Finite Group Databasebeta

← #29580559

Subgroups — #29580559

Order 15616.

830 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 113 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×#16839901C211yesmaximal, characteristic
2#21451753C211yesmaximal, characteristic
3C2×#16839499C211yesmaximal, characteristic, complemented
4C2×#16839497C211yesmaximal, characteristic
5#21444090C211yesmaximal, characteristic, complemented
6#21443661C211yesmaximal, characteristic, complemented
7C2×C61×#451C211yesmaximal, characteristic, complemented
8C22×#11758201—21no—
9C2×#14519751C2211yescharacteristic
10#16839901—22no—
11#16839901—22no—
12C2×#14519701—21no—
13C2×C22⋊#92418C2211yescharacteristic
14#16839701—21no—
15#16839688—21no—
16#16839679—21no—
17#16839673—21no—
18#16839639—22no—
19C2×#14519684—22no—
20C2×#14519684C2211yescharacteristic
21C2×#14519684C2211yescharacteristic
22#16839499—22no—
23#16839497—22no—
24C23×Q8×C61C2211yescharacteristic
25C22×Q16×C61—21no—
26C2×C22⋊Q8×C61C2211yescharacteristic
27C61×#451—22no—
28C61×#451—22no—
29C2×#103×C61—21no—
30C2×C22⋊C8×C61C2211yescharacteristic
31C22×D244D411yescharacteristic
32C22×C22⋊D61—21no—
33C23×Dic61—21no—
34C23×Dic61D412yes—
35C2×#11758201—24no—
36C2×#11758201—44no—
37C22×#92418—21no—
38#14519751—21no—
39#14519751—21no—
40#14519751—21no—
41#14519751—21no—
42C2×C4⋊Dic61D411yescharacteristic
43C2×#11758198—21no—
44C2×#11758198—22no—
45C2×#11758198—22no—
46#14519701—24no—
47C22⋊#92418—21no—
48C22⋊#92418D412yes—
49#14519684—22no—
50#14519684—22no—
51#14519684—21no—
52#14519684—21no—
53#14519684—21no—
54#14519684—21no—
55#14519684—44no—
56C22×Q8×C61—21no—
57C22×Q8×C61—21no—
58C22×Q8×C61—21no—
59C22×Q8×C61—21no—
60C22×Q8×C61—21no—
61C22×Q8×C61—44no—
62C22×Q8×C61D411yescharacteristic
63C22×Q8×C61D412yes—
64C23×C4×C61—21no—
65C23×C4×C61C2311yescharacteristic
66C2×Q16×C61—24no—
67C2×Q16×C61—44no—
68C22×C8×C61—21no—
69C22⋊Q8×C61—21no—
70C22⋊Q8×C61—21no—
71C22⋊Q8×C61—21no—
72C22⋊Q8×C61—21no—
73C2×C4⋊C4×C61—21no—
74C2×C4⋊C4×C61D411yescharacteristic
75C2×C22⋊C4×C61—21no—
76#103×C61—24no—
77C22⋊C8×C61—21no—
78C22⋊C8×C61D412yes—
79C24×C61C2×D411yescharacteristic
80C23×D61—21no—
81C2×Q8×C61—24no—
82C2×Q8×C61—24no—
83C2×Q8×C61—22no—
84C2×Q8×C61—22no—
85C2×Q8×C61—21no—
86C2×Q8×C61—21no—
87C2×Q8×C61—44no—
88C2×Q8×C61—42no—
89C2×Q8×C61—42no—
90C2×Q8×C61—44no—
91C2×Q8×C61—42no—
92C2×Q8×C61—44no—
93C2×Q8×C61—44no—
94C2×Q8×C61—41no—
95C2×Q8×C61—41no—
96C22×C4×C61—24no—
97C22×C4×C61—22no—
98C22×C4×C61—21no—
99C22×C4×C61—21no—
100C22×C4×C61—42no—
101C22×C4×C61—42no—
102C22×C4×C61—41no—
103C22×C4×C61—41no—
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61#4111yescharacteristic
110C2×C22⋊D61—41no—
111C2×C22⋊D61—41no—
112C2×C22⋊D61—42no—
113C2×C22⋊D61—42no—
114C22×Dic61—22no—
115C22×Dic61—22no—
116C22×Dic61—21no—
117C22×Dic61—22no—
118C22×Dic61—22no—
119C22×Dic61—42no—
120C22×Dic61—42no—
121C22×Dic61—41no—
122C22×Dic61—42no—
123C22×Dic61—41no—
124C22×Dic61—42no—
125C2×D244—21no—
126C2×D244—21no—
127C2×D244—42no—
128Q16×C61—48no—
129C2×C8×C61—22no—
130C2×C8×C61—42no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—41no—
134C4⋊C4×C61—41no—
135C22⋊C4×C61—41no—
136C22⋊C4×C61—41no—
137#11758201—48no—
138#11758198—41no—
139#11758198—41no—
140#11758198—82no—
141#11758198—82no—
142C4⋊Dic61—21no—
143C4⋊Dic61—21no—
144C2×#92418—22no—
145C2×#92418—42no—
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—24no—
160Q8×C61—24no—
161Q8×C61—42no—
162Q8×C61—42no—
163Q8×C61—44no—
164Q8×C61—44no—
165Q8×C61—42no—
166Q8×C61—81no—
167C2×C4×C61—22no—
168C2×C4×C61—22no—
169C2×C4×C61—24no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—44no—
175C2×C4×C61—42no—
176C2×C4×C61—42no—
177C2×C4×C61—44no—
178C2×C4×C61—41no—
179C2×C4×C61—41no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—42no—
183C2×C4×C61#12011yescharacteristic
184C2×C4×C61#12011yescharacteristic
185C2×C4×C61#12111yescharacteristic
186C2×C4×C61#12111yescharacteristic
187C22⋊D61—82no—
188C22⋊D61—81no—
189C22⋊D61—81no—
190C2×Dic61—44no—
191C2×Dic61—42no—
192C2×Dic61—44no—
193C2×Dic61—41no—
194C2×Dic61—42no—
195C2×Dic61—42no—
196C2×Dic61—42no—
197C2×Dic61—42no—
198C2×Dic61—82no—
199C2×Dic61—81no—
200C2×Dic61—81no—
201C2×Dic61—82no—
202D244—42no—
203D244—81no—
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—44no—
233C4×C61—81no—
234C4×C61—81no—
235Dic61—82no—
236Dic61—82no—
237Dic61—81no—
238C2×#451D6111yescharacteristic, complemented
239C2×#448—611no—
240#2639—611no—
241C2×#328—611no—
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×Q8C2×D6111yescharacteristic
255C22×Q16—21no—
256C22×SD16—1221no—
257C2×C22⋊Q8C2×D6111yescharacteristic
258C2×#121—611no—
259#451—22no—
260#451—22no—
261#448—1222no—
262#448—1222no—
263C2×#103—21no—
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—611no—
274C2×#95—1222no—
275#328—1222no—
276#327—1222no—
277C61#996511yescharacteristic, complemented, Sylow
278C22×Q8—21no—
279C22×Q8—21no—
280C22×Q8—21no—
281C22×Q8—21no—
282C22×Q8—21no—
283C22×Q8—44no—
284C22×Q8C22⋊D6111yescharacteristic
285C22×Q8C22⋊D6112yes—
286C22×D4—611no—
287C22×D4—1221no—
288C23×C4—21no—
289C23×C4—612no—
290C23×C4—1221no—
291C23×C4C22×D6111yescharacteristic
292C2×Q16—24no—
293C2×Q16—44no—
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⋊C4C22⋊D6111yescharacteristic
309C2×C22⋊C4—21no—
310C2×C22⋊C4—1222no—
311C2×C22⋊C4—1222no—
312C2×C22⋊C4—1221no—
313#103—24no—
314#103—1224no—
315C22⋊C8—21no—
316C22⋊C8—612no—
317C22⋊C8—1221no—
318C22⋊C8C22⋊D6112yes—
319#95—1222no—
320#95—1222no—
321#95—1221no—
322#95—1221no—
323#95—1221no—
324#95—1221no—
325#95—2444no—
326C24—1221no—
327C24D4×D6111yescharacteristic
328C2×Q8—24no—
329C2×Q8—24no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—44no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—44no—
338C2×Q8—42no—
339C2×Q8—44no—
340C2×Q8—44no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×D4—1221no—
344C2×D4—1221no—
345C2×D4—2442no—
346C2×D4—2442no—
347C2×D4—2441no—
348C2×D4—2441no—
349C2×D4—2442no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—1222no—
359C22×C4—1222no—
360C22×C4—1222no—
361C22×C4—1222no—
362C22×C4—1221no—
363C22×C4—2442no—
364C22×C4—2441no—
365C22×C4—2441no—
366C22×C4—2442no—
367C22×C4—2442no—
368C22×C4—2442no—
369C22×C4D4×D6111yescharacteristic
370C22×C4D4×D6111yescharacteristic
371C22×C4#1175822411yescharacteristic
372C22×C4C2×C22⋊D6111yescharacteristic
373C22×C4C2×C22⋊D6111yescharacteristic
374C22×C4C2×C22⋊D6111yescharacteristic
375Q16—48no—
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—2442no—
400C23—2441no—
401C23#1451974511yescharacteristic
402C23#1451974911yescharacteristic
403C23#1451977311yescharacteristic
404C23#1451980012yes—
405C23#1451980912yes—
406Q8—24no—
407Q8—24no—
408Q8—42no—
409Q8—42no—
410Q8—44no—
411Q8—44no—
412Q8—42no—
413Q8—81no—
414D4—4881no—
415D4—4882no—
416D4—4881no—
417D4—4881no—
418D4—2442no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—4881no—
436C2×C4—4882no—
437C2×C4—4881no—
438C2×C4—4882no—
439C2×C4—2444no—
440C2×C4—2442no—
441C2×C4—2444no—
442C2×C4—2442no—
443C2×C4—2442no—
444C2×C4—2442no—
445C2×C4—2442no—
446C2×C4—2441no—
447C2×C4#1451974511yescharacteristic
448C2×C4#1451974611yescharacteristic
449C2×C4#1451974811yescharacteristic
450C2×C4#1451974911yescharacteristic
451C8—42no—
452C8—2442no—
453C22—21no—
454C22—21no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—21no—
460C22—21no—
461C22—4881no—
462C22—4882no—
463C22—4882no—
464C22—4881no—
465C22—4881no—
466C22#1683967011yescharacteristic
467C22#1683992312yes—
468C22#1683993811yescharacteristic
469C22#1683994311yescharacteristic
470C22#1683994812yes—
471C22#1683998612yes—
472C22#1684014711yescharacteristic
473C22#1684015511yescharacteristic
474C4—21no—
475C4—21no—
476C4—21no—
477C4—21no—
478C4—44no—
479C4—81no—
480C4—81no—
481C4—4882no—
482C4—4882no—
483C4—4881no—
484C2—21no—
485C2—22no—
486C2—21no—
487C2—9761no—
488C2#2144478112yes—
489C2#2144479812yes—
490C2#2144686411yescharacteristic
491C2#2144691011yescharacteristic
492C2#2144694211yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC2×C61×#451C2
socleC23×C61#120
prime radicalC2×#14519751C22
hypercenterC2×#451D61
hypocenterC61#9965
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
3. C2×#16839499C2semidirect
5. #21444090C2semidirect
6. #21443661C2semidirect
7. C2×C61×#451C2semidirect
238. C2×#451D61semidirect
277. C61#9965semidirect

Maximal subgroups

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

subgroupindex
C2×#168399012
#214517532
C2×#168394992
C2×#168394972
#214440902
#214436612
C2×C61×#4512
#996561