C61—21no—
126C23×C61—21no—
127C23×C61#13111yescharacteristic
128Q8×C61—21no—
129Q8×C61—21no—
130Q8×C61—21no—
131Q8×C61C22⋊C812yes—
132D4×C61—22no—
133D4×C61—22no—
134D4×C61—21no—
135D4×C61—21no—
136D4×C61—21no—
137D4×C61—41no—
138D4×C61—41no—
139D4×C61C22⋊C812yes—
140C2×C4×C61—21no—
141C2×C4×C61—22no—
142C2×C4×C61—22no—
143C2×C4×C61—21no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61C2×C22⋊C411yescharacteristic
148C2×C4×C61C4×D411yescharacteristic
149C2×C4×C61C4×D411yescharacteristic
150C2×C4×C61C22×C811yescharacteristic
151C2×C4×C61C2×M1611yescharacteristic
152C2×C4×C61#13111yescharacteristic
153C8×C61—22no—
154C8×C61—22no—
155C8×C61—41no—
156C8×C61#12011yescharacteristic
157C8×C61#12111yescharacteristic
158C8×C61#12111yescharacteristic
159C8×C61#12311yescharacteristic
160#12987—611nomaximal, Sylow
161C22×C61—22no—
162C22×C61—41no—
163C22×C61—42no—
164C22×C61—41no—
165C22×C61—81no—
166C22×C61#40811yescharacteristic
167C22×C61C8×D411yescharacteristic
168C22×C61#43511yescharacteristic
169C4×C61—22no—
170C4×C61—41no—
171C4×C61—81no—
172C4×C61#38611yescharacteristic
173C4×C61C2×C22⋊C811yescharacteristic
174C4×C61C8×D411yescharacteristic
175C4×C61#43511yescharacteristic
176C2×#504—611no—
177#2955D6111yescharacteristic
178#2117—611no—
179C2×#350—611no—
180#2115—611no—
181#1379—611no—
182#1378—611no—
183C2×C61—21no—
184C2×C61—21no—
185C2×C61—21no—
186C2×C61—42no—
187C2×C61—81no—
188C2×C61#186911yescharacteristic
189#575Dic6111yescharacteristic
190C2×#131C2×D6111yescharacteristic
191C2×#131C2×D6111yescharacteristic
192#504—1222no—
193#504—1222no—
194C2×M32—611no—
195C2×M32—611no—
196C2×M32—611no—
197C2×M32—1221no—
198C22×C16—611no—
199C22×C16—1221no—
200#444—22no—
201#443—22no—
202#430Dic6111yescharacteristic
203#421Dic6111yescharacteristic
204#421Dic6111yescharacteristic
205#405C2×D6111yescharacteristic
206#350—1221no—
207#350—1221no—
208#350—1222no—
209#349—1222no—
210C61#1298711yescharacteristic, complemented, Sylow
211C2×#41C2×Dic6111yescharacteristic
212C2×#41C2×Dic6111yescharacteristic
213#137—21no—
214#137#9241812yes—
215AGL(1,ℤ/8)—21no—
216AGL(1,ℤ/8)#9241812yes—
217#135—22no—
218#131—21no—
219#131—21no—
220#131—21no—
221#131—21no—
222#131—21no—
223#131—21no—
224#131C22⋊D6112yes—
225#131C22⋊D6112yes—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16C2×Dic6111yescharacteristic
229C2×M16C2×Dic6111yescharacteristic
230C2×M16C2×Dic6111yescharacteristic
231C22×C8—21no—
232C22×C8—21no—
233C22×C8C22×D6111yescharacteristic
234#117C2×Dic6111yescharacteristic
235M32—612no—
236M32—1224no—
237M32—1221no—
238M32—1221no—
239M32—1221no—
240M32—1221no—
241M32—1221no—
242C2×C16—611no—
243C2×C16—611no—
244C2×C16—1224no—
245C2×C16—1221no—
246C2×C16—1221no—
247C2×C16—1221no—
248C2×C16—1221no—
249C2×C16—1221no—
250#108—22no—
251#104—22no—
252#104—22no—
253#97—21no—
254C4×C8—21no—
255#41—21no—
256#41—21no—
257#41—21no—
258#41—21no—
259#41—21no—
260#41—21no—
261#41C2×#9241812yes—
262#41C22⋊Dic6112yes—
263C2×Q8C2×#9241811yescharacteristic
264C2×Q8#1175819411yescharacteristic
265C2×D4—21no—
266C2×D4—21no—
267C2×D4C2×#9241811yescharacteristic
268C2×D4#1175819411yescharacteristic
269C22×C4—21no—
270C22×C4—21no—
271C22×C4C22×Dic6111yescharacteristic
272Q16—22no—
273SD16—22no—
274SD16—22no—
275D8—22no—
276M16—21no—
277M16—21no—
278M16—22no—
279M16—22no—
280M16—21no—
281M16—21no—
282M16—41no—
283M16—41no—
284M16C2×#9241812yes—
285M16C22⋊Dic6112yes—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—21no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8D4×D6111yescharacteristic
294C2×C8D4×D6111yescharacteristic
295C2×C8#1175822411yescharacteristic
296C2×C8#1175822411yescharacteristic
297C2×C8C2×C22⋊D6111yescharacteristic
298C2×C8C2×C22⋊D6111yescharacteristic
299C4⋊C4—21no—
300C22⋊C4—21no—
301C42—21no—
302C16—1221no—
303C16—1221no—
304C16—1222no—
305C16—2441no—
306C16—2441no—
307C23—21no—
308C23—21no—
309C23#1451978411yescharacteristic
310Q8—21no—
311Q8—21no—
312Q8—21no—
313Q8C22⋊#9241812yes—
314D4—22no—
315D4—22no—
316D4—21no—
317D4—21no—
318D4—21no—
319D4—41no—
320D4—41no—
321D4C22⋊#9241812yes—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4C2×C22⋊Dic6111yescharacteristic
330C2×C4D4×Dic6111yescharacteristic
331C2×C4D4×Dic6111yescharacteristic
332C2×C4C22×#9241811yescharacteristic
333C2×C4C2×#1175819411yescharacteristic
334C2×C4#1451978411yescharacteristic
335C8—22no—
336C8—22no—
337C8—41no—
338C8#1451974511yescharacteristic
339C8#1451974911yescharacteristic
340C8#1451974911yescharacteristic
341C8#1451975811yescharacteristic
342C22—22no—
343C22—41no—
344C22—42no—
345C22—41no—
346C22—81no—
347C22#1683972911yescharacteristic
348C22D4×#9241811yescharacteristic
349C22#1683983411yescharacteristic
350C4—22no—
351C4—41no—
352C4—81no—
353C4#1683964211yescharacteristic
354C4C2×C22⋊#9241811yescharacteristic
355C4D4×#9241811yescharacteristic
356C4#1683983411yescharacteristic
357C2—21no—
358C2—21no—
359C2—21no—
360C2—42no—
361C2—81no—
362C2#2144611211yescharacteristic
subgroupG/H
centerC8#14519749
commutatorC2×C4×C61C22×C8
FrattiniC22×C8C22×D61
FittingC61×#2955C2
socleC2×C61#1869
prime radicalC61×#575C4
cosocleC22×C8×C61C23
hypercenter#2955D61
hypocenterC61#12987
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2955

Complements

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

NH ≅ G/Ntype
1. C2×#16840179C2semidirect
2. #21447447C2semidirect
3. C2×#16839546C2semidirect
4. #21447440C2semidirect
5. #21443908C2semidirect
6. #21443905C2semidirect
210. C61#12987semidirect

Maximal subgroups

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

subgroupindex
C2×#168401792
#214474472
C2×#168395462
#214474402
#214439082
#214439052
C61×#29552
#1298761