Finite Group Databasebeta

← #29736751

Subgroups — #29736751

Order 15616.

1022 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1022 conjugacy classes of subgroups — too many to draw. Only its 191 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459622C211yesmaximal, characteristic
2#21456200C212yesmaximal, complemented
3#21450811C211yesmaximal, characteristic
4#21446981C214yesmaximal, complemented
5#21446763C212yesmaximal, complemented
6#21446205C214yesmaximal
7C4×#100×C61C211yesmaximal, characteristic
8#16840471—24no—
9#16840469—22no—
10D4×Dic122—22no—
11#16840393C2212yes—
12#16840388—22no—
13#16840319C2212yes—
14C2×#14519777C2211yescharacteristic
15C2×#14519764C2212yes—
16C2×#14519756—22no—
17#16840237C2212yes—
18#16840131—24no—
19#16840113—22no—
20#16840113—22no—
21#16839772C2218yes—
22#16839687—22no—
23#16839667C2212yes—
24#16839622—22no—
25C4×C4⋊Dic61C2211yescharacteristic
26#16839523C2218yes—
27C2×C4×D4×C61C2211yescharacteristic
28C2×#100×C61C2212yes—
29C4×M16×C61C2212yes—
30C61×#343C2212yes—
31C22×Dic122—22no—
32C22×Dic122C2312yes—
33#14519777—22no—
34#14519777—22no—
35#14519777D412yes—
36#14519777D412yes—
37#14519776—24no—
38#14519769D414yes—
39#14519768—24no—
40#14519764—28no—
41#14519756—24no—
42#14519756—24no—
43#14519756—22no—
44#14519756—24no—
45#14519756—22no—
46#14519754—28no—
47#14519754—28no—
48C4×Dic122—22no—
49C4×Dic122—22no—
50D4×Dic61—24no—
51D4×Dic61—24no—
52#14519736C2314yes—
53C2×C4⋊Dic61—22no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—21no—
57C2×#11758196—24no—
58C2×C22⋊Dic61—24no—
59C2×C22⋊Dic61—24no—
60C2×C4×Dic61C2312yes—
61#14519702—28no—
62#14519702—28no—
63#14519684—21no—
64#14519684—24no—
65C22×D4×C61C2311yescharacteristic
66C23×C4×C61D412yes—
67C2×M16×C61C2314yes—
68C4×D4×C61—22no—
69C4×D4×C61—22no—
70C2×C4⋊C4×C61C2311yescharacteristic
71C2×C22⋊C4×C61D412yes—
72C2×C42×C61C2311yescharacteristic
73#100×C61—24no—
74#97×C61—22no—
75C4×C8×C61—22no—
76C24×C61C2×D412yes—
77C2×D4×C61—22no—
78C2×D4×C61—22no—
79C2×D4×C61#4114yes—
80C22×C4×C61—22no—
81C22×C4×C61—22no—
82C22×C4×C61—44no—
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D412yes—
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2411yescharacteristic
87C22×Dic61—24no—
88C22×Dic61—24no—
89C22×Dic61—22no—
90C22×Dic61—22no—
91C2×Dic122—22no—
92C2×Dic122—24no—
93C2×Dic122—24no—
94C2×Dic122—24no—
95C2×Dic122—24no—
96C2×Dic122—22no—
97C2×Dic122—84no—
98M16×C61—24no—
99M16×C61—24no—
100C2×C8×C61—24no—
101C4⋊C4×C61—22no—
102C22⋊C4×C61—24no—
103C42×C61C2×D412yes—
104C42×C61C2×D412yes—
105C22⋊Dic61—28no—
106C22⋊Dic61—24no—
107C22⋊Dic61—48no—
108C4⋊Dic61—24no—
109C4⋊Dic61—22no—
110C4⋊Dic61—22no—
111C4⋊Dic61—42no—
112C4⋊Dic61—41no—
113C4⋊Dic61—42no—
114C4⋊Dic61—41no—
115C4⋊Dic61C2×D414yes—
116C4⋊Dic61C2×D414yes—
117#11758196—28no—
118#11758196—24no—
119#11758196—24no—
120C4×Dic61—24no—
121C4×Dic61—24no—
122C23×C61—22no—
123C23×C61—22no—
124C23×C61—42no—
125C23×C61—42no—
126C23×C61#12414yes—
127C23×C61C22×D411yescharacteristic
128D4×C61—28no—
129C2×C4×C61—22no—
130C2×C4×C61—22no—
131C2×C4×C61—41no—
132C2×C4×C61—44no—
133C2×C4×C61—42no—
134C2×C4×C61—82no—
135C2×C4×C61#12011yescharacteristic
136C2×C4×C61#12012yes—
137C2×C4×C61#12011yescharacteristic
138C2×C4×C61C22×D412yes—
139C2×C4×C61C2×#4112yes—
140C2×C4×C612−1+412yes—
141C2×Dic61—24no—
142C2×Dic61—24no—
143C2×Dic61—42no—
144C2×Dic61—42no—
145C2×Dic61—42no—
146C2×Dic61—42no—
147C2×Dic61—42no—
148C2×Dic61—42no—
149C2×Dic61—84no—
150C2×Dic61—84no—
151Dic122—44no—
152Dic122—48no—
153Dic122—82no—
154Dic122—84no—
155C8×C61—44no—
156#25917—611nomaximal, Sylow
157C22×C61—21no—
158C22×C61—21no—
159C22×C61—24no—
160C22×C61—44no—
161C22×C61—42no—
162C22×C61—42no—
163C22×C61—82no—
164C22×C61C2×#12011yescharacteristic
165C22×C61C2×#12411yescharacteristic
166C22×C61#53611yescharacteristic
167C4×C61—21no—
168C4×C61—22no—
169C4×C61—42no—
170C4×C61#45512yes—
171C4×C61#54814yes—
172Dic61—44no—
173Dic61—44no—
174Dic61—82no—
175Dic61—82no—
176#3501—611no—
177#3139—612no—
178#2547—611no—
179#2028—614no—
180#1986—612no—
181#1885—614no—
182C4×#100D6111yescharacteristic
183C2×C61—21no—
184C2×C61—21no—
185C2×C61—44no—
186C2×C61#243611yescharacteristic
187C2×C61C2×#45511yescharacteristic
188C2×C61#302211yescharacteristic
189#554—1224no—
190#554—1222no—
191D4×Q8—1222no—
192#544—612no—
193#543—1222no—
194#533—612no—
195C2×C4⋊Q8—611no—
196C2×#124—612no—
197C2×C22⋊Q8—1222no—
198#518—612no—
199C2×C4×D4C2×D6111yescharacteristic
200#491—1224no—
201#487—1222no—
202#487—1222no—
203#419—618no—
204C2×#100C2×D6112yes—
205C4×M16C2×D6112yes—
206#397—1222no—
207#391—612no—
208#380—1222no—
209C4×C4⋊C4—611no—
210#343C2×D6112yes—
211#339—618no—
212C61#2591711yescharacteristic, complemented, Sylow
213C22×Q8—612no—
214C22×Q8—1222no—
215C22×D4C22×D6111yescharacteristic
216C23×C4D24412yes—
217C2×M16C22×D6114yes—
218C4⋊Q8—612no—
219C4⋊Q8—612no—
220C4⋊Q8—1224no—
221C4⋊Q8—1222no—
222C4⋊Q8—1222no—
223#125—614no—
224#125—1224no—
225#124—1228no—
226C22⋊Q8—1228no—
227C22⋊Q8—1228no—
228C22⋊Q8—1224no—
229C22⋊Q8—1224no—
230C22⋊Q8—1222no—
231C22⋊Q8—1224no—
232C22⋊Q8—1222no—
233C4×Q8—1222no—
234C4×Q8—1222no—
235C4×D4—22no—
236C4×D4—22no—
237C4×D4—1224no—
238C4×D4—1224no—
239#117—614no—
240C2×C4⋊C4—1224no—
241C2×C4⋊C4—1222no—
242C2×C4⋊C4—1221no—
243C2×C4⋊C4—1221no—
244C2×C4⋊C4—1221no—
245C2×C4⋊C4C22×D6111yescharacteristic
246C2×C22⋊C4—1224no—
247C2×C22⋊C4—1224no—
248C2×C22⋊C4D24412yes—
249C2×C42—612no—
250C2×C42C22×D6111yescharacteristic
251#103—1228no—
252#103—1228no—
253#100—24no—
254#97—22no—
255C4×C8—22no—
256#95—1221no—
257#95—1224no—
258C24C2×D24412yes—
259C2×Q8—1224no—
260C2×Q8—1222no—
261C2×Q8—1222no—
262C2×Q8—1224no—
263C2×Q8—1224no—
264C2×Q8—1224no—
265C2×Q8—4884no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4#1175822214yes—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—44no—
272C22×C4—1224no—
273C22×C4—1224no—
274C22×C4—1222no—
275C22×C4—1222no—
276C22×C4C2×D24411yescharacteristic
277C22×C4C2×D24412yes—
278C22×C4C2×D24411yescharacteristic
279C22×C4C23×D6111yescharacteristic
280M16—24no—
281M16—24no—
282C2×C8—24no—
283C4⋊C4—22no—
284C4⋊C4—614no—
285C4⋊C4—614no—
286C4⋊C4—1228no—
287C4⋊C4—1224no—
288C4⋊C4—1224no—
289C4⋊C4—1222no—
290C4⋊C4—1224no—
291C4⋊C4—1222no—
292C4⋊C4—2442no—
293C4⋊C4—2442no—
294C4⋊C4—2441no—
295C4⋊C4—2441no—
296C22⋊C4—24no—
297C22⋊C4—1228no—
298C22⋊C4—1224no—
299C22⋊C4—2448no—
300C42—1224no—
301C42—1224no—
302C42D4×D6112yes—
303C42D4×D6112yes—
304C23—22no—
305C23—22no—
306C23—42no—
307C23—42no—
308C23#1451976514yes—
309C23C22×D24411yescharacteristic
310Q8—4882no—
311Q8—4884no—
312Q8—2448no—
313Q8—2444no—
314D4—28no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—44no—
319C2×C4—42no—
320C2×C4—82no—
321C2×C4—1224no—
322C2×C4—1224no—
323C2×C4—4884no—
324C2×C4—4884no—
325C2×C4—2442no—
326C2×C4—2442no—
327C2×C4—2442no—
328C2×C4—2442no—
329C2×C4—2442no—
330C2×C4—2442no—
331C2×C4#1451974712yes—
332C2×C4#1451974711yescharacteristic
333C2×C4#1451974711yescharacteristic
334C2×C4C2×D4×D6112yes—
335C2×C4C2×#1175822212yes—
336C2×C4#1451982412yes—
337C8—44no—
338C22—21no—
339C22—21no—
340C22—24no—
341C22—44no—
342C22—42no—
343C22—42no—
344C22—82no—
345C22C2×#1451974711yescharacteristic
346C22C2×#1451976511yescharacteristic
347C22#1684033811yescharacteristic
348C4—21no—
349C4—22no—
350C4—42no—
351C4—4882no—
352C4—4882no—
353C4—2444no—
354C4—2444no—
355C4#1683994212yes—
356C4#1684042114yes—
357C2—21no—
358C2—21no—
359C2—44no—
360C2#2144947711yescharacteristic
361C2C2×#1683994211yescharacteristic
362C2#2145520011yescharacteristic
subgroupG/H
centerC22C2×#14519747
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC4×#100×C61C2
socleC22×C61C2×#120
prime radicalC22×D4×C61C23
hypercenterC4×#100D61
hypocenterC61#25917
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
2. #21456200C2semidirect
4. #21446981C2semidirect
5. #21446763C2semidirect
212. C61#25917semidirect

Maximal subgroups

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

subgroupindex
#214596222
#214562002
#214508112
#214469812
#214467632
#214462052
C4×#100×C612
#2591761