Finite Group Databasebeta

← #29571371

Subgroups — #29571371

Order 15616.

1174 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1174 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C22×#14519755C211yesmaximal, characteristic
2#21445728C212yesmaximal
3C2×#16839488C211yesmaximal, characteristic, complemented
4C2×#16839487C212yesmaximal, complemented
5C2×C61×#324C211yesmaximal, characteristic, complemented
6C23×Dic122C411yescharacteristic
7C23×C4×D61C2211yescharacteristic
8C2×#14519755—24no—
9C2×#14519755—22no—
10C2×#14519755—22no—
11C2×#14519755—24no—
12C22×#11758196C411yescharacteristic
13C22×#11758196C2211yescharacteristic
14C22×#11758198C411yescharacteristic
15C22×#11758198C411yescharacteristic
16C2×C22⋊#92418C2212yes—
17C2×#14519689—24no—
18#16839488—22no—
19#16839487—24no—
20C22×C4⋊C4×C61C2211yescharacteristic
21C2×C22⋊C8×C61C2212yes—
22C61×#324—22no—
23C24×D61C2×C411yescharacteristic
24C22×Dic122—22no—
25C22×Dic122—22no—
26C22×Dic122—22no—
27C22×Dic122—44no—
28C23×Dic61C2×C411yescharacteristic
29C23×Dic61C2×C411yescharacteristic
30C23×Dic61C2×C411yescharacteristic
31C22×C4×D61—21no—
32C22×C4×D61—22no—
33C22×C4×D61—21no—
34C22×C4×D61—44no—
35C22×C4×D61D414yes—
36C22×#92418—22no—
37#14519755—216no—
38#14519755—216no—
39C2×#11758196—22no—
40C2×#11758196—21no—
41C2×#11758196—22no—
42C2×#11758196—21no—
43C2×#11758196—22no—
44C2×#11758196—22no—
45C2×#11758196D414yes—
46C2×#11758198—22no—
47C2×#11758198—22no—
48C2×#11758198—22no—
49C2×#11758198—22no—
50C2×#11758198—22no—
51C2×#11758198—22no—
52C22⋊#92418—22no—
53C22⋊#92418—22no—
54#14519689—48no—
55C23×C4×C61C2×C411yescharacteristic
56C23×C4×C61C2×C411yescharacteristic
57C23×C4×C61C2311yescharacteristic
58C22×C8×C61—22no—
59C2×C4⋊C4×C61—22no—
60C2×C4⋊C4×C61—22no—
61C2×C4⋊C4×C61D412yes—
62C2×C4⋊C4×C61D412yes—
63C22⋊C8×C61—22no—
64C22⋊C8×C61—22no—
65C24×C61C22×C411yescharacteristic
66C23×D61—21no—
67C23×D61—21no—
68C23×D61—24no—
69C23×D61—22no—
70C23×D61—22no—
71C23×D61—42no—
72C23×D61C22⋊C412yes—
73C22×C4×C61—21no—
74C22×C4×C61—21no—
75C22×C4×C61—21no—
76C22×C4×C61—21no—
77C22×C4×C61—22no—
78C22×C4×C61—22no—
79C22×C4×C61—42no—
80C22×C4×C61—42no—
81C22×C4×C61C22⋊C412yes—
82C22×C4×C61C22⋊C412yes—
83C22×C4×C61C2×D412yes—
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61C2×D412yes—
86C22×C4×C61C2×D411yescharacteristic
87C22×Dic61—21no—
88C22×Dic61—21no—
89C22×Dic61—24no—
90C22×Dic61—21no—
91C22×Dic61—21no—
92C22×Dic61—21no—
93C22×Dic61—21no—
94C22×Dic61—22no—
95C22×Dic61—22no—
96C22×Dic61—42no—
97C22×Dic61C22⋊C412yes—
98C22×Dic61C22⋊C412yes—
99C22×Dic61C22⋊C412yes—
100C2×C4×D61—24no—
101C2×C4×D61—24no—
102C2×C4×D61—28no—
103C2×C4×D61—44no—
104C2×C4×D61—44no—
105C2×C4×D61—48no—
106C2×C4×D61—44no—
107C2×Dic122—28no—
108C2×Dic122—48no—
109C2×Dic122—48no—
110C2×Dic122—44no—
111C2×Dic122—44no—
112C2×C8×C61—44no—
113C2×C8×C61—42no—
114C4⋊C4×C61—24no—
115C4⋊C4×C61—24no—
116#11758198—28no—
117#11758198—28no—
118#11758196—28no—
119#11758196—24no—
120#11758196—24no—
121C2×#92418—44no—
122C2×#92418—42no—
123C23×C61—22no—
124C23×C61—22no—
125C23×C61C2×C22⋊C411yescharacteristic
126C23×C61C2×C22⋊C411yescharacteristic
127C23×C61C2×C22⋊C411yescharacteristic
128C23×C61#13112yes—
129C23×C61#13112yes—
130C22×D61—24no—
131C22×D61—24no—
132C22×D61—22no—
133C22×D61—24no—
134C22×D61—24no—
135C22×D61—22no—
136C22×D61—48no—
137C22×D61—48no—
138C22×D61—42no—
139C22×D61—42no—
140C22×D61—42no—
141C22×D61—41no—
142C22×D61—41no—
143C22×D61#10114yes—
144C2×C4×C61—24no—
145C2×C4×C61—22no—
146C2×C4×C61—22no—
147C2×C4×C61—22no—
148C2×C4×C61—22no—
149C2×C4×C61—41no—
150C2×C4×C61—42no—
151C2×C4×C61—41no—
152C2×C4×C61—42no—
153C2×C4×C61—42no—
154C2×C4×C61—42no—
155C2×C4×C61—42no—
156C2×C4×C61—42no—
157C2×C4×C61#12012yes—
158C2×C4×C61#12012yes—
159C2×Dic61—24no—
160C2×Dic61—24no—
161C2×Dic61—22no—
162C2×Dic61—22no—
163C2×Dic61—22no—
164C2×Dic61—22no—
165C2×Dic61—24no—
166C2×Dic61—42no—
167C2×Dic61—42no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—42no—
171C2×Dic61—42no—
172C2×Dic61#9914yes—
173C4×D61—48no—
174C4×D61—48no—
175Dic122—416no—
176C8×C61—82no—
177#92418—82no—
178#8308—611nomaximal, Sylow
179C22×C61—24no—
180C22×C61—24no—
181C22×C61—21no—
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—21no—
185C22×C61#37911yescharacteristic
186C22×C61#40812yes—
187C22×C61#40812yes—
188C22×C61#40811yescharacteristic
189C22×C61#40811yescharacteristic
190C22×C61C2×#9912yes—
191C22×C61C2×#10112yes—
192C2×D61—28no—
193C2×D61—24no—
194C2×D61—48no—
195C2×D61—44no—
196C2×D61—44no—
197C2×D61—44no—
198C2×D61—42no—
199C2×D61—42no—
200C4×C61—22no—
201C4×C61—22no—
202C4×C61—42no—
203C4×C61—42no—
204Dic61—24no—
205Dic61—42no—
206Dic61—42no—
207C22×C22⋊Q8—611no—
208#1779—612no—
209C2×#324—612no—
210C2×#324—611no—
211C2×#324D6111yescharacteristic, complemented
212C2×C61—22no—
213C2×C61—21no—
214C2×C61—21no—
215C2×C61#147214yes—
216C2×C61#178011yescharacteristic
217C2×C61#178111yescharacteristic
218C2×C61#178411yescharacteristic
219D61—44no—
220C23×Q8—611no—
221C24×C4—611no—
222C2×C22⋊Q8—1224no—
223C2×C22⋊Q8—1224no—
224C2×C22⋊Q8—1222no—
225C2×C22⋊Q8—1222no—
226C22×C4⋊C4—611no—
227C22×C4⋊C4—611no—
228C22×C4⋊C4C2×D6111yescharacteristic
229C22×C22⋊C4—611no—
230C22×C22⋊C4—611no—
231C2×C22⋊C8—612no—
232C2×C22⋊C8—1224no—
233C2×C22⋊C8C2×D6112yes—
234#324—22no—
235#324—1224no—
236#324—1222no—
237C61#830811yescharacteristic, complemented, Sylow
238C25—611no—
239C22×Q8—1222no—
240C22×Q8—1222no—
241C22×Q8—1222no—
242C22×Q8—2444no—
243C23×C4—611no—
244C23×C4—611no—
245C23×C4—611no—
246C23×C4—614no—
247C23×C4—1221no—
248C23×C4—1222no—
249C23×C4—1221no—
250C23×C4—2444no—
251C23×C4C4×D6111yescharacteristic
252C23×C4C4×D6111yescharacteristic
253C23×C4C22×D6111yescharacteristic
254C22×C8—22no—
255C22×C8—1222no—
256C22⋊Q8—12216no—
257C22⋊Q8—12216no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—614no—
261C2×C4⋊C4—1222no—
262C2×C4⋊C4—1222no—
263C2×C4⋊C4—1222no—
264C2×C4⋊C4—1221no—
265C2×C4⋊C4—1222no—
266C2×C4⋊C4—1221no—
267C2×C4⋊C4D24412yes—
268C2×C4⋊C4C22⋊D6112yes—
269C2×C22⋊C4—1222no—
270C2×C22⋊C4—1222no—
271C2×C22⋊C4—1222no—
272C2×C22⋊C4—1222no—
273C2×C22⋊C4—1222no—
274C2×C22⋊C4—1222no—
275C22⋊C8—22no—
276C22⋊C8—22no—
277C22⋊C8—1222no—
278C22⋊C8—1222no—
279C22⋊C8—2448no—
280C24—612no—
281C24—1224no—
282C24—1221no—
283C24—1221no—
284C24—1222no—
285C24—1222no—
286C24—2442no—
287C24C2×C4×D6111yescharacteristic
288C2×Q8—1228no—
289C2×Q8—2444no—
290C2×Q8—2448no—
291C2×Q8—2444no—
292C2×Q8—2448no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—612no—
302C22×C4—612no—
303C22×C4—612no—
304C22×C4—1228no—
305C22×C4—1224no—
306C22×C4—1221no—
307C22×C4—1221no—
308C22×C4—1221no—
309C22×C4—1221no—
310C22×C4—1221no—
311C22×C4—1221no—
312C22×C4—1222no—
313C22×C4—1222no—
314C22×C4—1224no—
315C22×C4—1224no—
316C22×C4—2444no—
317C22×C4—2444no—
318C22×C4—2448no—
319C22×C4—2444no—
320C22×C4—2442no—
321C22×C4#1175819812yes—
322C22×C4#1175819812yes—
323C22×C4C2×D24411yescharacteristic
324C22×C4D4×D6112yes—
325C22×C4D4×D6112yes—
326C22×C4C2×C22⋊D6111yescharacteristic
327C2×C8—44no—
328C2×C8—42no—
329C2×C8—2444no—
330C2×C8—2442no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—1228no—
334C4⋊C4—1224no—
335C4⋊C4—1224no—
336C22⋊C4—1228no—
337C22⋊C4—1228no—
338C23—22no—
339C23—22no—
340C23—614no—
341C23—1224no—
342C23—1224no—
343C23—1224no—
344C23—1224no—
345C23—1222no—
346C23—1222no—
347C23—2448no—
348C23—2448no—
349C23—2442no—
350C23—2442no—
351C23—2442no—
352C23—2441no—
353C23—2441no—
354C23C22⋊C4×D6111yescharacteristic
355C23C22⋊C4×D6111yescharacteristic
356C23C2×#1175819811yescharacteristic
357C23#1451978312yes—
358C23#1451978512yes—
359Q8—24416no—
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—614no—
374C2×C4—1224no—
375C2×C4—1224no—
376C2×C4—1222no—
377C2×C4—1222no—
378C2×C4—1224no—
379C2×C4—1222no—
380C2×C4—1222no—
381C2×C4—2448no—
382C2×C4—2448no—
383C2×C4—2442no—
384C2×C4—2442no—
385C2×C4—2441no—
386C2×C4—2441no—
387C2×C4—2442no—
388C2×C4—2442no—
389C2×C4#1451974512yes—
390C2×C4#1451974712yes—
391C8—82no—
392C8—4882no—
393C22—24no—
394C22—24no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—1228no—
400C22—1224no—
401C22—2448no—
402C22—2444no—
403C22—2444no—
404C22—2444no—
405C22—2442no—
406C22—2442no—
407C22#1683961711yescharacteristic
408C22#1683972712yes—
409C22#1683972712yes—
410C22#1683972811yescharacteristic
411C22#1683973011yescharacteristic
412C22#99×D6112yes—
413C22#101×D6112yes—
414C4—22no—
415C4—22no—
416C4—42no—
417C4—42no—
418C4—1224no—
419C4—2442no—
420C4—2442no—
421C2—22no—
422C2—21no—
423C2—21no—
424C2—2444no—
425C2#2144418114yes—
426C2#2144573011yescharacteristic
427C2#2144573211yescharacteristic
428C2#2144574111yescharacteristic
subgroupG/H
centerC23C22⋊C4×D61
commutatorC24×C61C22×C4
FrattiniC23×C4C22×D61
FittingC2×C61×#324C2
socleC23×C61C2×C22⋊C4
prime radicalC24×D61C2×C4
cosocleC23×C4×C61C23
hypercenterC2×#324D61
hypocenterC61#8308
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
3. C2×#16839488C2semidirect
4. C2×#16839487C2semidirect
5. C2×C61×#324C2semidirect
211. C2×#324D61semidirect
237. C61#8308semidirect

Maximal subgroups

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

subgroupindex
C22×#145197552
#214457282
C2×#168394882
C2×#168394872
C2×C61×#3242
#830861