Finite Group Databasebeta

← #29736752

Subgroups — #29736752

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#21456199C212yesmaximal, complemented
3#21446980C214yesmaximal, complemented
4#21446762C212yesmaximal, complemented
5#21446208C214yesmaximal
6C4×#14519694C211yesmaximal, characteristic
7C61×#2547C211yesmaximal, characteristic
8#16840471—24no—
9#16840469—22no—
10D4×Dic122—22no—
11#16840393C2212yes—
12#16840388—22no—
13#16840319C2212yes—
14C2×#14519756—22no—
15#16840237C2212yes—
16#16840130—24no—
17#16840112—22no—
18#16840112—22no—
19#16839771C2218yes—
20C2×#14519694C2212yes—
21C4×#11758194C2212yes—
22#16839531C2212yes—
23#16839522C2218yes—
24C2×C4⋊Q8×C61C2211yescharacteristic
25C2×#124×C61C2212yes—
26C2×C4×D4×C61C2211yescharacteristic
27C61×#397—22no—
28C61×#391C2212yes—
29C61×#380—22no—
30C4×C4⋊C4×C61C2211yescharacteristic
31C22×Dic122—22no—
32C2×#11758194C2314yes—
33#14519777D412yes—
34#14519777D412yes—
35#14519776—24no—
36#14519769D414yes—
37#14519768—24no—
38#14519756—24no—
39#14519756—24no—
40#14519756—22no—
41#14519756—24no—
42#14519756—22no—
43#14519754—28no—
44#14519754—28no—
45C4×Dic122—22no—
46C4×Dic122—22no—
47D4×Dic61—24no—
48D4×Dic61—24no—
49#14519736C2314yes—
50C2×C4⋊Dic61—22no—
51C2×#11758196—24no—
52C2×C22⋊Dic61—24no—
53#14519701—28no—
54#14519701—28no—
55#14519694—24no—
56#14519688—22no—
57C4×#92418—22no—
58C22×Q8×C61C2312yes—
59C22×D4×C61C2311yescharacteristic
60C23×C4×C61D412yes—
61C4⋊Q8×C61—22no—
62C4⋊Q8×C61—22no—
63#124×C61—28no—
64C4×D4×C61—22no—
65C4×D4×C61—22no—
66C2×C4⋊C4×C61—21no—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61C2311yescharacteristic
70C2×C22⋊C4×C61—24no—
71C2×C22⋊C4×C61D412yes—
72C2×C42×C61C2312yes—
73C2×C42×C61C2311yescharacteristic
74#95×C61—21no—
75#95×C61—24no—
76C24×C61C2×D412yes—
77C2×Q8×C61—24no—
78C2×Q8×C61—24no—
79C2×Q8×C61—24no—
80C2×D4×C61—22no—
81C2×D4×C61—22no—
82C2×D4×C61#4114yes—
83C22×C4×C61—22no—
84C22×C4×C61—22no—
85C22×C4×C61—22no—
86C22×C4×C61—22no—
87C22×C4×C61—44no—
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2×D412yes—
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2411yescharacteristic
92C22×Dic61—24no—
93C22×Dic61—24no—
94C2×Dic122—22no—
95C2×Dic122—24no—
96C2×Dic122—22no—
97C2×Dic122—84no—
98C4⋊C4×C61—22no—
99C4⋊C4×C61—42no—
100C4⋊C4×C61—41no—
101C4⋊C4×C61—42no—
102C4⋊C4×C61—41no—
103C22⋊C4×C61—24no—
104C22⋊C4×C61—48no—
105C42×C61—24no—
106C42×C61C2×D412yes—
107C42×C61C2×D412yes—
108C22⋊Dic61—28no—
109C22⋊Dic61—24no—
110C4⋊Dic61—24no—
111C4⋊Dic61—22no—
112C4⋊Dic61—22no—
113C4⋊Dic61C2×D414yes—
114C4⋊Dic61C2×D414yes—
115#11758196—28no—
116#11758196—24no—
117#11758196—24no—
118C4×Dic61—24no—
119#11758194—24no—
120#11758194—24no—
121C2×#92418—24no—
122C23×C61—22no—
123C23×C61—22no—
124C23×C61—42no—
125C23×C61—42no—
126C23×C61#12414yes—
127C23×C61C22×D411yescharacteristic
128Q8×C61—48no—
129D4×C61—28no—
130C2×C4×C61—22no—
131C2×C4×C61—22no—
132C2×C4×C61—41no—
133C2×C4×C61—44no—
134C2×C4×C61—42no—
135C2×C4×C61—42no—
136C2×C4×C61—42no—
137C2×C4×C61—42no—
138C2×C4×C61—42no—
139C2×C4×C61—42no—
140C2×C4×C61—42no—
141C2×C4×C61—82no—
142C2×C4×C61#12011yescharacteristic
143C2×C4×C61#12012yes—
144C2×C4×C61#12011yescharacteristic
145C2×C4×C61C22×D412yes—
146C2×C4×C61C2×#4112yes—
147C2×C4×C612−1+412yes—
148C2×Dic61—24no—
149C2×Dic61—24no—
150C2×Dic61—84no—
151C2×Dic61—84no—
152Dic122—44no—
153Dic122—82no—
154Dic122—84no—
155#92418—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—82no—
171C4×C61—82no—
172C4×C61#45512yes—
173C4×C61#54814yes—
174Dic61—44no—
175Dic61—44no—
176#3501—611no—
177#3139—612no—
178#2547D6111yescharacteristic
179#2028—614no—
180#1986—612no—
181#1885—614no—
182C4×#100—611no—
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⋊Q8C2×D6111yescharacteristic
196C2×#124C2×D6112yes—
197C2×C22⋊Q8—1222no—
198#518—612no—
199C2×C4×D4C2×D6111yescharacteristic
200#491—1224no—
201#487—1222no—
202#487—1222no—
203#419—618no—
204C2×#100—612no—
205C4×M16—612no—
206#397—22no—
207#391C2×D6112yes—
208#380—22no—
209C4×C4⋊C4C2×D6111yescharacteristic
210#343—612no—
211#339—618no—
212C61#2591711yescharacteristic, complemented, Sylow
213C22×Q8—1222no—
214C22×Q8C22×D6112yes—
215C22×D4C22×D6111yescharacteristic
216C23×C4C22⋊D6112yes—
217C2×M16—614no—
218C4⋊Q8—22no—
219C4⋊Q8—22no—
220C4⋊Q8—612no—
221C4⋊Q8—612no—
222C4⋊Q8—1224no—
223#125—614no—
224#125—1224no—
225#124—28no—
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—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—1224no—
244C2×C4⋊C4—1222no—
245C2×C4⋊C4C22×D6111yescharacteristic
246C2×C22⋊C4—24no—
247C2×C22⋊C4—1224no—
248C2×C22⋊C4C22⋊D6112yes—
249C2×C42C22×D6112yes—
250C2×C42C22×D6111yescharacteristic
251#103—1228no—
252#103—1228no—
253#100—1224no—
254#97—1222no—
255C4×C8—1222no—
256#95—21no—
257#95—24no—
258C24C2×C22⋊D6112yes—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—1224no—
263C2×Q8—1222no—
264C2×Q8—1222no—
265C2×Q8—4884no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4#1175822414yes—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—44no—
274C22×C4—1224no—
275C22×C4—1224no—
276C22×C4C2×C22⋊D6111yescharacteristic
277C22×C4C2×C22⋊D6112yes—
278C22×C4C2×C22⋊D6111yescharacteristic
279C22×C4C23×D6111yescharacteristic
280M16—1224no—
281M16—1224no—
282C2×C8—1224no—
283C4⋊C4—22no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—614no—
289C4⋊C4—614no—
290C4⋊C4—1228no—
291C4⋊C4—1224no—
292C4⋊C4—1224no—
293C4⋊C4—1222no—
294C4⋊C4—1224no—
295C4⋊C4—1222no—
296C22⋊C4—24no—
297C22⋊C4—48no—
298C22⋊C4—1228no—
299C22⋊C4—1224no—
300C42—24no—
301C42—1224no—
302C42D4×D6112yes—
303C42D4×D6112yes—
304C23—22no—
305C23—22no—
306C23—42no—
307C23—42no—
308C23#1451976414yes—
309C23C22×C22⋊D6111yescharacteristic
310Q8—48no—
311Q8—4882no—
312Q8—4884no—
313Q8—2444no—
314D4—28no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—44no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—82no—
327C2×C4—1224no—
328C2×C4—1224no—
329C2×C4—4884no—
330C2×C4—4884no—
331C2×C4#1451974512yes—
332C2×C4#1451974511yescharacteristic
333C2×C4#1451974511yescharacteristic
334C2×C4C2×D4×D6112yes—
335C2×C4C2×#1175822412yes—
336C2×C4#1451982412yes—
337C8—2444no—
338C22—21no—
339C22—21no—
340C22—24no—
341C22—44no—
342C22—42no—
343C22—42no—
344C22—82no—
345C22C2×#1451974511yescharacteristic
346C22C2×#1451976411yescharacteristic
347C22#1684033311yescharacteristic
348C4—21no—
349C4—22no—
350C4—42no—
351C4—82no—
352C4—82no—
353C4—2444no—
354C4—2444no—
355C4#1683993912yes—
356C4#1684041814yes—
357C2—21no—
358C2—21no—
359C2—44no—
360C2#2144947811yescharacteristic
361C2C2×#1683993911yescharacteristic
362C2#2145519711yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2547C2
socleC22×C61C2×#120
prime radicalC22×D4×C61C23
hypercenter#2547D61
hypocenterC61#25917
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2547

Complements

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

NH ≅ G/Ntype
2. #21456199C2semidirect
3. #21446980C2semidirect
4. #21446762C2semidirect
212. C61#25917semidirect

Maximal subgroups

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

subgroupindex
#214596222
#214561992
#214469802
#214467622
#214462082
C4×#145196942
C61×#25472
#2591761