Finite Group Databasebeta

← #30601015

Subgroups — #30601015

Order 15616.

558 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 96 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#21462668C211yesmaximal, characteristic
2#21461684C211yesmaximal, characteristic
3#21461328C212yesmaximal, complemented
4#21460717C212yesmaximal, complemented
5#21450810C211yesmaximal, characteristic
6#16840789—24no—
7#16840779C2212yes—
8#16840765C2212yes—
9C2×#14519837—21no—
10D61⋊(C22⋊C8)—21no—
11C2×C22⋊#92420—21no—
12#16840642—21no—
13#16840621—24no—
14C2×#14519776C2211yescharacteristic
15C2×#14519763—21no—
16C2×#14519737C2211yescharacteristic
17#16839689C411yescharacteristic
18#16839688C411yescharacteristic
19#16839687C411yescharacteristic
20#16839666—21no—
21#16839621—21no—
22#16839620—21no—
23#16839613C2211yescharacteristic
24C61×#397C411yescharacteristic
25C2×#11758217—22no—
26C2×D61⋊C8—22no—
27C22×#92420—22no—
28#14519844—22no—
29#14519844D414yes—
30C4⋊#92420—22no—
31#14519837—42no—
32#14519834—22no—
33C22⋊#92420—22no—
34C22⋊#92420—22no—
35C4×#92420—22no—
36C22×Dic122—21no—
37C22×C22⋊D61D411yescharacteristic
38C23×Dic61D411yescharacteristic
39C22×C4×D61D411yescharacteristic
40#14519776—22no—
41#14519776—22no—
42#14519763—42no—
43#14519763—42no—
44#14519737—22no—
45#14519737—22no—
46C2×C4⋊Dic61C2×C411yescharacteristic
47C2×#11758196—21no—
48C2×#11758196—21no—
49C2×#11758196D411yescharacteristic
50C2×C22⋊Dic61—21no—
51C2×#11758198—21no—
52C2×#11758198—21no—
53C2×#11758198C2×C411yescharacteristic
54C2×C4×Dic61—21no—
55C2×C4×Dic61C2311yescharacteristic
56#14519684—21no—
57#14519684—21no—
58#14519684—21no—
59#14519684C2×C411yescharacteristic
60C2×C4⋊C4×C61C2×C411yescharacteristic
61C2×C22⋊C4×C61—21no—
62C2×C22⋊C4×C61C2×C411yescharacteristic
63#95×C61—21no—
64#95×C61C2×C411yescharacteristic
65C24×C61C22⋊C411yescharacteristic
66C23×D61C22⋊C411yescharacteristic
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61C22⋊C411yescharacteristic
70C22×C4×C61C22×C411yescharacteristic
71C2×C22⋊D61—22no—
72C2×C22⋊D61—22no—
73C2×C22⋊D61D812yes—
74C2×C22⋊D61SD1612yes—
75C22×Dic61—21no—
76C22×Dic61—21no—
77C22×Dic61—21no—
78C22×Dic61—21no—
79C22×Dic61—42no—
80C22×Dic61C22⋊C411yescharacteristic
81C22×Dic61C2×D411yescharacteristic
82C22×Dic61C2×D411yescharacteristic
83C2×C4×D61—21no—
84C2×C4×D61—21no—
85C2×C4×D61—42no—
86C2×Dic122—42no—
87C2×Dic122—41no—
88C2×Dic122—41no—
89C2×Dic122—42no—
90#11758217—42no—
91#11758217—42no—
92C2×#92420—24no—
93C2×#92420—42no—
94C2×#92420—82no—
95D61⋊C8—82no—
96C4⋊C4×C61—22no—
97C22⋊C4×C61—22no—
98C22⋊C4×C61—42no—
99C22⋊Dic61—42no—
100#11758198—22no—
101#11758198—42no—
102#11758198—42no—
103C4⋊Dic61—22no—
104#11758196—22no—
105#11758196—42no—
106#11758196—41no—
107#11758196—41no—
108C4×Dic61—41no—
109C4×Dic61—41no—
110C4×Dic61C2×D412yes—
111C4×Dic61#4112yes—
112C23×C61—21no—
113C23×C61—21no—
114C23×C61—41no—
115C23×C61—41no—
116C23×C61#10212yes—
117C23×C61C2×C22⋊C411yescharacteristic
118C22×D61—21no—
119C22×D61—21no—
120C22×D61—41no—
121C22×D61—41no—
122C22×D61#10212yes—
123C2×C4×C61—21no—
124C2×C4×C61—41no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61C4×D412yes—
133C22⋊D61—24no—
134C22⋊D61—24no—
135C2×Dic61—22no—
136C2×Dic61—21no—
137C2×Dic61—42no—
138C2×Dic61—41no—
139C2×Dic61—41no—
140C2×Dic61—41no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143C2×Dic61—41no—
144C2×Dic61—41no—
145C2×Dic61—81no—
146C2×Dic61#12011yescharacteristic
147C2×Dic61#12112yes—
148C2×Dic61#12311yescharacteristic
149C2×Dic61C2×D811yescharacteristic
150C2×Dic61C2×SD1611yescharacteristic
151C2×Dic61#13512yes—
152C4×D61—42no—
153C4×D61—81no—
154Dic122—82no—
155Dic122—82no—
156#92420—44no—
157#92420—82no—
158#9393—611nomaximal, Sylow
159C22×C61—21no—
160C22×C61—21no—
161C22×C61—22no—
162C22×C61—42no—
163C22×C61—41no—
164C22×C61—41no—
165C22×C61—81no—
166C22×C61#38611yescharacteristic
167C22×C61C2×#10211yescharacteristic
168C22×C61#41611yescharacteristic
169C2×D61—22no—
170C2×D61—41no—
171C2×D61—41no—
172C2×D61—42no—
173C2×D61—81no—
174C4×C61—22no—
175C4×C61—41no—
176C4×C61—81no—
177C4×C61—81no—
178Dic61—21no—
179Dic61—41no—
180Dic61—81no—
181Dic61—81no—
182Dic61#45512yes—
183Dic61#46512yes—
184Dic61#46612yes—
185#2972—611no—
186#2547—611no—
187#1800—611no—
188#1546—612no—
189#1303—612no—
190C2×C61—21no—
191C2×C61—21no—
192C2×C61—42no—
193C2×C61#188511yescharacteristic
194C2×C61#189311yescharacteristic
195C2×C61#191511yescharacteristic
196D61—42no—
197C2×C4⋊Q8—611no—
198C2×#124—1221no—
199C2×C4×D4—611no—
200C8×D4—1224no—
201#431—612no—
202#423—612no—
203C2×#100—1221no—
204C2×C22⋊C8—1221no—
205C2×C22⋊C8—1221no—
206#397—611no—
207#397—611no—
208#397—611no—
209#397#692311yescharacteristic
210#391—1221no—
211#380—1221no—
212#380—1221no—
213C4×C4⋊C4—611no—
214#343—1221no—
215#332—1224no—
216C61#939311yescharacteristic, complemented, Sylow
217C22×Q8—1221no—
218C22×D4—611no—
219C23×C4—611no—
220C23×C4—611no—
221C2×M16—1222no—
222C22×C8—1222no—
223C22×C8—1222no—
224C4⋊Q8—1222no—
225C4⋊Q8—1222no—
226#124—2442no—
227#124—2442no—
228C4×D4—1222no—
229C4×D4—1222no—
230C2×C4⋊C4—611no—
231C2×C4⋊C4—611no—
232C2×C4⋊C4—1221no—
233C2×C4⋊C4—1221no—
234C2×C4⋊C4C2×#692311yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—611no—
237C2×C22⋊C4—1221no—
238C2×C22⋊C4—1221no—
239C2×C22⋊C4—1221no—
240C2×C22⋊C4C2×#692311yescharacteristic
241C2×C42—611no—
242C2×C42—1221no—
243C4⋊C8—614no—
244C4⋊C8—1222no—
245C4⋊C8—1222no—
246#100—2442no—
247C22⋊C8—1222no—
248C22⋊C8—1222no—
249C22⋊C8—1222no—
250C4×C8—1222no—
251#95—21no—
252#95—611no—
253#95—1221no—
254#95—1221no—
255#95—1221no—
256#95C2×#692311yescharacteristic
257C24—611no—
258C24C22⋊#692311yescharacteristic
259C2×Q8—2441no—
260C2×Q8—2441no—
261C2×Q8—2442no—
262C2×Q8—2442no—
263C2×D4—612no—
264C2×D4—612no—
265C2×D4—1222no—
266C2×D4—1222no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—611no—
270C22×C4—611no—
271C22×C4—611no—
272C22×C4—1221no—
273C22×C4—1221no—
274C22×C4—1221no—
275C22×C4—1221no—
276C22×C4—1221no—
277C22×C4—1221no—
278C22×C4—2442no—
279C22×C4—2442no—
280C22×C4C22⋊#692311yescharacteristic
281C22×C4C22×#692311yescharacteristic
282M16—2442no—
283M16—2442no—
284C2×C8—1224no—
285C2×C8—4882no—
286C2×C8—4882no—
287C2×C8—2442no—
288C4⋊C4—22no—
289C4⋊C4—1222no—
290C4⋊C4—1222no—
291C4⋊C4—2441no—
292C4⋊C4—2441no—
293C4⋊C4—2442no—
294C22⋊C4—22no—
295C22⋊C4—42no—
296C22⋊C4—1222no—
297C22⋊C4—2442no—
298C22⋊C4—2442no—
299C22⋊C4—2442no—
300C42—612no—
301C42—612no—
302C42—2441no—
303C42—2441no—
304C23—21no—
305C23—21no—
306C23—41no—
307C23—41no—
308C23—612no—
309C23—1221no—
310C23—1221no—
311C23—2441no—
312C23—2441no—
313C23#1451983912yes—
314C23C2×C22⋊#692311yescharacteristic
315Q8—4882no—
316Q8—4882no—
317D4—1224no—
318D4—1224no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—611no—
329C2×C4—612no—
330C2×C4—611no—
331C2×C4—611no—
332C2×C4—612no—
333C2×C4—611no—
334C2×C4—1222no—
335C2×C4—1221no—
336C2×C4—4881no—
337C2×C4—4881no—
338C2×C4—2442no—
339C2×C4—2442no—
340C2×C4—2441no—
341C2×C4—2441no—
342C2×C4—2441no—
343C2×C4—2441no—
344C2×C4—2441no—
345C2×C4—2441no—
346C2×C4—2441no—
347C2×C4D4×#692312yes—
348C8—4882no—
349C8—2444no—
350C22—21no—
351C22—21no—
352C22—22no—
353C22—42no—
354C22—41no—
355C22—41no—
356C22—81no—
357C22—1222no—
358C22—4881no—
359C22—2442no—
360C22—2441no—
361C22—2441no—
362C22#1684071011yescharacteristic
363C22C2×#1451983911yescharacteristic
364C22#1684075211yescharacteristic
365C4—22no—
366C4—41no—
367C4—81no—
368C4—81no—
369C4—612no—
370C4—612no—
371C4—612no—
372C4—1221no—
373C4—4881no—
374C4—4881no—
375C4—2441no—
376C2—21no—
377C2—21no—
378C2—42no—
379C2—2442no—
380C2#2146182411yescharacteristic
381C2#2146183311yescharacteristic
382C2#2146187111yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#397C4
socleC22×C61#386
prime radicalC22×C22⋊D61D4
cosocleC2×C4×Dic61C23
hypercenter#397#6923
hypocenterC61#9393
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #397

Complements

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

NH ≅ G/Ntype
3. #21461328C2semidirect
4. #21460717C2semidirect
216. C61#9393semidirect

Maximal subgroups

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

subgroupindex
#214626682
#214616842
#214613282
#214607172
#214508102
#939361