Finite Group Databasebeta

← #29708128

Subgroups — #29708128

Order 15616.

1074 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454408C212yesmaximal
2#21448443C211yesmaximal, characteristic
3#21447166C214yesmaximal, complemented
4#21447161C214yesmaximal, complemented
5C2×#16839531C211yesmaximal, characteristic
6#21445528C212yesmaximal
7C2×C61×#544C211yesmaximal, characteristic
8C22×#11758194C2212yes—
9C2×#14519736C2211yescharacteristic
10C2×#14519736C2211yescharacteristic
11#16839805C2214yes—
12#16839805C2214yes—
13#16839802—22no—
14#16839802—22no—
15#16839798—24no—
16#16839658—22no—
17#16839635—22no—
18#16839631C2212yes—
19#16839627C2212yes—
20#16839531C2214yes—
21#16839521C2218yes—
22C61×#544—24no—
23C61×#544—22no—
24C61×#544C2214yes—
25C2×C4⋊Q8×C61—22no—
26C2×#125×C61—22no—
27C2×C22⋊Q8×C61—22no—
28C2×#117×C61C2212yes—
29C22×C4⋊C4×C61C2211yescharacteristic
30C2×#11758194—22no—
31C2×#11758194—24no—
32C2×#11758194Q818yescomplemented
33C2×#11758194C2314yes—
34C22×#92418—22no—
35#14519736—22no—
36#14519736—22no—
37#14519736—22no—
38#14519736Q814yescomplemented
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61C2311yescharacteristic
41C2×C4⋊Dic61C2311yescharacteristic
42C2×#11758196—24no—
43C2×C22⋊Dic61—21no—
44C2×C22⋊Dic61—21no—
45C2×C4×Dic61—21no—
46C2×C4×Dic61C2312yes—
47#14519710—24no—
48#14519710—24no—
49#14519708—24no—
50#14519708—24no—
51#14519684—24no—
52C22×Q8×C61—22no—
53C23×C4×C61D412yes—
54C23×C4×C61C2311yescharacteristic
55C4⋊Q8×C61—28no—
56C4⋊Q8×C61—28no—
57#125×C61—28no—
58#125×C61—28no—
59C22⋊Q8×C61—28no—
60C22⋊Q8×C61—28no—
61#117×C61—22no—
62#117×C61—22no—
63#117×C61D414yes—
64#117×C61C2314yes—
65C2×C4⋊C4×C61—24no—
66C2×C4⋊C4×C61—22no—
67C2×C4⋊C4×C61—22no—
68C2×C4⋊C4×C61—22no—
69C2×C4⋊C4×C61D412yes—
70C2×C4⋊C4×C61D412yes—
71C2×C4⋊C4×C61D412yes—
72C2×C4⋊C4×C61C2312yes—
73C2×C22⋊C4×C61—22no—
74C2×C42×C61—22no—
75C24×C61C2×D411yescharacteristic
76C2×Q8×C61—28no—
77C2×Q8×C61—84no—
78C22×C4×C61—22no—
79C22×C4×C61—22no—
80C22×C4×C61—22no—
81C22×C4×C61—22no—
82C22×C4×C61—82no—
83C22×C4×C61C2×D414yes—
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61C2×D412yes—
86C22×C4×C61C2×D412yes—
87C22×C4×C61C2×Q814yes—
88C22×C4×C61C2×Q812yes—
89C22×C4×C61#4112yes—
90C22×C4×C61C2411yescharacteristic
91C22×Dic61—22no—
92C22×Dic61—21no—
93C22×Dic61—21no—
94C4⋊C4×C61—28no—
95C4⋊C4×C61—216no—
96C4⋊C4×C61—24no—
97C4⋊C4×C61—22no—
98C4⋊C4×C61—22no—
99C4⋊C4×C61—24no—
100C4⋊C4×C61C2×D414yes—
101C4⋊C4×C61C2×D414yes—
102C22⋊C4×C61—24no—
103C22⋊C4×C61—24no—
104C42×C61—24no—
105C42×C61—24no—
106C22⋊Dic61—24no—
107C22⋊Dic61—41no—
108C22⋊Dic61—41no—
109C4⋊Dic61—24no—
110C4⋊Dic61—22no—
111C4⋊Dic61—22no—
112#11758196—84no—
113C4×Dic61—22no—
114C4×Dic61—42no—
115C4×Dic61C2×Q814yescomplemented
116#11758194—28no—
117#11758194—24no—
118#11758194C2×Q818yescomplemented
119C2×#92418—28no—
120C2×#92418—24no—
121C23×C61—41no—
122C23×C61—41no—
123C23×C61#12011yescharacteristic
124C23×C61#12011yescharacteristic
125C23×C61C22⋊Q812yes—
126C23×C61C4⋊Q812yes—
127C23×C61C22×D411yescharacteristic
128Q8×C61—88no—
129C2×C4×C61—24no—
130C2×C4×C61—24no—
131C2×C4×C61—24no—
132C2×C4×C61—44no—
133C2×C4×C61—44no—
134C2×C4×C61—44no—
135C2×C4×C61—44no—
136C2×C4×C61—82no—
137C2×C4×C61#12012yes—
138C2×C4×C61C22⋊Q814yes—
139C2×C4×C61C22⋊Q814yes—
140C2×C4×C61C22⋊Q812yes—
141C2×C4×C61C4⋊Q814yes—
142C2×C4×C61C4⋊Q814yes—
143C2×C4×C61C4⋊Q812yes—
144C2×C4×C61C22×D412yes—
145C2×C4×C61C22×Q812yes—
146C2×C4×C61C22×Q811yescharacteristic
147C2×C4×C61C2×#4111yescharacteristic
148C2×Dic61—24no—
149C2×Dic61—42no—
150C2×Dic61—42no—
151C2×Dic61—42no—
152C2×Dic61—42no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155#92418—28no—
156#23561—611nomaximal, Sylow
157C22×C61—21no—
158C22×C61—21no—
159C22×C61—21no—
160C22×C61—21no—
161C22×C61—21no—
162C22×C61—21no—
163C22×C61—42no—
164C22×C61—42no—
165C22×C61#39512yes—
166C22×C61#39512yes—
167C22×C61C2×#12011yescharacteristic
168C22×C61C2×C22⋊Q811yescharacteristic
169C22×C61C2×C4⋊Q811yescharacteristic
170C4×C61—44no—
171C4×C61—44no—
172C4×C61—44no—
173C4×C61#39514yes—
174C4×C61C2×C22⋊Q812yes—
175C4×C61C2×C4⋊Q812yes—
176Dic61—44no—
177Dic61—81no—
178Dic61—81no—
179C2×#544D6111yescharacteristic
180#2911—612no—
181#2317—611no—
182#2061—614no—
183#2060—614no—
184C2×#343—611no—
185#1733—612no—
186C2×C61—21no—
187C2×C61—21no—
188C2×C61—22no—
189C2×C61—22no—
190C2×C61C2×#39511yescharacteristic
191C2×C61#301911yescharacteristic
192C2×C61#302011yescharacteristic
193C22×M16—612no—
194#544—24no—
195#544—22no—
196#544C2×D6114yes—
197C2×C4⋊Q8—22no—
198C2×#125—22no—
199C2×C22⋊Q8—22no—
200C2×#117—611no—
201C2×#117—611no—
202C2×#117C2×D6112yes—
203C22×C4⋊C4C2×D6111yescharacteristic
204#428—614no—
205#428—614no—
206#428—1222no—
207#428—1222no—
208#427—1224no—
209#389—1222no—
210#385—1222no—
211#384—612no—
212#382—612no—
213#343—614no—
214#339—618no—
215C61#2356111yescharacteristic, complemented, Sylow
216C22×Q8—22no—
217C23×C4C22⋊D6112yes—
218C23×C4C22×D6111yescharacteristic
219C2×M16—618no—
220C2×M16—614no—
221C2×M16—1224no—
222C2×M16—1222no—
223C22×C8—1222no—
224C4⋊Q8—28no—
225C4⋊Q8—28no—
226#125—28no—
227#125—28no—
228C22⋊Q8—28no—
229C22⋊Q8—28no—
230#117—22no—
231#117—22no—
232#117—614no—
233#117—1222no—
234#117—1222no—
235#117—1222no—
236#117C22⋊D6114yes—
237#117C22×D6114yes—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—611no—
243C2×C4⋊C4—611no—
244C2×C4⋊C4—1221no—
245C2×C4⋊C4—1224no—
246C2×C4⋊C4C22⋊D6112yes—
247C2×C4⋊C4C22⋊D6112yes—
248C2×C4⋊C4C22⋊D6112yes—
249C2×C4⋊C4C22×D6112yes—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—1221no—
252C2×C22⋊C4—1221no—
253C2×C42—22no—
254C2×C42—612no—
255C2×C42—1221no—
256#107—1224no—
257#107—1224no—
258#106—1224no—
259#106—1224no—
260#95—1224no—
261C24C2×C22⋊D6111yescharacteristic
262C2×Q8—28no—
263C2×Q8—84no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—82no—
269C22×C4—1222no—
270C22×C4—1221no—
271C22×C4—1221no—
272C22×C4Q8×D6114yes—
273C22×C4Q8×D6112yes—
274C22×C4#1175822612yes—
275C22×C4C2×C22⋊D6114yes—
276C22×C4C2×C22⋊D6112yes—
277C22×C4C2×C22⋊D6112yes—
278C22×C4C2×C22⋊D6111yescharacteristic
279C22×C4C23×D6111yescharacteristic
280M16—618no—
281M16—1224no—
282M16—1228no—
283C2×C8—1224no—
284C2×C8—1228no—
285C4⋊C4—28no—
286C4⋊C4—216no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—1224no—
292C4⋊C4—1222no—
293C4⋊C4—1222no—
294C4⋊C4—4884no—
295C4⋊C4C2×C22⋊D6114yes—
296C4⋊C4C2×C22⋊D6114yes—
297C22⋊C4—24no—
298C22⋊C4—24no—
299C22⋊C4—1224no—
300C22⋊C4—2441no—
301C22⋊C4—2441no—
302C42—24no—
303C42—24no—
304C42—614no—
305C42—1222no—
306C42—2442no—
307C23—41no—
308C23—41no—
309C23#1451974611yescharacteristic
310C23#1451974611yescharacteristic
311C23#1451975312yes—
312C23#1451977512yes—
313C23C22×C22⋊D6111yescharacteristic
314Q8—88no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—82no—
323C2×C4—1224no—
324C2×C4—2442no—
325C2×C4—2442no—
326C2×C4—2441no—
327C2×C4—2441no—
328C2×C4—2442no—
329C2×C4—2442no—
330C2×C4#1451974612yes—
331C2×C4#1451975312yes—
332C2×C4#1451975314yes—
333C2×C4#1451975314yes—
334C2×C4#1451977512yes—
335C2×C4#1451977514yes—
336C2×C4#1451977514yes—
337C2×C4C22×C22⋊D6112yes—
338C2×C4C2×Q8×D6112yes—
339C2×C4C2×Q8×D6111yescharacteristic
340C2×C4C2×#1175822611yescharacteristic
341C8—1228no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—42no—
349C22—42no—
350C22#1683968112yes—
351C22#1683968112yes—
352C22C2×#1451974611yescharacteristic
353C22C2×#1451975311yescharacteristic
354C22C2×#1451977511yescharacteristic
355C4—44no—
356C4—44no—
357C4—44no—
358C4—4881no—
359C4—4881no—
360C4—2444no—
361C4#1683968114yes—
362C4C2×#1451975312yes—
363C4C2×#1451977512yes—
364C2—21no—
365C2—22no—
366C2—22no—
367C2—21no—
368C2C2×#1683968111yescharacteristic
369C2#2145517611yescharacteristic
370C2#2145518311yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#544C2
socleC22×C61C2×#120
prime radicalC24×C61C2×D4
hypercenterC2×#544D61
hypocenterC61#23561
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
3. #21447166C2semidirect
4. #21447161C2semidirect
32. C2×#11758194Q8semidirect
38. #14519736Q8semidirect
115. C4×Dic61C2×Q8semidirect
118. #11758194C2×Q8semidirect
215. C61#23561semidirect

Maximal subgroups

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

subgroupindex
#214544082
#214484432
#214471662
#214471612
C2×#168395312
#214455282
C2×C61×#5442
#2356161