Finite Group Databasebeta

← #30599881

Subgroups — #30599881

Order 15616.

540 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 102 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
1#21462738C211yesmaximal, characteristic, complemented
2#21461940C211yesmaximal, characteristic, complemented
3#21461118C211yesmaximal, characteristic, complemented
4#21461110C211yesmaximal, characteristic, complemented
5#21460665C211yesmaximal, characteristic, complemented
6#21460652C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×Dic61C211yesmaximal, characteristic
8#16840813—21no—
9#16840797—21no—
10#16840793—22no—
11#16840783—21no—
12#16840778C2211yescharacteristic
13#16840778C2211yescharacteristic
14#16840765—21no—
15C2×C4⋊#92420C2211yescharacteristic
16C2×C22⋊#92420—22no—
17C2×C4×#92420C2211yescharacteristic
18Q8⋊#92420—21no—
19D4⋊#92420—21no—
20D4⋊#92420—21no—
21D4⋊#92420—21no—
22#41×Dic61C2211yescharacteristic
23C2×D4×Dic61C2211yescharacteristic
24#16839874—21no—
25#16839863—21no—
26SD16×Dic61—21no—
27D8×Dic61—21no—
28#16839804C411yescharacteristic
29#16839767C411yescharacteristic
30#16839763C411yescharacteristic
31M16×Dic61C2211yescharacteristic
32C2×AGL(1,ℤ/8)×C61C411yescharacteristic
33C2×#11758217—22no—
34C22×#92420—21no—
35C22×#92420D412yes—
36#14519844—22no—
37C4⋊#92420—21no—
38C4⋊#92420—21no—
39C4⋊#92420—21no—
40C4⋊#92420D411yescharacteristic
41C4⋊#92420D411yescharacteristic
42C22⋊#92420—22no—
43C22⋊#92420—21no—
44C22⋊#92420—21no—
45C22⋊#92420—44no—
46#14519832—22no—
47C4×#92420—21no—
48C4×#92420—21no—
49C4×#92420—21no—
50C4×#92420—21no—
51C23×Dic61—21no—
52C2×#11758194C2×C411yescharacteristic
53Q8×Dic61D411yescharacteristic
54Q8×Dic61D411yescharacteristic
55D4×Dic61—21no—
56D4×Dic61—21no—
57D4×Dic61—21no—
58D4×Dic61—21no—
59D4×Dic61—21no—
60D4×Dic61—21no—
61D4×Dic61D411yescharacteristic
62D4×Dic61D411yescharacteristic
63#14519736—21no—
64#14519736C2×C411yescharacteristic
65C2×C4⋊Dic61C2×C411yescharacteristic
66C2×C22⋊Dic61—21no—
67C2×C4×Dic61—21no—
68C2×C4×Dic61C2311yescharacteristic
69#14519711—21no—
70#14519709—21no—
71#14519701—21no—
72#14519698—21no—
73#14519698—21no—
74#14519698—21no—
75#14519687—21no—
76C8×Dic61—21no—
77C2×#41×C61C2×C411yescharacteristic
78C22×D4×C61C2×C411yescharacteristic
79AGL(1,ℤ/8)×C61—22no—
80AGL(1,ℤ/8)×C61—22no—
81C2×SD16×C61—21no—
82C2×D8×C61—21no—
83C2×M16×C61C2×C411yescharacteristic
84C24×C61—21no—
85#41×C61—21no—
86#41×C61—21no—
87#41×C61—21no—
88#41×C61—21no—
89C2×Q8×C61C22⋊C411yescharacteristic
90C2×D4×C61—21no—
91C2×D4×C61—21no—
92C2×D4×C61—21no—
93C2×D4×C61—21no—
94C2×D4×C61—21no—
95C2×D4×C61—21no—
96C2×D4×C61C22⋊C411yescharacteristic
97C2×D4×C61M1612yescomplemented
98C22×C4×C61—21no—
99C22×C4×C61C22×C411yescharacteristic
100C22×Dic61—21no—
101C22×Dic61—21no—
102C22×Dic61—21no—
103C22×Dic61—42no—
104C22×Dic61—42no—
105C22×Dic61—41no—
106C22×Dic61—41no—
107C22×Dic61C2×D411yescharacteristic
108C22×Dic61#4111yescharacteristic
109#11758217—82no—
110C2×#92420—22no—
111C2×#92420—21no—
112C2×#92420—21no—
113C2×#92420—22no—
114C2×#92420—41no—
115C2×#92420—41no—
116C2×#92420—42no—
117C2×#92420—41no—
118SD16×C61—24no—
119D8×C61—24no—
120M16×C61—21no—
121M16×C61M1612yescomplemented
122C2×C8×C61—21no—
123C22⋊Dic61—21no—
124C22⋊Dic61—21no—
125C22⋊Dic61—21no—
126C22⋊Dic61—41no—
127C22⋊Dic61—41no—
128C4⋊Dic61—21no—
129C4⋊Dic61—21no—
130C4⋊Dic61—21no—
131C4⋊Dic61—21no—
132C4⋊Dic61C22⋊C411yescharacteristic
133C4⋊Dic61C22⋊C411yescharacteristic
134C4×Dic61—21no—
135C4×Dic61—21no—
136C4×Dic61—21no—
137C4×Dic61C2×D411yescharacteristic
138C4×Dic61C2×D411yescharacteristic
139C4×Dic61C2×D411yescharacteristic
140C4×Dic61#4111yescharacteristic
141#11758194—21no—
142#11758194—21no—
143C2×#92418—21no—
144C23×C61—21no—
145C23×C61—22no—
146C23×C61—42no—
147C23×C61—42no—
148C23×C61—41no—
149C23×C61—41no—
150C23×C61C4×D411yescharacteristic
151Q8×C61—21no—
152Q8×C61—21no—
153D4×C61—21no—
154D4×C61—22no—
155D4×C61—21no—
156D4×C61—22no—
157D4×C61—41no—
158D4×C61—41no—
159D4×C61—41no—
160D4×C61—41no—
161C2×C4×C61—21no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61C2×C22⋊C411yescharacteristic
165C2×C4×C61C4×D411yescharacteristic
166C2×C4×C61C2×M1611yescharacteristic
167C2×C4×C61C2×M1611yescharacteristic
168C2×C4×C61#13111yescharacteristic
169C2×C4×C61#13111yescharacteristic
170C2×Dic61—21no—
171C2×Dic61—21no—
172C2×Dic61—21no—
173C2×Dic61—21no—
174C2×Dic61—41no—
175C2×Dic61—41no—
176C2×Dic61—41no—
177C2×Dic61—42no—
178C2×Dic61—41no—
179C2×Dic61—81no—
180C2×Dic61—82no—
181C2×Dic61—81no—
182C2×Dic61—81no—
183C2×Dic61#12011yescharacteristic
184C2×Dic61#12111yescharacteristic
185C2×Dic61#12111yescharacteristic
186C2×Dic61#12311yescharacteristic
187#92420—42no—
188#92420—42no—
189#92420—41no—
190#92420—41no—
191#92420—81no—
192C8×C61—22no—
193#92418—41no—
194#8956—611nomaximal, Sylow
195C22×C61—21no—
196C22×C61—21no—
197C22×C61—44no—
198C22×C61—41no—
199C22×C61—42no—
200C22×C61—41no—
201C22×C61—41no—
202C22×C61—82no—
203C22×C61—81no—
204C22×C61—81no—
205C22×C61#38611yescharacteristic
206C22×C61#43511yescharacteristic
207C22×C61#43611yescharacteristic
208C4×C61—41no—
209C4×C61#40711yescharacteristic
210C4×C61#40811yescharacteristic
211C4×C61#43511yescharacteristic
212C4×C61#43611yescharacteristic
213Dic61—21no—
214Dic61—21no—
215Dic61—21no—
216Dic61—41no—
217Dic61—41no—
218Dic61—81no—
219Dic61#45311yescharacteristic
220Dic61#45411yescharacteristic
221#2989—611no—
222C4×AGL(1,ℤ/8)—611no—
223#1954—611no—
224#1478—611no—
225#1476—611no—
226#1283—611no—
227#1278—611no—
228C2×C61—21no—
229C2×C61—21no—
230C2×C61—81no—
231C2×C61—82no—
232C2×C61#187011yescharacteristic
233C2×C61#188111yescharacteristic
234C2×C61#188311yescharacteristic
235C2×AGL(1,ℤ/8)#692311yescharacteristic
236C4×#41—611no—
237C2×C4×D4—611no—
238#446—1221no—
239#442—1221no—
240#440—1221no—
241C4×SD16—1221no—
242C4×D8—1221no—
243#436—1221no—
244#435—1222no—
245#432—1221no—
246#431—611no—
247#431—611no—
248#428—611no—
249#423—1221no—
250C2×C4⋊C8—611no—
251#418—611no—
252#417—611no—
253C2×C22⋊C8—1222no—
254C4×M16—611no—
255C2×C4×C8—611no—
256Q8⋊C8—1221no—
257D4⋊C8—1221no—
258D4⋊C8—1221no—
259D4⋊C8—1221no—
260C61#895611yescharacteristic, complemented, Sylow
261C2×#41C2×#692311yescharacteristic
262C22×D4C2×#692311yescharacteristic
263C23×C4—1221no—
264AGL(1,ℤ/8)—22no—
265AGL(1,ℤ/8)—22no—
266C2×SD16—21no—
267C2×D8—21no—
268C2×M16—611no—
269C2×M16—1222no—
270C2×M16C2×#692311yescharacteristic
271C22×C8—612no—
272C22×C8—1221no—
273C4×Q8—611no—
274C4×Q8—611no—
275C4×D4—611no—
276C4×D4—611no—
277C4×D4—1221no—
278C4×D4—1221no—
279C4×D4—1221no—
280C4×D4—1221no—
281C4×D4—1221no—
282C4×D4—1221no—
283#117—611no—
284#117—1221no—
285C2×C4⋊C4—611no—
286C2×C22⋊C4—1221no—
287C2×C42—611no—
288C2×C42—1221no—
289#107—1221no—
290#106—1221no—
291C4⋊C8—611no—
292C4⋊C8—611no—
293C4⋊C8—1221no—
294C4⋊C8—1221no—
295C4⋊C8—1221no—
296C4⋊C8—1222no—
297#103—1221no—
298#102—1221no—
299#102—1221no—
300#102—1221no—
301C22⋊C8—1221no—
302C22⋊C8—1222no—
303C22⋊C8—1221no—
304C22⋊C8—2444no—
305#97—1221no—
306#97—1222no—
307C4×C8—1221no—
308C4×C8—1221no—
309C4×C8—1221no—
310C4×C8—1221no—
311C4×C8—1221no—
312C24—21no—
313#41—21no—
314#41—21no—
315#41—21no—
316#41—21no—
317C2×Q8C22⋊#692311yescharacteristic
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4#1175821712yescomplemented
325C2×D4C22⋊#692311yescharacteristic
326C22×C4—21no—
327C22×C4—611no—
328C22×C4—611no—
329C22×C4—1221no—
330C22×C4—1221no—
331C22×C4—1221no—
332C22×C4—2441no—
333C22×C4—2441no—
334C22×C4—2442no—
335C22×C4—2442no—
336C22×C4C22×#692311yescharacteristic
337SD16—24no—
338D8—24no—
339M16—21no—
340M16—1221no—
341M16—1221no—
342M16—4882no—
343M16#1175821712yescomplemented
344C2×C8—21no—
345C2×C8—1221no—
346C2×C8—1221no—
347C2×C8—1222no—
348C2×C8—1222no—
349C2×C8—1221no—
350C2×C8—2441no—
351C2×C8—2441no—
352C2×C8—2441no—
353C2×C8—2442no—
354C4⋊C4—611no—
355C4⋊C4—611no—
356C4⋊C4—1221no—
357C4⋊C4—1221no—
358C4⋊C4—1221no—
359C4⋊C4—1221no—
360C22⋊C4—1221no—
361C22⋊C4—1221no—
362C22⋊C4—1221no—
363C22⋊C4—2441no—
364C22⋊C4—2441no—
365C42—611no—
366C42—611no—
367C42—611no—
368C42—611no—
369C42—1221no—
370C42—1221no—
371C42—1221no—
372C23—21no—
373C23—22no—
374C23—42no—
375C23—42no—
376C23—41no—
377C23—41no—
378C23D4×#692311yescharacteristic
379Q8—21no—
380Q8—21no—
381D4—21no—
382D4—22no—
383D4—21no—
384D4—22no—
385D4—41no—
386D4—41no—
387D4—41no—
388D4—41no—
389C2×C4—21no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—611no—
393C2×C4—611no—
394C2×C4—611no—
395C2×C4—611no—
396C2×C4—1221no—
397C2×C4—1221no—
398C2×C4—1221no—
399C2×C4—1221no—
400C2×C4—4881no—
401C2×C4—4881no—
402C2×C4—4882no—
403C2×C4—4881no—
404C2×C4—2441no—
405C2×C4—2441no—
406C2×C4—2441no—
407C2×C4—2442no—
408C2×C4—2441no—
409C2×C4C2×C22⋊#692311yescharacteristic
410C2×C4D4×#692311yescharacteristic
411C2×C4C2×#1175821711yescharacteristic
412C2×C4C2×#1175821711yescharacteristic
413C2×C4#1451986411yescharacteristic
414C2×C4#1451986411yescharacteristic
415C8—22no—
416C8—4881no—
417C8—2441no—
418C8—2442no—
419C8—2441no—
420C8—2442no—
421C8—2441no—
422C22—21no—
423C22—21no—
424C22—44no—
425C22—41no—
426C22—42no—
427C22—41no—
428C22—41no—
429C22—82no—
430C22—81no—
431C22—81no—
432C22#1684071011yescharacteristic
433C22#1684079311yescharacteristic
434C22#1684079711yescharacteristic
435C4—41no—
436C4—611no—
437C4—611no—
438C4—1221no—
439C4—1221no—
440C4—1221no—
441C4—4881no—
442C4—2441no—
443C4—2441no—
444C4#1684073211yescharacteristic
445C4#1684073511yescharacteristic
446C4#1684079311yescharacteristic
447C4#1684079711yescharacteristic
448C2—21no—
449C2—21no—
450C2—82no—
451C2—81no—
452C2#2146179611yescharacteristic
453C2#2146181711yescharacteristic
454C2#2146181911yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
Fitting, prime radicalC2×AGL(1,ℤ/8)×C61C4
socleC22×C61#386
cosocleC2×C4×Dic61C23
hypercenterC2×AGL(1,ℤ/8)#6923
hypocenterC61#8956
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #21462738C2semidirect
2. #21461940C2semidirect
3. #21461118C2semidirect
4. #21461110C2semidirect
5. #21460665C2semidirect
6. #21460652C2semidirect
97. C2×D4×C61M16semidirect
121. M16×C61M16semidirect
260. C61#8956semidirect
324. C2×D4#11758217semidirect
343. M16#11758217semidirect

Maximal subgroups

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

subgroupindex
#214627382
#214619402
#214611182
#214611102
#214606652
#214606522
AGL(1,ℤ/8)×Dic612
#895661