Finite Group Databasebeta

← #29733773

Subgroups — #29733773

Order 15616.

1058 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456279C211yesmaximal, characteristic
2#21456201C211yesmaximal, characteristic, complemented
3#21450804C211yesmaximal, characteristic, complemented
4#21446853C214yesmaximal, complemented
5#21446764C211yesmaximal, characteristic, complemented
6#21446739C211yesmaximal, characteristic
7#21446188C214yesmaximal, complemented
8C4×#14519696C211yesmaximal, characteristic, complemented
9C61×#3507C211yesmaximal, characteristic
10C2×#14519807C2214yes—
11C2×#14519777C2211yescharacteristic
12C2×#14519775C2212yes—
13#16840157—22no—
14#16840141—21no—
15#16840134—21no—
16#16840132—22no—
17#16840114—21no—
18#16840114—21no—
19C2×#14519704C2214yes—
20#16839770C2214yes—
21C2×#14519696C2212yes—
22C4×#11758194C2211yescharacteristic
23C4×#11758194C2211yescharacteristic
24#16839692—22no—
25#16839667C2212yes—
26#16839660—22no—
27C4×C4⋊Dic61C2211yescharacteristic
28#16839531C2212yes—
29#16839521C2214yes—
30C61×#559—21no—
31C61×#557—21no—
32C61×#556—21no—
33C61×#556—22no—
34C61×#554—21no—
35C61×#552—22no—
36C61×#539—22no—
37C61×#536C2212yes—
38C61×#533C2211yescharacteristic
39C61×#532C2211yescharacteristic
40C61×#529—22no—
41C61×#520C2212yes—
42C2×C4×Q8×C61C2211yescharacteristic
43C22×Dic122C2312yes—
44#14519807—28no—
45#14519807—44no—
46C2×#11758202—24no—
47C2×#11758200—24no—
48C2×#11758194C2312yes—
49C2×#11758194C2312yes—
50#14519777—21no—
51#14519777—21no—
52#14519777—21no—
53#14519777—21no—
54#14519775—44no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—21no—
58C2×#11758196—24no—
59C2×C4×Dic61C2312yes—
60#14519704—44no—
61#14519700—24no—
62#14519700—24no—
63#14519696—24no—
64#14519688—21no—
65#14519688—21no—
66C4×#92418—21no—
67C4×#92418—21no—
68#14519684—21no—
69#14519684—24no—
70C2×#41×C61C2312yes—
71C22×Q8×C61C2311yescharacteristic
72C4⋊Q8×C61—22no—
73C4⋊Q8×C61D411yescharacteristic
74C4⋊Q8×C61D411yescharacteristic
75C4⋊Q8×C61D411yescharacteristic
76#127×C61D411yescharacteristic
77#126×C61—24no—
78#126×C61—24no—
79#125×C61—24no—
80#125×C61—22no—
81#125×C61D412yes—
82#124×C61—22no—
83#124×C61—22no—
84#124×C61D412yes—
85#123×C61—24no—
86#123×C61—24no—
87C22⋊Q8×C61—24no—
88C22⋊Q8×C61—21no—
89C22⋊Q8×C61—21no—
90C22⋊Q8×C61—22no—
91C22⋊Q8×C61—22no—
92#121×C61—21no—
93#121×C61—21no—
94C4×Q8×C61—22no—
95C4×Q8×C61—22no—
96C4×Q8×C61—21no—
97C4×Q8×C61—21no—
98C4×Q8×C61—21no—
99C4×Q8×C61—21no—
100C4×Q8×C61—21no—
101C4×Q8×C61—21no—
102C4×D4×C61—22no—
103C4×D4×C61—22no—
104C4×D4×C61—24no—
105C4×D4×C61—21no—
106C4×D4×C61—21no—
107#117×C61—24no—
108#117×C61—22no—
109#117×C61C2312yes—
110C2×C4⋊C4×C61—22no—
111C2×C4⋊C4×C61D412yes—
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C42×C61D412yes—
114C2×C42×C61C2311yescharacteristic
115#41×C61—24no—
116#41×C61—24no—
117C2×Q8×C61—21no—
118C2×Q8×C61—21no—
119C2×Q8×C61—21no—
120C2×Q8×C61—21no—
121C2×Q8×C61—22no—
122C2×Q8×C61—22no—
123C2×Q8×C61C2×D412yes—
124C2×D4×C61—22no—
125C2×D4×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61C2×D412yes—
128C22×C4×C61—24no—
129C22×C4×C61—22no—
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D412yes—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2411yescharacteristic
135C22×Dic61—22no—
136C22×Dic61—22no—
137C2×Dic122—22no—
138C2×Dic122—22no—
139C2×Dic122—24no—
140C2×Dic122—24no—
141C4⋊C4×C61—24no—
142C4⋊C4×C61—24no—
143C4⋊C4×C61—24no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61C2×D412yes—
155C4⋊C4×C61C2×D412yes—
156C22⋊C4×C61—24no—
157C22⋊C4×C61—24no—
158C22⋊C4×C61—24no—
159C22⋊C4×C61—22no—
160C22⋊C4×C61—22no—
161C22⋊C4×C61—22no—
162C42×C61—24no—
163C42×C61—22no—
164C42×C61—22no—
165C42×C61—22no—
166C42×C61C2×D411yescharacteristic
167C42×C61C2×D411yescharacteristic
168C42×C61C2×D411yescharacteristic
169C42×C61C2×D411yescharacteristic
170#11758202—48no—
171#11758200—48no—
172C4⋊Dic61—24no—
173C4⋊Dic61—41no—
174C4⋊Dic61—41no—
175C4⋊Dic61—41no—
176C4⋊Dic61—41no—
177#11758196—48no—
178C4×Dic61—24no—
179#11758194—22no—
180#11758194—22no—
181#11758194—22no—
182#11758194#4114yes—
183C2×#92418—22no—
184C2×#92418—22no—
185C23×C61—22no—
186C23×C61C22×D411yescharacteristic
187Q8×C61—24no—
188Q8×C61—42no—
189Q8×C61—42no—
190Q8×C61—42no—
191Q8×C61—81no—
192D4×C61—24no—
193D4×C61—42no—
194D4×C61—42no—
195D4×C61—42no—
196D4×C61—81no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—41no—
206C2×C4×C61—44no—
207C2×C4×C61—42no—
208C2×C4×C61—84no—
209C2×C4×C61#12011yescharacteristic
210C2×C4×C61#12011yescharacteristic
211C2×C4×C61#12011yescharacteristic
212C2×C4×C61#12011yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C2×#4112yes—
216C2×C4×C61H(5,𝔽2)12yes—
217C2×Dic61—24no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223Dic122—44no—
224Dic122—44no—
225#92418—24no—
226#92418—42no—
227#25623—611nomaximal, Sylow
228C22×C61—21no—
229C22×C61—21no—
230C22×C61—42no—
231C22×C61—42no—
232C22×C61—42no—
233C22×C61C2×#12011yescharacteristic
234C22×C61#53211yescharacteristic
235C22×C61#53511yescharacteristic
236C4×C61—21no—
237C4×C61—21no—
238C4×C61—21no—
239C4×C61—42no—
240C4×C61—42no—
241C4×C61—42no—
242C4×C61—42no—
243C4×C61—42no—
244C4×C61#45612yes—
245C4×C61#54612yes—
246C4×C61#55012yes—
247Dic61—44no—
248Dic61—82no—
249#3507D6111yescharacteristic
250#3152—611no—
251#3139—611no—
252#2546—611no—
253#2005—614no—
254#1986—611no—
255#1980—611no—
256#1882—614no—
257C4×#101—611no—
258C2×C61—21no—
259C2×C61—21no—
260C2×C61—82no—
261C2×C61#243111yescharacteristic
262C2×C61C2×#45611yescharacteristic
263C2×C61#302211yescharacteristic
264C2×#137—614no—
265#559—21no—
266#557—21no—
267#556—21no—
268#556—22no—
269#554—21no—
270#552—22no—
271#539—22no—
272#536C2×D6112yes—
273#533C2×D6111yescharacteristic
274#532C2×D6111yescharacteristic
275C2×C4⋊Q8—612no—
276C2×C4⋊Q8—611no—
277#529—22no—
278#520C2×D6112yes—
279C2×C4×Q8C2×D6111yescharacteristic
280#497—1222no—
281#494—1221no—
282#492—1221no—
283#491—1222no—
284#487—1221no—
285#487—1221no—
286C2×#104—614no—
287#419—614no—
288C2×#101—612no—
289C4×M16—611no—
290C4×M16—611no—
291#398—1222no—
292#391—612no—
293#389—1222no—
294C4×C4⋊C4—611no—
295#343—612no—
296#339—614no—
297C61#2562311yescharacteristic, complemented, Sylow
298C2×#41C22×D6112yes—
299C22×Q8—612no—
300C22×Q8C22×D6111yescharacteristic
301#137—1228no—
302#137—2444no—
303C2×Q16—1224no—
304C2×SD16—1224no—
305C2×M16—612no—
306C2×M16—612no—
307C4⋊Q8—22no—
308C4⋊Q8—1221no—
309C4⋊Q8—1221no—
310C4⋊Q8—1221no—
311C4⋊Q8—1221no—
312C4⋊Q8—2444no—
313C4⋊Q8C22⋊D6111yescharacteristic
314C4⋊Q8C22⋊D6111yescharacteristic
315C4⋊Q8C22⋊D6111yescharacteristic
316#127C22⋊D6111yescharacteristic
317#126—24no—
318#126—24no—
319#125—22no—
320#125—24no—
321#125C22⋊D6112yes—
322#124—22no—
323#124—22no—
324#124C22⋊D6112yes—
325#123—24no—
326#123—24no—
327C22⋊Q8—24no—
328C22⋊Q8—21no—
329C22⋊Q8—21no—
330C22⋊Q8—22no—
331C22⋊Q8—22no—
332#121—21no—
333#121—21no—
334C4×Q8—22no—
335C4×Q8—22no—
336C4×Q8—21no—
337C4×Q8—21no—
338C4×Q8—21no—
339C4×Q8—21no—
340C4×Q8—21no—
341C4×Q8—21no—
342C4×D4—22no—
343C4×D4—22no—
344C4×D4—24no—
345C4×D4—21no—
346C4×D4—21no—
347#117—24no—
348#117—22no—
349#117C22×D6112yes—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—1221no—
352C2×C4⋊C4—1221no—
353C2×C4⋊C4—1224no—
354C2×C4⋊C4—1221no—
355C2×C4⋊C4C22⋊D6112yes—
356C2×C4⋊C4C22×D6111yescharacteristic
357C2×C42—612no—
358C2×C42C22⋊D6112yes—
359C2×C42C22×D6111yescharacteristic
360#104—2444no—
361#103—1224no—
362#103—1224no—
363#101—1224no—
364#97—1221no—
365#97—1221no—
366C4×C8—1221no—
367C4×C8—1221no—
368#95—1221no—
369#95—1224no—
370#41—24no—
371#41—24no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—1222no—
379C2×Q8—1222no—
380C2×Q8—1224no—
381C2×Q8—1224no—
382C2×Q8C2×C22⋊D6112yes—
383C2×D4—22no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4C2×C22⋊D6112yes—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—1222no—
390C22×C4—1222no—
391C22×C4C2×C22⋊D6112yes—
392C22×C4C2×C22⋊D6111yescharacteristic
393C22×C4C2×C22⋊D6112yes—
394C22×C4C2×C22⋊D6111yescharacteristic
395C22×C4C23×D6111yescharacteristic
396Q16—2448no—
397SD16—2448no—
398M16—614no—
399M16—1222no—
400M16—1222no—
401M16—1222no—
402C2×C8—1222no—
403C2×C8—1222no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—22no—
416C4⋊C4—21no—
417C4⋊C4—1224no—
418C4⋊C4—2448no—
419C4⋊C4—2441no—
420C4⋊C4—2441no—
421C4⋊C4—2441no—
422C4⋊C4—2441no—
423C4⋊C4C2×C22⋊D6112yes—
424C4⋊C4C2×C22⋊D6112yes—
425C22⋊C4—24no—
426C22⋊C4—24no—
427C22⋊C4—24no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C42—24no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—1224no—
436C42C2×C22⋊D6111yescharacteristic
437C42C2×C22⋊D6111yescharacteristic
438C42C2×C22⋊D6111yescharacteristic
439C42C2×C22⋊D6111yescharacteristic
440C23—22no—
441C23C22×C22⋊D6111yescharacteristic
442Q8—24no—
443Q8—42no—
444Q8—42no—
445Q8—42no—
446Q8—81no—
447Q8—2444no—
448Q8—2444no—
449D4—24no—
450D4—42no—
451D4—42no—
452D4—42no—
453D4—81no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—41no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—84no—
466C2×C4—1224no—
467C2×C4—2442no—
468C2×C4—2442no—
469C2×C4—2442no—
470C2×C4—2442no—
471C2×C4—2442no—
472C2×C4#1451974611yescharacteristic
473C2×C4#1451974611yescharacteristic
474C2×C4#1451974611yescharacteristic
475C2×C4#1451974611yescharacteristic
476C2×C4C22×C22⋊D6111yescharacteristic
477C2×C4C22×C22⋊D6111yescharacteristic
478C2×C4#41×D6112yes—
479C2×C4#1451982312yes—
480C8—1224no—
481C8—2442no—
482C22—21no—
483C22—21no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22C2×#1451974611yescharacteristic
488C22#1684030811yescharacteristic
489C22#1684032611yescharacteristic
490C4—21no—
491C4—21no—
492C4—21no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—4882no—
499C4—2444no—
500C4#1683994512yes—
501C4#1684040012yes—
502C4#1684043112yes—
503C2—21no—
504C2—21no—
505C2—82no—
506C2#2144943411yescharacteristic
507C2C2×#1683994511yescharacteristic
508C2#2145519811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3507C2
socleC22×C61C2×#120
prime radicalC61×#532C22
hypercenter#3507D61
hypocenterC61#25623
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3507

Complements

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

NH ≅ G/Ntype
2. #21456201C2semidirect
3. #21450804C2semidirect
4. #21446853C2semidirect
5. #21446764C2semidirect
7. #21446188C2semidirect
8. C4×#14519696C2semidirect
297. C61#25623semidirect

Maximal subgroups

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

subgroupindex
#214562792
#214562012
#214508042
#214468532
#214467642
#214467392
#214461882
C4×#145196962
C61×#35072
#2562361