Finite Group Databasebeta

← #29572743

Subgroups — #29572743

Order 15616.

698 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14519751C211yesmaximal, characteristic
2#21445799C211yesmaximal, characteristic, complemented
3C2×#16839495C212yesmaximal
4#21443698C212yesmaximal, complemented
5C61×#1434C211yesmaximal, characteristic, complemented
6C2×#14519751C2211yescharacteristic
7C2×C4×D244C411yescharacteristic, complemented
8C2×C4×C22⋊D61—21no—
9C2×C4⋊#92418—21no—
10C2×C22⋊#92418C2212yes—
11#16839664C411yescharacteristic, complemented
12#16839660C411yescharacteristic, complemented
13#16839653—21no—
14#16839641—21no—
15#16839622C2211yescharacteristic
16C4×C4⋊Dic61C411yescharacteristic, complemented
17C4×#11758198—21no—
18#16839517—21no—
19#16839495—24no—
20C22×C42×C61C2211yescharacteristic
21C2×#95×C61—21no—
22C2×#95×C61—21no—
23C2×#95×C61C2212yes—
24C22×D244C2×C411yescharacteristic
25C22×C22⋊D61—21no—
26C22×C4×D61—21no—
27C22×#92418—22no—
28#14519751—21no—
29#14519751—21no—
30#14519751D412yes—
31#14519751Q812yes—
32C4×D244—42no—
33C4×C22⋊D61—44no—
34C2×C4⋊Dic61C2×C411yescharacteristic
35C2×C4⋊Dic61C2×C411yescharacteristic
36C2×C4⋊Dic61C2×C411yescharacteristic
37C2×#11758196—21no—
38C2×C22⋊Dic61—21no—
39C2×#11758198—21no—
40C2×#11758198—21no—
41C2×#11758198—21no—
42C2×#11758198—21no—
43C2×C4×Dic61—21no—
44C4⋊#92418—42no—
45C22⋊#92418—22no—
46C22⋊#92418—22no—
47#14519684—21no—
48#14519684—21no—
49#14519684—21no—
50C23×C4×C61—22no—
51C23×C4×C61D411yescharacteristic
52C23×C4×C61D411yescharacteristic
53C23×C4×C61C2311yescharacteristic
54C2×C42×C61—42no—
55C2×C42×C61C2×C411yescharacteristic
56C2×C42×C61C2×C411yescharacteristic
57C2×C42×C61D411yescharacteristic
58C2×C42×C61D411yescharacteristic
59#95×C61—24no—
60#95×C61—22no—
61#95×C61—22no—
62#95×C61—42no—
63C24×C61C2×D411yescharacteristic
64C23×D61—21no—
65C22×C4×C61—22no—
66C22×C4×C61—22no—
67C22×C4×C61—22no—
68C22×C4×C61—22no—
69C22×C4×C61—24no—
70C22×C4×C61—41no—
71C22×C4×C61—41no—
72C22×C4×C61—41no—
73C22×C4×C61—41no—
74C22×C4×C61—42no—
75C22×C4×C61—42no—
76C22×C4×C61—42no—
77C22×C4×C61—42no—
78C22×C4×C61—42no—
79C22×C4×C61—42no—
80C22×C4×C61C22⋊C411yescharacteristic
81C22×C4×C61C22⋊C411yescharacteristic
82C22×C4×C61C22⋊C411yescharacteristic
83C22×C4×C61C22⋊C411yescharacteristic
84C22×C4×C61C22×C411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×Q811yescharacteristic
88C22×C4×C61#4111yescharacteristic
89C22×C4×C61#4111yescharacteristic
90C2×C22⋊D61—22no—
91C2×C22⋊D61—22no—
92C2×C22⋊D61—42no—
93C2×C22⋊D61—42no—
94C22×Dic61—21no—
95C22×Dic61—21no—
96C22×Dic61—21no—
97C2×D244—21no—
98C2×D244—41no—
99C2×D244—41no—
100C2×D244C4⋊C412yes—
101C2×C4×D61—42no—
102C2×C4×D61—42no—
103C2×C4×D61—42no—
104C42×C61—22no—
105C42×C61—22no—
106C42×C61—41no—
107C42×C61—41no—
108C22⋊Dic61—42no—
109#11758198—22no—
110#11758198—22no—
111#11758198—42no—
112#11758198—42no—
113#11758198—42no—
114C4⋊Dic61—21no—
115C4⋊Dic61—41no—
116C4⋊Dic61—41no—
117C4⋊Dic61C4⋊C412yes—
118#11758196—42no—
119C4×Dic61—42no—
120C2×#92418—42no—
121C2×#92418—42no—
122C2×#92418—42no—
123C23×C61—21no—
124C23×C61—21no—
125C23×C61—21no—
126C23×C61—21no—
127C23×C61C2×C22⋊C411yescharacteristic
128C23×C61#12111yescharacteristic
129C23×C61C22⋊Q811yescharacteristic
130C23×C61#13512yes—
131C23×C61AGL(1,ℤ/8)12yes—
132C22×D61—22no—
133C22×D61—41no—
134C22×D61—41no—
135C22×D61—41no—
136C22×D61—41no—
137C22×D61—41no—
138C22×D61—41no—
139C2×C4×C61—21no—
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—44no—
158C2×C4×C61—44no—
159C2×C4×C61—44no—
160C2×C4×C61—44no—
161C2×C4×C61—44no—
162C2×C4×C61—82no—
163C2×C4×C61—82no—
164C2×C4×C61—82no—
165C2×C4×C61—82no—
166C2×C4×C61#9912yes—
167C2×C4×C61#10412yes—
168C2×C4×C61C2×C4⋊C411yescharacteristic
169C2×C4×C61#11711yescharacteristic
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61C22⋊Q811yescharacteristic
172C22⋊D61—44no—
173C22⋊D61—44no—
174C2×Dic61—22no—
175C2×Dic61—41no—
176C2×Dic61—41no—
177C2×Dic61—41no—
178C2×Dic61—41no—
179C2×Dic61—41no—
180C2×Dic61—41no—
181C2×Dic61—41no—
182C2×Dic61—41no—
183D244—42no—
184D244—42no—
185C4×D61—84no—
186#92418—82no—
187#8573—611nomaximal, Sylow
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—22no—
193C22×C61—22no—
194C22×C61—22no—
195C22×C61—22no—
196C22×C61#38011yescharacteristic
197C22×C61C2×#9911yescharacteristic
198C22×C61#41011yescharacteristic
199C22×C61#41611yescharacteristic
200C22×C61#41811yescharacteristic
201C22×C61C2×#10411yescharacteristic
202C22×C61#42111yescharacteristic
203C22×C61#46312yes—
204C22×C61#47812yes—
205C2×D61—42no—
206C2×D61—42no—
207C2×D61—42no—
208C2×D61—81no—
209C2×D61—81no—
210C2×D61—81no—
211C2×D61—81no—
212C4×C61—21no—
213C4×C61—21no—
214C4×C61—21no—
215C4×C61—21no—
216C4×C61—22no—
217C4×C61—22no—
218C4×C61—41no—
219C4×C61—41no—
220C4×C61—84no—
221Dic61—42no—
222Dic61—81no—
223Dic61—81no—
224C4×#121—611no—
225#1797—611no—
226C2×#327—612no—
227#1434D6111yescharacteristic, complemented
228#1294—612no—
229C2×C61—22no—
230C2×C61—22no—
231C2×C61#150011yescharacteristic
232C2×C61#150711yescharacteristic
233C2×C61#151411yescharacteristic
234C2×C61#151611yescharacteristic
235C2×C61#179911yescharacteristic
236C2×C61#180411yescharacteristic
237C2×C61#180611yescharacteristic
238D61—82no—
239C2×#121—611no—
240C2×C4×D4—611no—
241C2×C4×D4—1221no—
242C22×C42C2×D6111yescharacteristic
243C2×C4⋊C8—1221no—
244C2×C22⋊C8—612no—
245#390—611no—
246#389—611no—
247#388—1221no—
248#386—1221no—
249#380—611no—
250C4×C4⋊C4—611no—
251C4×C22⋊C4—1221no—
252C2×#95—21no—
253C2×#95—21no—
254C2×#95C2×D6112yes—
255#336—1221no—
256#327—1224no—
257C61#857311yescharacteristic, complemented, Sylow
258C22×D4—611no—
259C22×D4—1221no—
260C23×C4—22no—
261C23×C4—1221no—
262C23×C4C22⋊D6111yescharacteristic
263C23×C4C22⋊D6111yescharacteristic
264C23×C4C22×D6111yescharacteristic
265C22×C8—1222no—
266#121—612no—
267#121—612no—
268#121—1221no—
269#121—1221no—
270C4×D4—2442no—
271C4×D4—2444no—
272C2×C4⋊C4—611no—
273C2×C4⋊C4—611no—
274C2×C4⋊C4—611no—
275C2×C4⋊C4—1221no—
276C2×C22⋊C4—1221no—
277C2×C22⋊C4—1221no—
278C2×C22⋊C4—1221no—
279C2×C22⋊C4—1221no—
280C2×C22⋊C4—1221no—
281C2×C42—42no—
282C2×C42—1221no—
283C2×C42C4×D6111yescharacteristic
284C2×C42C4×D6111yescharacteristic
285C2×C42D24411yescharacteristic
286C2×C42D24411yescharacteristic
287C4⋊C8—2442no—
288C22⋊C8—1222no—
289C22⋊C8—1222no—
290#95—24no—
291#95—22no—
292#95—22no—
293#95—42no—
294#95—1221no—
295#95—1221no—
296#95—1221no—
297C24—1221no—
298C24C2×C22⋊D6111yescharacteristic
299C2×D4—612no—
300C2×D4—1222no—
301C2×D4—1222no—
302C2×D4—1221no—
303C2×D4—2441no—
304C2×D4—2441no—
305C2×D4—2442no—
306C2×D4—2442no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—1221no—
323C22×C4—1221no—
324C22×C4—1221no—
325C22×C4—2442no—
326C22×C4—2442no—
327C22×C4—2442no—
328C22×C4#1175819811yescharacteristic
329C22×C4#1175819811yescharacteristic
330C22×C4#1175819811yescharacteristic
331C22×C4#1175819811yescharacteristic
332C22×C4C2×C4×D6111yescharacteristic
333C22×C4C2×D24411yescharacteristic
334C22×C4D4×D6111yescharacteristic
335C22×C4#1175822411yescharacteristic
336C22×C4Q8×D6111yescharacteristic
337C22×C4#1175822611yescharacteristic
338C2×C8—2442no—
339C2×C8—2442no—
340C2×C8—2442no—
341C4⋊C4—612no—
342C4⋊C4—1221no—
343C4⋊C4—2441no—
344C4⋊C4—2441no—
345C4⋊C4—2442no—
346C22⋊C4—1222no—
347C22⋊C4—1222no—
348C22⋊C4—2442no—
349C22⋊C4—2442no—
350C22⋊C4—2442no—
351C22⋊C4—2442no—
352C42—22no—
353C42—22no—
354C42—41no—
355C42—41no—
356C42—2442no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—1222no—
362C23—2441no—
363C23—2441no—
364C23—2441no—
365C23—2441no—
366C23—2441no—
367C23—2441no—
368C23C2×#1175819811yescharacteristic
369C23#1451974811yescharacteristic
370C23#1451975311yescharacteristic
371C23#1451979712yes—
372C23#1451980212yes—
373D4—2442no—
374D4—2442no—
375D4—2444no—
376D4—2444no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—1222no—
405C2×C4—4884no—
406C2×C4—2441no—
407C2×C4—2441no—
408C2×C4—2441no—
409C2×C4—2441no—
410C2×C4—2441no—
411C2×C4—2441no—
412C2×C4—2441no—
413C2×C4—2441no—
414C2×C4#1451969112yes—
415C2×C4#1451970512yes—
416C2×C4C4⋊C4×D6111yescharacteristic
417C2×C4#1451973511yescharacteristic
418C2×C4#1451975211yescharacteristic
419C2×C4#1451975711yescharacteristic
420C8—4882no—
421C22—22no—
422C22—22no—
423C22—21no—
424C22—21no—
425C22—22no—
426C22—21no—
427C22—21no—
428C22—22no—
429C22—4881no—
430C22—4881no—
431C22—4881no—
432C22—4881no—
433C22—2442no—
434C22—2442no—
435C22—2442no—
436C22#1683961911yescharacteristic
437C22C2×#1451969111yescharacteristic
438C22#1683973511yescharacteristic
439C22#1683975611yescharacteristic
440C22#1683976611yescharacteristic
441C22C2×#1451970511yescharacteristic
442C22#1683978011yescharacteristic
443C22#1683997412yes—
444C22#1684005912yes—
445C4—22no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—22no—
451C4—41no—
452C4—41no—
453C4—84no—
454C4—4881no—
455C4—4881no—
456C4—2442no—
457C2—22no—
458C2—22no—
459C2—4882no—
460C2#2144430311yescharacteristic
461C2#2144433411yescharacteristic
462C2#2144436511yescharacteristic
463C2#2144437311yescharacteristic
464C2#2144580411yescharacteristic
465C2#2144582611yescharacteristic
466C2#2144584011yescharacteristic
subgroupG/H
centerC23C2×#11758198
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1434C2
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519751C22
cosocleC23×C4×C61C23
hypercenter#1434D61
hypocenterC61#8573
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1434

Complements

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

NH ≅ G/Ntype
2. #21445799C2semidirect
4. #21443698C2semidirect
5. C61×#1434C2semidirect
7. C2×C4×D244C4semidirect
11. #16839664C4semidirect
12. #16839660C4semidirect
16. C4×C4⋊Dic61C4semidirect
227. #1434D61semidirect
257. C61#8573semidirect

Maximal subgroups

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

subgroupindex
C4×#145197512
#214457992
C2×#168394952
#214436982
C61×#14342
#857361