Finite Group Databasebeta

← #28483555

Subgroups — #28483555

Order 15104.

500 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434127C211yesmaximal, characteristic
2#21425015C211yesmaximal, characteristic
3C4×#14273651C211yesmaximal, characteristic, complemented
4#21422546C211yesmaximal, characteristic
5#21421997C211yesmaximal, characteristic, complemented
6#21421934C211yesmaximal, characteristic, complemented
7Q8×M16×C59C211yesmaximal, characteristic, complemented
8#16837952C2211yescharacteristic
9#16837720—21no—
10#16837687C411yescharacteristic, complemented
11#16837687C411yescharacteristic, complemented
12#16837602C411yescharacteristic, complemented
13#16837602C411yescharacteristic, complemented
14#16837541—21no—
15C4×#11745671—21no—
16C4×#11745671—21no—
17#16837455C2211yescharacteristic
18C2×#14273653—21no—
19C2×#14273653—21no—
20C2×#14273651—21no—
21C2×C4×C59⋊C8C2211yescharacteristic
22C4×C4⋊Dic59C2211yescharacteristic
23#16837164—21no—
24#16837129—22no—
25C2×C4×Q8×C59C2211yescharacteristic
26C59×#446—21no—
27C59×#446—22no—
28C8×Q8×C59—21no—
29C59×#423—21no—
30C59×#423C2211yescharacteristic
31C4×M16×C59—21no—
32C4×M16×C59C2211yescharacteristic
33C2×#11745685—21no—
34C2×#11745671—21no—
35C2×#11745671—21no—
36C22×C59⋊C8—21no—
37#14273727C2×C411yescharacteristic
38#14273724C2×C411yescharacteristic
39#14273715—21no—
40#14273701—21no—
41#14273701—21no—
42C4×D236—21no—
43C4×D236—21no—
44C2×C4⋊Dic59—21no—
45C2×C4×Dic59—21no—
46#14273658—21no—
47#14273658—21no—
48C4⋊(C59⋊C8)C2×C411yescharacteristic
49C4⋊(C59⋊C8)C2×C411yescharacteristic
50#14273653—42no—
51#14273653—42no—
52#14273651—24no—
53#14273651—21no—
54#14273651—21no—
55#14273647—21no—
56#14273647—21no—
57C22⋊(C59⋊C8)—21no—
58C4×C59⋊C8—21no—
59C4×C59⋊C8—21no—
60C4×C59⋊C8—21no—
61C59⋊#95—21no—
62C22×Q8×C59D411yescharacteristic
63C2×M16×C59—21no—
64C2×M16×C59D411yescharacteristic
65C2×M16×C59D411yescharacteristic
66C4×Q8×C59—21no—
67C4×Q8×C59—21no—
68C4×Q8×C59C2×C411yescharacteristic
69C4×Q8×C59C2×C411yescharacteristic
70C4×Q8×C59D411yescharacteristic
71C4×Q8×C59D411yescharacteristic
72C2×C4⋊C4×C59—21no—
73C2×C4⋊C4×C59D411yescharacteristic
74C2×C42×C59—21no—
75C2×C42×C59C2311yescharacteristic
76C4⋊C8×C59—22no—
77C4⋊C8×C59—22no—
78C4⋊C8×C59—21no—
79C4⋊C8×C59D412yes—
80#97×C59—22no—
81#97×C59—21no—
82C4×C8×C59—22no—
83C4×C8×C59—21no—
84C2×Q8×C59—21no—
85C2×Q8×C59—22no—
86C2×Q8×C59—41no—
87C2×Q8×C59C22⋊C411yescharacteristic
88C2×Q8×C59C22⋊C411yescharacteristic
89C22×C4×C59—21no—
90C22×C4×C59—21no—
91C22×C4×C59C2×D411yescharacteristic
92C22×C4×C59C2×D411yescharacteristic
93C22×C4×C59#4111yescharacteristic
94C2×C22⋊D59—41no—
95C22×Dic59—41no—
96#11745685—41no—
97#11745685—41no—
98#11745685—41no—
99#11745685—41no—
100C2×D236—21no—
101C2×D236—21no—
102C2×D236—21no—
103C2×C4×D59—41no—
104C2×Dic118—21no—
105M16×C59—21no—
106M16×C59—21no—
107M16×C59—21no—
108M16×C59—41no—
109M16×C59—41no—
110M16×C59SD1612yescomplemented
111C2×C8×C59—21no—
112C2×C8×C59—21no—
113C2×C8×C59—22no—
114C4⋊C4×C59—21no—
115C4⋊C4×C59—21no—
116C4⋊C4×C59—21no—
117C4⋊C4×C59—21no—
118C4⋊C4×C59—21no—
119C4⋊C4×C59C22⋊C411yescharacteristic
120C4⋊C4×C59C22⋊C411yescharacteristic
121C42×C59—21no—
122C42×C59—21no—
123C42×C59—21no—
124C42×C59—21no—
125C42×C59C22×C411yescharacteristic
126C42×C59C2×D411yescharacteristic
127C42×C59C2×D411yescharacteristic
128C42×C59#4111yescharacteristic
129#11745671—42no—
130#11745671—42no—
131(C2×D59)⋊C4—41no—
132(C2×D59)⋊C4—41no—
133C4⋊Dic59—22no—
134C4⋊Dic59—21no—
135C4⋊Dic59—21no—
136C4⋊Dic59—41no—
137C4×Dic59—41no—
138C4×Dic59—41no—
139C2×C59⋊C8—22no—
140C2×C59⋊C8—21no—
141C2×C59⋊C8—21no—
142C2×C59⋊C8—41no—
143C2×C59⋊C8—41no—
144C23×C59#12111yescharacteristic
145C22×D59—41no—
146Q8×C59—21no—
147Q8×C59—21no—
148Q8×C59—21no—
149Q8×C59—42no—
150Q8×C59#10412yescomplemented
151C2×C4×C59—21no—
152C2×C4×C59—21no—
153C2×C4×C59—21no—
154C2×C4×C59—21no—
155C2×C4×C59—21no—
156C2×C4×C59—21no—
157C2×C4×C59—41no—
158C2×C4×C59—42no—
159C2×C4×C59—81no—
160C2×C4×C59C2×C22⋊C411yescharacteristic
161C2×C4×C59C4×D411yescharacteristic
162C2×C4×C59C4×D411yescharacteristic
163C2×C4×C59#12011yescharacteristic
164C2×C4×C59#12111yescharacteristic
165C2×C4×C59#12311yescharacteristic
166C2×C4×C59C2×SD1611yescharacteristic
167C2×C4×C59#13511yescharacteristic
168C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C59#13711yescharacteristic
170C22⋊D59—81no—
171C22⋊D59—81no—
172C2×Dic59—42no—
173C2×Dic59—41no—
174C2×Dic59—81no—
175C2×Dic59—81no—
176D236—42no—
177D236—41no—
178D236—41no—
179D236—41no—
180D236—41no—
181C4×D59—81no—
182C4×D59—81no—
183Dic118—41no—
184Dic118—41no—
185C8×C59—22no—
186C8×C59—41no—
187C8×C59—42no—
188C59⋊C8—42no—
189C59⋊C8—41no—
190C59⋊C8—41no—
191#8992—591nomaximal, Sylow
192C22×C59—21no—
193C22×C59—21no—
194C22×C59#38611yescharacteristic
195C22×C59#46011yescharacteristic
196C22×C59#46411yescharacteristic
197C2×D59—81no—
198C2×D59—81no—
199C2×D59—81no—
200C4×C59—21no—
201C4×C59—22no—
202C4×C59—21no—
203C4×C59—21no—
204C4×C59—41no—
205C4×C59—41no—
206C4×C59—41no—
207C4×C59C2×#10411yescharacteristic
208C4×C59#42111yescharacteristic
209C4×C59C4×SD1611yescharacteristic
210C4×C59#44011yescharacteristic
211C4×C59#44811yescharacteristic
212C4×C59#44911yescharacteristic
213Dic59—82no—
214Dic59—81no—
215#3103—591no—
216Q8×M16D5911yescharacteristic, complemented
217#1980—591no—
218C4×#103—591no—
219#1492—591no—
220#1303—591no—
221#1278—591no—
222C2×C59—21no—
223C2×C59—21no—
224C2×C59#187111yescharacteristic
225C2×C59#188011yescharacteristic
226C2×C59#188111yescharacteristic
227D59—161no—
228#532—591no—
229C2×C4×Q8C2×D5911yescharacteristic
230#481—1181no—
231#475—591no—
232#475—591no—
233#460—591no—
234#460—591no—
235#448—1181no—
236#446—21no—
237#446—22no—
238C8×Q8—21no—
239C4×SD16—1181no—
240C4×SD16—1181no—
241#431—591no—
242#423—21no—
243#423C2×D5911yescharacteristic
244C2×#104—1181no—
245C2×#104—1181no—
246C2×#103—1181no—
247C4×M16—21no—
248C4×M16C2×D5911yescharacteristic
249C2×C4×C8—591no—
250C4×C4⋊C4—591no—
251#340—1181no—
252D4⋊C8—1182no—
253C59#899211yescharacteristic, complemented, Sylow
254C2×#41—1181no—
255C22×Q8D23611yescharacteristic
256C2×SD16—1181no—
257C2×SD16—1181no—
258C2×M16—21no—
259C2×M16C22⋊D5911yescharacteristic
260C2×M16C22⋊D5911yescharacteristic
261C22×C8—1181no—
262C4⋊Q8—591no—
263#127—591no—
264#124—1181no—
265#121—1181no—
266#121—1181no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8C4×D5911yescharacteristic
270C4×Q8C4×D5911yescharacteristic
271C4×Q8C22⋊D5911yescharacteristic
272C4×Q8C22⋊D5911yescharacteristic
273C4×D4—1181no—
274C4×D4—1181no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—1181no—
277C2×C4⋊C4D23611yescharacteristic
278C2×C42—21no—
279C2×C42—1181no—
280C2×C42C22×D5911yescharacteristic
281#106—1181no—
282#106—1181no—
283C4⋊C8—22no—
284C4⋊C8—22no—
285C4⋊C8—21no—
286C4⋊C8—591no—
287C4⋊C8—591no—
288C4⋊C8C22⋊D5912yes—
289#104—2362no—
290#104—2362no—
291#103—1184no—
292#103—1181no—
293#103—1181no—
294#102—1181no—
295#102—1181no—
296C22⋊C8—1181no—
297#97—22no—
298#97—21no—
299C4×C8—22no—
300C4×C8—21no—
301C4×C8—1181no—
302C4×C8—1181no—
303C4×C8—1181no—
304#95—1181no—
305#41—2361no—
306#41—2361no—
307#41—2361no—
308#41—2361no—
309C2×Q8—21no—
310C2×Q8—22no—
311C2×Q8—41no—
312C2×Q8—1181no—
313C2×Q8(C2×D59)⋊C411yescharacteristic
314C2×Q8(C2×D59)⋊C411yescharacteristic
315C2×D4—1181no—
316C2×D4—1181no—
317C2×D4—1181no—
318C2×D4—2361no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—2361no—
322C22×C4—2361no—
323C22×C4C2×D23611yescharacteristic
324C22×C4#1174568511yescharacteristic
325C22×C4C2×C22⋊D5911yescharacteristic
326SD16—2362no—
327SD16—2362no—
328M16—21no—
329M16—21no—
330M16—21no—
331M16—41no—
332M16—41no—
333M16#1174567112yescomplemented
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—22no—
337C2×C8—1182no—
338C2×C8—1181no—
339C2×C8—1181no—
340C2×C8—2361no—
341C2×C8—2361no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—1182no—
348C4⋊C4—1181no—
349C4⋊C4—1181no—
350C4⋊C4—2361no—
351C4⋊C4(C2×D59)⋊C411yescharacteristic
352C4⋊C4(C2×D59)⋊C411yescharacteristic
353C22⋊C4—2361no—
354C22⋊C4—2361no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—2361no—
360C42—2361no—
361C42C2×C4×D5911yescharacteristic
362C42#1174568511yescharacteristic
363C42C2×C22⋊D5911yescharacteristic
364C42C2×C22⋊D5911yescharacteristic
365C23—2361no—
366C23#1427370111yescharacteristic
367Q8—21no—
368Q8—21no—
369Q8—21no—
370Q8—42no—
371Q8—2361no—
372Q8—2361no—
373Q8#1427365312yescomplemented
374D4—4721no—
375D4—4721no—
376D4—2362no—
377D4—2361no—
378D4—2361no—
379D4—2361no—
380D4—2361no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—81no—
390C2×C4—4721no—
391C2×C4—4721no—
392C2×C4—4721no—
393C2×C4—4721no—
394C2×C4—2362no—
395C2×C4—2361no—
396C2×C4C2×(C2×D59)⋊C411yescharacteristic
397C2×C4C4×C22⋊D5911yescharacteristic
398C2×C4C4×C22⋊D5911yescharacteristic
399C2×C4#1427369611yescharacteristic
400C2×C4#1427370111yescharacteristic
401C2×C4#1427371011yescharacteristic
402C2×C4C2×#1174567111yescharacteristic
403C2×C4#1427374711yescharacteristic
404C2×C4#1427375211yescharacteristic
405C2×C4#1427375811yescharacteristic
406C8—22no—
407C8—41no—
408C8—42no—
409C8—2362no—
410C8—2361no—
411C8—2361no—
412C22—21no—
413C22—21no—
414C22—4721no—
415C22—4721no—
416C22—4721no—
417C22#1683728111yescharacteristic
418C22#1683760211yescharacteristic
419C22#1683762411yescharacteristic
420C4—21no—
421C4—22no—
422C4—21no—
423C4—21no—
424C4—41no—
425C4—41no—
426C4—41no—
427C4—4722no—
428C4—4721no—
429C4C2×#1427365311yescharacteristic
430C4#1683741811yescharacteristic
431C4C4×#1174567111yescharacteristic
432C4#1683750011yescharacteristic
433C4#1683754111yescharacteristic
434C4#1683754711yescharacteristic
435C2—21no—
436C2—21no—
437C2—9441no—
438C2#2142439811yescharacteristic
439C2#2142445111yescharacteristic
440C2#2142445811yescharacteristic
subgroupG/H
centerC2×C4#14273696
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingQ8×M16×C59C2
socleC22×C59#386
prime radical#16837952C22
cosocleC2×C42×C59C23
hypercenterQ8×M16D59
hypocenterC59#8992
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
3. C4×#14273651C2semidirect
5. #21421997C2semidirect
6. #21421934C2semidirect
7. Q8×M16×C59C2semidirect
10. #16837687C4semidirect
11. #16837687C4semidirect
12. #16837602C4semidirect
13. #16837602C4semidirect
110. M16×C59SD16semidirect
150. Q8×C59#104semidirect
216. Q8×M16D59semidirect
253. C59#8992semidirect
333. M16#11745671semidirect
373. Q8#14273653semidirect

Maximal subgroups

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

subgroupindex
#214341272
#214250152
C4×#142736512
#214225462
#214219972
#214219342
Q8×M16×C592
#899259