Finite Group Databasebeta

← #29577777

Subgroups — #29577777

Order 15616.

934 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 934 conjugacy classes of subgroups — too many to draw. Only its 135 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#21449175C211yesmaximal, characteristic
2#21445807C211yesmaximal, characteristic, complemented
3#21445751C212yesmaximal, complemented
4#21443687C212yesmaximal, complemented
5C61×#2972C211yesmaximal, characteristic, complemented
6C23×D244C2211yescharacteristic
7C2×#14519751—21no—
8C2×#14519745—21no—
9C2×#14519699—24no—
10C2×#14519692—21no—
11#16839679C412yescomplemented
12#16839671C412yescomplemented
13#16839664—21no—
14#16839622C2212yes—
15#16839622C2211yescharacteristic
16#16839617—21no—
17#16839529—21no—
18C2×C4×D4×C61C2211yescharacteristic
19C8×D4×C61—24no—
20C61×#431—21no—
21C2×C22⋊C8×C61—21no—
22C2×C22⋊C8×C61C2212yes—
23C24×D61—21no—
24C22×D244—22no—
25C22×D244—21no—
26C22×D244—22no—
27C22×D244—44no—
28C22×D244C2×C412yes—
29C22×C22⋊D61—22no—
30#14519751—44no—
31#14519745—42no—
32#14519745—42no—
33C2×C4⋊Dic61—21no—
34C2×C4⋊Dic61—22no—
35C2×C22⋊Dic61—21no—
36C2×C22⋊Dic61—22no—
37C2×#11758198—22no—
38C2×#11758198—22no—
39C2×#11758198—22no—
40#14519699—48no—
41#14519692—42no—
42#14519684—22no—
43#14519684C2×C412yes—
44C22×D4×C61D411yescharacteristic
45C23×C4×C61D411yescharacteristic
46C23×C4×C61C2311yescharacteristic
47C22×C8×C61—22no—
48C22×C8×C61D414yescomplemented
49C4×D4×C61—22no—
50C4×D4×C61—22no—
51C2×C4⋊C4×C61C2×C411yescharacteristic
52C2×C22⋊C4×C61C2×C411yescharacteristic
53C2×C22⋊C4×C61D411yescharacteristic
54C2×C42×C61D411yescharacteristic
55C4⋊C8×C61—22no—
56C22⋊C8×C61—22no—
57C22⋊C8×C61—22no—
58C22⋊C8×C61—22no—
59C4×C8×C61—22no—
60C24×C61C22⋊C411yescharacteristic
61C24×C61C2×D411yescharacteristic
62C23×D61—22no—
63C23×D61—44no—
64C23×D61—42no—
65C23×D61—41no—
66C23×D61—41no—
67C23×D61—42no—
68C23×D61—82no—
69C2×D4×C61—22no—
70C2×D4×C61—22no—
71C2×D4×C61D812yes—
72C2×D4×C61SD1612yes—
73C22×C4×C61—21no—
74C22×C4×C61—21no—
75C22×C4×C61—42no—
76C22×C4×C61—42no—
77C22×C4×C61C22⋊C411yescharacteristic
78C22×C4×C61C22⋊C411yescharacteristic
79C22×C4×C61C22⋊C411yescharacteristic
80C22×C4×C61C22×C411yescharacteristic
81C22×C4×C61C2×D412yes—
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61#4112yes—
84C2×C22⋊D61—44no—
85C2×C22⋊D61—44no—
86C2×C22⋊D61—82no—
87C22×Dic61—22no—
88C22×Dic61—42no—
89C2×D244—28no—
90C2×D244—24no—
91C2×D244—24no—
92C2×D244—44no—
93C2×D244—48no—
94C2×D244—44no—
95C2×D244—42no—
96C2×D244—41no—
97C2×D244—41no—
98C2×C8×C61—24no—
99C2×C8×C61—44no—
100C2×C8×C61—82no—
101C4⋊C4×C61—22no—
102C22⋊C4×C61—22no—
103C22⋊C4×C61—22no—
104C42×C61—21no—
105C42×C61—21no—
106C22⋊Dic61—41no—
107C22⋊Dic61—41no—
108C22⋊Dic61—42no—
109C22⋊Dic61—42no—
110#11758198—44no—
111#11758198—44no—
112#11758198—82no—
113C4⋊Dic61—42no—
114C4⋊Dic61—44no—
115C23×C61—21no—
116C23×C61—21no—
117C23×C61—41no—
118C23×C61—41no—
119C23×C61—41no—
120C23×C61—41no—
121C23×C61#10212yescomplemented
122C23×C61C2×C22⋊C411yescharacteristic
123C23×C61#12112yes—
124C23×C61C2×D811yescharacteristic
125C23×C61C2×SD1611yescharacteristic
126C23×C61#13512yes—
127C22×D61—24no—
128C22×D61—42no—
129C22×D61—48no—
130C22×D61—42no—
131C22×D61—42no—
132C22×D61—42no—
133C22×D61—42no—
134C22×D61—42no—
135C22×D61—84no—
136C22×D61—84no—
137C22×D61—88no—
138C22×D61—84no—
139C22×D61—82no—
140C22×D61—82no—
141D4×C61—24no—
142D4×C61—24no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—24no—
146C2×C4×C61—22no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—41no—
150C2×C4×C61—42no—
151C2×C4×C61—42no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—81no—
155C2×C4×C61#10212yescomplemented
156C2×C4×C61C4×D412yes—
157C2×C4×C61#12011yescharacteristic
158C2×C4×C61#12311yescharacteristic
159C22⋊D61—88no—
160C2×Dic61—42no—
161C2×Dic61—42no—
162C2×Dic61—42no—
163C2×Dic61—82no—
164C2×Dic61—82no—
165C2×Dic61—82no—
166D244—28no—
167D244—44no—
168D244—44no—
169D244—44no—
170C8×C61—44no—
171#9468—611nomaximal, Sylow
172C22×C61—21no—
173C22×C61—22no—
174C22×C61—21no—
175C22×C61—22no—
176C22×C61—22no—
177C22×C61—41no—
178C22×C61—41no—
179C22×C61—42no—
180C22×C61—42no—
181C22×C61—41no—
182C22×C61—41no—
183C22×C61—81no—
184C22×C61#38611yescharacteristic
185C22×C61C2×#10211yescharacteristic
186C22×C61#41611yescharacteristic
187C22×C61#46512yes—
188C22×C61#46612yes—
189C2×D61—44no—
190C2×D61—44no—
191C2×D61—44no—
192C2×D61—82no—
193C2×D61—82no—
194C2×D61—88no—
195C2×D61—88no—
196C2×D61—82no—
197C2×D61—82no—
198C4×C61—22no—
199C4×C61—21no—
200C4×C61—22no—
201C4×C61—41no—
202C4×C61—41no—
203C4×C61B(4,𝔽2)12yes—
204Dic61—82no—
205Dic61—162no—
206#2972D6111yescharacteristic, complemented
207#2408—611no—
208#1799—611no—
209#1786—612no—
210#1289—612no—
211C2×C61—21no—
212C2×C61—22no—
213C2×C61—21no—
214C2×C61—22no—
215C2×C61—42no—
216C2×C61#188911yescharacteristic
217C2×C61#189311yescharacteristic
218C2×C61#191011yescharacteristic
219D61—84no—
220C23×D4—611no—
221C2×#121—1221no—
222C2×#120—1221no—
223C2×C4×D4C2×D6111yescharacteristic
224C8×D4—24no—
225#431—21no—
226C2×#102—1224no—
227C2×#99—1221no—
228C2×C22⋊C8—21no—
229C2×C22⋊C8C2×D6112yes—
230#394—612no—
231#392—612no—
232#390—1221no—
233#380—611no—
234#380—612no—
235#379—1221no—
236#342—1221no—
237C61#946811yescharacteristic, complemented, Sylow
238C25—1221no—
239C22×D4—612no—
240C22×D4—1222no—
241C22×D4—1222no—
242C22×D4—1222no—
243C22×D4—1221no—
244C22×D4—2444no—
245C22×D4D24411yescharacteristic
246C23×C4C22⋊D6111yescharacteristic
247C23×C4C22×D6111yescharacteristic
248C22×C8—22no—
249C22×C8C22⋊D6114yescomplemented
250#121—2444no—
251#120—2442no—
252#120—2442no—
253C4×D4—22no—
254C4×D4—22no—
255C2×C4⋊C4—1221no—
256C2×C4⋊C4—1222no—
257C2×C4⋊C4C4×D6111yescharacteristic
258C2×C22⋊C4—1222no—
259C2×C22⋊C4—1221no—
260C2×C22⋊C4—1222no—
261C2×C22⋊C4—1222no—
262C2×C22⋊C4—1222no—
263C2×C22⋊C4C4×D6111yescharacteristic
264C2×C22⋊C4D24411yescharacteristic
265C2×C42C22⋊D6111yescharacteristic
266C4⋊C8—22no—
267#102—2448no—
268#99—2442no—
269C22⋊C8—22no—
270C22⋊C8—22no—
271C22⋊C8—22no—
272C4×C8—22no—
273#95—612no—
274#95—1222no—
275C24—1222no—
276C24—4882no—
277C24—2444no—
278C24—2441no—
279C24—2441no—
280C24—2442no—
281C24—2442no—
282C24#1175819811yescharacteristic
283C24C2×D24411yescharacteristic
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—1228no—
287C2×D4—1224no—
288C2×D4—1224no—
289C2×D4—4882no—
290C2×D4—2444no—
291C2×D4—2442no—
292C2×D4—2444no—
293C2×D4—2448no—
294C2×D4—2444no—
295C2×D4—2444no—
296C2×D4—2441no—
297C2×D4—2441no—
298C2×D4#1175819012yes—
299C2×D4D48812yes—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—1222no—
305C22×C4—2442no—
306C22×C4#1175819811yescharacteristic
307C22×C4#1175819811yescharacteristic
308C22×C4#1175819811yescharacteristic
309C22×C4C2×C4×D6111yescharacteristic
310C22×C4#1175822212yes—
311C22×C4C2×C22⋊D6112yes—
312C22×C4C2×C22⋊D6111yescharacteristic
313C2×C8—24no—
314C2×C8—44no—
315C2×C8—82no—
316C4⋊C4—22no—
317C4⋊C4—2444no—
318C4⋊C4—2442no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—4882no—
322C22⋊C4—2444no—
323C22⋊C4—2442no—
324C22⋊C4—2442no—
325C22⋊C4—2441no—
326C22⋊C4—2441no—
327C22⋊C4—2444no—
328C42—21no—
329C42—21no—
330C23—21no—
331C23—21no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—1224no—
337C23—4884no—
338C23—4884no—
339C23—4884no—
340C23—4888no—
341C23—4882no—
342C23—4882no—
343C23—2442no—
344C23—2448no—
345C23—2442no—
346C23—2442no—
347C23—2442no—
348C23—2442no—
349C23—2442no—
350C23#1451969912yescomplemented
351C23C2×#1175819811yescharacteristic
352C23#1451975112yes—
353C23C2×D48811yescharacteristic
354C23C2×#1175819011yescharacteristic
355C23#1451979912yes—
356D4—24no—
357D4—24no—
358D4—1228no—
359D4—4888no—
360D4—2444no—
361D4—2444no—
362D4—2444no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—81no—
375C2×C4—4882no—
376C2×C4—4882no—
377C2×C4—4882no—
378C2×C4—2442no—
379C2×C4—2442no—
380C2×C4—2442no—
381C2×C4#1451969912yescomplemented
382C2×C4C4×C22⋊D6112yes—
383C2×C4#1451974611yescharacteristic
384C2×C4#1451976011yescharacteristic
385C8—44no—
386C22—21no—
387C22—22no—
388C22—21no—
389C22—22no—
390C22—22no—
391C22—41no—
392C22—41no—
393C22—42no—
394C22—42no—
395C22—41no—
396C22—41no—
397C22—81no—
398C22—4888no—
399C22—4888no—
400C22—4882no—
401C22—4882no—
402C22—4882no—
403C22—4882no—
404C22—2444no—
405C22—2444no—
406C22—2444no—
407C22#1683964111yescharacteristic
408C22C2×#1451969911yescharacteristic
409C22#1683975911yescharacteristic
410C22#1683999112yes—
411C22#1683999812yes—
412C4—21no—
413C4—22no—
414C4—22no—
415C4—41no—
416C4—41no—
417C4—9762no—
418C4—4882no—
419C4#1683994912yes—
420C2—22no—
421C2—21no—
422C2—21no—
423C2—22no—
424C2—42no—
425C2—4884no—
426C2#2144622711yescharacteristic
427C2#2144624711yescharacteristic
428C2#2144634611yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2972C2
socleC22×C61#386
prime radical#21449175C2
cosocleC23×C4×C61C23
hypercenter#2972D61
hypocenterC61#9468
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #21445807C2semidirect
3. #21445751C2semidirect
4. #21443687C2semidirect
5. C61×#2972C2semidirect
11. #16839679C4semidirect
12. #16839671C4semidirect
48. C22×C8×C61D4semidirect
121. C23×C61#102semidirect
155. C2×C4×C61#102semidirect
206. #2972D61semidirect
237. C61#9468semidirect
249. C22×C8C22⋊D61semidirect
350. C23#14519699semidirect
381. C2×C4#14519699semidirect

Maximal subgroups

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

subgroupindex
#214491752
#214458072
#214457512
#214436872
C61×#29722
#946861