Finite Group Databasebeta

← #30600373

Subgroups — #30600373

Order 15616.

644 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519844C211yesmaximal, characteristic
2#21461675C211yesmaximal, characteristic
3C2×#16840622C212yesmaximal, complemented
4#21460697C212yesmaximal, complemented
5#21454167C211yesmaximal, characteristic
6C23×#92420—21no—
7C2×#14519844—21no—
8C2×#14519844—24no—
9C2×#14519844C2212yes—
10C2×#14519844C2212yes—
11C2×C22⋊#92420—21no—
12#16840628—21no—
13#16840622—24no—
14C2×#14519775C2211yescharacteristic
15C2×#14519756C411yescharacteristic
16C22×C4×Dic61C2211yescharacteristic
17#16839669C411yescharacteristic
18#16839667C411yescharacteristic
19#16839657—21no—
20#16839657—21no—
21#16839631C2211yescharacteristic
22#16839620—21no—
23#16839620—21no—
24C2×C22⋊Q8×C61C411yescharacteristic
25C22×#92420—24no—
26C22×#92420—22no—
27C22×#92420—22no—
28C22×#92420—42no—
29C22×#92420—42no—
30#14519844—24no—
31#14519844—22no—
32#14519844—22no—
33#14519844—22no—
34#14519844D414yes—
35C22⋊#92420—42no—
36C22×Dic122C2×C411yescharacteristic
37C23×Dic61D411yescharacteristic
38C23×Dic61D411yescharacteristic
39#14519775—22no—
40#14519775—22no—
41#14519756—42no—
42C2×C4⋊Dic61C2×C411yescharacteristic
43C2×#11758196—21no—
44C2×#11758196—21no—
45C2×#11758196—21no—
46C2×#11758196—21no—
47C2×#11758196—21no—
48C2×C22⋊Dic61—21no—
49C2×C4×Dic61—22no—
50C2×C4×Dic61—22no—
51C2×C4×Dic61C2×C411yescharacteristic
52C2×C4×Dic61D411yescharacteristic
53C2×C4×Dic61D411yescharacteristic
54C2×C4×Dic61C2311yescharacteristic
55#14519684—21no—
56#14519684—21no—
57#14519684—21no—
58#14519684—21no—
59C22×Q8×C61C2×C411yescharacteristic
60C23×C4×C61C2×C411yescharacteristic
61C22⋊Q8×C61—42no—
62C2×C4⋊C4×C61—21no—
63C2×C4⋊C4×C61C2×C411yescharacteristic
64C2×C22⋊C4×C61—21no—
65C24×C61C22⋊C411yescharacteristic
66C2×Q8×C61—22no—
67C2×Q8×C61—41no—
68C2×Q8×C61—41no—
69C22×C4×C61—22no—
70C22×C4×C61—22no—
71C22×C4×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61—41no—
74C22×C4×C61C22⋊C411yescharacteristic
75C22×C4×C61C22×C411yescharacteristic
76C22×Dic61—21no—
77C22×Dic61—22no—
78C22×Dic61—21no—
79C22×Dic61—22no—
80C22×Dic61—21no—
81C22×Dic61—21no—
82C22×Dic61—21no—
83C22×Dic61—21no—
84C22×Dic61C22⋊C411yescharacteristic
85C22×Dic61C22⋊C411yescharacteristic
86C22×Dic61SD1612yes—
87C22×Dic61SD1612yes—
88C22×Dic61Q1612yes—
89C22×Dic61Q1612yes—
90C22×Dic61C2×D411yescharacteristic
91C22×Dic61C2×D411yescharacteristic
92C2×Dic122—22no—
93C2×Dic122—41no—
94C2×Dic122—41no—
95C2×#92420—24no—
96C2×#92420—48no—
97C2×#92420—42no—
98C2×#92420—42no—
99C2×#92420—42no—
100C2×#92420—42no—
101C2×#92420—42no—
102C4⋊C4×C61—42no—
103C4⋊C4×C61—41no—
104C22⋊C4×C61—42no—
105C22⋊Dic61—42no—
106C4⋊Dic61—41no—
107#11758196—42no—
108#11758196—41no—
109#11758196—42no—
110#11758196—41no—
111#11758196—42no—
112#11758196—41no—
113#11758196—41no—
114C4×Dic61—22no—
115C4×Dic61—21no—
116C4×Dic61—22no—
117C4×Dic61—21no—
118C4×Dic61C2×D412yes—
119C4×Dic61#4112yes—
120C23×C61—21no—
121C23×C61—21no—
122C23×C61—21no—
123C23×C61—21no—
124C23×C61#10012yes—
125C23×C61#10312yes—
126C23×C61#10312yes—
127C23×C61C2×C22⋊C411yescharacteristic
128Q8×C61—82no—
129C2×C4×C61—21no—
130C2×C4×C61—21no—
131C2×C4×C61—21no—
132C2×C4×C61—21no—
133C2×C4×C61—21no—
134C2×C4×C61—44no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61C4×D412yes—
142C2×Dic61—21no—
143C2×Dic61—21no—
144C2×Dic61—21no—
145C2×Dic61—21no—
146C2×Dic61—22no—
147C2×Dic61—22no—
148C2×Dic61—24no—
149C2×Dic61—24no—
150C2×Dic61—22no—
151C2×Dic61—22no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61#10312yes—
159C2×Dic61#10312yes—
160C2×Dic61#12011yescharacteristic
161C2×Dic61#12112yes—
162C2×Dic61#12311yescharacteristic
163C2×Dic61C2×SD1611yescharacteristic
164C2×Dic61C2×SD1611yescharacteristic
165C2×Dic61C2×Q1611yescharacteristic
166C2×Dic61C2×Q1611yescharacteristic
167C2×Dic61#13512yes—
168C2×Dic61AGL(1,ℤ/8)11yescharacteristic
169C2×Dic61#13711yescharacteristic
170Dic122—82no—
171#92420—44no—
172#92420—82no—
173#9127—611nomaximal, Sylow
174C22×C61—22no—
175C22×C61—22no—
176C22×C61—22no—
177C22×C61—22no—
178C22×C61—22no—
179C22×C61—22no—
180C22×C61#33314yescomplemented
181C22×C61#38611yescharacteristic
182C22×C61C2×#10011yescharacteristic
183C22×C61#41311yescharacteristic
184C22×C61C2×#10311yescharacteristic
185C22×C61C2×#10311yescharacteristic
186C22×C61#41611yescharacteristic
187C22×C61#41711yescharacteristic
188C4×C61—22no—
189C4×C61—41no—
190C4×C61—81no—
191C4×C61—81no—
192Dic61—22no—
193Dic61—22no—
194Dic61—81no—
195Dic61—81no—
196Dic61#45011yescharacteristic
197Dic61#45111yescharacteristic
198Dic61#46512yes—
199Dic61#46712yes—
200Dic61#48311yescharacteristic
201Dic61#48411yescharacteristic
202C22×C4⋊C8—611no—
203#2857—611no—
204#1797—611no—
205C2×#333—612no—
206#1296—612no—
207C2×C61—24no—
208C2×C61C2×#33311yescharacteristic
209C2×C61#154211yescharacteristic
210C2×C61#155011yescharacteristic
211C2×C61#155111yescharacteristic
212C2×C61#187211yescharacteristic
213C2×C61#189211yescharacteristic
214C2×C61#190011yescharacteristic
215C23×C8—1221no—
216C2×C4⋊Q8—611no—
217C2×C22⋊Q8—611no—
218C2×C22⋊Q8#692311yescharacteristic
219C22×C42—611no—
220C2×C4⋊C8—612no—
221C2×C4⋊C8—612no—
222C2×C4⋊C8—1224no—
223C2×C4⋊C8—1221no—
224C2×C22⋊C8—1221no—
225#391—611no—
226#391—611no—
227#389—1221no—
228#389—1221no—
229#384—611no—
230#380—1221no—
231#380—1221no—
232#336—1221no—
233#333—1224no—
234C61#912711yescharacteristic, complemented, Sylow
235C22×Q8—611no—
236C22×Q8C2×#692311yescharacteristic
237C23×C4—611no—
238C23×C4—611no—
239C23×C4C2×#692311yescharacteristic
240C22×C8—1222no—
241C22×C8—1224no—
242C22×C8—1222no—
243C22×C8—2442no—
244C22×C8—2442no—
245C4⋊Q8—1222no—
246C4⋊Q8—1222no—
247C22⋊Q8—42no—
248C22⋊Q8—2442no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—611no—
251C2×C4⋊C4—1221no—
252C2×C4⋊C4—1221no—
253C2×C4⋊C4—1221no—
254C2×C4⋊C4—1221no—
255C2×C4⋊C4—1221no—
256C2×C4⋊C4C2×#692311yescharacteristic
257C2×C22⋊C4—21no—
258C2×C22⋊C4—1221no—
259C2×C42—611no—
260C2×C42—611no—
261C2×C42—611no—
262C2×C42—611no—
263C2×C42—1222no—
264C2×C42—1222no—
265C4⋊C8—614no—
266C4⋊C8—1222no—
267C4⋊C8—1222no—
268C4⋊C8—1222no—
269C4⋊C8—1224no—
270C22⋊C8—2442no—
271#95—1221no—
272#95—1221no—
273#95—1221no—
274#95—1221no—
275C24C22⋊#692311yescharacteristic
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—1222no—
280C2×Q8—2441no—
281C2×Q8—2441no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—41no—
287C22×C4—611no—
288C22×C4—611no—
289C22×C4—612no—
290C22×C4—612no—
291C22×C4—612no—
292C22×C4—612no—
293C22×C4—611no—
294C22×C4—611no—
295C22×C4—1221no—
296C22×C4—1222no—
297C22×C4—1221no—
298C22×C4—1221no—
299C22×C4—1222no—
300C22×C4—1221no—
301C22×C4—1221no—
302C22×C4—1221no—
303C22×C4C22⋊#692311yescharacteristic
304C22×C4C22×#692311yescharacteristic
305C2×C8—1224no—
306C2×C8—2442no—
307C2×C8—2442no—
308C2×C8—2442no—
309C2×C8—2442no—
310C2×C8—2442no—
311C2×C8—2448no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—2441no—
315C4⋊C4—2442no—
316C4⋊C4—2442no—
317C4⋊C4—2441no—
318C4⋊C4—2441no—
319C4⋊C4—2441no—
320C4⋊C4—2442no—
321C4⋊C4—2441no—
322C22⋊C4—42no—
323C22⋊C4—2442no—
324C42—612no—
325C42—612no—
326C42—1221no—
327C42—1222no—
328C42—1221no—
329C42—1222no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23#1451983712yes—
335C23#1451984012yes—
336C23#1451984012yes—
337C23C2×C22⋊#692311yescharacteristic
338Q8—82no—
339Q8—4882no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—44no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—611no—
353C2×C4—612no—
354C2×C4—611no—
355C2×C4—611no—
356C2×C4—611no—
357C2×C4—611no—
358C2×C4—611no—
359C2×C4—612no—
360C2×C4—611no—
361C2×C4—611no—
362C2×C4—612no—
363C2×C4—612no—
364C2×C4—1222no—
365C2×C4—1222no—
366C2×C4—1222no—
367C2×C4—1222no—
368C2×C4—1224no—
369C2×C4—1224no—
370C2×C4—1221no—
371C2×C4—1221no—
372C2×C4—1221no—
373C2×C4—1221no—
374C2×C4—2441no—
375C2×C4—2441no—
376C2×C4—2441no—
377C2×C4—2441no—
378C2×C4—2441no—
379C2×C4—2441no—
380C2×C4D4×#692312yes—
381C8—4882no—
382C8—2444no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22#1684062214yescomplemented
390C22#1684071011yescharacteristic
391C22C2×#1451983711yescharacteristic
392C22#1684074911yescharacteristic
393C22C2×#1451984011yescharacteristic
394C22C2×#1451984011yescharacteristic
395C22#1684075211yescharacteristic
396C22#1684075311yescharacteristic
397C4—22no—
398C4—41no—
399C4—81no—
400C4—81no—
401C4—611no—
402C4—611no—
403C4—611no—
404C4—611no—
405C4—612no—
406C4—612no—
407C4—1222no—
408C4—1222no—
409C4—4881no—
410C4—4881no—
411C2—24no—
412C2C2×#1684062211yescharacteristic
413C2#2146132111yescharacteristic
414C2#2146133711yescharacteristic
415C2#2146133811yescharacteristic
416C2#2146180411yescharacteristic
417C2#2146183211yescharacteristic
418C2#2146184411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C22⋊Q8×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenterC2×C22⋊Q8#6923
hypocenterC61#9127
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#16840622C2semidirect
4. #21460697C2semidirect
180. C22×C61#333semidirect
234. C61#9127semidirect
389. C22#16840622semidirect

Maximal subgroups

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

subgroupindex
C22×#145198442
#214616752
C2×#168406222
#214606972
#214541672
#912761