Finite Group Databasebeta

← #29568196

Subgroups — #29568196

Order 15616.

550 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#21445799C211yesmaximal, characteristic
2#21445513C211yesmaximal, characteristic
3#21443645C212yesmaximal, complemented
4#21443642C212yesmaximal, complemented
5C4×C22⋊Q8×C61C211yesmaximal, characteristic
6C22×#11758194—21no—
7C2×C4⋊#92418—21no—
8C2×C22⋊#92418—21no—
9#16839517—21no—
10#16839517C2212yes—
11#16839517C2212yes—
12C2×C22⋊Q8×C61C411yescharacteristic
13C2×C4×Q8×C61C411yescharacteristic
14C22×C42×C61C2211yescharacteristic
15C61×#391C2211yescharacteristic
16C61×#389C2211yescharacteristic
17C61×#387—21no—
18C61×#385—21no—
19C61×#380C411yescharacteristic
20C4×C4⋊C4×C61—21no—
21C4×C4⋊C4×C61C411yescharacteristic
22C4×C22⋊C4×C61—21no—
23C2×#11758194—24no—
24C2×#11758194—42no—
25C2×#11758194—42no—
26C22×#92418—22no—
27C22×#92418—22no—
28C4⋊#92418—42no—
29C22⋊#92418—42no—
30C22×Q8×C61C2×C411yescharacteristic
31C23×C4×C61C2×C411yescharacteristic
32C23×C4×C61D411yescharacteristic
33C23×C4×C61D411yescharacteristic
34C22⋊Q8×C61—22no—
35C22⋊Q8×C61—22no—
36C4×Q8×C61—42no—
37C2×C4⋊C4×C61—21no—
38C2×C4⋊C4×C61—21no—
39C2×C4⋊C4×C61—21no—
40C2×C4⋊C4×C61C2×C411yescharacteristic
41C2×C4⋊C4×C61C2×C411yescharacteristic
42C2×C4⋊C4×C61C2×C411yescharacteristic
43C2×C22⋊C4×C61—21no—
44C2×C22⋊C4×C61—21no—
45C2×C42×C61—21no—
46C2×C42×C61—21no—
47C2×C42×C61—42no—
48C2×C42×C61C2×C411yescharacteristic
49C2×C42×C61D411yescharacteristic
50C2×C42×C61D411yescharacteristic
51C2×C42×C61C2311yescharacteristic
52#95×C61—21no—
53#95×C61—21no—
54#95×C61—21no—
55#95×C61—21no—
56#95×C61—21no—
57C24×C61C22⋊C411yescharacteristic
58C2×Q8×C61—21no—
59C2×Q8×C61—41no—
60C2×Q8×C61—41no—
61C2×Q8×C61M1612yescomplemented
62C22×C4×C61—22no—
63C22×C4×C61—22no—
64C22×C4×C61—21no—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—21no—
72C22×C4×C61—41no—
73C22×C4×C61—41no—
74C22×C4×C61—41no—
75C22×C4×C61—41no—
76C22×C4×C61C22⋊C411yescharacteristic
77C22×C4×C61C22⋊C411yescharacteristic
78C22×C4×C61C22⋊C411yescharacteristic
79C22×C4×C61SD1612yes—
80C22×C4×C61Q1612yes—
81C22×C4×C61C22×C411yescharacteristic
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C4⋊C4×C61—24no—
85C4⋊C4×C61—21no—
86C4⋊C4×C61—42no—
87C4⋊C4×C61—41no—
88C4⋊C4×C61—41no—
89C4⋊C4×C61—41no—
90C4⋊C4×C61—41no—
91C4⋊C4×C61M1612yescomplemented
92C22⋊C4×C61—24no—
93C22⋊C4×C61—42no—
94C42×C61—22no—
95C42×C61—22no—
96C42×C61—41no—
97C42×C61—41no—
98C42×C61—42no—
99C42×C61—42no—
100#11758194—48no—
101C2×#92418—24no—
102C2×#92418—42no—
103C2×#92418—42no—
104C2×#92418—42no—
105C2×#92418—42no—
106C2×#92418—42no—
107C23×C61—21no—
108C23×C61—21no—
109C23×C61—21no—
110C23×C61—21no—
111C23×C61—21no—
112C23×C61—21no—
113C23×C61#10312yes—
114C23×C61C2×C22⋊C411yescharacteristic
115Q8×C61—42no—
116Q8×C61—42no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—24no—
124C2×C4×C61—24no—
125C2×C4×C61—22no—
126C2×C4×C61—22no—
127C2×C4×C61—21no—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
130C2×C4×C61—21no—
131C2×C4×C61—21no—
132C2×C4×C61—21no—
133C2×C4×C61—42no—
134C2×C4×C61—42no—
135C2×C4×C61—42no—
136C2×C4×C61—42no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61#9912yes—
148C2×C4×C61#10312yes—
149C2×C4×C61#10412yes—
150C2×C4×C61C2×M1611yescharacteristic
151C2×C4×C61C2×M1611yescharacteristic
152C2×C4×C61C2×SD1611yescharacteristic
153C2×C4×C61C2×Q1611yescharacteristic
154C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C61#13711yescharacteristic
156#92418—44no—
157#92418—82no—
158#7617—611nomaximal, Sylow
159C22×C61—22no—
160C22×C61—22no—
161C22×C61—22no—
162C22×C61—41no—
163C22×C61—41no—
164C22×C61—41no—
165C22×C61—41no—
166C22×C61#40711yescharacteristic
167C22×C61C2×#9911yescharacteristic
168C22×C61#41311yescharacteristic
169C22×C61C2×#10311yescharacteristic
170C22×C61#41711yescharacteristic
171C22×C61C2×#10411yescharacteristic
172C22×C61#42111yescharacteristic
173C4×C61—21no—
174C4×C61—21no—
175C4×C61—21no—
176C4×C61—21no—
177C4×C61—21no—
178C4×C61—21no—
179C4×C61—41no—
180C4×C61—42no—
181C4×C61—42no—
182C4×C61—41no—
183C4×C61—81no—
184C4×C61—81no—
185C4×C61#32814yes—
186C4×C22⋊Q8D6111yescharacteristic
187#1797—611no—
188#1728—611no—
189#1272—612no—
190#1271—612no—
191C2×C61—42no—
192C2×C61#146511yescharacteristic
193C2×C61#148711yescharacteristic
194C2×C61#148911yescharacteristic
195C2×C61C2×#32811yescharacteristic
196C2×C61#151311yescharacteristic
197C2×C61#152911yescharacteristic
198C2×C61#155011yescharacteristic
199C22×M16—1221no—
200C2×C22⋊Q8Dic6111yescharacteristic
201C2×C4×Q8Dic6111yescharacteristic
202C22×C42C2×D6111yescharacteristic
203C2×C4⋊C8—1221no—
204C2×C22⋊C8—1221no—
205#391C2×D6111yescharacteristic
206#389C2×D6111yescharacteristic
207#387—21no—
208#385—21no—
209#380Dic6111yescharacteristic
210C4×C4⋊C4—21no—
211C4×C4⋊C4Dic6111yescharacteristic
212C4×C22⋊C4—21no—
213#336—612no—
214#336—612no—
215#336—1221no—
216C61#761711yescharacteristic, complemented, Sylow
217C22×Q8C2×Dic6111yescharacteristic
218C23×C4C2×Dic6111yescharacteristic
219C23×C4C22⋊D6111yescharacteristic
220C23×C4C22⋊D6111yescharacteristic
221C2×M16—1224no—
222C2×M16—2442no—
223C2×M16—2442no—
224C22×C8—1222no—
225C22×C8—1222no—
226C22⋊Q8—22no—
227C22⋊Q8—22no—
228C4×Q8—42no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4C2×Dic6111yescharacteristic
233C2×C4⋊C4C2×Dic6111yescharacteristic
234C2×C4⋊C4C2×Dic6111yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—42no—
240C2×C42C2×Dic6111yescharacteristic
241C2×C42C22⋊D6111yescharacteristic
242C2×C42C22⋊D6111yescharacteristic
243C2×C42C22×D6111yescharacteristic
244C4⋊C8—2442no—
245C22⋊C8—2442no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251C24C22⋊Dic6111yescharacteristic
252C2×Q8—21no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8#1175819412yescomplemented
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4#1175820112yes—
271C22×C4#1175820212yes—
272C22×C4C22⋊Dic6111yescharacteristic
273C22×C4C22⋊Dic6111yescharacteristic
274C22×C4C22⋊Dic6111yescharacteristic
275C22×C4C22×Dic6111yescharacteristic
276C22×C4C2×C22⋊D6111yescharacteristic
277C22×C4C2×C22⋊D6111yescharacteristic
278M16—2448no—
279C2×C8—1224no—
280C2×C8—2442no—
281C2×C8—2442no—
282C2×C8—2442no—
283C2×C8—2442no—
284C2×C8—2442no—
285C4⋊C4—24no—
286C4⋊C4—21no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—42no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4#1175819412yescomplemented
293C22⋊C4—24no—
294C22⋊C4—42no—
295C42—22no—
296C42—22no—
297C42—41no—
298C42—41no—
299C42—42no—
300C42—42no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23#1451970112yes—
308C23C2×C22⋊Dic6111yescharacteristic
309Q8—42no—
310Q8—42no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4#1451969212yes—
342C2×C4#1451970112yes—
343C2×C4#1451970412yes—
344C2×C4C2×#1175819411yescharacteristic
345C2×C4C2×#1175819411yescharacteristic
346C2×C4C2×#1175820111yescharacteristic
347C2×C4C2×#1175820211yescharacteristic
348C2×C4#1451980211yescharacteristic
349C2×C4#1451980711yescharacteristic
350C8—4882no—
351C8—2444no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22#1683972611yescharacteristic
360C22C2×#1451969211yescharacteristic
361C22#1683974611yescharacteristic
362C22C2×#1451970111yescharacteristic
363C22#1683976311yescharacteristic
364C22C2×#1451970411yescharacteristic
365C22#1683977911yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—41no—
373C4—41no—
374C4—42no—
375C4—42no—
376C4—81no—
377C4—81no—
378C4#1683949914yes—
379C2—42no—
380C2#2144415711yescharacteristic
381C2#2144424811yescharacteristic
382C2#2144425811yescharacteristic
383C2C2×#1683949911yescharacteristic
384C2#2144436211yescharacteristic
385C2#2144442611yescharacteristic
386C2#2144450911yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic61
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC4×C22⋊Q8×C61C2
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C22⋊C4
cosocleC2×C42×C61C23
hypercenterC4×C22⋊Q8D61
hypocenterC61#7617
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #21443645C2semidirect
4. #21443642C2semidirect
61. C2×Q8×C61M16semidirect
91. C4⋊C4×C61M16semidirect
216. C61#7617semidirect
255. C2×Q8#11758194semidirect
292. C4⋊C4#11758194semidirect

Maximal subgroups

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

subgroupindex
#214457992
#214455132
#214436452
#214436422
C4×C22⋊Q8×C612
#761761