Finite Group Databasebeta

← #30603595

Subgroups — #30603595

Order 15616.

822 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840773C212yesmaximal
2#21462299C211yesmaximal, characteristic
3C2×#16840634C212yesmaximal
4#21460996C211yesmaximal, characteristic
5#343×D61C211yesmaximal, characteristic
6C2×#14519856C2212yes—
7C22×C4⋊#6923C2211yescharacteristic
8C22×C4×#6923C2211yescharacteristic
9#16840773—24no—
10#16840773—24no—
11C2×#14519846—22no—
12C2×#14519845—22no—
13#16840713—22no—
14#16840713—22no—
15#16840704—22no—
16C2×(C2×#6923)⋊C4—22no—
17#16840634—24no—
18C2×M16×D61C2212yes—
19C2×C4⋊C4×D61C2211yescharacteristic
20#16839531C411yescharacteristic, complemented
21#16839530C412yescomplemented
22C61×#343C411yescharacteristic, complemented
23C23×#6923—21no—
24C23×#6923—21no—
25#14519856—22no—
26#14519856—22no—
27#14519856—22no—
28#14519856—22no—
29C2×C4⋊#6923—22no—
30C2×C4⋊#6923—22no—
31C2×C4⋊#6923—21no—
32C2×C4⋊#6923—22no—
33C2×C4⋊#6923—22no—
34C2×C4⋊#6923D412yes—
35C2×C4⋊#6923D412yes—
36C2×C4⋊#6923D412yes—
37C2×C22⋊#6923—22no—
38C2×C4×#6923—22no—
39C2×C4×#6923—22no—
40C2×C4×#6923—22no—
41C2×C4×#6923—21no—
42C2×C4×#6923Q812yes—
43#14519846—28no—
44#14519845—28no—
45(C2×#6923)⋊C4—24no—
46(C2×#6923)⋊C4—44no—
47C22×C4×D61D412yes—
48C22×C4×D61C2311yescharacteristic
49C2×#11758194C2×C412yes—
50C2×#11758189—22no—
51M16×D61—22no—
52M16×D61—22no—
53M16×D61D414yes—
54M16×D61Q814yes—
55C2×C8×D61—22no—
56C2×C4⋊Dic61C2×C411yescharacteristic
57C2×#11758196Q812yes—
58C4⋊C4×D61—22no—
59C4⋊C4×D61—22no—
60C2×M16×C61C2×C412yes—
61C2×C4⋊C4×C61C2×C411yescharacteristic
62C23×D61C2×D411yescharacteristic
63C22×#6923—22no—
64C22×#6923—22no—
65C22×#6923—22no—
66C22×#6923—22no—
67C22×#6923—22no—
68C22×#6923—22no—
69C22×#6923—41no—
70C22×#6923—41no—
71C22×#6923—41no—
72C22×#6923—41no—
73C22×#6923—42no—
74C22×#6923—42no—
75C22×#6923—41no—
76C22×#6923—41no—
77C22×C4×C61C4⋊C412yes—
78C22×C4×C61C22×C411yescharacteristic
79C22×Dic61C4⋊C412yes—
80C22×Dic61C2×Q811yescharacteristic
81C2×C4×D61—22no—
82C2×C4×D61—22no—
83C2×C4×D61—42no—
84C2×C4×D61—42no—
85C2×C4×D61C2×D411yescharacteristic
86C2×C4×D61C2×D412yes—
87C2×C4×D61C2×D412yes—
88C2×C4×D61C2×Q812yes—
89C2×C4×D61C2×Q811yescharacteristic
90C2×C4×D61#4111yescharacteristic
91C2×C4×D61#4111yescharacteristic
92C2×C4×D61#4112yes—
93C22⋊#6923—42no—
94C22⋊#6923—42no—
95C4⋊#6923—22no—
96C4⋊#6923—24no—
97C4⋊#6923—22no—
98C4⋊#6923—22no—
99C4⋊#6923—22no—
100C4⋊#6923—42no—
101C4⋊#6923—82no—
102C4⋊#6923—82no—
103C4×#6923—24no—
104C4×#6923—42no—
105C4×#6923—42no—
106C4×#6923—42no—
107M16×C61—22no—
108M16×C61C4⋊C414yescomplemented
109C2×C8×C61—22no—
110C4⋊C4×C61—22no—
111C4⋊Dic61—22no—
112#11758196—24no—
113#11758194—22no—
114#11758194C4⋊C414yescomplemented
115C2×#92418—22no—
116#11758189—24no—
117#11758189—24no—
118C8×D61—24no—
119C8×D61—24no—
120C23×C61C2×C4⋊C411yescharacteristic
121C22×D61—21no—
122C22×D61—21no—
123C22×D61—21no—
124C22×D61—21no—
125C22×D61#12011yescharacteristic
126C22×D61#12111yescharacteristic
127C22×D61C22⋊Q811yescharacteristic
128C22×D61#12311yescharacteristic
129C22×D61#12711yescharacteristic
130C22×D61C4⋊Q811yescharacteristic
131C2×#6923—24no—
132C2×#6923—42no—
133C2×#6923—42no—
134C2×#6923—42no—
135C2×#6923—42no—
136C2×#6923—42no—
137C2×#6923—42no—
138C2×#6923—44no—
139C2×#6923—42no—
140C2×#6923—42no—
141C2×#6923—42no—
142C2×#6923—42no—
143C2×#6923—42no—
144C2×#6923—42no—
145C2×#6923—42no—
146C2×#6923—82no—
147C2×#6923—81no—
148C2×#6923—81no—
149C2×C4×C61—22no—
150C2×C4×C61—42no—
151C2×C4×C61C2×C4⋊C412yes—
152C2×C4×C61C4×D411yescharacteristic
153C2×C4×C61C4×D412yes—
154C2×C4×C61C4×Q811yescharacteristic
155C2×Dic61—22no—
156C2×Dic61—42no—
157C2×Dic61C22⋊Q812yes—
158C2×Dic61#12511yescharacteristic
159C2×Dic61C4⋊Q811yescharacteristic
160C2×Dic61C4⋊Q812yes—
161C4×D61—42no—
162C4×D61—42no—
163C4×D61—82no—
164C4×D61#12012yes—
165C4×D61#12112yes—
166C4×D61C22⋊Q812yes—
167C4×D61C22⋊Q812yes—
168C4×D61C22⋊Q812yes—
169C4×D61#12312yes—
170C4×D61#12512yes—
171C4×D61C4⋊Q812yes—
172C8×C61—24no—
173#92418—24no—
174#11342—611nomaximal, Sylow
175C22×C61—21no—
176C22×C61—21no—
177C22×C61#38411yescharacteristic
178C22×C61#38511yescharacteristic
179C22×C61#38911yescharacteristic
180C2×D61—21no—
181C2×D61—21no—
182C2×D61—21no—
183C2×D61—21no—
184C2×D61—21no—
185C2×D61—21no—
186C2×D61—21no—
187C2×D61—21no—
188C2×D61—21no—
189C2×D61—21no—
190C2×D61#39211yescharacteristic
191C2×D61#39511yescharacteristic
192C2×D61#39811yescharacteristic
193C2×D61#39911yescharacteristic
194C2×D61#45311yescharacteristic
195C2×D61#45411yescharacteristic
196C2×D61#45511yescharacteristic
197C2×D61#45611yescharacteristic
198#6923—44no—
199#6923—82no—
200#6923—82no—
201#6923—82no—
202C4×C61—42no—
203C4×C61#38512yes—
204C4×C61#38912yes—
205Dic61—42no—
206Dic61#39512yes—
207Dic61#39812yes—
208C2×#428—612no—
209#2316—611no—
210C2×#343—611no—
211C2×#339—612no—
212C4⋊#95—611no—
213C2×C61—21no—
214C2×C61—21no—
215C2×C61#144411yescharacteristic
216C2×C61#185611yescharacteristic
217C2×C61#185711yescharacteristic
218D61—21no—
219D61—21no—
220D61#200611yescharacteristic
221D61#200711yescharacteristic
222D61#206011yescharacteristic
223D61#206111yescharacteristic
224C22×M16—612no—
225C2×#117—612no—
226C22×C4⋊C4—611no—
227C22×C4⋊C4—611no—
228C22×C42—611no—
229#428—1224no—
230#428—1224no—
231C2×#107—1222no—
232C2×#106—1222no—
233#389—1222no—
234#389—1222no—
235#380—1222no—
236C2×#95—1222no—
237#343—612no—
238#343—611no—
239#343#692311yescharacteristic, complemented
240#339—1224no—
241C61#1134211yescharacteristic, complemented, Sylow
242C23×C4—611no—
243C23×C4—612no—
244C23×C4—1221no—
245C23×C4—1221no—
246C2×M16—612no—
247C2×M16—614no—
248C2×M16—614no—
249C2×M16—1222no—
250C2×M16—1222no—
251C2×M16—1222no—
252C2×M16C2×#692312yes—
253C22×C8—1222no—
254#117—1222no—
255#117—1222no—
256#117—1222no—
257#117—1222no—
258C2×C4⋊C4—612no—
259C2×C4⋊C4—611no—
260C2×C4⋊C4—612no—
261C2×C4⋊C4—612no—
262C2×C4⋊C4—612no—
263C2×C4⋊C4—1222no—
264C2×C4⋊C4—1222no—
265C2×C4⋊C4—1222no—
266C2×C4⋊C4—1222no—
267C2×C4⋊C4—1222no—
268C2×C4⋊C4—1222no—
269C2×C4⋊C4—1221no—
270C2×C4⋊C4C2×#692311yescharacteristic
271C2×C22⋊C4—1222no—
272C2×C42—612no—
273C2×C42—1222no—
274C2×C42—1222no—
275C2×C42—1222no—
276C2×C42—1221no—
277#107—1228no—
278#106—1228no—
279#95—1224no—
280#95—2444no—
281C24—611no—
282C22×C4—611no—
283C22×C4—612no—
284C22×C4—612no—
285C22×C4—612no—
286C22×C4—612no—
287C22×C4—611no—
288C22×C4—611no—
289C22×C4—611no—
290C22×C4—611no—
291C22×C4—612no—
292C22×C4—1222no—
293C22×C4—1222no—
294C22×C4—1222no—
295C22×C4—1222no—
296C22×C4—1222no—
297C22×C4—1222no—
298C22×C4—1222no—
299C22×C4—1222no—
300C22×C4—2442no—
301C22×C4—2442no—
302C22×C4—2441no—
303C22×C4—2441no—
304C22×C4—2442no—
305C22×C4—2442no—
306C22×C4—2441no—
307C22×C4—2441no—
308C22×C4—2441no—
309C22×C4—2441no—
310C22×C4C4⋊#692312yes—
311C22×C4C22×#692311yescharacteristic
312M16—22no—
313M16—614no—
314M16—1224no—
315M16—1224no—
316M16—1222no—
317M16C4⋊#692314yescomplemented
318C2×C8—22no—
319C2×C8—1224no—
320C2×C8—1224no—
321C2×C8—1222no—
322C4⋊C4—22no—
323C4⋊C4—1224no—
324C4⋊C4—1222no—
325C4⋊C4—1224no—
326C4⋊C4—1222no—
327C4⋊C4—1222no—
328C4⋊C4—1222no—
329C4⋊C4—1222no—
330C4⋊C4—4882no—
331C4⋊C4—4882no—
332C4⋊C4—2442no—
333C22⋊C4—2442no—
334C22⋊C4—2442no—
335C42—1224no—
336C42—2442no—
337C42—2442no—
338C42—2442no—
339C23—611no—
340C23—611no—
341C23—611no—
342C23—611no—
343C23—611no—
344C23—611no—
345C23—1221no—
346C23—1221no—
347C23—1221no—
348C23—1221no—
349C23C2×C4⋊#692311yescharacteristic
350C2×C4—22no—
351C2×C4—42no—
352C2×C4—612no—
353C2×C4—612no—
354C2×C4—611no—
355C2×C4—611no—
356C2×C4—612no—
357C2×C4—612no—
358C2×C4—612no—
359C2×C4—612no—
360C2×C4—612no—
361C2×C4—612no—
362C2×C4—612no—
363C2×C4—612no—
364C2×C4—1224no—
365C2×C4—1222no—
366C2×C4—4882no—
367C2×C4—4881no—
368C2×C4—4881no—
369C2×C4—4882no—
370C2×C4—2442no—
371C2×C4—2442no—
372C2×C4—2442no—
373C2×C4—2442no—
374C2×C4—2442no—
375C2×C4—2442no—
376C2×C4—2442no—
377C2×C4—2442no—
378C2×C4—2442no—
379C2×C4—2444no—
380C2×C4—2442no—
381C2×C4—2442no—
382C2×C4—2442no—
383C2×C4—2442no—
384C2×C4—2442no—
385C2×C4—2442no—
386C2×C4—2442no—
387C2×C4C2×C4⋊#692312yes—
388C2×C4D4×#692311yescharacteristic
389C2×C4D4×#692312yes—
390C2×C4Q8×#692311yescharacteristic
391C8—24no—
392C8—1224no—
393C22—21no—
394C22—21no—
395C22—611no—
396C22—611no—
397C22—611no—
398C22—611no—
399C22—611no—
400C22—611no—
401C22—611no—
402C22—611no—
403C22—1221no—
404C22—1221no—
405C22—1221no—
406C22—1221no—
407C22—1221no—
408C22—1221no—
409C22—1221no—
410C22—1221no—
411C22—1221no—
412C22—1221no—
413C22#1684070811yescharacteristic
414C22#1684070911yescharacteristic
415C22#1684071311yescharacteristic
416C4—42no—
417C4—612no—
418C4—612no—
419C4—4882no—
420C4—4882no—
421C4—4882no—
422C4—2444no—
423C4—2442no—
424C4#1684070912yes—
425C4#1684071312yes—
426C2—21no—
427C2—21no—
428C2—611no—
429C2—611no—
430C2—611no—
431C2—611no—
432C2—1221no—
433C2—1221no—
434C2#2146100711yescharacteristic
435C2#2146177411yescharacteristic
436C2#2146177511yescharacteristic
subgroupG/H
centerC22#16840709
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#343C4
socleC22×C61#385
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#343#6923
hypocenterC61#11342
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
20. #16839531C4semidirect
21. #16839530C4semidirect
22. C61×#343C4semidirect
108. M16×C61C4⋊C4semidirect
114. #11758194C4⋊C4semidirect
239. #343#6923semidirect
241. C61#11342semidirect
317. M16C4⋊#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168407732
#214622992
C2×#168406342
#214609962
#343×D612
#1134261