Finite Group Databasebeta

← #29580561

Subgroups — #29580561

Order 15616.

830 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 111 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×#16839917C211yesmaximal, characteristic, complemented
2C2×#16839903C211yesmaximal, characteristic
3#21451753C211yesmaximal, characteristic
4C2×#16839495C211yesmaximal, characteristic
5#21444089C211yesmaximal, characteristic, complemented
6#21443661C211yesmaximal, characteristic, complemented
7C2×C61×#328C211yesmaximal, characteristic, complemented
8C23×Dic122C2211yescharacteristic
9C22×#11758202—21no—
10C22×#11758190—21no—
11C2×#14519751C2211yescharacteristic
12#16839917—22no—
13#16839917—22no—
14#16839903—22no—
15#16839903—22no—
16C2×#14519702—21no—
17C2×#14519700—21no—
18C2×C22⋊#92418C2211yescharacteristic
19#16839701—21no—
20#16839688—21no—
21#16839679—21no—
22#16839673—21no—
23#16839639—22no—
24C2×#14519684—22no—
25C2×#14519684C2211yescharacteristic
26#16839495—22no—
27C2×C22⋊Q8×C61C2211yescharacteristic
28C2×C22⋊C8×C61C2211yescharacteristic
29C2×#95×C61C2211yescharacteristic
30C61×#328—22no—
31C22×Dic122—21no—
32C22×Dic122—21no—
33C22×Dic122—21no—
34C22×Dic122—21no—
35C22×Dic122—21no—
36C22×Dic122—44no—
37C22×Dic122D412yes—
38C22×D244D411yescharacteristic
39C22×C22⋊D61—21no—
40C23×Dic61—21no—
41C23×Dic61D412yes—
42C2×#11758202—24no—
43C2×#11758202—44no—
44C2×#11758190—24no—
45C2×#11758190—44no—
46C22×#92418—21no—
47#14519751—21no—
48#14519751—21no—
49#14519751—21no—
50#14519751—21no—
51C2×C4⋊Dic61D411yescharacteristic
52C2×#11758198—21no—
53C2×#11758198—22no—
54C2×#11758198—22no—
55#14519702—24no—
56#14519700—24no—
57C22⋊#92418—21no—
58C22⋊#92418D412yes—
59#14519684—22no—
60#14519684—22no—
61#14519684—21no—
62#14519684—21no—
63#14519684—44no—
64C22×Q8×C61D411yescharacteristic
65C23×C4×C61—21no—
66C23×C4×C61C2311yescharacteristic
67C22×C8×C61—21no—
68C22⋊Q8×C61—21no—
69C22⋊Q8×C61—21no—
70C22⋊Q8×C61—21no—
71C22⋊Q8×C61—21no—
72C2×C4⋊C4×C61—21no—
73C2×C4⋊C4×C61D411yescharacteristic
74C2×C22⋊C4×C61—21no—
75C22⋊C8×C61—21no—
76C22⋊C8×C61D412yes—
77#95×C61—21no—
78#95×C61—21no—
79C24×C61C2×D411yescharacteristic
80C23×D61—21no—
81C2×Q8×C61—21no—
82C2×Q8×C61—21no—
83C2×Q8×C61—42no—
84C22×C4×C61—22no—
85C22×C4×C61—21no—
86C22×C4×C61—21no—
87C22×C4×C61—42no—
88C22×C4×C61—41no—
89C22×C4×C61—42no—
90C22×C4×C61—41no—
91C22×C4×C61—42no—
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61#4111yescharacteristic
98C2×C22⋊D61—41no—
99C2×C22⋊D61—41no—
100C2×C22⋊D61—42no—
101C2×C22⋊D61—42no—
102C22×Dic61—24no—
103C22×Dic61—22no—
104C22×Dic61—22no—
105C22×Dic61—21no—
106C22×Dic61—22no—
107C22×Dic61—22no—
108C22×Dic61—42no—
109C22×Dic61—41no—
110C22×Dic61—41no—
111C22×Dic61—42no—
112C22×Dic61—42no—
113C2×D244—21no—
114C2×D244—21no—
115C2×D244—42no—
116C2×Dic122—24no—
117C2×Dic122—24no—
118C2×Dic122—22no—
119C2×Dic122—22no—
120C2×Dic122—44no—
121C2×Dic122—44no—
122C2×Dic122—44no—
123C2×Dic122—44no—
124C2×Dic122—42no—
125C2×Dic122—42no—
126C2×Dic122—41no—
127C2×Dic122—41no—
128C2×C8×C61—22no—
129C2×C8×C61—42no—
130C4⋊C4×C61—21no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—41no—
133C4⋊C4×C61—41no—
134C22⋊C4×C61—41no—
135C22⋊C4×C61—41no—
136#11758202—48no—
137#11758198—41no—
138#11758198—41no—
139#11758198—82no—
140#11758198—82no—
141C4⋊Dic61—21no—
142C4⋊Dic61—21no—
143C2×#92418—22no—
144C2×#92418—42no—
145#11758190—48no—
146C23×C61—21no—
147C23×C61—21no—
148C23×C61—22no—
149C23×C61#12011yescharacteristic
150C23×C61#12111yescharacteristic
151C23×C61#12711yescharacteristic
152C23×C61#13512yes—
153C23×C61#13712yes—
154C22×D61—41no—
155C22×D61—42no—
156C22×D61—41no—
157C22×D61—41no—
158C22×D61—42no—
159Q8×C61—42no—
160Q8×C61—81no—
161C2×C4×C61—22no—
162C2×C4×C61—22no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—44no—
168C2×C4×C61—42no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61—42no—
174C2×C4×C61—82no—
175C2×C4×C61—81no—
176C2×C4×C61—81no—
177C2×C4×C61#12011yescharacteristic
178C2×C4×C61#12011yescharacteristic
179C2×C4×C61#12111yescharacteristic
180C2×C4×C61#12111yescharacteristic
181C22⋊D61—82no—
182C22⋊D61—81no—
183C22⋊D61—81no—
184C2×Dic61—24no—
185C2×Dic61—42no—
186C2×Dic61—44no—
187C2×Dic61—42no—
188C2×Dic61—44no—
189C2×Dic61—44no—
190C2×Dic61—41no—
191C2×Dic61—42no—
192C2×Dic61—42no—
193C2×Dic61—42no—
194C2×Dic61—42no—
195C2×Dic61—82no—
196D244—42no—
197D244—81no—
198Dic122—24no—
199Dic122—24no—
200Dic122—42no—
201Dic122—42no—
202Dic122—44no—
203Dic122—44no—
204C8×C61—42no—
205#92418—42no—
206#9965—611nomaximal, Sylow
207C22×C61—22no—
208C22×C61—22no—
209C22×C61—22no—
210C22×C61—22no—
211C22×C61—21no—
212C22×C61—21no—
213C22×C61—21no—
214C22×C61—21no—
215C22×C61#39211yescharacteristic
216C22×C61#45212yes—
217C22×C61#45511yescharacteristic
218C22×C61#45611yescharacteristic
219C22×C61B(4,𝔽2)12yes—
220C22×C61#46412yes—
221C22×C61#49511yescharacteristic
222C22×C61#49711yescharacteristic
223C2×D61—82no—
224C2×D61—82no—
225C2×D61—81no—
226C2×D61—81no—
227C2×D61—81no—
228C4×C61—21no—
229C4×C61—21no—
230C4×C61—21no—
231C4×C61—21no—
232C4×C61—82no—
233C4×C61—81no—
234C4×C61—81no—
235Dic61—44no—
236Dic61—82no—
237Dic61—81no—
238C2×#451—611no—
239C2×#448—611no—
240#2639—611no—
241C2×#328D6111yescharacteristic, complemented
242C2×#327—611no—
243#1446—611no—
244#1280—611no—
245C2×C61—21no—
246C2×C61—22no—
247C2×C61—21no—
248C2×C61#160312yes—
249C2×C61#160612yes—
250C2×C61#200711yescharacteristic
251C2×C61#201511yescharacteristic
252C2×C61#202111yescharacteristic
253D61—161no—
254C23×Q8—611no—
255C22×Q16—1221no—
256C22×SD16—1221no—
257C2×C22⋊Q8C2×D6111yescharacteristic
258C2×#121—611no—
259#451—1222no—
260#451—1222no—
261#448—1222no—
262#448—1222no—
263C2×#103—1221no—
264C2×#103—1221no—
265C2×C22⋊C8—611no—
266C2×C22⋊C8C2×D6111yescharacteristic
267#400—1221no—
268#397—1221no—
269#394—1221no—
270#393—1221no—
271#386—1222no—
272C2×#95—611no—
273C2×#95—1222no—
274C2×#95C2×D6111yescharacteristic
275#328—22no—
276#327—1222no—
277C61#996511yescharacteristic, complemented, Sylow
278C22×Q8—612no—
279C22×Q8—1221no—
280C22×Q8—1221no—
281C22×Q8—1221no—
282C22×Q8—1221no—
283C22×Q8—1221no—
284C22×Q8—2444no—
285C22×Q8D24411yescharacteristic
286C22×D4—611no—
287C22×D4—1221no—
288C23×C4—21no—
289C23×C4—612no—
290C23×C4—1221no—
291C23×C4C22×D6111yescharacteristic
292C2×Q16—1224no—
293C2×Q16—2444no—
294C2×SD16—1224no—
295C2×SD16—2444no—
296C22×C8—21no—
297C22×C8—1221no—
298C22⋊Q8—21no—
299C22⋊Q8—21no—
300C22⋊Q8—21no—
301C22⋊Q8—21no—
302#121—1221no—
303#121—1221no—
304#121—1221no—
305#121—1221no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—611no—
308C2×C4⋊C4D24411yescharacteristic
309C2×C22⋊C4—21no—
310C2×C22⋊C4—1222no—
311C2×C22⋊C4—1222no—
312C2×C22⋊C4—1221no—
313#103—1224no—
314#103—1224no—
315C22⋊C8—21no—
316C22⋊C8—612no—
317C22⋊C8—1221no—
318C22⋊C8C22⋊D6112yes—
319#95—21no—
320#95—21no—
321#95—1222no—
322#95—1222no—
323#95—1221no—
324#95—1221no—
325#95—2444no—
326C24—1221no—
327C24D4×D6111yescharacteristic
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—42no—
331C2×Q8—1224no—
332C2×Q8—1224no—
333C2×Q8—1222no—
334C2×Q8—1222no—
335C2×Q8—2442no—
336C2×Q8—2442no—
337C2×Q8—2444no—
338C2×Q8—2444no—
339C2×Q8—2444no—
340C2×Q8—2444no—
341C2×Q8—2441no—
342C2×Q8—2441no—
343C2×D4—1221no—
344C2×D4—1221no—
345C2×D4—2442no—
346C2×D4—2442no—
347C2×D4—2441no—
348C2×D4—2441no—
349C2×D4—2442no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—1224no—
359C22×C4—1222no—
360C22×C4—1222no—
361C22×C4—1222no—
362C22×C4—1222no—
363C22×C4—1221no—
364C22×C4—2442no—
365C22×C4—2442no—
366C22×C4—2442no—
367C22×C4—2441no—
368C22×C4—2441no—
369C22×C4C2×D24411yescharacteristic
370C22×C4#1175822211yescharacteristic
371C22×C4D4×D6111yescharacteristic
372C22×C4D4×D6111yescharacteristic
373C22×C4D4×D6111yescharacteristic
374C22×C4C2×C22⋊D6111yescharacteristic
375Q16—2448no—
376SD16—2448no—
377C2×C8—22no—
378C2×C8—42no—
379C2×C8—1222no—
380C2×C8—2442no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—1221no—
386C4⋊C4—1221no—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—4882no—
390C22⋊C4—4882no—
391C22⋊C4—2441no—
392C22⋊C4—2441no—
393C23—21no—
394C23—21no—
395C23—22no—
396C23—2441no—
397C23—2441no—
398C23—2442no—
399C23—2441no—
400C23—2442no—
401C23#1451974711yescharacteristic
402C23#1451975011yescharacteristic
403C23#1451977311yescharacteristic
404C23#1451979812yes—
405C23#1451980912yes—
406Q8—42no—
407Q8—81no—
408Q8—1224no—
409Q8—1224no—
410Q8—2442no—
411Q8—2442no—
412Q8—2444no—
413Q8—2444no—
414D4—4881no—
415D4—4882no—
416D4—4881no—
417D4—4881no—
418D4—2442no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—81no—
433C2×C4—82no—
434C2×C4—81no—
435C2×C4—1224no—
436C2×C4—4882no—
437C2×C4—2444no—
438C2×C4—2442no—
439C2×C4—2442no—
440C2×C4—2441no—
441C2×C4—2442no—
442C2×C4—2442no—
443C2×C4—2444no—
444C2×C4—2442no—
445C2×C4—2444no—
446C2×C4—2442no—
447C2×C4#1451974511yescharacteristic
448C2×C4#1451974711yescharacteristic
449C2×C4#1451975011yescharacteristic
450C2×C4#1451975111yescharacteristic
451C8—42no—
452C8—2442no—
453C22—21no—
454C22—21no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—21no—
460C22—21no—
461C22—4881no—
462C22—4881no—
463C22—4882no—
464C22—4882no—
465C22—4881no—
466C22#1683967111yescharacteristic
467C22#1683992712yes—
468C22#1683994111yescharacteristic
469C22#1683994611yescharacteristic
470C22#1683995012yes—
471C22#1683998712yes—
472C22#1684014311yescharacteristic
473C22#1684015511yescharacteristic
474C4—21no—
475C4—21no—
476C4—21no—
477C4—21no—
478C4—82no—
479C4—81no—
480C4—81no—
481C4—4881no—
482C4—4882no—
483C4—2444no—
484C2—21no—
485C2—22no—
486C2—21no—
487C2—9761no—
488C2#2144478512yes—
489C2#2144480212yes—
490C2#2144686711yescharacteristic
491C2#2144691411yescharacteristic
492C2#2144694411yescharacteristic
subgroupG/H
centerC23#14519747
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC2×C61×#328C2
socleC23×C61#120
prime radicalC2×#14519751C22
hypercenterC2×#328D61
hypocenterC61#9965
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
1. C2×#16839917C2semidirect
5. #21444089C2semidirect
6. #21443661C2semidirect
7. C2×C61×#328C2semidirect
241. C2×#328D61semidirect
277. C61#9965semidirect

Maximal subgroups

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

subgroupindex
C2×#168399172
C2×#168399032
#214517532
C2×#168394952
#214440892
#214436612
C2×C61×#3282
#996561