Finite Group Databasebeta

← #29574788

Subgroups — #29574788

Order 15616.

660 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839962C211yesmaximal, characteristic
2#21446746C211yesmaximal, characteristic
3#21446624C211yesmaximal, characteristic
4C2×Q8⋊#92418C211yesmaximal, characteristic, complemented
5#21443702C211yesmaximal, characteristic, complemented
6#21443660C211yesmaximal, characteristic, complemented
7Q8×C4⋊C4×C61C211yesmaximal, characteristic, complemented
8C22×#11758201—21no—
9C2×#14519774C2211yescharacteristic
10#16839962—22no—
11#16839962C414yescomplemented
12C2×C4⋊#92418C2211yescharacteristic
13C2×C4⋊#92418C2211yescharacteristic
14C2×#14519697—21no—
15C2×#14519697—21no—
16C2×#14519697—21no—
17C2×#14519697—22no—
18C2×C4×#92418C2211yescharacteristic
19Q8⋊#92418—22no—
20C2×C4×Q8×C61—21no—
21C2×C4×Q8×C61C2211yescharacteristic
22C61×#389—21no—
23C61×#389—22no—
24C61×#384—21no—
25C61×#384C2211yescharacteristic
26C4×C4⋊C4×C61—21no—
27C4×C4⋊C4×C61C2211yescharacteristic
28C22×D244—21no—
29C22×D244—21no—
30C22×D244—21no—
31C2×#11758201—24no—
32C2×#11758201—42no—
33C2×#11758201—42no—
34C22×#92418—21no—
35C22×#92418—21no—
36C22×#92418—21no—
37#14519774—21no—
38#14519774—21no—
39#14519774—21no—
40#14519774C2×C412yes—
41C4⋊#92418—21no—
42C4⋊#92418—21no—
43C4⋊#92418—21no—
44C4⋊#92418C2×C412yes—
45#14519697—24no—
46#14519697—44no—
47#14519697—42no—
48#14519697—42no—
49C4×#92418—21no—
50C4×#92418—21no—
51C22×Q8×C61D411yescharacteristic
52C4×Q8×C61—21no—
53C4×Q8×C61—21no—
54C4×Q8×C61—42no—
55C4×Q8×C61—42no—
56C4×Q8×C61C2×C412yes—
57C4×Q8×C61D412yes—
58C2×C4⋊C4×C61—21no—
59C2×C4⋊C4×C61—22no—
60C2×C4⋊C4×C61—21no—
61C2×C4⋊C4×C61—21no—
62C2×C4⋊C4×C61—22no—
63C2×C4⋊C4×C61D411yescharacteristic
64C2×C4⋊C4×C61D411yescharacteristic
65C2×C4⋊C4×C61D411yescharacteristic
66C2×C4⋊C4×C61D412yes—
67C2×C42×C61—21no—
68C2×C42×C61—22no—
69C2×C42×C61—21no—
70C2×C42×C61C2311yescharacteristic
71#95×C61—22no—
72#95×C61—21no—
73C23×D61—41no—
74C2×Q8×C61—21no—
75C2×Q8×C61—21no—
76C2×Q8×C61—21no—
77C2×Q8×C61C22⋊C412yes—
78C2×Q8×C61D812yes—
79C2×Q8×C61SD1612yes—
80C22×C4×C61—21no—
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61—22no—
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D411yescharacteristic
88C2×D244—22no—
89C2×D244—22no—
90C2×D244—22no—
91C2×D244—42no—
92C2×D244—42no—
93C2×D244—41no—
94C2×D244—41no—
95C2×D244—41no—
96C2×D244—41no—
97C2×D244—41no—
98C2×D244—41no—
99C2×D244—41no—
100C2×D244—41no—
101C2×D244—41no—
102C2×D244—41no—
103C2×D244—41no—
104C4⋊C4×C61—22no—
105C4⋊C4×C61—22no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—44no—
111C4⋊C4×C61—42no—
112C4⋊C4×C61—41no—
113C4⋊C4×C61—41no—
114C4⋊C4×C61—41no—
115C4⋊C4×C61—41no—
116C4⋊C4×C61—82no—
117C4⋊C4×C61C22⋊C412yes—
118C4⋊C4×C61SD1612yescomplemented
119C42×C61—22no—
120C42×C61—22no—
121C42×C61—44no—
122C42×C61—41no—
123C42×C61—41no—
124C42×C61C22×C411yescharacteristic
125C42×C61C2×D411yescharacteristic
126C42×C61#4111yescharacteristic
127C42×C61#4111yescharacteristic
128#11758201—48no—
129C2×#92418—22no—
130C2×#92418—22no—
131C2×#92418—41no—
132C2×#92418—41no—
133C2×#92418—41no—
134C2×#92418—41no—
135C2×#92418—41no—
136C2×#92418—41no—
137C2×#92418—41no—
138C23×C61#12011yescharacteristic
139C22×D61—42no—
140C22×D61—81no—
141C22×D61—81no—
142C22×D61—81no—
143C22×D61—81no—
144C22×D61—81no—
145C22×D61—81no—
146Q8×C61—22no—
147Q8×C61—22no—
148Q8×C61—22no—
149Q8×C61#10214yescomplemented
150C2×C4×C61—22no—
151C2×C4×C61—22no—
152C2×C4×C61—22no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—22no—
156C2×C4×C61—22no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—42no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—42no—
165C2×C4×C61—42no—
166C2×C4×C61—41no—
167C2×C4×C61C2×C22⋊C411yescharacteristic
168C2×C4×C61C4×D411yescharacteristic
169C2×C4×C61C4×D411yescharacteristic
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61#12111yescharacteristic
172C2×C4×C61#12311yescharacteristic
173C2×C4×C61C2×D811yescharacteristic
174C2×C4×C61C2×SD1611yescharacteristic
175C2×C4×C61C2×SD1611yescharacteristic
176C2×C4×C61#13511yescharacteristic
177C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
178C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
179C2×C4×C61#13711yescharacteristic
180C2×C4×C61#13711yescharacteristic
181D244—44no—
182D244—42no—
183D244—42no—
184D244—42no—
185D244—42no—
186D244—82no—
187D244—81no—
188D244—81no—
189D244—81no—
190D244—81no—
191#92418—42no—
192#92418—42no—
193#92418—81no—
194#8979—611nomaximal, Sylow
195C22×C61#38611yescharacteristic
196C22×C61#44711yescharacteristic
197C22×C61#44811yescharacteristic
198C22×C61#44811yescharacteristic
199C22×C61#44911yescharacteristic
200C22×C61#45311yescharacteristic
201C22×C61#45411yescharacteristic
202C2×D61—82no—
203C2×D61—82no—
204C2×D61—82no—
205C2×D61—161no—
206C2×D61—161no—
207C2×D61—161no—
208C2×D61—161no—
209C4×C61—24no—
210C4×C61—21no—
211C4×C61—21no—
212C4×C61—42no—
213C4×C61—42no—
214C4×C61—81no—
215C4×C61—82no—
216C4×C61C2×#10211yescharacteristic
217C4×C61#41711yescharacteristic
218C4×C61C4×SD1611yescharacteristic
219C4×C61#44011yescharacteristic
220C4×C61#46011yescharacteristic
221C4×C61#46411yescharacteristic
222C4×C61#48011yescharacteristic
223C4×C61#48111yescharacteristic
224C2×#460—611no—
225Q8×C4⋊C4D6111yescharacteristic, complemented
226#1982—611no—
227#1957—611no—
228C2×Q8⋊C8—611no—
229#1296—611no—
230#1279—611no—
231C2×C61#162111yescharacteristic
232C2×C61#162311yescharacteristic
233C2×C61#162611yescharacteristic
234C2×C61#162711yescharacteristic
235C2×C61#187411yescharacteristic
236C2×C61#188011yescharacteristic
237C2×C61#188311yescharacteristic
238D61—162no—
239C22×SD16—1221no—
240C2×#127—611no—
241C2×C4×Q8—21no—
242C2×C4×Q8C2×D6111yescharacteristic
243#460—614no—
244#460—1222no—
245C2×C4⋊C8—611no—
246C2×C4⋊C8—611no—
247C2×#102—1221no—
248C2×#102—1222no—
249C2×#102—1221no—
250C2×#102—1221no—
251C2×C4×C8—611no—
252#389—21no—
253#389—22no—
254#384—21no—
255#384C2×D6111yescharacteristic
256C4×C4⋊C4—21no—
257C4×C4⋊C4C2×D6111yescharacteristic
258Q8⋊C8—1222no—
259C61#897911yescharacteristic, complemented, Sylow
260C22×Q8C22⋊D6111yescharacteristic
261C22×D4—1221no—
262C22×D4—1221no—
263C22×D4—1221no—
264C2×SD16—1224no—
265C2×SD16—2442no—
266C2×SD16—2442no—
267C22×C8—1221no—
268C22×C8—1221no—
269C22×C8—1221no—
270#127—612no—
271#127—1221no—
272#127—1221no—
273#127—1221no—
274C4×Q8—21no—
275C4×Q8—21no—
276C4×Q8—42no—
277C4×Q8—42no—
278C4×Q8C4×D6112yes—
279C4×Q8D24412yes—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4C22⋊D6111yescharacteristic
286C2×C4⋊C4C22⋊D6111yescharacteristic
287C2×C4⋊C4C22⋊D6111yescharacteristic
288C2×C4⋊C4C22⋊D6112yes—
289C2×C42—21no—
290C2×C42—22no—
291C2×C42—21no—
292C2×C42C22×D6111yescharacteristic
293C4⋊C8—612no—
294C4⋊C8—1221no—
295C4⋊C8—1221no—
296C4⋊C8—1221no—
297#102—1224no—
298#102—2442no—
299#102—2444no—
300#102—2442no—
301C4×C8—1221no—
302C4×C8—1221no—
303#95—22no—
304#95—21no—
305C24—2441no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8#1175819812yes—
310C2×Q8#1175819912yes—
311C2×Q8#1175820112yes—
312C2×D4—1222no—
313C2×D4—1222no—
314C2×D4—1222no—
315C2×D4—2442no—
316C2×D4—2442no—
317C2×D4—2441no—
318C2×D4—2441no—
319C2×D4—2441no—
320C2×D4—2441no—
321C2×D4—2441no—
322C2×D4—2441no—
323C2×D4—2441no—
324C2×D4—2441no—
325C2×D4—2441no—
326C2×D4—2441no—
327C2×D4—2441no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4C2×C22⋊D6111yescharacteristic
334C22×C4C2×C22⋊D6111yescharacteristic
335C22×C4C2×C22⋊D6111yescharacteristic
336SD16—2448no—
337C2×C8—1222no—
338C2×C8—1222no—
339C2×C8—2441no—
340C2×C8—2441no—
341C2×C8—2441no—
342C2×C8—2441no—
343C2×C8—2441no—
344C2×C8—2441no—
345C2×C8—2441no—
346C4⋊C4—21no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—42no—
353C4⋊C4—44no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—82no—
359C4⋊C4#1175819812yes—
360C4⋊C4#1175820112yescomplemented
361C42—22no—
362C42—22no—
363C42—44no—
364C42—41no—
365C42—41no—
366C42C2×C4×D6111yescharacteristic
367C42C2×D24411yescharacteristic
368C42#1175822211yescharacteristic
369C42#1175822211yescharacteristic
370C23—4881no—
371C23—4881no—
372C23—4881no—
373C23—4881no—
374C23—4881no—
375C23—4881no—
376C23—2442no—
377C23#1451974611yescharacteristic
378Q8—22no—
379Q8—22no—
380Q8—22no—
381Q8#1451969714yescomplemented
382D4—4882no—
383D4—4881no—
384D4—4881no—
385D4—4881no—
386D4—4881no—
387D4—2444no—
388D4—2442no—
389D4—2442no—
390D4—2442no—
391D4—2442no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4C2×#1175819811yescharacteristic
410C2×C4C4×C22⋊D6111yescharacteristic
411C2×C4C4×C22⋊D6111yescharacteristic
412C2×C4#1451975111yescharacteristic
413C2×C4#1451975111yescharacteristic
414C2×C4#1451976011yescharacteristic
415C2×C4C2×#1175819911yescharacteristic
416C2×C4C2×#1175820111yescharacteristic
417C2×C4C2×#1175820111yescharacteristic
418C2×C4#1451979711yescharacteristic
419C2×C4#1451980211yescharacteristic
420C2×C4#1451980311yescharacteristic
421C2×C4#1451980811yescharacteristic
422C2×C4#1451980811yescharacteristic
423C8—4881no—
424C8—2442no—
425C8—2442no—
426C22—9761no—
427C22—9761no—
428C22—9761no—
429C22—9761no—
430C22—4882no—
431C22—4882no—
432C22—4882no—
433C22#1683964111yescharacteristic
434C22#1683989611yescharacteristic
435C22#1683990111yescharacteristic
436C22#1683990111yescharacteristic
437C22#1683990711yescharacteristic
438C22#1683993011yescharacteristic
439C22#1683993511yescharacteristic
440C4—21no—
441C4—21no—
442C4—24no—
443C4—42no—
444C4—42no—
445C4—81no—
446C4—82no—
447C4C2×#1451969711yescharacteristic
448C4#1683976111yescharacteristic
449C4C4×#1175820111yescharacteristic
450C4#1683986011yescharacteristic
451C4#1683996211yescharacteristic
452C4#1683998411yescharacteristic
453C4#1684007511yescharacteristic
454C4#1684008011yescharacteristic
455C2—9762no—
456C2#2144488411yescharacteristic
457C2#2144489711yescharacteristic
458C2#2144491511yescharacteristic
459C2#2144492211yescharacteristic
460C2#2144613611yescharacteristic
461C2#2144617411yescharacteristic
462C2#2144619511yescharacteristic
subgroupG/H
centerC23#14519746
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingQ8×C4⋊C4×C61C2
socleC23×C61#120
prime radicalC2×#14519774C22
cosocleC2×C42×C61C23
hypercenterQ8×C4⋊C4D61
hypocenterC61#8979
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×Q8⋊#92418C2semidirect
5. #21443702C2semidirect
6. #21443660C2semidirect
7. Q8×C4⋊C4×C61C2semidirect
11. #16839962C4semidirect
118. C4⋊C4×C61SD16semidirect
149. Q8×C61#102semidirect
225. Q8×C4⋊C4D61semidirect
259. C61#8979semidirect
360. C4⋊C4#11758201semidirect
381. Q8#14519697semidirect

Maximal subgroups

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

subgroupindex
C2×#168399622
#214467462
#214466242
C2×Q8⋊#924182
#214437022
#214436602
Q8×C4⋊C4×C612
#897961