Finite Group Databasebeta

← #29571649

Subgroups — #29571649

Order 15616.

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

normal subgroups only

This group has 808 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×#16840497C211yesmaximal, characteristic
2#21444416C212yesmaximal, complemented
3#21444263C212yesmaximal, complemented
4#21444163C211yesmaximal, characteristic, complemented
5C4×#101×C61C211yesmaximal, characteristic, complemented
6#16840497—24no—
7#16840497—24no—
8C2×#14519774C411yescharacteristic
9C2×#14519766—21no—
10C2×#14519765C411yescharacteristic
11C2×#14519765C2211yescharacteristic
12C2×#14519762—21no—
13C2×#14519762—21no—
14C2×C4×Dic122C411yescharacteristic
15C2×C4×D244C411yescharacteristic
16C2×C4×D244C2211yescharacteristic
17C2×#14519739—21no—
18#16839819C2212yes—
19#16839501—24no—
20#16839491—24no—
21#16839488—21no—
22#16839488—21no—
23C2×C4×Q8×C61C2211yescharacteristic
24C2×#101×C61—21no—
25C4×M16×C61C2212yes—
26C61×#343—21no—
27C22×Dic122C2×C411yescharacteristic
28C22×D244—21no—
29C22×D244—21no—
30C22×D244C2×C411yescharacteristic
31C22×D244C2×C411yescharacteristic
32C22×C4×D61—21no—
33C22×C4×D61—21no—
34#14519774—22no—
35#14519774—22no—
36#14519766—28no—
37#14519765—22no—
38#14519765—22no—
39#14519765—22no—
40#14519765D414yes—
41#14519762—28no—
42#14519762—28no—
43C4×Dic122—22no—
44C4×Dic122—22no—
45C4×D244—22no—
46C4×D244—22no—
47C4×D244—22no—
48C4×D244D414yes—
49#14519739—28no—
50C2×C4⋊Dic61C2×C411yescharacteristic
51C2×#11758196—21no—
52C2×#11758198—21no—
53C2×#11758198—21no—
54C2×#11758198—21no—
55C2×#11758198—21no—
56C2×#11758198—21no—
57C2×#11758198—21no—
58C2×C4×Dic61—21no—
59C4⋊#92418—22no—
60C22⋊#92418—22no—
61#14519688—22no—
62C22×Q8×C61D411yescharacteristic
63C2×M16×C61—22no—
64C4×Q8×C61—22no—
65C4×Q8×C61—22no—
66C2×C4⋊C4×C61D411yescharacteristic
67C2×C4⋊C4×C61D411yescharacteristic
68C2×C4⋊C4×C61D411yescharacteristic
69C2×C42×C61C2×C411yescharacteristic
70C2×C42×C61C2×C411yescharacteristic
71C2×C42×C61C2311yescharacteristic
72#101×C61—24no—
73#97×C61—22no—
74C4×C8×C61—22no—
75C23×D61—21no—
76C23×D61—21no—
77C2×Q8×C61—22no—
78C2×Q8×C61—22no—
79C2×Q8×C61—41no—
80C22×C4×C61C22⋊C411yescharacteristic
81C22×C4×C61C22⋊C411yescharacteristic
82C22×C4×C61C22⋊C411yescharacteristic
83C22×C4×C61C22⋊C411yescharacteristic
84C22×C4×C61C22×C411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×Dic61—21no—
88C22×Dic61—21no—
89C2×D244—24no—
90C2×D244—21no—
91C2×D244—21no—
92C2×D244—24no—
93C2×D244—42no—
94C2×D244—42no—
95C2×D244—42no—
96C2×D244—42no—
97C2×D244—82no—
98C2×D244C22⋊C412yes—
99C2×D244C22⋊C412yes—
100C2×C4×D61—24no—
101C2×C4×D61—24no—
102C2×C4×D61—82no—
103C2×C4×D61—82no—
104C2×Dic122—21no—
105C2×Dic122—42no—
106C2×Dic122C22⋊C412yes—
107M16×C61—24no—
108M16×C61—42no—
109C2×C8×C61—42no—
110C4⋊C4×C61—22no—
111C4⋊C4×C61—22no—
112C4⋊C4×C61—22no—
113C42×C61—21no—
114C42×C61—21no—
115C42×C61—21no—
116C42×C61—21no—
117C42×C61C2×D412yes—
118C42×C61C2×D412yes—
119#11758198—24no—
120#11758198—24no—
121#11758198—24no—
122#11758198—24no—
123#11758198—24no—
124#11758198—24no—
125C4⋊Dic61—21no—
126C4⋊Dic61C22⋊C412yes—
127#11758196—24no—
128C4×Dic61—24no—
129C2×#92418—42no—
130C23×C61C2×C22⋊C411yescharacteristic
131C22×D61—22no—
132C22×D61—22no—
133C22×D61—42no—
134C22×D61—42no—
135C22×D61—42no—
136C22×D61—42no—
137C22×D61—41no—
138C22×D61—41no—
139C22×D61—41no—
140C22×D61—41no—
141Q8×C61—44no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—21no—
146C2×C4×C61—22no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61#10412yes—
156C2×C4×C61C2×C22⋊C411yescharacteristic
157C2×C4×C61C2×C22⋊C411yescharacteristic
158C2×C4×C61#12012yes—
159C2×C4×C61#12012yes—
160C2×Dic61—22no—
161C2×Dic61—22no—
162C2×Dic61—42no—
163C2×Dic61—42no—
164D244—24no—
165D244—42no—
166D244—42no—
167D244—48no—
168D244—42no—
169D244—84no—
170C4×D61—84no—
171C4×D61—84no—
172Dic122—42no—
173Dic122—42no—
174C8×C61—44no—
175#92418—82no—
176#8370—611nomaximal, Sylow
177C22×C61—21no—
178C22×C61—21no—
179C22×C61#37911yescharacteristic
180C22×C61C2×#10411yescharacteristic
181C22×C61#42111yescharacteristic
182C2×D61—42no—
183C2×D61—42no—
184C2×D61—42no—
185C2×D61—42no—
186C2×D61—42no—
187C2×D61—42no—
188C2×D61—82no—
189C2×D61—82no—
190C2×D61—82no—
191C2×D61—82no—
192C4×C61—22no—
193C4×C61—21no—
194C4×C61—22no—
195C4×C61—22no—
196C4×C61—41no—
197C4×C61—41no—
198C4×C61—41no—
199C4×C61#41312yes—
200Dic61—42no—
201Dic61—42no—
202C2×#559—611no—
203C4×#101D6111yescharacteristic, complemented
204#1527—612no—
205#1490—612no—
206#1467—611no—
207C2×C61—21no—
208C2×C61—21no—
209C2×C61#178511yescharacteristic
210C2×C61#178911yescharacteristic
211C2×C61#179511yescharacteristic
212D61—82no—
213D61—82no—
214#559—1224no—
215#559—1224no—
216C2×#127—611no—
217C2×#124—611no—
218C2×#124—611no—
219C2×#124—1221no—
220C2×#123—1221no—
221C2×#123—1221no—
222C2×C4×Q8—611no—
223C2×C4×Q8C2×D6111yescharacteristic
224C2×C4×D4—611no—
225C2×C4×D4—611no—
226C2×C4×D4—1221no—
227#432—612no—
228C2×#101—21no—
229C4×M16C2×D6112yes—
230#343—21no—
231#329—1224no—
232D4⋊C8—1224no—
233#324—1221no—
234#324—1221no—
235C61#837011yescharacteristic, complemented, Sylow
236C22×Q8—611no—
237C22×Q8C22⋊D6111yescharacteristic
238C22×D4—611no—
239C22×D4—611no—
240C22×D4—1221no—
241C22×D4—1221no—
242C23×C4—1221no—
243C23×C4—1221no—
244C2×M16—22no—
245#127—1222no—
246#127—1222no—
247#124—614no—
248#124—1222no—
249#124—1222no—
250#124—1228no—
251#124—1222no—
252#123—1228no—
253#123—1228no—
254C4×Q8—22no—
255C4×Q8—22no—
256C4×Q8—1222no—
257C4×Q8—1222no—
258C4×D4—614no—
259C4×D4—1222no—
260C4×D4—1222no—
261C4×D4—1228no—
262C4×D4—1222no—
263C2×C4⋊C4—611no—
264C2×C4⋊C4—1221no—
265C2×C4⋊C4D24411yescharacteristic
266C2×C4⋊C4D24411yescharacteristic
267C2×C4⋊C4C22⋊D6111yescharacteristic
268C2×C22⋊C4—1221no—
269C2×C22⋊C4—1221no—
270C2×C22⋊C4—1221no—
271C2×C22⋊C4—1221no—
272C2×C22⋊C4—1221no—
273C2×C22⋊C4—1221no—
274C2×C42—1221no—
275C2×C42C4×D6111yescharacteristic
276C2×C42C4×D6111yescharacteristic
277C2×C42C22×D6111yescharacteristic
278C4⋊C8—1222no—
279#101—24no—
280C22⋊C8—1222no—
281#97—22no—
282#97—1222no—
283C4×C8—22no—
284C24—1221no—
285C24—1221no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—41no—
289C2×Q8—612no—
290C2×Q8—1221no—
291C2×Q8—2442no—
292C2×D4—612no—
293C2×D4—612no—
294C2×D4—1224no—
295C2×D4—1224no—
296C2×D4—1221no—
297C2×D4—1221no—
298C2×D4—4882no—
299C2×D4—2442no—
300C2×D4—2442no—
301C2×D4—2442no—
302C2×D4—2442no—
303C22×C4—1224no—
304C22×C4—1221no—
305C22×C4—1221no—
306C22×C4—1224no—
307C22×C4—4882no—
308C22×C4—4882no—
309C22×C4#1175819811yescharacteristic
310C22×C4#1175819811yescharacteristic
311C22×C4#1175819811yescharacteristic
312C22×C4#1175819811yescharacteristic
313C22×C4C2×C4×D6111yescharacteristic
314C22×C4C2×D24411yescharacteristic
315C22×C4C2×C22⋊D6111yescharacteristic
316M16—24no—
317M16—42no—
318C2×C8—42no—
319C2×C8—2442no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—612no—
324C4⋊C4—1224no—
325C4⋊C4—1221no—
326C22⋊C4—1224no—
327C22⋊C4—1224no—
328C22⋊C4—1224no—
329C22⋊C4—1224no—
330C22⋊C4—1224no—
331C22⋊C4—1224no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42—21no—
336C42—1224no—
337C42D4×D6112yes—
338C42D4×D6112yes—
339C23—1222no—
340C23—1222no—
341C23—2442no—
342C23—2442no—
343C23—2442no—
344C23—2442no—
345C23—2441no—
346C23—2441no—
347C23—2441no—
348C23—2441no—
349C23C2×#1175819811yescharacteristic
350Q8—44no—
351Q8—2442no—
352Q8—2442no—
353D4—1224no—
354D4—4884no—
355D4—2442no—
356D4—2442no—
357D4—2448no—
358D4—2442no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—1222no—
373C2×C4—1222no—
374C2×C4—4884no—
375C2×C4—4884no—
376C2×C4—2442no—
377C2×C4—2442no—
378C2×C4#1451970512yes—
379C2×C4C22⋊C4×D6111yescharacteristic
380C2×C4C22⋊C4×D6111yescharacteristic
381C2×C4#1451974512yes—
382C2×C4#1451974712yes—
383C8—44no—
384C8—4882no—
385C22—21no—
386C22—21no—
387C22—4882no—
388C22—4882no—
389C22—4882no—
390C22—4882no—
391C22—2442no—
392C22—2442no—
393C22—2442no—
394C22—2442no—
395C22—2442no—
396C22—2442no—
397C22#1683961711yescharacteristic
398C22C2×#1451970511yescharacteristic
399C22#1683978011yescharacteristic
400C4—21no—
401C4—22no—
402C4—22no—
403C4—22no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—2442no—
408C4—2442no—
409C4#1683974512yes—
410C2—21no—
411C2—21no—
412C2—4882no—
413C2—4882no—
414C2#2144574611yescharacteristic
415C2#2144576411yescharacteristic
416C2#2144579011yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC4×#101×C61C2
socleC22×C61#379
prime radicalC2×#14519774C4
cosocleC2×C42×C61C23
hypercenterC4×#101D61
hypocenterC61#8370
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
2. #21444416C2semidirect
3. #21444263C2semidirect
4. #21444163C2semidirect
5. C4×#101×C61C2semidirect
203. C4×#101D61semidirect
235. C61#8370semidirect

Maximal subgroups

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

subgroupindex
C2×#168404972
#214444162
#214442632
#214441632
C4×#101×C612
#837061