Finite Group Databasebeta

← #29595494

Subgroups — #29595494

Order 15616.

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

normal subgroups only

This group has 976 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×#16840333C211yesmaximal, characteristic
2C2×#16839551C214yesmaximal, complemented
3#21445747C211yesmaximal, characteristic
4C61×#1782C211yesmaximal, characteristic
5C23×Dic122C411yescharacteristic, complemented
6#16840333—28no—
7C2×#14519775—21no—
8C2×#14519764—21no—
9C2×#14519758C412yescomplemented
10C2×#14519758C2212yes—
11C2×#14519756—21no—
12C2×#14519756—21no—
13C2×#14519736C411yescharacteristic, complemented
14C2×#14519696—21no—
15C2×#14519694C2212yes—
16#16839726—21no—
17#16839551—28no—
18C22×#41×C61C2211yescharacteristic
19C2×#99×C61—21no—
20C2×#99×C61C2212yes—
21C61×#381—21no—
22C22×Dic122—21no—
23C22×Dic122—21no—
24C22×Dic122—24no—
25C22×Dic122—42no—
26C22×Dic122—42no—
27C23×Dic61C2×C412yes—
28C2×#11758194—22no—
29#14519775—28no—
30#14519764—28no—
31#14519758—24no—
32#14519758—24no—
33#14519758—24no—
34#14519758D418yes—
35#14519756—28no—
36#14519756—28no—
37#14519736—24no—
38C2×C4⋊Dic61—21no—
39C2×#11758196—22no—
40C2×#11758196—22no—
41C2×C22⋊Dic61—22no—
42C2×C22⋊Dic61—22no—
43C2×C22⋊Dic61C2×C412yes—
44C2×C4×Dic61—21no—
45#14519696—42no—
46#14519694—24no—
47C22⋊#92418—42no—
48C2×#41×C61—24no—
49C2×#41×C61—42no—
50C2×#41×C61—42no—
51C22×Q8×C61D411yescharacteristic
52C22×D4×C61C2×C411yescharacteristic
53C22×D4×C61D411yescharacteristic
54C22×D4×C61C2311yescharacteristic
55C23×C4×C61C2×C411yescharacteristic
56C23×C4×C61D411yescharacteristic
57C23×C4×C61D411yescharacteristic
58C2×C22⋊C4×C61—22no—
59#99×C61—24no—
60#99×C61—42no—
61#95×C61—42no—
62C24×C61C22⋊C411yescharacteristic
63C24×C61C22×C411yescharacteristic
64C24×C61C2×D411yescharacteristic
65#41×C61—416no—
66C2×Q8×C61—22no—
67C2×Q8×C61—22no—
68C2×Q8×C61—41no—
69C2×D4×C61—22no—
70C2×D4×C61—22no—
71C2×D4×C61—22no—
72C2×D4×C61—41no—
73C2×D4×C61—41no—
74C2×D4×C61—41no—
75C2×D4×C61—41no—
76C2×D4×C61—41no—
77C2×D4×C61C2×D414yes—
78C22×C4×C61—22no—
79C22×C4×C61—22no—
80C22×C4×C61—22no—
81C22×C4×C61—22no—
82C22×C4×C61—22no—
83C22×C4×C61—22no—
84C22×C4×C61—22no—
85C22×C4×C61—41no—
86C22×C4×C61—41no—
87C22×C4×C61—42no—
88C22×C4×C61C22⋊C411yescharacteristic
89C22×C4×C61C22⋊C411yescharacteristic
90C22×C4×C61C22⋊C411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×Dic61—22no—
93C22×Dic61—22no—
94C22×Dic61—22no—
95C22×Dic61—42no—
96C22×Dic61—42no—
97C22×Dic61C22⋊C414yes—
98C2×Dic122—24no—
99C2×Dic122—24no—
100C2×Dic122—416no—
101C2×Dic122—42no—
102C2×Dic122—42no—
103C2×Dic122—42no—
104C2×Dic122—42no—
105C22⋊C4×C61—42no—
106C22⋊C4×C61—42no—
107C22⋊Dic61—28no—
108C22⋊Dic61—22no—
109C22⋊Dic61—28no—
110C22⋊Dic61C22⋊C414yes—
111C4⋊Dic61—24no—
112#11758196—28no—
113#11758196—28no—
114C4×Dic61—24no—
115#11758194—42no—
116#11758194—42no—
117C2×#92418—42no—
118C23×C61—21no—
119C23×C61—21no—
120C23×C61—21no—
121C23×C61—21no—
122C23×C61—21no—
123C23×C61—21no—
124C23×C61—21no—
125C23×C61—42no—
126C23×C61—42no—
127C23×C61—42no—
128C23×C61C2×C22⋊C411yescharacteristic
129C23×C61C2×C22⋊C412yes—
130C23×C61#12012yes—
131Q8×C61—44no—
132D4×C61—44no—
133D4×C61—44no—
134D4×C61—44no—
135C2×C4×C61—28no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—44no—
146C2×C4×C61—44no—
147C2×C4×C61—82no—
148C2×C4×C61—84no—
149C2×C4×C61#9912yes—
150C2×C4×C61#9912yes—
151C2×C4×C61#12012yes—
152C2×Dic61—24no—
153C2×Dic61—24no—
154C2×Dic61—44no—
155C2×Dic61—44no—
156C2×Dic61—42no—
157C2×Dic61—42no—
158C2×Dic61—44no—
159Dic122—48no—
160Dic122—48no—
161#92418—82no—
162#12692—611nomaximal, Sylow
163C22×C61—22no—
164C22×C61—22no—
165C22×C61—22no—
166C22×C61—42no—
167C22×C61—42no—
168C22×C61—42no—
169C22×C61—42no—
170C22×C61—42no—
171C22×C61—42no—
172C22×C61—42no—
173C22×C61—41no—
174C22×C61—81no—
175C22×C61—81no—
176C22×C61#37911yescharacteristic
177C22×C61C2×#9911yescharacteristic
178C22×C61C2×#9911yescharacteristic
179C4×C61—22no—
180C4×C61—22no—
181C4×C61—22no—
182C4×C61—41no—
183C4×C61—162no—
184Dic61—44no—
185Dic61—44no—
186C2×#536—611no—
187C2×#352—614no—
188#1785—611no—
189#1782D6111yescharacteristic
190C2×C61—21no—
191C2×C61—21no—
192C2×C61—42no—
193C2×C61—42no—
194C2×C61—81no—
195C2×C61#178111yescharacteristic
196C2×C61#212312yes—
197C22×#41C2×D6111yescharacteristic
198C23×Q8—611no—
199#536—1228no—
200C2×C4⋊Q8—1221no—
201C2×#124—1221no—
202C2×#123—612no—
203C2×#123—612no—
204C2×C22⋊Q8—1221no—
205C2×C22⋊Q8—1221no—
206C2×#117—611no—
207C2×#101—1221no—
208C2×#100—612no—
209C2×#99—21no—
210C2×#99C2×D6112yes—
211#407—1221no—
212#381—21no—
213#352—1228no—
214C61#1269211yescharacteristic, complemented, Sylow
215C2×#41—24no—
216C2×#41—42no—
217C2×#41—42no—
218C22×Q8—1224no—
219C22×Q8—1221no—
220C22×Q8—1221no—
221C22×Q8—2442no—
222C22×Q8—2442no—
223C22×Q8D24411yescharacteristic
224C22×D4C4×D6111yescharacteristic
225C22×D4C22⋊D6111yescharacteristic
226C22×D4C22×D6111yescharacteristic
227C23×C4—612no—
228C23×C4C4×D6111yescharacteristic
229C23×C4D24411yescharacteristic
230C23×C4C22⋊D6111yescharacteristic
231C2×M16—1222no—
232C4⋊Q8—1228no—
233#124—1228no—
234#123—618no—
235#123—1224no—
236#123—1224no—
237#123—1224no—
238C22⋊Q8—1228no—
239C22⋊Q8—1228no—
240#117—1224no—
241C2×C4⋊C4—1222no—
242C2×C4⋊C4—1221no—
243C2×C4⋊C4—1222no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—612no—
246C2×C22⋊C4—1222no—
247C2×C22⋊C4—1222no—
248C2×C42—1221no—
249#101—2442no—
250#100—1224no—
251#99—24no—
252#99—42no—
253C22⋊C8—2442no—
254#95—42no—
255C24#1175819811yescharacteristic
256C24C2×C4×D6111yescharacteristic
257C24C2×C22⋊D6111yescharacteristic
258#41—416no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—41no—
262C2×Q8—1224no—
263C2×Q8—1224no—
264C2×Q8—24416no—
265C2×Q8—2442no—
266C2×Q8—2442no—
267C2×Q8—2442no—
268C2×Q8—2442no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4D4×D6114yes—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—614no—
289C22×C4—1222no—
290C22×C4—1222no—
291C22×C4—1222no—
292C22×C4—2442no—
293C22×C4—2442no—
294C22×C4#1175819811yescharacteristic
295C22×C4#1175819811yescharacteristic
296C22×C4#1175819811yescharacteristic
297C22×C4C2×D24411yescharacteristic
298M16—2442no—
299M16—2442no—
300C2×C8—2442no—
301C4⋊C4—1228no—
302C4⋊C4—1224no—
303C4⋊C4—1228no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4—614no—
307C22⋊C4—1228no—
308C22⋊C4—1222no—
309C22⋊C4—1228no—
310C42—1224no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23C22⋊C4×D6112yes—
322C23C2×#1175819811yescharacteristic
323C23#1451974512yes—
324Q8—44no—
325Q8—2448no—
326Q8—2448no—
327D4—44no—
328D4—44no—
329D4—44no—
330C2×C4—28no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—82no—
343C2×C4—84no—
344C2×C4—1224no—
345C2×C4—1224no—
346C2×C4—2444no—
347C2×C4—2444no—
348C2×C4—2442no—
349C2×C4—2442no—
350C2×C4—2444no—
351C2×C4#1451969112yes—
352C2×C4#1451969112yes—
353C2×C4#1451974712yes—
354C8—4882no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—41no—
366C22—81no—
367C22—81no—
368C22#1683961711yescharacteristic
369C22C2×#1451969111yescharacteristic
370C22C2×#1451969111yescharacteristic
371C4—22no—
372C4—22no—
373C4—22no—
374C4—41no—
375C4—162no—
376C4—2444no—
377C4—2444no—
378C2—21no—
379C2—21no—
380C2—42no—
381C2—42no—
382C2—81no—
383C2#2144573311yescharacteristic
384C2#2144747212yes—
subgroupG/H
centerC22C2×#14519691
commutatorC24×C61C22×C4
FrattiniC22×D4C22×D61
FittingC61×#1782C2
socleC22×C61C2×#99
prime radicalC22×#41×C61C22
cosocleC22×D4×C61C23
hypercenter#1782D61
hypocenterC61#12692
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #1782

Complements

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

NH ≅ G/Ntype
2. C2×#16839551C2semidirect
5. C23×Dic122C4semidirect
9. C2×#14519758C4semidirect
13. C2×#14519736C4semidirect
214. C61#12692semidirect

Maximal subgroups

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

subgroupindex
C2×#168403332
C2×#168395512
#214457472
C61×#17822
#1269261