Finite Group Databasebeta

← #29583242

Subgroups — #29583242

Order 15616.

602 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 77 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#21445764C212yesmaximal
2#21445736C211yesmaximal, characteristic
3#21443727C211yesmaximal, characteristic, complemented
4#21443672C212yesmaximal, complemented
5C61×#2607C211yesmaximal, characteristic, complemented
6C22×#11758222C2211yescharacteristic
7C2×#14519705—22no—
8C2×#14519705—22no—
9C2×#14519691—21no—
10C2×#14519691—21no—
11#16839726C2212yes—
12#16839625C2211yescharacteristic
13C61×#399—21no—
14C61×#393—21no—
15C61×#391—22no—
16C61×#387—22no—
17C61×#381C2211yescharacteristic
18C4×C22⋊C4×C61C2212yes—
19C2×#11758222—22no—
20C2×#11758222—22no—
21C2×#11758222—42no—
22C2×#11758222—42no—
23C22×Dic122Q811yescharacteristic, complemented
24C22×D244Q811yescharacteristic, complemented
25C22×C22⋊D61—21no—
26C22×C4×D61—21no—
27C2×#11758194—22no—
28C2×C22⋊Dic61—21no—
29#14519705—44no—
30#14519705—44no—
31#14519691—42no—
32#14519691—42no—
33C22⋊#92418—22no—
34C22⋊#92418—22no—
35#14519684—22no—
36C22×Q8×C61—21no—
37C23×C4×C61C2311yescharacteristic
38C2×C4⋊C4×C61—22no—
39C2×C4⋊C4×C61—21no—
40C2×C22⋊C4×C61—22no—
41C2×C22⋊C4×C61D411yescharacteristic
42C2×C22⋊C4×C61D411yescharacteristic
43C2×C42×C61D412yes—
44C2×C42×C61D412yes—
45#95×C61—21no—
46#95×C61—22no—
47#95×C61—21no—
48#95×C61—22no—
49#95×C61—22no—
50C24×C61C2×D411yescharacteristic
51C23×D61—21no—
52C2×Q8×C61—44no—
53C2×Q8×C61—81no—
54C22×C4×C61—22no—
55C22×C4×C61—21no—
56C22×C4×C61—22no—
57C22×C4×C61—21no—
58C22×C4×C61—41no—
59C22×C4×C61—41no—
60C22×C4×C61C2×D412yes—
61C22×C4×C61C2×Q811yescharacteristic
62C22×C4×C61#4112yes—
63C22×C4×C61#4111yescharacteristic
64C2×C22⋊D61—22no—
65C2×C22⋊D61—22no—
66C2×C22⋊D61—41no—
67C2×C22⋊D61—41no—
68C2×C22⋊D61—41no—
69C2×C22⋊D61—41no—
70C22×Dic61—21no—
71C22×Dic61—41no—
72#11758222—24no—
73#11758222—24no—
74#11758222—44no—
75#11758222—44no—
76#11758222—42no—
77#11758222—42no—
78C2×D244—22no—
79C2×D244—42no—
80C2×C4×D61—42no—
81C2×C4×D61—42no—
82C2×C4×D61—42no—
83C2×Dic122—22no—
84C2×Dic122—42no—
85C4⋊C4×C61—44no—
86C4⋊C4×C61—81no—
87C22⋊C4×C61—44no—
88C22⋊C4×C61—41no—
89C22⋊C4×C61—41no—
90C42×C61—42no—
91C42×C61—42no—
92C22⋊Dic61—41no—
93C22⋊Dic61—41no—
94#11758194—42no—
95#11758194—42no—
96C2×#92418—42no—
97C23×C61—21no—
98C23×C61—21no—
99C23×C61—42no—
100C23×C61#12011yescharacteristic
101C23×C61C22⋊Q811yescharacteristic
102C23×C61#12411yescharacteristic
103C22×D61—22no—
104C22×D61—41no—
105C22×D61—41no—
106C22×D61—82no—
107Q8×C61—84no—
108C2×C4×C61—22no—
109C2×C4×C61—22no—
110C2×C4×C61—24no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61—42no—
114C2×C4×C61—42no—
115C2×C4×C61—42no—
116C2×C4×C61—42no—
117C2×C4×C61—42no—
118C2×C4×C61—42no—
119C2×C4×C61—42no—
120C2×C4×C61—42no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61C22⋊Q812yes—
124C2×C4×C61#12412yes—
125C22⋊D61—42no—
126C22⋊D61—42no—
127C22⋊D61—42no—
128C22⋊D61—42no—
129C2×Dic61—22no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—81no—
133C2×Dic61—82no—
134D244—24no—
135D244—42no—
136C4×D61—44no—
137C4×D61—44no—
138Dic122—24no—
139Dic122—42no—
140#92418—82no—
141#10386—611nomaximal, Sylow
142C22×C61—22no—
143C22×C61—21no—
144C22×C61—21no—
145C22×C61—22no—
146C22×C61—22no—
147C22×C61—81no—
148C22×C61—81no—
149C22×C61#39311yescharacteristic
150C22×C61B(4,𝔽2)11yescharacteristic
151C22×C61B(4,𝔽2)11yescharacteristic
152C2×D61—42no—
153C2×D61—82no—
154C2×D61—82no—
155C2×D61—82no—
156C4×C61—22no—
157C4×C61—22no—
158C4×C61—44no—
159C4×C61—82no—
160C4×C61—81no—
161C4×C61—81no—
162Dic61—42no—
163Dic61—161no—
164#2607D6111yescharacteristic, complemented
165#1789—612no—
166#1782—611no—
167#1306—611no—
168#1284—612no—
169C2×C61—21no—
170C2×C61—21no—
171C2×C61—41no—
172C2×C61—41no—
173C2×C61#202911yescharacteristic
174C2×C61#203112yes—
175D61—82no—
176C22×#41—611no—
177C2×#104—1222no—
178C2×#104—1222no—
179C2×#99—1221no—
180C2×#99—1221no—
181#407—612no—
182#399—21no—
183#393—21no—
184#391—22no—
185#387—22no—
186#381—611no—
187#381C2×D6111yescharacteristic
188C4×C22⋊C4C2×D6112yes—
189C61#1038611yescharacteristic, complemented, Sylow
190C2×#41—1222no—
191C2×#41—1222no—
192C2×#41—2442no—
193C2×#41—2442no—
194C22×Q8—21no—
195C22×Q8—611no—
196C22×D4—611no—
197C22×D4—1221no—
198C23×C4—1221no—
199C23×C4C22×D6111yescharacteristic
200C2×M16—1222no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—1221no—
205C2×C22⋊C4C22⋊D6111yescharacteristic
206C2×C22⋊C4C22⋊D6111yescharacteristic
207C2×C42C22⋊D6112yes—
208C2×C42C22⋊D6112yes—
209#104—2444no—
210#104—2444no—
211#99—2442no—
212#99—2442no—
213C22⋊C8—1222no—
214C22⋊C8—1222no—
215#95—21no—
216#95—22no—
217#95—21no—
218#95—22no—
219#95—22no—
220#95—1222no—
221C24—1221no—
222C24C2×C22⋊D6111yescharacteristic
223#41—1224no—
224#41—1224no—
225#41—2442no—
226#41—2442no—
227#41—2444no—
228#41—2444no—
229C2×Q8—44no—
230C2×Q8—81no—
231C2×Q8—1222no—
232C2×Q8—2442no—
233C2×D4—1222no—
234C2×D4—1222no—
235C2×D4—1222no—
236C2×D4—2442no—
237C2×D4—2441no—
238C2×D4—2441no—
239C2×D4—2441no—
240C2×D4—2441no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—1221no—
248C22×C4—2442no—
249C22×C4—2442no—
250C22×C4—2441no—
251C22×C4—2442no—
252C22×C4Q8×D6111yescharacteristic
253C22×C4#1175822612yes—
254C22×C4#1175822611yescharacteristic
255C22×C4C2×C22⋊D6112yes—
256M16—2442no—
257M16—2442no—
258C2×C8—2442no—
259C4⋊C4—44no—
260C4⋊C4—81no—
261C22⋊C4—44no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—2441no—
265C22⋊C4—2441no—
266C42—42no—
267C42—42no—
268C23—21no—
269C23—21no—
270C23—42no—
271C23—1222no—
272C23—4882no—
273C23—2441no—
274C23—2441no—
275C23#1451974611yescharacteristic
276C23#1451975311yescharacteristic
277C23#1451976611yescharacteristic
278Q8—84no—
279Q8—1224no—
280Q8—2442no—
281D4—1224no—
282D4—2442no—
283D4—2442no—
284D4—2442no—
285D4—2442no—
286D4—2442no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—1222no—
303C2×C4—4881no—
304C2×C4—4882no—
305C2×C4—2444no—
306C2×C4—2444no—
307C2×C4—2441no—
308C2×C4—2441no—
309C2×C4#1451975312yes—
310C2×C4#1451976612yes—
311C8—4882no—
312C22—22no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—81no—
318C22—81no—
319C22—4882no—
320C22—4882no—
321C22—4882no—
322C22—2442no—
323C22#1683967311yescharacteristic
324C22#1683994911yescharacteristic
325C22#1683994911yescharacteristic
326C4—22no—
327C4—22no—
328C4—44no—
329C4—82no—
330C4—81no—
331C4—81no—
332C4—9761no—
333C4—2442no—
334C2—21no—
335C2—21no—
336C2—41no—
337C2—41no—
338C2—4882no—
339C2#2144698211yescharacteristic
340C2#2144698812yes—
subgroupG/H
centerC22#16839673
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#2607C2
socleC22×C61#393
prime radicalC22×#11758222C22
hypercenter#2607D61
hypocenterC61#10386
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2607

Complements

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

NH ≅ G/Ntype
3. #21443727C2semidirect
4. #21443672C2semidirect
5. C61×#2607C2semidirect
23. C22×Dic122Q8semidirect
24. C22×D244Q8semidirect
164. #2607D61semidirect
189. C61#10386semidirect

Maximal subgroups

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

subgroupindex
#214457642
#214457362
#214437272
#214436722
C61×#26072
#1038661