Finite Group Databasebeta

← #29574987

Subgroups — #29574987

Order 15616.

782 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 107 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#21449034C211yesmaximal, characteristic
2#21445762C212yesmaximal, complemented
3C4×#14519692C211yesmaximal, characteristic, complemented
4#21443704C212yesmaximal, complemented
5C61×#2975C211yesmaximal, characteristic, complemented
6C22×#11758222C2211yescharacteristic
7C2×C4×C22⋊D61—21no—
8C2×#14519703—24no—
9C2×#14519692—21no—
10#16839682C411yescharacteristic, complemented
11#16839679C411yescharacteristic, complemented
12#16839677—21no—
13#16839674C411yescharacteristic, complemented
14#16839671C411yescharacteristic, complemented
15#16839670—21no—
16C4×#11758198—21no—
17C4×C22⋊Dic61C2212yes—
18C4×C22⋊Dic61C2211yescharacteristic
19#16839529—21no—
20C2×C4×D4×C61C2211yescharacteristic
21C61×#436—24no—
22C61×#431—21no—
23C61×#407—21no—
24C61×#407C2212yes—
25C2×#11758222—22no—
26C2×#11758222—22no—
27C2×#11758222—44no—
28C22×Dic122C2×C411yescharacteristic
29C22×D244C2×C411yescharacteristic
30C22×C22⋊D61—21no—
31C22×C22⋊D61—21no—
32C22×C4×D61—21no—
33C4×C22⋊D61—44no—
34C2×#11758196—21no—
35C2×C22⋊Dic61—22no—
36C2×C22⋊Dic61—21no—
37C2×C22⋊Dic61—21no—
38C2×#11758198—21no—
39C2×#11758198—21no—
40C2×#11758198—21no—
41C2×C4×Dic61—21no—
42C2×C4×Dic61—22no—
43#14519703—48no—
44#14519692—24no—
45#14519684—21no—
46#14519684—21no—
47#14519684—22no—
48#14519684C2×C411yescharacteristic
49#14519684C2×C411yescharacteristic
50C22×D4×C61D411yescharacteristic
51C23×C4×C61D411yescharacteristic
52C23×C4×C61C2311yescharacteristic
53C2×M16×C61—22no—
54C2×M16×C61D414yescomplemented
55C4×D4×C61—22no—
56C4×D4×C61—22no—
57C2×C4⋊C4×C61C2×C411yescharacteristic
58C2×C22⋊C4×C61C2×C411yescharacteristic
59C2×C22⋊C4×C61D411yescharacteristic
60C2×C42×C61D411yescharacteristic
61C4⋊C8×C61—22no—
62C22⋊C8×C61—22no—
63C22⋊C8×C61—22no—
64C22⋊C8×C61—22no—
65C4×C8×C61—22no—
66C24×C61C22⋊C411yescharacteristic
67C24×C61C2×D411yescharacteristic
68C23×D61—21no—
69C2×D4×C61—22no—
70C2×D4×C61—22no—
71C2×D4×C61—41no—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61—42no—
75C22×C4×C61—42no—
76C22×C4×C61C22⋊C411yescharacteristic
77C22×C4×C61C22⋊C411yescharacteristic
78C22×C4×C61C22⋊C411yescharacteristic
79C22×C4×C61C22×C411yescharacteristic
80C22×C4×C61C2×D412yes—
81C22×C4×C61C2×D411yescharacteristic
82C22×C4×C61#4112yes—
83C2×C22⋊D61—42no—
84C2×C22⋊D61—42no—
85C2×C22⋊D61—42no—
86C2×C22⋊D61—42no—
87C2×C22⋊D61—42no—
88C2×C22⋊D61—42no—
89C22×Dic61—21no—
90C22×Dic61—21no—
91C22×Dic61—21no—
92C22×Dic61—42no—
93#11758222—28no—
94#11758222—44no—
95#11758222—44no—
96#11758222—44no—
97C2×D244—22no—
98C2×D244—42no—
99C2×C4×D61—24no—
100C2×C4×D61—42no—
101C2×C4×D61—41no—
102C2×C4×D61—41no—
103C2×Dic122—22no—
104C2×Dic122—42no—
105M16×C61—44no—
106M16×C61—82no—
107C2×C8×C61—24no—
108C4⋊C4×C61—22no—
109C22⋊C4×C61—22no—
110C22⋊C4×C61—22no—
111C42×C61—21no—
112C42×C61—21no—
113C22⋊Dic61—24no—
114C22⋊Dic61—42no—
115C22⋊Dic61—42no—
116C22⋊Dic61—42no—
117#11758198—42no—
118#11758198—42no—
119#11758198—42no—
120#11758196—42no—
121C4×Dic61—42no—
122C4×Dic61—44no—
123C23×C61—21no—
124C23×C61—21no—
125C23×C61—21no—
126C23×C61—21no—
127C23×C61—21no—
128C23×C61—41no—
129C23×C61—41no—
130C23×C61—41no—
131C23×C61—41no—
132C23×C61C2×C22⋊C411yescharacteristic
133C23×C61#12112yes—
134C22×D61—22no—
135C22×D61—41no—
136C22×D61—41no—
137C22×D61—41no—
138C22×D61—41no—
139C22×D61—41no—
140C22×D61—41no—
141D4×C61—44no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—24no—
145C2×C4×C61—22no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—41no—
149C2×C4×C61—42no—
150C2×C4×C61—42no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—81no—
154C2×C4×C61#10412yescomplemented
155C2×C4×C61C4×D412yes—
156C2×C4×C61#12011yescharacteristic
157C2×C4×C61#12311yescharacteristic
158C22⋊D61—48no—
159C22⋊D61—48no—
160C2×Dic61—22no—
161C2×Dic61—22no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61—44no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—82no—
171C2×Dic61—82no—
172D244—24no—
173D244—42no—
174C4×D61—44no—
175C4×D61—44no—
176Dic122—24no—
177Dic122—42no—
178C8×C61—44no—
179#9010—611nomaximal, Sylow
180C22×C61—21no—
181C22×C61—21no—
182C22×C61—22no—
183C22×C61—22no—
184C22×C61—22no—
185C22×C61—42no—
186C22×C61—41no—
187C22×C61—41no—
188C22×C61—41no—
189C22×C61—81no—
190C22×C61—81no—
191C22×C61—81no—
192C22×C61#38611yescharacteristic
193C22×C61C2×#10411yescharacteristic
194C22×C61#42111yescharacteristic
195C2×D61—42no—
196C2×D61—42no—
197C2×D61—42no—
198C2×D61—81no—
199C2×D61—81no—
200C2×D61—81no—
201C2×D61—81no—
202C4×C61—22no—
203C4×C61—22no—
204C4×C61—21no—
205C4×C61—41no—
206C4×C61—41no—
207C4×C61B(4,𝔽2)12yes—
208Dic61—42no—
209Dic61—42no—
210Dic61—81no—
211Dic61—84no—
212#2975D6111yescharacteristic, complemented
213#2388—611no—
214#1789—612no—
215C4×#99—611no—
216#1297—612no—
217C2×C61—21no—
218C2×C61—21no—
219C2×C61—42no—
220C2×C61—81no—
221C2×C61#187111yescharacteristic
222C2×C61#188411yescharacteristic
223C2×C61#188911yescharacteristic
224D61—82no—
225C22×#41—611no—
226C2×C4×D4—1221no—
227C2×C4×D4C2×D6111yescharacteristic
228#436—24no—
229#431—21no—
230C2×#104—1224no—
231C2×#99—1221no—
232#407—21no—
233#407C2×D6112yes—
234#395—611no—
235#394—611no—
236#394—1221no—
237#393—611no—
238#392—611no—
239#392—1221no—
240C4×C22⋊C4—611no—
241C4×C22⋊C4—612no—
242C4×C22⋊C4—1221no—
243#342—1221no—
244C61#901011yescharacteristic, complemented, Sylow
245C2×#41—1222no—
246C2×#41—1222no—
247C2×#41—2444no—
248C22×Q8—611no—
249C22×D4—611no—
250C22×D4—1221no—
251C22×D4—1221no—
252C22×D4D24411yescharacteristic
253C23×C4—1221no—
254C23×C4C22⋊D6111yescharacteristic
255C23×C4C22×D6111yescharacteristic
256C2×M16—22no—
257C2×M16C22⋊D6114yescomplemented
258C4×D4—22no—
259C4×D4—22no—
260C4×D4—2444no—
261C2×C4⋊C4—1221no—
262C2×C4⋊C4C4×D6111yescharacteristic
263C2×C22⋊C4—1222no—
264C2×C22⋊C4—1221no—
265C2×C22⋊C4—1221no—
266C2×C22⋊C4—1221no—
267C2×C22⋊C4—1221no—
268C2×C22⋊C4—1221no—
269C2×C22⋊C4C4×D6111yescharacteristic
270C2×C22⋊C4D24411yescharacteristic
271C2×C42—1221no—
272C2×C42—1222no—
273C2×C42C22⋊D6111yescharacteristic
274C4⋊C8—22no—
275#104—2448no—
276#99—1224no—
277C22⋊C8—22no—
278C22⋊C8—22no—
279C22⋊C8—22no—
280C4×C8—22no—
281#95—611no—
282#95—611no—
283#95—1221no—
284#95—1221no—
285#95—1222no—
286C24—1221no—
287C24#1175819811yescharacteristic
288C24C2×D24411yescharacteristic
289#41—1228no—
290#41—2444no—
291#41—2444no—
292#41—2444no—
293C2×Q8—1222no—
294C2×Q8—2442no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—41no—
298C2×D4—1222no—
299C2×D4—2442no—
300C2×D4—2442no—
301C2×D4—2442no—
302C2×D4—2442no—
303C2×D4—2442no—
304C2×D4—2442no—
305C2×D4—2442no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—1224no—
311C22×C4—1221no—
312C22×C4—1221no—
313C22×C4—1221no—
314C22×C4—2442no—
315C22×C4—2442no—
316C22×C4—2441no—
317C22×C4—2441no—
318C22×C4#1175819811yescharacteristic
319C22×C4#1175819811yescharacteristic
320C22×C4#1175819811yescharacteristic
321C22×C4C2×C4×D6111yescharacteristic
322C22×C4#1175822212yes—
323C22×C4C2×C22⋊D6112yes—
324C22×C4C2×C22⋊D6111yescharacteristic
325M16—44no—
326M16—82no—
327C2×C8—24no—
328C4⋊C4—22no—
329C4⋊C4—2442no—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—1224no—
333C22⋊C4—2442no—
334C22⋊C4—2442no—
335C22⋊C4—2442no—
336C22⋊C4—2442no—
337C22⋊C4—2442no—
338C22⋊C4—2442no—
339C42—21no—
340C42—21no—
341C42—2444no—
342C42—2442no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—1222no—
353C23—2441no—
354C23—2441no—
355C23—2441no—
356C23—2441no—
357C23—2441no—
358C23—2441no—
359C23C2×#1175819811yescharacteristic
360C23#1451975112yes—
361Q8—1224no—
362Q8—2442no—
363D4—44no—
364D4—1224no—
365D4—2442no—
366D4—2448no—
367D4—2448no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—81no—
380C2×C4—1222no—
381C2×C4—1222no—
382C2×C4—4882no—
383C2×C4—4882no—
384C2×C4—2444no—
385C2×C4—2441no—
386C2×C4—2444no—
387C2×C4—2441no—
388C2×C4—2441no—
389C2×C4—2441no—
390C2×C4—2441no—
391C2×C4—2441no—
392C2×C4—2441no—
393C2×C4—2444no—
394C2×C4#1451970312yescomplemented
395C2×C4C4×C22⋊D6112yes—
396C2×C4#1451974611yescharacteristic
397C2×C4#1451976011yescharacteristic
398C8—44no—
399C22—21no—
400C22—21no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—42no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—4881no—
412C22—4881no—
413C22—4881no—
414C22—4881no—
415C22—2442no—
416C22—2442no—
417C22—2442no—
418C22#1683964111yescharacteristic
419C22C2×#1451970311yescharacteristic
420C22#1683977811yescharacteristic
421C4—21no—
422C4—22no—
423C4—22no—
424C4—41no—
425C4—41no—
426C4—4881no—
427C4—4884no—
428C4—2442no—
429C4—2442no—
430C4#1683994912yes—
431C2—21no—
432C2—21no—
433C2—42no—
434C2—81no—
435C2—4882no—
436C2#2144612111yescharacteristic
437C2#2144620211yescharacteristic
438C2#2144622711yescharacteristic
subgroupG/H
centerC2×C4#14519746
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2975C2
socleC22×C61#386
prime radical#21449034C2
cosocleC23×C4×C61C23
hypercenter#2975D61
hypocenterC61#9010
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2975

Complements

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

NH ≅ G/Ntype
2. #21445762C2semidirect
3. C4×#14519692C2semidirect
4. #21443704C2semidirect
5. C61×#2975C2semidirect
10. #16839682C4semidirect
11. #16839679C4semidirect
13. #16839674C4semidirect
14. #16839671C4semidirect
54. C2×M16×C61D4semidirect
154. C2×C4×C61#104semidirect
212. #2975D61semidirect
244. C61#9010semidirect
257. C2×M16C22⋊D61semidirect
394. C2×C4#14519703semidirect

Maximal subgroups

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

subgroupindex
#214490342
#214457622
C4×#145196922
#214437042
C61×#29752
#901061