Finite Group Databasebeta

← #30599823

Subgroups — #30599823

Order 15616.

604 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 124 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
1C2×#16840783C211yesmaximal, characteristic
2C4×C22⋊#92420C211yesmaximal, characteristic
3C2×#16840612C212yesmaximal, complemented
4#21460655C212yesmaximal, complemented
5#21448347C211yesmaximal, characteristic
6#16840783—24no—
7C2×C4⋊#92420C2212yes—
8C2×C22⋊#92420—21no—
9C2×C22⋊#92420—21no—
10C2×#14519832—21no—
11C2×C4×#92420C2212yes—
12#16840628—21no—
13#16840612—24no—
14C2×#14519756C411yescharacteristic
15C2×#14519742C2211yescharacteristic
16C22×C4×Dic61C2211yescharacteristic
17#16839667C411yescharacteristic
18#16839658C411yescharacteristic
19#16839644—21no—
20#16839634—21no—
21C4⋊C4×Dic61C2211yescharacteristic
22#16839613—21no—
23C22⋊C4×Dic61—21no—
24C61×#380C411yescharacteristic
25C22×#92420—22no—
26C22×#92420—22no—
27C4⋊#92420—22no—
28C4⋊#92420D414yes—
29C22⋊#92420—24no—
30C22⋊#92420—24no—
31#14519832—24no—
32C4×#92420—22no—
33C4×#92420—22no—
34C22×Dic122C2×C411yescharacteristic
35C23×Dic61D411yescharacteristic
36C23×Dic61D411yescharacteristic
37#14519756—22no—
38#14519756—22no—
39#14519742—22no—
40#14519742—22no—
41C2×C4⋊Dic61C2×C411yescharacteristic
42C2×#11758196—21no—
43C2×#11758196—21no—
44C2×#11758196—21no—
45C2×C22⋊Dic61—21no—
46C2×C4×Dic61—22no—
47C2×C4×Dic61—22no—
48C2×C4×Dic61—21no—
49C2×C4×Dic61—21no—
50C2×C4×Dic61C2×C411yescharacteristic
51C2×C4×Dic61D411yescharacteristic
52C2×C4×Dic61D411yescharacteristic
53C2×C4×Dic61C2311yescharacteristic
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57#14519684—21no—
58C23×C4×C61C2×C411yescharacteristic
59C2×C4⋊C4×C61C2×C411yescharacteristic
60C2×C4⋊C4×C61C2×C411yescharacteristic
61C2×C22⋊C4×C61—21no—
62#95×C61—21no—
63C24×C61C22⋊C411yescharacteristic
64C22×C4×C61—22no—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61—22no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61C22⋊C411yescharacteristic
71C22×C4×C61C22×C411yescharacteristic
72C22×Dic61—22no—
73C22×Dic61—21no—
74C22×Dic61—21no—
75C22×Dic61—21no—
76C22×Dic61—22no—
77C22×Dic61—21no—
78C22×Dic61—22no—
79C22×Dic61—22no—
80C22×Dic61—41no—
81C22×Dic61C22⋊C411yescharacteristic
82C22×Dic61C22⋊C411yescharacteristic
83C22×Dic61C2×D411yescharacteristic
84C22×Dic61C2×D411yescharacteristic
85C2×Dic122—22no—
86C2×Dic122—22no—
87C2×Dic122—21no—
88C2×Dic122M1612yes—
89C2×#92420—24no—
90C2×#92420—24no—
91C2×#92420—42no—
92C2×#92420—42no—
93C2×#92420—42no—
94C2×#92420—42no—
95C4⋊C4×C61—22no—
96C4⋊C4×C61—22no—
97C22⋊C4×C61—42no—
98C22⋊Dic61—24no—
99C4⋊Dic61—21no—
100C4⋊Dic61M1612yes—
101#11758196—24no—
102#11758196—42no—
103#11758196—42no—
104C4×Dic61—22no—
105C4×Dic61—21no—
106C4×Dic61—22no—
107C4×Dic61—21no—
108C4×Dic61—41no—
109C4×Dic61—41no—
110C4×Dic61—42no—
111C4×Dic61C2×D412yes—
112C4×Dic61#4112yes—
113C23×C61—21no—
114C23×C61—21no—
115C23×C61—21no—
116C23×C61—21no—
117C23×C61—21no—
118C23×C61—21no—
119C23×C61#10412yes—
120C23×C61C2×C22⋊C411yescharacteristic
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—42no—
125C2×C4×C61—42no—
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—
133C2×C4×C61C2×M1611yescharacteristic
134C2×C4×C61C2×M1611yescharacteristic
135C2×C4×C61#13112yes—
136C2×Dic61—21no—
137C2×Dic61—21no—
138C2×Dic61—21no—
139C2×Dic61—21no—
140C2×Dic61—22no—
141C2×Dic61—22no—
142C2×Dic61—21no—
143C2×Dic61—21no—
144C2×Dic61—21no—
145C2×Dic61—21no—
146C2×Dic61—44no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—42no—
152C2×Dic61—42no—
153C2×Dic61#12011yescharacteristic
154C2×Dic61#12112yes—
155C2×Dic61#12311yescharacteristic
156C2×Dic61#13512yes—
157C2×Dic61#13712yes—
158Dic122—24no—
159Dic122—42no—
160#92420—44no—
161#92420—44no—
162#8941—611nomaximal, Sylow
163C22×C61—22no—
164C22×C61—22no—
165C22×C61—22no—
166C22×C61—41no—
167C22×C61—41no—
168C22×C61—41no—
169C22×C61—41no—
170C22×C61#38611yescharacteristic
171C22×C61#40711yescharacteristic
172C22×C61#40811yescharacteristic
173C22×C61#41611yescharacteristic
174C22×C61#41911yescharacteristic
175C22×C61C2×#10411yescharacteristic
176C22×C61#42111yescharacteristic
177C4×C61—41no—
178C4×C61—81no—
179C4×C61—81no—
180C4×C61#43512yes—
181C4×C61#43612yes—
182Dic61—22no—
183Dic61—42no—
184Dic61—41no—
185Dic61—41no—
186Dic61—42no—
187Dic61#45212yes—
188Dic61#48512yes—
189C2×#432—611no—
190C4×C22⋊Q8—611no—
191C4×C22⋊C8—611no—
192C2×Q8⋊C8—612no—
193#1279—612no—
194C2×C61—42no—
195C2×C61#148811yescharacteristic
196C2×C61#149111yescharacteristic
197C2×C61#149411yescharacteristic
198C2×C61#149711yescharacteristic
199C2×C61#187011yescharacteristic
200C2×C61#187111yescharacteristic
201C2×C61#187311yescharacteristic
202C2×C22⋊Q8—611no—
203C2×C4×Q8—611no—
204C22×C42—611no—
205#432—1224no—
206C2×C4⋊C8—612no—
207C2×C22⋊C8—1221no—
208C2×C22⋊C8—1221no—
209C2×#97—1221no—
210C2×C4×C8—612no—
211#391—611no—
212#389—611no—
213#387—1221no—
214#385—1221no—
215#380#692311yescharacteristic
216C4×C4⋊C4—611no—
217C4×C4⋊C4—1221no—
218C4×C22⋊C4—1221no—
219#336—1221no—
220Q8⋊C8—1224no—
221C61#894111yescharacteristic, complemented, Sylow
222C22×Q8—611no—
223C23×C4—611no—
224C23×C4—611no—
225C23×C4C2×#692311yescharacteristic
226C22×C8—1222no—
227C22×C8—1222no—
228C22⋊Q8—1222no—
229C22⋊Q8—1222no—
230C4×Q8—1222no—
231C4×Q8—1222no—
232C2×C4⋊C4—611no—
233C2×C4⋊C4—1221no—
234C2×C4⋊C4—1221no—
235C2×C4⋊C4—1221no—
236C2×C4⋊C4C2×#692311yescharacteristic
237C2×C4⋊C4C2×#692311yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—1221no—
240C2×C42—611no—
241C2×C42—611no—
242C2×C42—611no—
243C2×C42—611no—
244C2×C42—1222no—
245C2×C42—1222no—
246C2×C42—1221no—
247C2×C42—1221no—
248C4⋊C8—614no—
249C4⋊C8—1222no—
250C22⋊C8—1224no—
251C22⋊C8—1224no—
252#97—1224no—
253C4×C8—1222no—
254C4×C8—1222no—
255#95—21no—
256#95—1221no—
257#95—1221no—
258#95—1221no—
259#95—1221no—
260C24C22⋊#692311yescharacteristic
261C2×Q8—612no—
262C2×Q8—1221no—
263C2×Q8—1222no—
264C2×Q8—1222no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—611no—
272C22×C4—611no—
273C22×C4—611no—
274C22×C4—611no—
275C22×C4—1221no—
276C22×C4—1221no—
277C22×C4—1222no—
278C22×C4—1222no—
279C22×C4—1222no—
280C22×C4—1222no—
281C22×C4—1221no—
282C22×C4—1221no—
283C22×C4—2441no—
284C22×C4C22⋊#692311yescharacteristic
285C22×C4C22×#692311yescharacteristic
286C2×C8—1224no—
287C2×C8—1224no—
288C2×C8—2442no—
289C2×C8—2442no—
290C2×C8—2442no—
291C2×C8—2442no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—612no—
295C4⋊C4—1224no—
296C4⋊C4—1221no—
297C4⋊C4—2442no—
298C4⋊C4—2442no—
299C22⋊C4—42no—
300C22⋊C4—1224no—
301C42—612no—
302C42—612no—
303C42—1221no—
304C42—1221no—
305C42—1222no—
306C42—1222no—
307C42—2442no—
308C42—2441no—
309C42—2441no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23#1451984112yes—
317C23C2×C22⋊#692311yescharacteristic
318Q8—1224no—
319Q8—2442no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—611no—
332C2×C4—611no—
333C2×C4—612no—
334C2×C4—612no—
335C2×C4—612no—
336C2×C4—1221no—
337C2×C4—1221no—
338C2×C4—1221no—
339C2×C4—1221no—
340C2×C4—1222no—
341C2×C4—1222no—
342C2×C4—1221no—
343C2×C4—1221no—
344C2×C4—1221no—
345C2×C4—1221no—
346C2×C4—2442no—
347C2×C4—2442no—
348C2×C4—2444no—
349C2×C4—2441no—
350C2×C4—2441no—
351C2×C4—2441no—
352C2×C4—2441no—
353C2×C4D4×#692312yes—
354C2×C4D61⋊M1611yescharacteristic
355C2×C4D61⋊M1611yescharacteristic
356C2×C4#1451986412yes—
357C8—2444no—
358C8—2444no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22#1684071011yescharacteristic
367C22#1684073411yescharacteristic
368C22#1684073511yescharacteristic
369C22#1684075211yescharacteristic
370C22#1684075511yescharacteristic
371C22C2×#1451984111yescharacteristic
372C22#1684075911yescharacteristic
373C4—41no—
374C4—81no—
375C4—81no—
376C4—612no—
377C4—612no—
378C4—1222no—
379C4—2442no—
380C4—2441no—
381C4—2441no—
382C4—2442no—
383C4#1684079412yes—
384C4#1684079512yes—
385C2—42no—
386C2#2146115211yescharacteristic
387C2#2146116311yescharacteristic
388C2#2146117011yescharacteristic
389C2#2146118111yescharacteristic
390C2#2146179811yescharacteristic
391C2#2146180211yescharacteristic
392C2#2146180511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#380C4
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#380#6923
hypocenterC61#8941
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. C2×#16840612C2semidirect
4. #21460655C2semidirect
221. C61#8941semidirect

Maximal subgroups

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

subgroupindex
C2×#168407832
C4×C22⋊#924202
C2×#168406122
#214606552
#214483472
#894161