Finite Group Databasebeta

← #29679402

Subgroups — #29679402

Order 15616.

1498 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839789C212yesmaximal, complemented
2C2×#16839729C212yesmaximal
3#21445803C214yesmaximal, complemented
4#21445799C214yesmaximal, complemented
5#21445513C211yesmaximal, characteristic, complemented
6C2×#16839517C211yesmaximal, characteristic, complemented
7C2×C4×#41×C61C211yesmaximal, characteristic
8C22×#11758194C2212yes—
9C23×#92418C2212yes—
10#16839789—28no—
11C2×C4⋊#92418C2218yes—
12#16839729—28no—
13C2×C22⋊#92418C2218yes—
14#16839517C2212yes—
15#16839517C2212yes—
16#16839517C2212yes—
17#16839517C2212yes—
18C22×#41×C61C2211yescharacteristic
19C4×#41×C61—28no—
20C2×C4×Q8×C61C412yes—
21C2×C4×D4×C61C412yes—
22C2×C4×D4×C61C414yes—
23C2×#117×C61C2212yes—
24C2×#117×C61C2211yescharacteristic
25C22×C42×C61C2211yescharacteristic
26C22×C42×C61C2212yes—
27C2×#11758194—24no—
28C2×#11758194—24no—
29C2×#11758194—24no—
30C22×#92418—24no—
31C22×#92418—24no—
32C22×#92418—24no—
33C22×#92418C2314yes—
34C22×#92418C2314yes—
35C4⋊#92418—216no—
36C22⋊#92418—216no—
37C2×#41×C61—24no—
38C2×#41×C61—24no—
39C2×#41×C61D414yes—
40C2×#41×C61Q814yes—
41C22×Q8×C61C2×C411yescharacteristic
42C22×D4×C61C2×C412yes—
43C22×D4×C61C2×C411yescharacteristic
44C23×C4×C61C2×C412yes—
45C23×C4×C61C2×C414yes—
46C23×C4×C61C2312yes—
47C23×C4×C61C2311yescharacteristic
48C4×Q8×C61—28no—
49C4×D4×C61—28no—
50C4×D4×C61—216no—
51#117×C61—28no—
52#117×C61D418yes—
53C2×C4⋊C4×C61C2×C414yes—
54C2×C4⋊C4×C61C2×C412yes—
55C2×C22⋊C4×C61C2×C414yes—
56C2×C22⋊C4×C61C2×C412yes—
57C2×C42×C61—24no—
58C2×C42×C61—28no—
59C2×C42×C61C2×C412yes—
60C2×C42×C61C2×C414yes—
61C2×C42×C61C2311yescharacteristic
62C2×C42×C61C2311yescharacteristic
63C2×C42×C61C2312yes—
64C24×C61C22×C412yes—
65C24×C61C22×C411yescharacteristic
66#41×C61—216no—
67#41×C61—216no—
68C2×Q8×C61—22no—
69C2×Q8×C61—22no—
70C2×Q8×C61C4⋊C414yes—
71C2×D4×C61—24no—
72C2×D4×C61—24no—
73C2×D4×C61—22no—
74C2×D4×C61—22no—
75C2×D4×C61C4⋊C418yes—
76C2×D4×C61C4⋊C414yes—
77C22×C4×C61—24no—
78C22×C4×C61—24no—
79C22×C4×C61—24no—
80C22×C4×C61—28no—
81C22×C4×C61—28no—
82C22×C4×C61—24no—
83C22×C4×C61—44no—
84C22×C4×C61—42no—
85C22×C4×C61C22×C412yes—
86C22×C4×C61C22×C414yes—
87C22×C4×C61C22×C412yes—
88C22×C4×C61C22×C412yes—
89C22×C4×C61C22×C411yescharacteristic
90C22×C4×C61C2×D414yes—
91C22×C4×C61C2×D414yes—
92C22×C4×C61C2×D412yes—
93C22×C4×C61C2×Q814yes—
94C22×C4×C61C2×Q812yes—
95C22×C4×C61#4114yes—
96C22×C4×C61C2411yescharacteristic
97C4⋊C4×C61—28no—
98C4⋊C4×C61C22⋊C418yes—
99C22⋊C4×C61—28no—
100C22⋊C4×C61C22⋊C418yes—
101C42×C61—28no—
102C42×C61—24no—
103C42×C61—22no—
104C42×C61—22no—
105C42×C61C2×D418yes—
106#11758194—48no—
107C2×#92418—24no—
108C2×#92418—24no—
109C2×#92418—24no—
110C2×#92418—24no—
111C2×#92418—24no—
112C2×#92418—24no—
113C2×#92418—48no—
114C23×C61—22no—
115C23×C61—22no—
116C23×C61—22no—
117C23×C61—22no—
118C23×C61—24no—
119C23×C61—24no—
120C23×C61C2×C22⋊C412yes—
121C23×C61C2×C4⋊C414yes—
122C23×C61C2×C4⋊C412yes—
123C23×C61#11712yes—
124C23×C61C23×C411yescharacteristic
125Q8×C61—28no—
126D4×C61—216no—
127D4×C61—28no—
128C2×C4×C61—22no—
129C2×C4×C61—22no—
130C2×C4×C61—22no—
131C2×C4×C61—24no—
132C2×C4×C61—22no—
133C2×C4×C61—24no—
134C2×C4×C61—28no—
135C2×C4×C61—28no—
136C2×C4×C61—22no—
137C2×C4×C61—22no—
138C2×C4×C61—22no—
139C2×C4×C61—416no—
140C2×C4×C61—48no—
141C2×C4×C61C2×C22⋊C418yes—
142C2×C4×C61C2×C22⋊C412yes—
143C2×C4×C61C2×C4⋊C414yes—
144C2×C4×C61C2×C4⋊C412yes—
145C2×C4×C61#11712yes—
146C2×C4×C61#12118yes—
147C2×C4×C61C22⋊Q818yes—
148C2×C4×C61#13114yes—
149C2×C4×C61C22×D411yescharacteristic
150C2×C4×C61C22×D412yes—
151C2×C4×C61C22×Q811yescharacteristic
152C2×C4×C61C2×#4112yes—
153#92418—44no—
154#92418—44no—
155#20630—611nomaximal, Sylow
156C22×C61—24no—
157C22×C61—24no—
158C22×C61—22no—
159C22×C61—22no—
160C22×C61—28no—
161C22×C61—24no—
162C22×C61—44no—
163C22×C61—42no—
164C22×C61—42no—
165C22×C61#38014yes—
166C22×C61C22×C22⋊C411yescharacteristic
167C22×C61C22×C4⋊C411yescharacteristic
168C22×C61C2×#11711yescharacteristic
169C22×C61C2×#13112yes—
170C22×C61#56812yes—
171C4×C61—24no—
172C4×C61—28no—
173C4×C61—24no—
174C4×C61—42no—
175C4×C61#38014yes—
176C4×C61C2×#12112yes—
177C4×C61C2×C22⋊Q812yes—
178C2×C4×#41D6111yescharacteristic
179C2×#424—612no—
180C2×#408—612no—
181#1798—614no—
182#1797—614no—
183#1728—611no—
184C2×#336—611no—
185C2×C61—24no—
186C2×C61—44no—
187C2×C61C2×#38011yescharacteristic
188C2×C61#288012yes—
189C2×C61#290612yes—
190C2×C61#292211yescharacteristic
191C2×C61#292311yescharacteristic
192C22×#41C2×D6111yescharacteristic
193C22×M16—612no—
194C23×C8—612no—
195C4×#41—28no—
196C2×C4×Q8Dic6112yes—
197C2×C4×D4Dic6112yes—
198C2×C4×D4Dic6114yes—
199C2×#117C2×D6112yes—
200C2×#117C2×D6111yescharacteristic
201C22×C42C2×D6112yes—
202C22×C42C2×D6111yescharacteristic
203#424—1228no—
204C2×C4⋊C8—618no—
205#408—1228no—
206C2×C22⋊C8—618no—
207#336—612no—
208#336—612no—
209#336—612no—
210#336—612no—
211C61#2063011yescharacteristic, complemented, Sylow
212C2×#41—24no—
213C2×#41—24no—
214C2×#41Dic12214yes—
215C2×#41D24414yes—
216C22×Q8C2×Dic6111yescharacteristic
217C22×D4C2×Dic6112yes—
218C22×D4C2×Dic6111yescharacteristic
219C23×C4C2×Dic6112yes—
220C23×C4C2×Dic6114yes—
221C23×C4C22×D6112yes—
222C23×C4C22×D6111yescharacteristic
223C2×M16—1224no—
224C2×M16—1224no—
225C2×M16—1224no—
226C22×C8—614no—
227C22×C8—614no—
228C22×C8—1224no—
229C22×C8—1224no—
230C22×C8—1224no—
231C4×Q8—28no—
232C4×D4—28no—
233C4×D4—216no—
234#117—28no—
235#117C22⋊D6118yes—
236C2×C4⋊C4C2×Dic6114yes—
237C2×C4⋊C4C2×Dic6112yes—
238C2×C22⋊C4C2×Dic6114yes—
239C2×C22⋊C4C2×Dic6112yes—
240C2×C42—24no—
241C2×C42—28no—
242C2×C42C2×Dic6112yes—
243C2×C42C2×Dic6114yes—
244C2×C42C22×D6111yescharacteristic
245C2×C42C22×D6111yescharacteristic
246C2×C42C22×D6112yes—
247C4⋊C8—12216no—
248C22⋊C8—12216no—
249C24C22×Dic6112yes—
250C24C22×Dic6111yescharacteristic
251#41—216no—
252#41—216no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8C4⋊Dic6114yes—
256C2×D4—24no—
257C2×D4—24no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4C4⋊Dic6118yes—
261C2×D4C4⋊Dic6114yes—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—28no—
266C22×C4—28no—
267C22×C4—24no—
268C22×C4—44no—
269C22×C4—42no—
270C22×C4C2×Dic12214yes—
271C22×C4C2×Dic12212yes—
272C22×C4C2×D24414yes—
273C22×C4C2×D24412yes—
274C22×C4#1175822214yes—
275C22×C4C22×Dic6112yes—
276C22×C4C22×Dic6114yes—
277C22×C4C22×Dic6112yes—
278C22×C4C22×Dic6112yes—
279C22×C4C22×Dic6111yescharacteristic
280C22×C4C2×C22⋊D6114yes—
281C22×C4C23×D6111yescharacteristic
282M16—2448no—
283C2×C8—1224no—
284C2×C8—1224no—
285C2×C8—1224no—
286C2×C8—1224no—
287C2×C8—1224no—
288C2×C8—1224no—
289C2×C8—2448no—
290C4⋊C4—28no—
291C4⋊C4C22⋊Dic6118yes—
292C22⋊C4—28no—
293C22⋊C4C22⋊Dic6118yes—
294C42—22no—
295C42—22no—
296C42—28no—
297C42—24no—
298C42C2×C22⋊D6118yes—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—22no—
303C23—24no—
304C23—24no—
305C23C2×C22⋊Dic6112yes—
306C23C2×C4⋊Dic6114yes—
307C23C2×C4⋊Dic6112yes—
308C23#1451973612yes—
309C23C23×Dic6111yescharacteristic
310Q8—28no—
311D4—216no—
312D4—28no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—24no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—28no—
320C2×C4—28no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—416no—
325C2×C4—48no—
326C2×C4C2×C22⋊Dic6118yes—
327C2×C4C2×C22⋊Dic6112yes—
328C2×C4C2×C4⋊Dic6114yes—
329C2×C4C2×C4⋊Dic6112yes—
330C2×C4#1451973612yes—
331C2×C4#1451975118yes—
332C2×C4#1451975618yes—
333C2×C4#1451978414yes—
334C2×C4C22×C22⋊D6112yes—
335C2×C4C22×D24411yescharacteristic
336C2×C4C22×Dic12211yescharacteristic
337C2×C4C2×#1175822212yes—
338C8—2444no—
339C8—2444no—
340C22—24no—
341C22—22no—
342C22—22no—
343C22—24no—
344C22—24no—
345C22—28no—
346C22—42no—
347C22—42no—
348C22—44no—
349C22#1683962214yes—
350C22C22×C22⋊Dic6111yescharacteristic
351C22C22×C4⋊Dic6111yescharacteristic
352C22C2×#1451973611yescharacteristic
353C22C2×#1451978412yes—
354C22#1684052912yes—
355C4—24no—
356C4—24no—
357C4—28no—
358C4—42no—
359C4#1683962214yes—
360C4C2×#1451975112yes—
361C4C2×#1451975612yes—
362C2—24no—
363C2—44no—
364C2C2×#1683962211yescharacteristic
365C2#2145426812yes—
366C2#2145438712yes—
367C2#2145446311yescharacteristic
368C2#2145447011yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D61
commutator, socleC23×C61C23×C4
FittingC2×C4×#41×C61C2
prime radicalC22×#41×C61C22
cosocleC22×C4×C61C24
hypercenterC2×C4×#41D61
hypocenterC61#20630
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16839789C2semidirect
3. #21445803C2semidirect
4. #21445799C2semidirect
5. #21445513C2semidirect
6. C2×#16839517C2semidirect
211. C61#20630semidirect

Maximal subgroups

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

subgroupindex
C2×#168397892
C2×#168397292
#214458032
#214457992
#214455132
C2×#168395172
C2×C4×#41×C612
#2063061