Finite Group Databasebeta

← #29576481

Subgroups — #29576481

Order 15616.

750 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839901C211yesmaximal, characteristic
2#21446655C211yesmaximal, characteristic
3#21445754C211yesmaximal, characteristic, complemented
4#21444386C211yesmaximal, characteristic
5#21443686C211yesmaximal, characteristic, complemented
6#21443667C211yesmaximal, characteristic, complemented
7Q8×C22⋊C4×C61C211yesmaximal, characteristic, complemented
8C22×#11758201C412yescomplemented
9C2×#14519751C2211yescharacteristic
10#16839901—22no—
11#16839901—21no—
12#16839901—21no—
13C2×#14519701—22no—
14C2×#14519701C412yescomplemented
15C2×#14519691—21no—
16C2×#14519691—21no—
17C2×C22⋊#92418C2211yescharacteristic
18C2×C22⋊#92418C2211yescharacteristic
19#16839622C2211yescharacteristic
20#16839495—22no—
21C23×Q8×C61C2211yescharacteristic
22C2×C4×Q8×C61—21no—
23C61×#391—21no—
24C61×#391—22no—
25C61×#380—21no—
26C61×#380C2211yescharacteristic
27C4×C22⋊C4×C61—21no—
28C4×C22⋊C4×C61C2211yescharacteristic
29C22×D244C2×C411yescharacteristic
30C22×C22⋊D61—21no—
31C2×#11758201—22no—
32C2×#11758201—22no—
33C2×#11758201—44no—
34C22×#92418—21no—
35C22×#92418C2×C412yes—
36#14519751—21no—
37#14519751—21no—
38#14519751—21no—
39#14519751—21no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61C2×C411yescharacteristic
42C2×C22⋊Dic61—21no—
43C2×#11758198—21no—
44#14519701—22no—
45#14519701—22no—
46#14519701—44no—
47#14519691—42no—
48#14519691—42no—
49C22⋊#92418—21no—
50C22⋊#92418—21no—
51C22⋊#92418—21no—
52C22⋊#92418—21no—
53#14519684—21no—
54C22×Q8×C61—21no—
55C22×Q8×C61—21no—
56C22×Q8×C61—21no—
57C22×Q8×C61—44no—
58C22×Q8×C61C2×C412yes—
59C22×Q8×C61D412yes—
60C22×Q8×C61D411yescharacteristic
61C22×Q8×C61D411yescharacteristic
62C23×C4×C61—21no—
63C23×C4×C61C2311yescharacteristic
64C4×Q8×C61—44no—
65C2×C4⋊C4×C61—22no—
66C2×C4⋊C4×C61—21no—
67C2×C22⋊C4×C61—21no—
68C2×C22⋊C4×C61D411yescharacteristic
69C2×C22⋊C4×C61D411yescharacteristic
70C2×C42×C61—22no—
71C2×C42×C61—21no—
72#95×C61—22no—
73#95×C61—22no—
74#95×C61—21no—
75#95×C61D412yes—
76C24×C61C2×D411yescharacteristic
77C23×D61—21no—
78C2×Q8×C61—24no—
79C2×Q8×C61—22no—
80C2×Q8×C61—22no—
81C2×Q8×C61—21no—
82C2×Q8×C61—21no—
83C2×Q8×C61—24no—
84C2×Q8×C61—24no—
85C2×Q8×C61—42no—
86C2×Q8×C61—42no—
87C2×Q8×C61—42no—
88C2×Q8×C61—44no—
89C2×Q8×C61—44no—
90C2×Q8×C61—44no—
91C2×Q8×C61C22⋊C412yes—
92C2×Q8×C61C22⋊C412yes—
93C22×C4×C61—21no—
94C22×C4×C61—22no—
95C22×C4×C61—21no—
96C22×C4×C61—21no—
97C22×C4×C61—22no—
98C22×C4×C61—21no—
99C22×C4×C61—22no—
100C22×C4×C61—42no—
101C22×C4×C61—44no—
102C22×C4×C61C22×C411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61#4111yescharacteristic
107C22×C4×C61#4111yescharacteristic
108C2×C22⋊D61—41no—
109C2×C22⋊D61—41no—
110C2×C22⋊D61—41no—
111C2×C22⋊D61—41no—
112C2×C22⋊D61—41no—
113C2×C22⋊D61—41no—
114C22×Dic61—21no—
115C22×Dic61—41no—
116C2×D244—21no—
117C2×D244—21no—
118C2×D244—42no—
119C4⋊C4×C61—44no—
120C4⋊C4×C61—42no—
121C22⋊C4×C61—21no—
122C22⋊C4×C61—21no—
123C22⋊C4×C61—21no—
124C22⋊C4×C61—41no—
125C22⋊C4×C61—41no—
126C22⋊C4×C61SD1612yescomplemented
127C42×C61—44no—
128C42×C61—42no—
129C22⋊Dic61—41no—
130C22⋊Dic61—41no—
131#11758201—44no—
132#11758201—44no—
133#11758198—41no—
134#11758198—41no—
135C4⋊Dic61—21no—
136C4⋊Dic61—21no—
137C4⋊Dic61—42no—
138C2×#92418—22no—
139C2×#92418—42no—
140C2×#92418—42no—
141C2×#92418—41no—
142C23×C61—41no—
143C23×C61—41no—
144C23×C61C4×D411yescharacteristic
145C23×C61#12011yescharacteristic
146C23×C61#12111yescharacteristic
147C23×C61C2×SD1611yescharacteristic
148C23×C61#13511yescharacteristic
149C23×C61AGL(1,ℤ/8)11yescharacteristic
150C23×C61#13711yescharacteristic
151C22×D61—41no—
152C22×D61—41no—
153C22×D61—41no—
154C22×D61—81no—
155C22×D61—81no—
156Q8×C61—24no—
157Q8×C61—22no—
158Q8×C61—22no—
159Q8×C61—22no—
160Q8×C61—44no—
161Q8×C61—44no—
162Q8×C61—42no—
163Q8×C61#9914yescomplemented
164C2×C4×C61—24no—
165C2×C4×C61—22no—
166C2×C4×C61—22no—
167C2×C4×C61—22no—
168C2×C4×C61—22no—
169C2×C4×C61—22no—
170C2×C4×C61—22no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—44no—
175C2×C4×C61—44no—
176C2×C4×C61—41no—
177C2×C4×C61—41no—
178C2×C4×C61—42no—
179C2×C4×C61—41no—
180C2×C4×C61—42no—
181C2×C4×C61—41no—
182C2×C4×C61—42no—
183C2×C4×C61—41no—
184C2×C4×C61—42no—
185C2×C4×C61C2×C22⋊C411yescharacteristic
186C2×C4×C61C4×D411yescharacteristic
187C2×C4×C61#12111yescharacteristic
188C2×C4×C61#12311yescharacteristic
189C22⋊D61—81no—
190C22⋊D61—81no—
191C22⋊D61—81no—
192C22⋊D61—81no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61—81no—
197C2×Dic61—81no—
198C2×Dic61—81no—
199D244—42no—
200D244—81no—
201#92418—42no—
202#92418—81no—
203#9262—611nomaximal, Sylow
204C22×C61—21no—
205C22×C61—21no—
206C22×C61—21no—
207C22×C61—21no—
208C22×C61—21no—
209C22×C61—21no—
210C22×C61—41no—
211C22×C61—41no—
212C22×C61—41no—
213C22×C61—41no—
214C22×C61#38611yescharacteristic
215C22×C61C4×SD1611yescharacteristic
216C22×C61#44011yescharacteristic
217C22×C61#44811yescharacteristic
218C22×C61#44911yescharacteristic
219C22×C61#46011yescharacteristic
220C22×C61#46411yescharacteristic
221C2×D61—81no—
222C2×D61—81no—
223C2×D61—81no—
224C2×D61—161no—
225C2×D61—161no—
226C4×C61—21no—
227C4×C61—21no—
228C4×C61—24no—
229C4×C61—21no—
230C4×C61—42no—
231C4×C61—42no—
232C4×C61—81no—
233C4×C61—82no—
234C4×C61C2×#9912yes—
235Dic61—81no—
236Dic61—161no—
237C2×#448—611no—
238Q8×C22⋊C4D6111yescharacteristic, complemented
239#1964—611no—
240#1787—611no—
241#1519—611no—
242#1289—611no—
243#1282—611no—
244C2×C61—21no—
245C2×C61—21no—
246C2×C61—21no—
247C2×C61—21no—
248C2×C61—21no—
249C2×C61—21no—
250C2×C61#188011yescharacteristic
251C2×C61#189811yescharacteristic
252C2×C61#190511yescharacteristic
253D61—161no—
254C23×Q8C2×D6111yescharacteristic
255C22×SD16—612no—
256C2×#121—611no—
257C2×C4×Q8—21no—
258#448—1222no—
259#448—1221no—
260#448—1221no—
261C2×#103—612no—
262C2×#103—1222no—
263C2×#99—1221no—
264C2×#99—1221no—
265C2×C22⋊C8—611no—
266C2×C22⋊C8—611no—
267#391—21no—
268#391—22no—
269#380—21no—
270#380—611no—
271#380C2×D6111yescharacteristic
272C4×C22⋊C4—21no—
273C4×C22⋊C4C2×D6111yescharacteristic
274#327—1222no—
275C61#926211yescharacteristic, complemented, Sylow
276C22×Q8—21no—
277C22×Q8—21no—
278C22×Q8—21no—
279C22×Q8—44no—
280C22×Q8C4×D6112yes—
281C22×Q8D24412yes—
282C22×Q8C22⋊D6111yescharacteristic
283C22×Q8C22⋊D6111yescharacteristic
284C22×D4—611no—
285C22×D4—1221no—
286C23×C4—21no—
287C23×C4C22×D6111yescharacteristic
288C2×SD16—1222no—
289C2×SD16—1222no—
290C2×SD16—2444no—
291C22×C8—612no—
292C22×C8—1221no—
293#121—1221no—
294#121—1221no—
295#121—1221no—
296#121—1221no—
297C4×Q8—44no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—611no—
301C2×C4⋊C4—1221no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—1221no—
304C2×C22⋊C4—1221no—
305C2×C22⋊C4C22⋊D6111yescharacteristic
306C2×C22⋊C4C22⋊D6111yescharacteristic
307C2×C42—22no—
308C2×C42—21no—
309#103—1222no—
310#103—1222no—
311#103—2444no—
312#99—2442no—
313#99—2442no—
314C22⋊C8—1221no—
315C22⋊C8—1221no—
316C22⋊C8—1221no—
317C22⋊C8—1221no—
318#95—22no—
319#95—22no—
320#95—21no—
321#95—1221no—
322#95C22⋊D6112yes—
323C24—1221no—
324C24C2×C22⋊D6111yescharacteristic
325C2×Q8—24no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—24no—
331C2×Q8—24no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—44no—
336C2×Q8—44no—
337C2×Q8—44no—
338C2×Q8#1175819812yes—
339C2×Q8#1175819812yes—
340C2×D4—1221no—
341C2×D4—1221no—
342C2×D4—2442no—
343C2×D4—2441no—
344C2×D4—2441no—
345C2×D4—2441no—
346C2×D4—2441no—
347C2×D4—2441no—
348C2×D4—2441no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4—1221no—
359C22×C4—2441no—
360C22×C4C2×C4×D6111yescharacteristic
361C22×C4C2×D24411yescharacteristic
362C22×C4#1175822211yescharacteristic
363C22×C4#1175822211yescharacteristic
364C22×C4C2×C22⋊D6111yescharacteristic
365C22×C4C2×C22⋊D6111yescharacteristic
366SD16—2444no—
367SD16—2444no—
368C2×C8—1222no—
369C2×C8—2442no—
370C2×C8—2441no—
371C2×C8—2442no—
372C4⋊C4—44no—
373C4⋊C4—42no—
374C4⋊C4—1221no—
375C4⋊C4—1221no—
376C4⋊C4—2442no—
377C22⋊C4—21no—
378C22⋊C4—21no—
379C22⋊C4—21no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—2441no—
383C22⋊C4—2441no—
384C22⋊C4—2441no—
385C22⋊C4—2441no—
386C22⋊C4#1175820112yescomplemented
387C42—44no—
388C42—42no—
389C23—41no—
390C23—41no—
391C23—4881no—
392C23—4881no—
393C23—2441no—
394C23—2441no—
395C23—2441no—
396C23C4×C22⋊D6111yescharacteristic
397C23#1451974611yescharacteristic
398C23#1451975111yescharacteristic
399C23C2×#1175820111yescharacteristic
400C23#1451979711yescharacteristic
401C23#1451980211yescharacteristic
402C23#1451980811yescharacteristic
403Q8—24no—
404Q8—22no—
405Q8—22no—
406Q8—22no—
407Q8—44no—
408Q8—44no—
409Q8—42no—
410Q8#1451969114yescomplemented
411D4—4881no—
412D4—4881no—
413D4—4881no—
414D4—4881no—
415D4—4881no—
416D4—2442no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—4881no—
439C2×C4—4881no—
440C2×C4—4881no—
441C2×C4—2441no—
442C2×C4—2441no—
443C2×C4—2441no—
444C2×C4C2×#1175819811yescharacteristic
445C2×C4C4×C22⋊D6111yescharacteristic
446C2×C4#1451975111yescharacteristic
447C2×C4#1451976011yescharacteristic
448C8—4881no—
449C8—2442no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—9761no—
461C22—9761no—
462C22—4881no—
463C22—4881no—
464C22—4881no—
465C22#1683964111yescharacteristic
466C22C4×#1175820111yescharacteristic
467C22#1683986011yescharacteristic
468C22#1683990111yescharacteristic
469C22#1683990711yescharacteristic
470C22#1683996211yescharacteristic
471C22#1683998411yescharacteristic
472C4—21no—
473C4—24no—
474C4—21no—
475C4—21no—
476C4—42no—
477C4—42no—
478C4—81no—
479C4—82no—
480C4—9761no—
481C4—4881no—
482C4C2×#1451969112yes—
483C2—21no—
484C2—21no—
485C2—21no—
486C2—21no—
487C2—21no—
488C2—21no—
489C2—9761no—
490C2#2144617411yescharacteristic
491C2#2144627211yescharacteristic
492C2#2144631311yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingQ8×C22⋊C4×C61C2
socleC22×C61#386
prime radicalC2×#14519751C22
cosocleC23×C4×C61C23
hypercenterQ8×C22⋊C4D61
hypocenterC61#9262
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ Q8×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #21445754C2semidirect
5. #21443686C2semidirect
6. #21443667C2semidirect
7. Q8×C22⋊C4×C61C2semidirect
8. C22×#11758201C4semidirect
14. C2×#14519701C4semidirect
126. C22⋊C4×C61SD16semidirect
163. Q8×C61#99semidirect
238. Q8×C22⋊C4D61semidirect
275. C61#9262semidirect
386. C22⋊C4#11758201semidirect
410. Q8#14519691semidirect

Maximal subgroups

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

subgroupindex
C2×#168399012
#214466552
#214457542
#214443862
#214436862
#214436672
Q8×C22⋊C4×C612
#926261