Finite Group Databasebeta

← #30599828

Subgroups — #30599828

Order 15616.

648 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840766C211yesmaximal, characteristic
2C4×#14519834C211yesmaximal, characteristic
3C2×#16840612C212yesmaximal, complemented
4#21460655C212yesmaximal, complemented
5#21448349C211yesmaximal, characteristic
6C22×#11758213—21no—
7#16840766—24no—
8C2×C4⋊#92420—21no—
9C2×C4⋊#92420C2212yes—
10C2×#14519834—21no—
11C2×C4×#92420C2212yes—
12#16840628—21no—
13#16840612—24no—
14C2×#14519757C411yescharacteristic
15C2×#14519742C2211yescharacteristic
16C2×C42×D61C2211yescharacteristic
17#16839668C411yescharacteristic
18#16839645—21no—
19#16839633—21no—
20#16839619C411yescharacteristic
21C4⋊C4×Dic61—21no—
22C4⋊C4×Dic61C2211yescharacteristic
23#16839609—21no—
24C61×#389C411yescharacteristic
25C2×#11758213—24no—
26C2×#11758213—42no—
27C2×#11758213—42no—
28C22×#92420—22no—
29C22×#92420—22no—
30C4⋊#92420—24no—
31C4⋊#92420—22no—
32C4⋊#92420D414yes—
33#14519834—24no—
34C4×#92420—22no—
35C4×#92420—22no—
36C22×Dic122C2×C411yescharacteristic
37C22×C4×D61C2×C411yescharacteristic
38C22×C4×D61D411yescharacteristic
39C22×C4×D61D411yescharacteristic
40#14519757—22no—
41#14519757—22no—
42#14519742—22no—
43#14519742—22no—
44C2×C4⋊Dic61—21no—
45C2×C4⋊Dic61C2×C411yescharacteristic
46C2×#11758196—21no—
47C2×#11758198—21no—
48C2×#11758198—21no—
49C2×C4×Dic61—21no—
50C2×C4×Dic61—21no—
51C2×C4×Dic61D411yescharacteristic
52C2×C4×Dic61D411yescharacteristic
53C2×C4×Dic61C2311yescharacteristic
54C42×D61—22no—
55C42×D61—22no—
56#14519684—21no—
57#14519684—21no—
58#14519684—21no—
59#14519684—21no—
60C2×C4⋊C4×C61—21no—
61C2×C4⋊C4×C61C2×C411yescharacteristic
62C2×C4⋊C4×C61C2×C411yescharacteristic
63C2×C42×C61C2×C411yescharacteristic
64#95×C61—21no—
65C23×D61C22⋊C411yescharacteristic
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61C22⋊C411yescharacteristic
69C22×C4×C61C22⋊C411yescharacteristic
70C22×C4×C61C22×C411yescharacteristic
71C22×Dic61—21no—
72C22×Dic61—21no—
73C22×Dic61C22⋊C411yescharacteristic
74C22×Dic61C2×D411yescharacteristic
75C22×Dic61C2×D411yescharacteristic
76C2×C4×D61—21no—
77C2×C4×D61—21no—
78C2×C4×D61—21no—
79C2×C4×D61—21no—
80C2×C4×D61—22no—
81C2×C4×D61—22no—
82C2×C4×D61—21no—
83C2×C4×D61—21no—
84C2×C4×D61—22no—
85C2×C4×D61—22no—
86C2×C4×D61—41no—
87C2×C4×D61SD1612yes—
88C2×C4×D61Q1612yes—
89C2×Dic122—22no—
90C2×Dic122—22no—
91C2×Dic122—21no—
92C2×Dic122M1612yescomplemented
93C2×#92420—24no—
94C2×#92420—24no—
95C2×#92420—42no—
96C2×#92420—42no—
97C2×#92420—42no—
98C2×#92420—42no—
99#11758213—48no—
100C4⋊C4×C61—21no—
101C4⋊C4×C61—22no—
102C4⋊C4×C61—42no—
103C4⋊C4×C61M1612yescomplemented
104C42×C61—22no—
105#11758198—24no—
106#11758198—42no—
107C4⋊Dic61—24no—
108C4⋊Dic61—22no—
109#11758196—42no—
110C4×Dic61—21no—
111C4×Dic61—22no—
112C4×Dic61—21no—
113C4×Dic61—41no—
114C4×Dic61—41no—
115C4×Dic61—41no—
116C4×Dic61—41no—
117C4×Dic61C2×D412yes—
118C4×Dic61#4112yes—
119C23×C61C2×C22⋊C411yescharacteristic
120C22×D61—21no—
121C22×D61—21no—
122C22×D61—21no—
123C22×D61—21no—
124C22×D61—21no—
125C22×D61—21no—
126C22×D61#10312yes—
127C2×C4×C61—21no—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
130C2×C4×C61—21no—
131C2×C4×C61—22no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61#10312yes—
138C2×C4×C61#10412yes—
139C2×C4×C61C4×D412yes—
140C2×C4×C61C2×M1611yescharacteristic
141C2×C4×C61C2×M1611yescharacteristic
142C2×C4×C61#13112yes—
143C2×Dic61—21no—
144C2×Dic61—21no—
145C2×Dic61—21no—
146C2×Dic61—21no—
147C2×Dic61—22no—
148C2×Dic61—21no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61#12011yescharacteristic
155C2×Dic61#12112yes—
156C2×Dic61#12311yescharacteristic
157C2×Dic61C2×SD1611yescharacteristic
158C2×Dic61C2×Q1611yescharacteristic
159C2×Dic61AGL(1,ℤ/8)11yescharacteristic
160C2×Dic61#13711yescharacteristic
161C4×D61—22no—
162C4×D61—22no—
163C4×D61—22no—
164C4×D61—22no—
165C4×D61—44no—
166C4×D61—42no—
167C4×D61—42no—
168Dic122—24no—
169Dic122—42no—
170#92420—44no—
171#92420—44no—
172#8942—611nomaximal, Sylow
173C22×C61#38611yescharacteristic
174C22×C61#40711yescharacteristic
175C22×C61#40811yescharacteristic
176C22×C61C2×#10311yescharacteristic
177C22×C61#41711yescharacteristic
178C22×C61C2×#10411yescharacteristic
179C22×C61#42111yescharacteristic
180C2×D61—22no—
181C2×D61—22no—
182C2×D61—22no—
183C2×D61—41no—
184C2×D61—41no—
185C2×D61—41no—
186C2×D61—41no—
187C4×C61—22no—
188C4×C61—22no—
189C4×C61—42no—
190C4×C61—81no—
191C4×C61#43512yes—
192C4×C61#43612yes—
193Dic61—22no—
194Dic61—42no—
195Dic61—41no—
196Dic61—81no—
197Dic61#45011yescharacteristic
198Dic61#45111yescharacteristic
199Dic61#48311yescharacteristic
200Dic61#48411yescharacteristic
201C2×#423—611no—
202C4×C22⋊Q8—611no—
203C4×C22⋊C8—611no—
204C2×Q8⋊C8—612no—
205#1279—612no—
206C2×C61#148711yescharacteristic
207C2×C61#148911yescharacteristic
208C2×C61#149311yescharacteristic
209C2×C61#149511yescharacteristic
210C2×C61#187011yescharacteristic
211C2×C61#187111yescharacteristic
212C2×C61#187211yescharacteristic
213D61—42no—
214C22×M16—1221no—
215C2×C22⋊Q8—611no—
216C2×C4×Q8—611no—
217C22×C42—611no—
218#423—1224no—
219C2×C4⋊C8—612no—
220C2×C4⋊C8—1221no—
221C2×C22⋊C8—1221no—
222C2×C4×C8—612no—
223#391—611no—
224#389#692311yescharacteristic
225#387—1221no—
226#385—1221no—
227#380—611no—
228C4×C4⋊C4—611no—
229C4×C4⋊C4—1221no—
230C4×C22⋊C4—1221no—
231#336—1221no—
232Q8⋊C8—1224no—
233C61#894211yescharacteristic, complemented, Sylow
234C22×Q8—611no—
235C23×C4—611no—
236C23×C4—611no—
237C23×C4—611no—
238C2×M16—1224no—
239C2×M16—2442no—
240C2×M16—2442no—
241C22×C8—1222no—
242C22×C8—1222no—
243C22⋊Q8—1222no—
244C22⋊Q8—1222no—
245C4×Q8—1222no—
246C4×Q8—1222no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—611no—
249C2×C4⋊C4—1221no—
250C2×C4⋊C4—1221no—
251C2×C4⋊C4C2×#692311yescharacteristic
252C2×C4⋊C4C2×#692311yescharacteristic
253C2×C22⋊C4—1221no—
254C2×C22⋊C4—1221no—
255C2×C42—611no—
256C2×C42—611no—
257C2×C42—611no—
258C2×C42—1222no—
259C2×C42—1222no—
260C2×C42—1221no—
261C2×C42—1221no—
262C2×C42C2×#692311yescharacteristic
263C4⋊C8—614no—
264C4⋊C8—1222no—
265C4⋊C8—1224no—
266C22⋊C8—1224no—
267C4×C8—1222no—
268C4×C8—1222no—
269#95—21no—
270#95—1221no—
271#95—1221no—
272#95—1221no—
273#95—1221no—
274C24—611no—
275C2×Q8—612no—
276C2×Q8—1221no—
277C2×Q8—1222no—
278C2×Q8—1222no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—611no—
282C22×C4—611no—
283C22×C4—612no—
284C22×C4—612no—
285C22×C4—611no—
286C22×C4—1221no—
287C22×C4—1221no—
288C22×C4—1221no—
289C22×C4—1221no—
290C22×C4—1222no—
291C22×C4—1221no—
292C22×C4—1222no—
293C22×C4—1222no—
294C22×C4—1221no—
295C22×C4—1221no—
296C22×C4—1222no—
297C22×C4—1221no—
298C22×C4—2441no—
299C22×C4C22⋊#692311yescharacteristic
300C22×C4C22⋊#692311yescharacteristic
301C22×C4C22×#692311yescharacteristic
302M16—2448no—
303C2×C8—1224no—
304C2×C8—1224no—
305C2×C8—2442no—
306C2×C8—2442no—
307C2×C8—2442no—
308C2×C8—2442no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—42no—
312C4⋊C4—1224no—
313C4⋊C4—1222no—
314C4⋊C4—2442no—
315C4⋊C4#1175821312yescomplemented
316C22⋊C4—1224no—
317C22⋊C4—2442no—
318C42—22no—
319C42—612no—
320C42—612no—
321C42—1221no—
322C42—1221no—
323C42—1222no—
324C42—2441no—
325C42—2441no—
326C42—2441no—
327C42—2441no—
328C23—612no—
329C23—1221no—
330C23—1221no—
331C23—1221no—
332C23—1221no—
333C23—1221no—
334C23—1221no—
335C23C2×C22⋊#692311yescharacteristic
336Q8—1224no—
337Q8—2442no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—611no—
349C2×C4—611no—
350C2×C4—611no—
351C2×C4—611no—
352C2×C4—611no—
353C2×C4—611no—
354C2×C4—612no—
355C2×C4—1222no—
356C2×C4—1222no—
357C2×C4—1222no—
358C2×C4—1222no—
359C2×C4—1221no—
360C2×C4—1221no—
361C2×C4—1221no—
362C2×C4—1222no—
363C2×C4—1221no—
364C2×C4—1221no—
365C2×C4—2444no—
366C2×C4—2441no—
367C2×C4—2441no—
368C2×C4—2441no—
369C2×C4—2441no—
370C2×C4—2441no—
371C2×C4—2442no—
372C2×C4—2442no—
373C2×C4#1451984012yes—
374C2×C4#1451984112yes—
375C2×C4D4×#692312yes—
376C2×C4D61⋊M1611yescharacteristic
377C2×C4C2×#1175821311yescharacteristic
378C2×C4#1451986512yes—
379C8—2444no—
380C8—2444no—
381C22—1222no—
382C22—1222no—
383C22—1222no—
384C22—2441no—
385C22—2441no—
386C22—2441no—
387C22—2441no—
388C22#1684071011yescharacteristic
389C22#1684073311yescharacteristic
390C22#1684073711yescharacteristic
391C22C2×#1451984011yescharacteristic
392C22#1684075311yescharacteristic
393C22C2×#1451984111yescharacteristic
394C22#1684075911yescharacteristic
395C4—22no—
396C4—22no—
397C4—42no—
398C4—81no—
399C4—611no—
400C4—611no—
401C4—611no—
402C4—611no—
403C4—1222no—
404C4—4881no—
405C4—2441no—
406C4—2442no—
407C4#1684079212yes—
408C4#1684079612yes—
409C2—2442no—
410C2#2146114811yescharacteristic
411C2#2146115511yescharacteristic
412C2#2146116911yescharacteristic
413C2#2146117711yescharacteristic
414C2#2146179911yescharacteristic
415C2#2146180211yescharacteristic
416C2#2146180411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#389C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#389#6923
hypocenterC61#8942
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
3. C2×#16840612C2semidirect
4. #21460655C2semidirect
92. C2×Dic122M16semidirect
103. C4⋊C4×C61M16semidirect
233. C61#8942semidirect
315. C4⋊C4#11758213semidirect

Maximal subgroups

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

subgroupindex
C2×#168407662
C4×#145198342
C2×#168406122
#214606552
#214483492
#894261