Finite Group Databasebeta

← #29572540

Subgroups — #29572540

Order 15616.

910 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C2×#16840338C211yesmaximal, characteristic
2#21444544C212yesmaximal, complemented
3#21444442C212yesmaximal, complemented
4#21444153C211yesmaximal, characteristic, complemented
5C61×#1464C211yesmaximal, characteristic, complemented
6C22×#11758222C411yescharacteristic
7C23×Dic122C411yescharacteristic
8#16840338—24no—
9#16840338—24no—
10C2×#14519777C411yescharacteristic
11C2×#14519777C2211yescharacteristic
12C2×#14519765C411yescharacteristic
13C2×#14519765C2211yescharacteristic
14C2×#14519761—21no—
15C2×#14519761—21no—
16C2×#14519757—21no—
17C2×#14519757—21no—
18#16839819C2212yes—
19#16839510—24no—
20#16839501—24no—
21#16839488—21no—
22#16839486—21no—
23C2×#117×C61C2211yescharacteristic
24C61×#432C2212yes—
25C61×#324—21no—
26C61×#323—21no—
27C2×#11758222—22no—
28C2×#11758222—22no—
29C2×#11758222—42no—
30C2×#11758222—42no—
31C22×Dic122—21no—
32C22×Dic122—22no—
33C22×Dic122—22no—
34C22×Dic122—42no—
35C22×Dic122—42no—
36C22×Dic122C2×C411yescharacteristic
37C22×Dic122C2×C411yescharacteristic
38C22×Dic122C2×C411yescharacteristic
39C22×D244C2×C411yescharacteristic
40C22×C22⋊D61—21no—
41C23×Dic61—21no—
42C22×C4×D61—21no—
43#14519777—22no—
44#14519777—22no—
45#14519777—22no—
46#14519777D414yes—
47#14519765—22no—
48#14519765—22no—
49#14519765—22no—
50#14519765D414yes—
51#14519761—28no—
52#14519761—28no—
53#14519757—28no—
54#14519757—28no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61—21no—
59C2×#11758198—21no—
60C2×#11758198—21no—
61C2×#11758198—21no—
62C2×#11758198—21no—
63C4⋊#92418—22no—
64C22⋊#92418—22no—
65#14519688—22no—
66C23×C4×C61C2×C411yescharacteristic
67#117×C61—22no—
68#117×C61—22no—
69C2×C4⋊C4×C61D411yescharacteristic
70C2×C4⋊C4×C61D411yescharacteristic
71C2×C22⋊C4×C61D411yescharacteristic
72C2×C22⋊C4×C61D411yescharacteristic
73C2×C42×C61C2×C411yescharacteristic
74C2×C42×C61C2311yescharacteristic
75C4⋊C8×C61—22no—
76C22⋊C8×C61—22no—
77#97×C61—22no—
78C24×C61C22⋊C411yescharacteristic
79C23×D61—21no—
80C22×C4×C61—22no—
81C22×C4×C61—22no—
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61C22⋊C411yescharacteristic
85C22×C4×C61C22⋊C411yescharacteristic
86C22×C4×C61C22⋊C411yescharacteristic
87C22×C4×C61C22×C411yescharacteristic
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C2×C22⋊D61—24no—
91C2×C22⋊D61—42no—
92C2×C22⋊D61—42no—
93C22×Dic61—24no—
94C22×Dic61—21no—
95C22×Dic61—21no—
96C22×Dic61—21no—
97C22×Dic61—42no—
98C22×Dic61—42no—
99#11758222—48no—
100#11758222—48no—
101C2×D244—21no—
102C2×D244—41no—
103C2×D244—41no—
104C2×D244C22⋊C412yes—
105C2×C4×D61—24no—
106C2×C4×D61—42no—
107C2×C4×D61—42no—
108C2×Dic122—24no—
109C2×Dic122—21no—
110C2×Dic122—21no—
111C2×Dic122—21no—
112C2×Dic122—42no—
113C2×Dic122—48no—
114C2×Dic122—48no—
115C2×Dic122—42no—
116C2×Dic122—41no—
117C2×Dic122—41no—
118C2×Dic122—41no—
119C2×Dic122—41no—
120C2×Dic122—41no—
121C2×Dic122—41no—
122C2×Dic122C22⋊C412yes—
123C2×Dic122C22⋊C412yes—
124C2×Dic122C22⋊C412yes—
125C2×C8×C61—42no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—22no—
128C22⋊C4×C61—22no—
129C22⋊C4×C61—22no—
130C42×C61—21no—
131C42×C61—21no—
132C42×C61C2×D412yes—
133C42×C61C2×D412yes—
134#11758198—24no—
135#11758198—24no—
136#11758198—24no—
137#11758198—24no—
138C4⋊Dic61—24no—
139C4⋊Dic61—24no—
140C4⋊Dic61—24no—
141C4⋊Dic61—24no—
142C2×#92418—42no—
143C23×C61—21no—
144C23×C61—21no—
145C23×C61—41no—
146C23×C61—41no—
147C23×C61#10012yes—
148C23×C61C2×C22⋊C411yescharacteristic
149C22×D61—22no—
150C22×D61—42no—
151C22×D61—42no—
152C22×D61—41no—
153C22×D61—41no—
154C2×C4×C61—24no—
155C2×C4×C61—24no—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—21no—
159C2×C4×C61—21no—
160C2×C4×C61—21no—
161C2×C4×C61—21no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61#10012yes—
167C2×C4×C61C2×C22⋊C411yescharacteristic
168C2×C4×C61C2×C22⋊C411yescharacteristic
169C2×C4×C61#12012yes—
170C2×C4×C61#12012yes—
171C22⋊D61—48no—
172C2×Dic61—22no—
173C2×Dic61—22no—
174C2×Dic61—22no—
175C2×Dic61—48no—
176C2×Dic61—41no—
177C2×Dic61—41no—
178C2×Dic61—41no—
179C2×Dic61—41no—
180C2×Dic61—41no—
181C2×Dic61—41no—
182D244—42no—
183D244—42no—
184C4×D61—48no—
185Dic122—42no—
186Dic122—42no—
187Dic122—48no—
188Dic122—42no—
189Dic122—42no—
190Dic122—42no—
191Dic122—42no—
192C8×C61—82no—
193#92418—82no—
194#8524—611nomaximal, Sylow
195C22×C61—21no—
196C22×C61—21no—
197C22×C61—22no—
198C22×C61—42no—
199C22×C61—42no—
200C22×C61—41no—
201C22×C61—41no—
202C22×C61#37911yescharacteristic
203C22×C61C2×#10011yescharacteristic
204C22×C61C2×#10011yescharacteristic
205C2×D61—42no—
206C2×D61—42no—
207C2×D61—42no—
208C2×D61—82no—
209C2×D61—82no—
210C4×C61—22no—
211C4×C61—22no—
212C4×C61—21no—
213C4×C61—21no—
214C4×C61—22no—
215C4×C61—41no—
216C4×C61—41no—
217Dic61—42no—
218Dic61—42no—
219Dic61—42no—
220C2×#536—611no—
221#1558—612no—
222#1533—612no—
223#1464—611no—
224#1464D6111yescharacteristic, complemented
225C2×C61—21no—
226C2×C61—21no—
227C2×C61—42no—
228C2×C61#178311yescharacteristic
229C2×C61#179611yescharacteristic
230C2×C61#179611yescharacteristic
231D61—82no—
232C22×#41—611no—
233C23×Q8—611no—
234#536—1224no—
235#536—1224no—
236C2×C4⋊Q8—611no—
237C2×C4⋊Q8—611no—
238C2×#124—611no—
239C2×#124—611no—
240C2×#123—1221no—
241C2×#123—1221no—
242C2×C22⋊Q8—1221no—
243C2×C22⋊Q8—1221no—
244C2×#117C2×D6111yescharacteristic
245#432—612no—
246#432C2×D6112yes—
247#333—1224no—
248#329—1224no—
249#324—21no—
250#324—1221no—
251#323—21no—
252#323—1221no—
253C61#852411yescharacteristic, complemented, Sylow
254C2×#41—1222no—
255C2×#41—1222no—
256C2×#41—2442no—
257C2×#41—2442no—
258C22×Q8—611no—
259C22×Q8—611no—
260C22×Q8—611no—
261C22×Q8—1222no—
262C22×Q8—1222no—
263C22×Q8—1221no—
264C22×Q8—2442no—
265C22×Q8—2442no—
266C22×D4—611no—
267C22×D4—1221no—
268C23×C4—1221no—
269C23×C4—1221no—
270C23×C4C4×D6111yescharacteristic
271C4⋊Q8—614no—
272C4⋊Q8—1222no—
273C4⋊Q8—1222no—
274C4⋊Q8—1222no—
275#124—614no—
276#124—1222no—
277#124—1222no—
278#124—1222no—
279#123—1228no—
280#123—1228no—
281C22⋊Q8—1228no—
282C22⋊Q8—1228no—
283#117—22no—
284#117—22no—
285C2×C4⋊C4—1221no—
286C2×C4⋊C4—1221no—
287C2×C4⋊C4—1221no—
288C2×C4⋊C4—1221no—
289C2×C4⋊C4D24411yescharacteristic
290C2×C4⋊C4C22⋊D6111yescharacteristic
291C2×C22⋊C4—1221no—
292C2×C22⋊C4—1221no—
293C2×C22⋊C4—1221no—
294C2×C22⋊C4—1221no—
295C2×C22⋊C4D24411yescharacteristic
296C2×C22⋊C4C22⋊D6111yescharacteristic
297C2×C42C4×D6111yescharacteristic
298C2×C42C22×D6111yescharacteristic
299C4⋊C8—22no—
300C4⋊C8—1222no—
301C22⋊C8—22no—
302C22⋊C8—1222no—
303#97—22no—
304#97—1222no—
305C24—1221no—
306C24#1175819811yescharacteristic
307#41—2448no—
308#41—2448no—
309C2×Q8—612no—
310C2×Q8—612no—
311C2×Q8—612no—
312C2×Q8—1224no—
313C2×Q8—1221no—
314C2×Q8—1221no—
315C2×Q8—1221no—
316C2×Q8—2448no—
317C2×Q8—2448no—
318C2×Q8—2442no—
319C2×Q8—2441no—
320C2×Q8—2441no—
321C2×Q8—2441no—
322C2×Q8—2441no—
323C2×Q8—2441no—
324C2×Q8—2441no—
325C2×Q8—2442no—
326C2×D4—612no—
327C2×D4—1224no—
328C2×D4—1221no—
329C2×D4—2441no—
330C2×D4—2441no—
331C2×D4—2442no—
332C2×D4—2442no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—1224no—
338C22×C4—1224no—
339C22×C4—1221no—
340C22×C4—1221no—
341C22×C4—1221no—
342C22×C4—2442no—
343C22×C4—2442no—
344C22×C4—2442no—
345C22×C4—2442no—
346C22×C4#1175819811yescharacteristic
347C22×C4#1175819811yescharacteristic
348C22×C4#1175819811yescharacteristic
349C22×C4C2×C4×D6111yescharacteristic
350C22×C4C2×D24411yescharacteristic
351C22×C4C2×C22⋊D6111yescharacteristic
352C2×C8—42no—
353C2×C8—2442no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—1224no—
357C4⋊C4—1224no—
358C4⋊C4—1224no—
359C4⋊C4—1224no—
360C22⋊C4—22no—
361C22⋊C4—22no—
362C22⋊C4—1224no—
363C22⋊C4—1224no—
364C22⋊C4—1224no—
365C22⋊C4—1224no—
366C42—21no—
367C42—21no—
368C42D4×D6112yes—
369C42D4×D6112yes—
370C23—21no—
371C23—21no—
372C23—41no—
373C23—41no—
374C23—1222no—
375C23—2442no—
376C23—2442no—
377C23—2441no—
378C23—2441no—
379C23#1451969312yes—
380C23C2×#1175819811yescharacteristic
381Q8—2442no—
382Q8—2442no—
383Q8—2442no—
384Q8—2448no—
385Q8—2442no—
386Q8—2442no—
387Q8—2442no—
388D4—2442no—
389D4—2442no—
390D4—2448no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—1222no—
404C2×C4—1222no—
405C2×C4—1222no—
406C2×C4—2448no—
407C2×C4—2448no—
408C2×C4—2441no—
409C2×C4—2441no—
410C2×C4—2441no—
411C2×C4—2441no—
412C2×C4—2441no—
413C2×C4—2441no—
414C2×C4#1451969312yes—
415C2×C4C22⋊C4×D6111yescharacteristic
416C2×C4C22⋊C4×D6111yescharacteristic
417C2×C4#1451974512yes—
418C2×C4#1451974712yes—
419C8—82no—
420C8—4882no—
421C22—21no—
422C22—21no—
423C22—22no—
424C22—42no—
425C22—42no—
426C22—41no—
427C22—41no—
428C22—4882no—
429C22—4882no—
430C22—2442no—
431C22—2442no—
432C22—2442no—
433C22#1683961711yescharacteristic
434C22C2×#1451969311yescharacteristic
435C22C2×#1451969311yescharacteristic
436C4—22no—
437C4—21no—
438C4—21no—
439C4—22no—
440C4—22no—
441C4—41no—
442C4—41no—
443C4—2442no—
444C4—2442no—
445C4—2442no—
446C2—21no—
447C2—21no—
448C2—42no—
449C2—4882no—
450C2#2144573911yescharacteristic
451C2#2144579211yescharacteristic
452C2#2144579511yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1464C2
socleC22×C61#379
prime radicalC22×#11758222C4
cosocleC2×C42×C61C23
hypercenter#1464D61
hypocenterC61#8524
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1464

Complements

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

NH ≅ G/Ntype
2. #21444544C2semidirect
3. #21444442C2semidirect
4. #21444153C2semidirect
5. C61×#1464C2semidirect
224. #1464D61semidirect
253. C61#8524semidirect

Maximal subgroups

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

subgroupindex
C2×#168403382
#214445442
#214444422
#214441532
C61×#14642
#852461