Finite Group Databasebeta

← #29590218

Subgroups — #29590218

Order 15616.

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

normal subgroups only

This group has 514 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#21451561C211yesmaximal, characteristic
2C2×#16839488C211yesmaximal, characteristic, complemented
3#21443824C212yesmaximal, complemented
4#21443821C212yesmaximal, complemented
5C2×C61×#343C211yesmaximal, characteristic, complemented
6C2×#14519760—21no—
7C2×#14519751C411yescharacteristic
8C2×C22⋊#92418—21no—
9#16839692C411yescharacteristic
10#16839692C2211yescharacteristic
11#16839684—21no—
12#16839684—21no—
13#16839679C411yescharacteristic
14#16839679C2211yescharacteristic
15#16839641—21no—
16#16839622C411yescharacteristic
17#16839488—21no—
18#16839488C2212yes—
19C22×M16×C61—21no—
20C22×C4⋊C4×C61C2211yescharacteristic
21C61×#343—21no—
22C61×#343C2212yes—
23C22×D244C2×C411yescharacteristic
24C22×C22⋊D61—21no—
25C22×#92418—41no—
26#14519760—42no—
27#14519760—42no—
28#14519751—41no—
29#14519751—41no—
30C2×C4⋊Dic61—21no—
31C2×C4⋊Dic61C2×C411yescharacteristic
32C2×#11758196—21no—
33C2×#11758196—21no—
34C2×C22⋊Dic61—21no—
35C2×#11758198—21no—
36C2×#11758198—21no—
37C2×#11758198—21no—
38C2×#11758198—21no—
39C2×#11758198—21no—
40C22⋊#92418—22no—
41C22⋊#92418—41no—
42#14519684—21no—
43#14519684—21no—
44#14519684C2×C411yescharacteristic
45#14519684C2×C411yescharacteristic
46C23×C4×C61C2×C411yescharacteristic
47C23×C4×C61D411yescharacteristic
48C23×C4×C61D411yescharacteristic
49C2×M16×C61—24no—
50C2×M16×C61—22no—
51C2×M16×C61—41no—
52C2×M16×C61—42no—
53C22×C8×C61—41no—
54C2×C4⋊C4×C61—41no—
55C2×C4⋊C4×C61—41no—
56C2×C4⋊C4×C61C2×C411yescharacteristic
57C2×C4⋊C4×C61D411yescharacteristic
58C2×C4⋊C4×C61D411yescharacteristic
59C2×C4⋊C4×C61C2311yescharacteristic
60C24×C61C22⋊C411yescharacteristic
61C23×D61—21no—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61—22no—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—81no—
70C22×C4×C61—81no—
71C22×C4×C61C22⋊C411yescharacteristic
72C22×C4×C61C22⋊C411yescharacteristic
73C22×C4×C61C22⋊C411yescharacteristic
74C22×C4×C61D812yes—
75C22×C4×C61SD1612yes—
76C22×C4×C61C22×C411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C2×C22⋊D61—41no—
80C2×C22⋊D61—41no—
81C2×C22⋊D61—42no—
82C2×C22⋊D61—42no—
83C22×Dic61—21no—
84C22×Dic61—21no—
85C22×Dic61—21no—
86C2×D244—41no—
87C2×D244—81no—
88M16×C61—24no—
89M16×C61—44no—
90M16×C61—42no—
91C2×C8×C61—44no—
92C2×C8×C61—42no—
93C4⋊C4×C61—41no—
94C4⋊C4×C61—41no—
95C4⋊C4×C61—41no—
96C4⋊C4×C61—41no—
97C22⋊Dic61—42no—
98#11758198—42no—
99#11758198—42no—
100#11758198—41no—
101#11758198—41no—
102#11758198—81no—
103#11758198—81no—
104C4⋊Dic61—41no—
105C4⋊Dic61—81no—
106#11758196—42no—
107#11758196—41no—
108#11758196—41no—
109C2×#92418—42no—
110C2×#92418—82no—
111C23×C61—41no—
112C23×C61—41no—
113C23×C61#9911yescharacteristic
114C23×C61#9911yescharacteristic
115C23×C61#10212yes—
116C23×C61#10412yes—
117C23×C61C2×C22⋊C411yescharacteristic
118C22×D61—41no—
119C22×D61—41no—
120C22×D61—41no—
121C22×D61—41no—
122C22×D61—41no—
123C22×D61—41no—
124C22×D61—41no—
125C2×C4×C61—22no—
126C2×C4×C61—22no—
127C2×C4×C61—22no—
128C2×C4×C61—22no—
129C2×C4×C61—21no—
130C2×C4×C61—21no—
131C2×C4×C61—21no—
132C2×C4×C61—21no—
133C2×C4×C61—21no—
134C2×C4×C61—21no—
135C2×C4×C61—21no—
136C2×C4×C61—21no—
137C2×C4×C61—42no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—81no—
143C2×C4×C61—81no—
144C2×C4×C61—81no—
145C2×C4×C61#10212yes—
146C2×C4×C61C2×D811yescharacteristic
147C2×C4×C61C2×SD1611yescharacteristic
148C22⋊D61—82no—
149C22⋊D61—82no—
150C2×Dic61—41no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159D244—82no—
160C8×C61—44no—
161#92418—82no—
162#11698—611nomaximal, Sylow
163C22×C61—21no—
164C22×C61—21no—
165C22×C61—21no—
166C22×C61—21no—
167C22×C61—21no—
168C22×C61—21no—
169C22×C61—42no—
170C22×C61—42no—
171C22×C61#32712yes—
172C22×C61#32712yes—
173C22×C61C2×#9911yescharacteristic
174C22×C61C2×#10211yescharacteristic
175C22×C61C2×#10411yescharacteristic
176C2×D61—81no—
177C2×D61—81no—
178C2×D61—81no—
179C2×D61—81no—
180C2×D61—81no—
181C2×D61—81no—
182C2×D61—81no—
183C4×C61—21no—
184C4×C61—22no—
185C4×C61—21no—
186C4×C61—41no—
187C4×C61—41no—
188C4×C61—41no—
189C4×C61—41no—
190Dic61—81no—
191Dic61—81no—
192Dic61—81no—
193#2618—611no—
194C2×#343D6111yescharacteristic, complemented
195C2×#324—611no—
196#1346—612no—
197#1345—612no—
198C2×C61—21no—
199C2×C61—21no—
200C2×C61—22no—
201C2×C61—22no—
202C2×C61C2×#32711yescharacteristic
203C2×C61#212111yescharacteristic
204C2×C61#213311yescharacteristic
205D61—161no—
206C22×M16—21no—
207C2×#123—1221no—
208C2×#121—611no—
209C22×C4⋊C4C2×D6111yescharacteristic
210C2×C22⋊C8—1221no—
211#398—611no—
212#398—611no—
213#396—1221no—
214#396—1221no—
215#394—611no—
216#394—611no—
217#386—1221no—
218#380—611no—
219#343—21no—
220#343C2×D6112yes—
221#324—612no—
222#324—1221no—
223C61#1169811yescharacteristic, complemented, Sylow
224C22×D4—611no—
225C22×D4—1221no—
226C23×C4C4×D6111yescharacteristic
227C23×C4D24411yescharacteristic
228C23×C4C22⋊D6111yescharacteristic
229C2×M16—24no—
230C2×M16—22no—
231C2×M16—41no—
232C2×M16—42no—
233C22×C8—41no—
234C22×C8—2441no—
235#123—2442no—
236#123—2442no—
237#121—2441no—
238#121—2441no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—611no—
242C2×C4⋊C4—1221no—
243C2×C4⋊C4—1221no—
244C2×C4⋊C4—1221no—
245C2×C4⋊C4C4×D6111yescharacteristic
246C2×C4⋊C4D24411yescharacteristic
247C2×C4⋊C4C22⋊D6111yescharacteristic
248C2×C4⋊C4C22×D6111yescharacteristic
249C2×C22⋊C4—1221no—
250C2×C22⋊C4—1221no—
251C2×C22⋊C4—1221no—
252C2×C22⋊C4—1221no—
253C2×C22⋊C4—1221no—
254C2×C22⋊C4—1221no—
255C22⋊C8—1222no—
256C22⋊C8—2441no—
257#95—611no—
258#95—611no—
259#95—1221no—
260#95—1221no—
261C24—1221no—
262C24#1175819811yescharacteristic
263C2×D4—4881no—
264C2×D4—2441no—
265C2×D4—2441no—
266C2×D4—2441no—
267C2×D4—2442no—
268C2×D4—2442no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—81no—
277C22×C4—81no—
278C22×C4—1221no—
279C22×C4—1221no—
280C22×C4—1221no—
281C22×C4#1175819811yescharacteristic
282C22×C4#1175819811yescharacteristic
283C22×C4#1175819811yescharacteristic
284C22×C4#1175819912yes—
285C22×C4#1175820112yes—
286C22×C4C2×C4×D6111yescharacteristic
287C22×C4C2×D24411yescharacteristic
288C22×C4C2×C22⋊D6111yescharacteristic
289M16—24no—
290M16—44no—
291M16—42no—
292C2×C8—42no—
293C2×C8—44no—
294C2×C8—4882no—
295C2×C8—2442no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—4881no—
301C4⋊C4—2441no—
302C4⋊C4—2442no—
303C4⋊C4—2441no—
304C4⋊C4—2441no—
305C22⋊C4—4881no—
306C22⋊C4—4881no—
307C22⋊C4—2442no—
308C22⋊C4—2442no—
309C22⋊C4—2442no—
310C22⋊C4—2441no—
311C22⋊C4—2441no—
312C23—41no—
313C23—41no—
314C23—2441no—
315C23—2441no—
316C23—2441no—
317C23—2441no—
318C23—2441no—
319C23—2441no—
320C23—2441no—
321C23#1451969111yescharacteristic
322C23#1451969111yescharacteristic
323C23#1451969712yes—
324C23#1451970512yes—
325C23C2×#1175819811yescharacteristic
326D4—4882no—
327D4—4882no—
328D4—4882no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—2441no—
350C2×C4—2441no—
351C2×C4—2441no—
352C2×C4—2441no—
353C2×C4—2441no—
354C2×C4—2441no—
355C2×C4—2441no—
356C2×C4—2441no—
357C2×C4—2441no—
358C2×C4#1451969712yes—
359C2×C4C2×#1175819911yescharacteristic
360C2×C4C2×#1175820111yescharacteristic
361C8—44no—
362C8—4882no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—42no—
370C22—42no—
371C22—4881no—
372C22—4881no—
373C22—4881no—
374C22—4881no—
375C22—4881no—
376C22—4881no—
377C22—4881no—
378C22#1683949512yes—
379C22#1683949512yes—
380C22C2×#1451969111yescharacteristic
381C22C2×#1451969711yescharacteristic
382C22C2×#1451970511yescharacteristic
383C4—21no—
384C4—22no—
385C4—21no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—41no—
390C4—4881no—
391C4—4881no—
392C4—4881no—
393C2—21no—
394C2—21no—
395C2—22no—
396C2—22no—
397C2—9761no—
398C2C2×#1683949511yescharacteristic
399C2#2144746611yescharacteristic
400C2#2144750611yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC2×C4⋊C4C22×D61
FittingC2×C61×#343C2
socleC22×C61C2×#99
prime radicalC2×#14519751C4
cosocleC2×C4⋊C4×C61C23
hypercenterC2×#343D61
hypocenterC61#11698
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
2. C2×#16839488C2semidirect
3. #21443824C2semidirect
4. #21443821C2semidirect
5. C2×C61×#343C2semidirect
194. C2×#343D61semidirect
223. C61#11698semidirect

Maximal subgroups

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

subgroupindex
#214515612
C2×#168394882
#214438242
#214438212
C2×C61×#3432
#1169861