Finite Group Databasebeta

← #30601197

Subgroups — #30601197

Order 15616.

542 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 94 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#21461147C212yesmaximal, complemented
4#21460672C212yesmaximal, complemented
5#21451172C211yesmaximal, characteristic
6#16840789—24no—
7#16840779C2212yes—
8#16840765C2212yes—
9C2×#14519837—21no—
10D61⋊(C22⋊C8)—21no—
11C2×C22⋊#92420—21no—
12#16840642—21no—
13Q8⋊#92420—24no—
14C2×Q8×Dic61C2211yescharacteristic
15C2×#14519737C2211yescharacteristic
16#16839699C411yescharacteristic
17#16839698C411yescharacteristic
18#16839673C411yescharacteristic
19#16839662—21no—
20#16839659—21no—
21#16839637—21no—
22#16839636—21no—
23#16839626C2211yescharacteristic
24C61×#393C411yescharacteristic
25C2×#11758217—22no—
26C2×D61⋊C8—22no—
27C22×#92420—22no—
28#14519844—22no—
29#14519844—22no—
30C4⋊#92420—22no—
31#14519837—42no—
32#14519834—22no—
33C22⋊#92420—22no—
34C22⋊#92420—22no—
35C4×#92420D414yes—
36C22×C22⋊D61D411yescharacteristic
37C23×Dic61D411yescharacteristic
38C22×C4×D61D411yescharacteristic
39Q8×Dic61—22no—
40Q8×Dic61—22no—
41#14519737—22no—
42#14519737—22no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61—21no—
45C2×C4⋊Dic61C2×C411yescharacteristic
46C2×#11758196—21no—
47C2×#11758196—21no—
48C2×#11758196D411yescharacteristic
49C2×C22⋊Dic61—21no—
50C2×#11758198—21no—
51C2×#11758198—21no—
52C2×#11758198C2×C411yescharacteristic
53C2×C4×Dic61—21no—
54C2×C4×Dic61C2311yescharacteristic
55#14519684—21no—
56#14519684—21no—
57#14519684C2×C411yescharacteristic
58C22×Q8×C61C2×C411yescharacteristic
59C2×C22⋊C4×C61—21no—
60C2×C22⋊C4×C61C2×C411yescharacteristic
61#95×C61—21no—
62#95×C61C2×C411yescharacteristic
63C24×C61C22⋊C411yescharacteristic
64C23×D61C22⋊C411yescharacteristic
65C2×Q8×C61—22no—
66C2×Q8×C61—42no—
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⋊D61SD1612yes—
74C2×C22⋊D61Q1612yes—
75C22×Dic61—21no—
76C22×Dic61—21no—
77C22×Dic61—21no—
78C22×Dic61—21no—
79C22×Dic61—81no—
80C22×Dic61C22⋊C411yescharacteristic
81C22×Dic61C2×D411yescharacteristic
82C22×Dic61C2×D411yescharacteristic
83C2×C4×D61—21no—
84C2×C4×D61—21no—
85C2×C4×D61—81no—
86#11758217—42no—
87#11758217—42no—
88C2×#92420—24no—
89C2×#92420—42no—
90C2×#92420—82no—
91D61⋊C8—82no—
92C22⋊C4×C61—22no—
93C22⋊C4×C61—42no—
94C22⋊Dic61—81no—
95#11758198—22no—
96#11758198—42no—
97#11758198—81no—
98C4⋊Dic61—22no—
99C4⋊Dic61—41no—
100C4⋊Dic61—41no—
101C4⋊Dic61—81no—
102#11758196—22no—
103#11758196—42no—
104#11758196—81no—
105C4×Dic61—42no—
106C4×Dic61C2×D412yes—
107C4×Dic61#4112yes—
108C23×C61—21no—
109C23×C61—21no—
110C23×C61—41no—
111C23×C61—41no—
112C23×C61#10312yes—
113C23×C61C2×C22⋊C411yescharacteristic
114C22×D61—21no—
115C22×D61—21no—
116C22×D61—41no—
117C22×D61—41no—
118C22×D61#10312yes—
119Q8×C61—24no—
120Q8×C61—42no—
121C2×C4×C61—22no—
122C2×C4×C61—21no—
123C2×C4×C61—41no—
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×C61C4×D412yes—
131C22⋊D61—24no—
132C22⋊D61—24no—
133C2×Dic61—21no—
134C2×Dic61—41no—
135C2×Dic61—42no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61—41no—
140C2×Dic61—41no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143C2×Dic61—81no—
144C2×Dic61#12011yescharacteristic
145C2×Dic61#12112yes—
146C2×Dic61#12311yescharacteristic
147C2×Dic61C2×SD1611yescharacteristic
148C2×Dic61C2×Q1611yescharacteristic
149C2×Dic61#13512yes—
150C4×D61—42no—
151C4×D61—81no—
152#92420—44no—
153#92420—82no—
154#9483—611nomaximal, Sylow
155C22×C61—21no—
156C22×C61—21no—
157C22×C61—22no—
158C22×C61—42no—
159C22×C61—41no—
160C22×C61—41no—
161C22×C61—81no—
162C22×C61#38611yescharacteristic
163C22×C61C2×#10311yescharacteristic
164C22×C61#41611yescharacteristic
165C2×D61—22no—
166C2×D61—41no—
167C2×D61—41no—
168C2×D61—42no—
169C2×D61—81no—
170C4×C61—22no—
171C4×C61—42no—
172C4×C61—41no—
173C4×C61—81no—
174Dic61—21no—
175Dic61—21no—
176Dic61—41no—
177Dic61—81no—
178Dic61—81no—
179Dic61#46512yes—
180Dic61#46712yes—
181#2972—611no—
182#2580—611no—
183#1800—611no—
184#1487—612no—
185#1286—612no—
186C2×C61—21no—
187C2×C61—21no—
188C2×C61—42no—
189C2×C61#189111yescharacteristic
190C2×C61#189211yescharacteristic
191C2×C61#191111yescharacteristic
192D61—42no—
193C2×C4×Q8—611no—
194C2×C4×D4—611no—
195C8×D4—1224no—
196#431—612no—
197#423—612no—
198C2×#100—1221no—
199C2×C22⋊C8—1221no—
200C2×C22⋊C8—1221no—
201#399—611no—
202#399—611no—
203#393—611no—
204#393#692311yescharacteristic
205#390—1221no—
206#389—1221no—
207#385—1221no—
208#385—1221no—
209#382—611no—
210#343—1221no—
211Q8⋊C8—1224no—
212C61#948311yescharacteristic, complemented, Sylow
213C22×Q8C2×#692311yescharacteristic
214C22×D4—611no—
215C23×C4—611no—
216C23×C4—611no—
217C2×M16—1222no—
218C22×C8—1222no—
219C22×C8—1222no—
220C4×Q8—1222no—
221C4×Q8—1222no—
222C4×D4—1222no—
223C4×D4—1222no—
224C2×C4⋊C4—611no—
225C2×C4⋊C4—611no—
226C2×C4⋊C4—1221no—
227C2×C4⋊C4—1221no—
228C2×C4⋊C4—1221no—
229C2×C4⋊C4—1221no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—611no—
232C2×C22⋊C4—1221no—
233C2×C22⋊C4—1221no—
234C2×C22⋊C4—1221no—
235C2×C22⋊C4C2×#692311yescharacteristic
236C2×C42—611no—
237C2×C42—1221no—
238C4⋊C8—1222no—
239C4⋊C8—1222no—
240C4⋊C8—1222no—
241#100—2442no—
242C22⋊C8—1222no—
243C22⋊C8—1222no—
244C22⋊C8—1222no—
245C4×C8—614no—
246#95—21no—
247#95—611no—
248#95—1221no—
249#95—1221no—
250#95C2×#692311yescharacteristic
251C24—611no—
252C24C22⋊#692311yescharacteristic
253C2×Q8—22no—
254C2×Q8—42no—
255C2×D4—612no—
256C2×D4—612no—
257C2×D4—1222no—
258C2×D4—1222no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—611no—
262C22×C4—611no—
263C22×C4—611no—
264C22×C4—1221no—
265C22×C4—1221no—
266C22×C4—1221no—
267C22×C4—1221no—
268C22×C4—1221no—
269C22×C4—1221no—
270C22×C4—4881no—
271C22×C4—4881no—
272C22×C4C22⋊#692311yescharacteristic
273C22×C4C22×#692311yescharacteristic
274M16—2442no—
275M16—2442no—
276C2×C8—1224no—
277C2×C8—4882no—
278C2×C8—4882no—
279C2×C8—2442no—
280C4⋊C4—1222no—
281C4⋊C4—1222no—
282C4⋊C4—4881no—
283C4⋊C4—4881no—
284C4⋊C4—2442no—
285C4⋊C4—2441no—
286C4⋊C4—2441no—
287C22⋊C4—22no—
288C22⋊C4—42no—
289C22⋊C4—1222no—
290C22⋊C4—4881no—
291C22⋊C4—4881no—
292C22⋊C4—2442no—
293C42—612no—
294C42—612no—
295C42—2442no—
296C23—21no—
297C23—21no—
298C23—41no—
299C23—41no—
300C23—612no—
301C23—1221no—
302C23—1221no—
303C23—2441no—
304C23—2441no—
305C23#1451984012yes—
306C23C2×C22⋊#692311yescharacteristic
307Q8—24no—
308Q8—42no—
309D4—1224no—
310D4—1224no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—612no—
321C2×C4—611no—
322C2×C4—611no—
323C2×C4—612no—
324C2×C4—611no—
325C2×C4—611no—
326C2×C4—1221no—
327C2×C4—4881no—
328C2×C4—4881no—
329C2×C4—2442no—
330C2×C4—2441no—
331C2×C4—2442no—
332C2×C4—2441no—
333C2×C4—2441no—
334C2×C4—2441no—
335C2×C4—2441no—
336C2×C4—2441no—
337C2×C4—2441no—
338C2×C4—2441no—
339C2×C4D4×#692312yes—
340C8—4882no—
341C8—2444no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—42no—
346C22—41no—
347C22—41no—
348C22—81no—
349C22—1222no—
350C22—4881no—
351C22—2442no—
352C22—2441no—
353C22—2441no—
354C22#1684071011yescharacteristic
355C22C2×#1451984011yescharacteristic
356C22#1684075211yescharacteristic
357C4—22no—
358C4—42no—
359C4—41no—
360C4—81no—
361C4—612no—
362C4—612no—
363C4—1221no—
364C4—1221no—
365C4—4881no—
366C4—4881no—
367C4—2441no—
368C2—21no—
369C2—21no—
370C2—42no—
371C2—2442no—
372C2#2146183011yescharacteristic
373C2#2146183211yescharacteristic
374C2#2146186411yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#393C4
socleC22×C61#386
prime radicalC22×C22⋊D61D4
cosocleC2×C4×Dic61C23
hypercenter#393#6923
hypocenterC61#9483
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #393

Complements

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

NH ≅ G/Ntype
3. #21461147C2semidirect
4. #21460672C2semidirect
212. C61#9483semidirect

Maximal subgroups

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

subgroupindex
#214626682
#214616842
#214611472
#214606722
#214511722
#948361