Finite Group Databasebeta

← #29591074

Subgroups — #29591074

Order 15616.

510 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 95 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#21450775C211yesmaximal, characteristic
2#21444137C211yesmaximal, characteristic, complemented
3#21443824C212yesmaximal, complemented
4#21443821C212yesmaximal, complemented
5C61×#1800C211yesmaximal, characteristic, complemented
6C2×#14519751C411yescharacteristic
7C2×#14519749—21no—
8C2×C22⋊#92418—21no—
9#16839692C2211yescharacteristic
10#16839687C411yescharacteristic
11#16839679C2211yescharacteristic
12#16839671C411yescharacteristic
13#16839653—21no—
14#16839641—21no—
15#16839639—21no—
16C2×#14519684C411yescharacteristic
17#16839488C2212yes—
18#16839486—21no—
19C2×C4×D4×C61C2211yescharacteristic
20C61×#423—21no—
21C2×#100×C61—21no—
22C61×#343C2212yes—
23C22×D244C2×C411yescharacteristic
24C22×C22⋊D61—21no—
25C22×C22⋊D61—21no—
26C23×Dic61—21no—
27C22×#92418—41no—
28#14519751—41no—
29#14519751—41no—
30#14519749—42no—
31#14519749—42no—
32C2×C4⋊Dic61C2×C411yescharacteristic
33C2×#11758196—21no—
34C2×C22⋊Dic61—21no—
35C2×#11758198—21no—
36C2×#11758198—21no—
37C2×#11758198—21no—
38C2×#11758198—21no—
39C22⋊#92418—22no—
40C22⋊#92418—41no—
41#14519684—21no—
42#14519684—21no—
43#14519684—21no—
44#14519684C2×C411yescharacteristic
45#14519684C2×C411yescharacteristic
46C22×D4×C61D411yescharacteristic
47C23×C4×C61C2×C411yescharacteristic
48C23×C4×C61D411yescharacteristic
49C2×M16×C61—22no—
50C4×D4×C61—41no—
51C4×D4×C61—41no—
52C2×C4⋊C4×C61C2311yescharacteristic
53C2×C22⋊C4×C61C2×C411yescharacteristic
54C2×C22⋊C4×C61D411yescharacteristic
55C2×C42×C61D411yescharacteristic
56C4⋊C8×C61—41no—
57C4⋊C8×C61—41no—
58#100×C61—42no—
59C24×C61C22⋊C411yescharacteristic
60C24×C61C22⋊C411yescharacteristic
61C23×D61—21no—
62C2×D4×C61—41no—
63C2×D4×C61—41no—
64C2×D4×C61D812yes—
65C2×D4×C61SD1612yes—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—81no—
71C22×C4×C61—81no—
72C22×C4×C61C22⋊C411yescharacteristic
73C22×C4×C61C22⋊C411yescharacteristic
74C22×C4×C61C22×C411yescharacteristic
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C2×C22⋊D61—42no—
78C2×C22⋊D61—42no—
79C2×C22⋊D61—42no—
80C2×C22⋊D61—42no—
81C2×C22⋊D61—42no—
82C2×C22⋊D61—81no—
83C22×Dic61—21no—
84C22×Dic61—21no—
85C22×Dic61—21no—
86C22×Dic61—42no—
87C22×Dic61—42no—
88C22×Dic61—81no—
89C2×D244—41no—
90C2×D244—81no—
91M16×C61—42no—
92M16×C61—42no—
93C2×C8×C61—42no—
94C4⋊C4×C61—41no—
95C22⋊C4×C61—41no—
96C22⋊C4×C61—41no—
97C42×C61—21no—
98C42×C61—21no—
99C22⋊Dic61—42no—
100#11758198—42no—
101#11758198—41no—
102#11758198—41no—
103#11758198—41no—
104#11758198—41no—
105#11758198—81no—
106C4⋊Dic61—41no—
107#11758196—41no—
108#11758196—41no—
109C2×#92418—42no—
110C2×#92418—82no—
111C23×C61—21no—
112C23×C61—21no—
113C23×C61—21no—
114C23×C61—21no—
115C23×C61—41no—
116C23×C61—41no—
117C23×C61—41no—
118C23×C61—41no—
119C23×C61#10212yes—
120C23×C61#10212yes—
121C23×C61C2×C22⋊C411yescharacteristic
122C22×D61—41no—
123C22×D61—41no—
124C22×D61—41no—
125C22×D61—41no—
126C22×D61—41no—
127C22×D61—41no—
128C22×D61—41no—
129D4×C61—42no—
130D4×C61—42no—
131C2×C4×C61—21no—
132C2×C4×C61—21no—
133C2×C4×C61—21no—
134C2×C4×C61—21no—
135C2×C4×C61—21no—
136C2×C4×C61—41no—
137C2×C4×C61—42no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—81no—
142C2×C4×C61—81no—
143C2×C4×C61—81no—
144C2×C4×C61#9911yescharacteristic
145C2×C4×C61#9911yescharacteristic
146C2×C4×C61C2×D811yescharacteristic
147C2×C4×C61C2×SD1611yescharacteristic
148C22⋊D61—84no—
149C22⋊D61—84no—
150C2×Dic61—22no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61—84no—
160D244—82no—
161C8×C61—82no—
162#92418—82no—
163#11884—611nomaximal, Sylow
164C22×C61—21no—
165C22×C61—21no—
166C22×C61—22no—
167C22×C61—22no—
168C22×C61—42no—
169C22×C61—42no—
170C22×C61—42no—
171C22×C61—41no—
172C22×C61—41no—
173C22×C61—41no—
174C22×C61—41no—
175C22×C61—81no—
176C22×C61C2×#9911yescharacteristic
177C22×C61C2×#10211yescharacteristic
178C22×C61#42111yescharacteristic
179C2×D61—81no—
180C2×D61—81no—
181C2×D61—81no—
182C2×D61—81no—
183C2×D61—81no—
184C2×D61—81no—
185C2×D61—81no—
186C4×C61—21no—
187C4×C61—21no—
188C4×C61—21no—
189C4×C61—21no—
190C4×C61—41no—
191C4×C61—41no—
192C4×C61—41no—
193Dic61—42no—
194Dic61—81no—
195Dic61—81no—
196#2543—611no—
197#1800D6111yescharacteristic, complemented
198#1459—611no—
199#1346—612no—
200#1345—612no—
201C2×C61—21no—
202C2×C61—21no—
203C2×C61—42no—
204C2×C61—42no—
205C2×C61#151911yescharacteristic
206C2×C61#212111yescharacteristic
207C2×C61#213211yescharacteristic
208D61—161no—
209C2×#121—611no—
210C2×#121—1221no—
211C2×C4×D4C2×D6111yescharacteristic
212#423—21no—
213C2×#100—21no—
214C2×C22⋊C8—1221no—
215#398—611no—
216#397—611no—
217#394—611no—
218#392—611no—
219#388—1221no—
220#386—1221no—
221#386—1221no—
222C2×#95—611no—
223#343C2×D6112yes—
224#324—612no—
225#323—1221no—
226C61#1188411yescharacteristic, complemented, Sylow
227C22×D4—611no—
228C22×D4—1221no—
229C22×D4—1221no—
230C22×D4C22⋊D6111yescharacteristic
231C23×C4—1221no—
232C23×C4C4×D6111yescharacteristic
233C23×C4D24411yescharacteristic
234C2×M16—22no—
235C22×C8—2441no—
236#121—2442no—
237#121—2442no—
238#121—2441no—
239#121—2441no—
240C4×D4—41no—
241C4×D4—41no—
242C2×C4⋊C4—611no—
243C2×C4⋊C4—1221no—
244C2×C4⋊C4C22×D6111yescharacteristic
245C2×C22⋊C4—1221no—
246C2×C22⋊C4—1221no—
247C2×C22⋊C4—1221no—
248C2×C22⋊C4—1221no—
249C2×C22⋊C4—1221no—
250C2×C22⋊C4C4×D6111yescharacteristic
251C2×C22⋊C4D24411yescharacteristic
252C2×C42C22⋊D6111yescharacteristic
253C4⋊C8—41no—
254C4⋊C8—41no—
255#100—42no—
256C22⋊C8—1222no—
257C22⋊C8—2441no—
258#95—611no—
259#95—611no—
260#95—1221no—
261#95—1221no—
262#95—1221no—
263C24—1221no—
264C24#1175819811yescharacteristic
265C24#1175819811yescharacteristic
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—4881no—
269C2×D4—4881no—
270C2×D4—2441no—
271C2×D4—2442no—
272C2×D4—2442no—
273C2×D4—2442no—
274C2×D4—2442no—
275C2×D4—2442no—
276C2×D4#1175819912yes—
277C2×D4#1175820112yes—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—81no—
283C22×C4—81no—
284C22×C4—1221no—
285C22×C4—1221no—
286C22×C4—1221no—
287C22×C4—4881no—
288C22×C4—2442no—
289C22×C4—2442no—
290C22×C4#1175819811yescharacteristic
291C22×C4#1175819811yescharacteristic
292C22×C4C2×C4×D6111yescharacteristic
293C22×C4C2×D24411yescharacteristic
294C22×C4C2×C22⋊D6111yescharacteristic
295M16—42no—
296M16—42no—
297C2×C8—42no—
298C2×C8—4882no—
299C2×C8—2442no—
300C4⋊C4—41no—
301C4⋊C4—2441no—
302C4⋊C4—2441no—
303C4⋊C4—2441no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—4881no—
307C22⋊C4—2442no—
308C22⋊C4—2442no—
309C22⋊C4—2441no—
310C22⋊C4—2441no—
311C22⋊C4—2441no—
312C22⋊C4—2441no—
313C42—21no—
314C42—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—2441no—
324C23—2441no—
325C23—2441no—
326C23—2441no—
327C23—2441no—
328C23—2441no—
329C23—2441no—
330C23#1451969712yes—
331C23#1451969712yes—
332C23C2×#1175819811yescharacteristic
333D4—42no—
334D4—42no—
335D4—4882no—
336D4—4884no—
337D4—4884no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—1222no—
352C2×C4—4884no—
353C2×C4—2441no—
354C2×C4—2441no—
355C2×C4—2441no—
356C2×C4—2441no—
357C2×C4—2441no—
358C2×C4—2441no—
359C2×C4—2441no—
360C2×C4—2441no—
361C2×C4#1451969111yescharacteristic
362C2×C4#1451969111yescharacteristic
363C2×C4C2×#1175819911yescharacteristic
364C2×C4C2×#1175820111yescharacteristic
365C8—82no—
366C8—4882no—
367C22—21no—
368C22—21no—
369C22—22no—
370C22—22no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—81no—
379C22—4881no—
380C22—4881no—
381C22—4881no—
382C22—4881no—
383C22—4881no—
384C22—4881no—
385C22—4881no—
386C22C2×#1451969111yescharacteristic
387C22C2×#1451969711yescharacteristic
388C22#1683978011yescharacteristic
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—4881no—
397C4—4881no—
398C4—2442no—
399C2—21no—
400C2—21no—
401C2—42no—
402C2—42no—
403C2—9761no—
404C2#2144438611yescharacteristic
405C2#2144746611yescharacteristic
406C2#2144750211yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC2×C4⋊C4C22×D61
FittingC61×#1800C2
socleC22×C61C2×#99
prime radical#21450775C2
cosocleC2×C4⋊C4×C61C23
hypercenter#1800D61
hypocenterC61#11884
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1800

Complements

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

NH ≅ G/Ntype
2. #21444137C2semidirect
3. #21443824C2semidirect
4. #21443821C2semidirect
5. C61×#1800C2semidirect
197. #1800D61semidirect
226. C61#11884semidirect

Maximal subgroups

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

subgroupindex
#214507752
#214441372
#214438242
#214438212
C61×#18002
#1188461