Finite Group Databasebeta

← #29779592

Subgroups — #29779592

Order 15616.

866 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448309C211yesmaximal, characteristic
2#21447238C211yesmaximal, characteristic, complemented
3#21447233C211yesmaximal, characteristic, complemented
4#21446967C211yesmaximal, characteristic, complemented
5#21446962C211yesmaximal, characteristic, complemented
6#21446515C212yesmaximal, complemented
7#21446508C212yesmaximal, complemented
8#21445911C212yesmaximal, complemented
9#21445757C212yesmaximal
10#21445581C211yesmaximal, characteristic, complemented
11C61×#2851C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C2×#14519736C2211yescharacteristic
14C2×#14519700C2212yes—
15C2×#14519700C2212yes—
16C2×#14519697C2212yes—
17C2×#14519697C2212yes—
18C2×C22⋊#92418C2212yes—
19C2×#14519688C2212yes—
20#16839668C2211yescharacteristic
21#16839668C2211yescharacteristic
22#16839664C2211yescharacteristic
23#16839664C2211yescharacteristic
24#16839641—22no—
25C4×#11758198—22no—
26#16839524C2212yes—
27#16839524C2212yes—
28#16839524C2212yes—
29#16839524C2212yes—
30#16839517C2211yescharacteristic
31#16839517C2211yescharacteristic
32C2×#125×C61C2211yescharacteristic
33C22×C42×C61C2211yescharacteristic
34C61×#389C2212yes—
35C61×#387—22no—
36C61×#385—22no—
37C61×#382C2211yescharacteristic
38C61×#380C2212yes—
39C2×#11758222—22no—
40C2×#11758222—24no—
41C2×#11758222—22no—
42C2×#11758222—42no—
43C22×Dic122C2311yescharacteristic
44C22×D244C2311yescharacteristic
45C22×C22⋊D61—21no—
46C22×C4×D61—21no—
47C22×#92418C2312yes—
48C22×#92418C2312yes—
49#14519736—24no—
50C2×C4⋊Dic61C2311yescharacteristic
51C2×C4⋊Dic61C2311yescharacteristic
52C2×C22⋊Dic61—21no—
53C2×#11758198—22no—
54C2×#11758198—22no—
55C2×C4×Dic61—21no—
56#14519700—24no—
57#14519700—42no—
58#14519697—24no—
59#14519697—42no—
60C22⋊#92418—24no—
61#14519688—42no—
62#14519684—22no—
63#14519684—22no—
64C23×C4×C61D412yes—
65C23×C4×C61C2311yescharacteristic
66#125×C61—42no—
67C2×C4⋊C4×C61—22no—
68C2×C4⋊C4×C61—22no—
69C2×C4⋊C4×C61C2312yes—
70C2×C4⋊C4×C61C2312yes—
71C2×C22⋊C4×C61—22no—
72C2×C42×C61—42no—
73C2×C42×C61D412yes—
74C2×C42×C61C2311yescharacteristic
75C2×C42×C61C2311yescharacteristic
76#95×C61—22no—
77#95×C61—22no—
78#95×C61—22no—
79C24×C61C2×D411yescharacteristic
80C23×D61—21no—
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—22no—
84C22×C4×C61—22no—
85C22×C4×C61—22no—
86C22×C4×C61—42no—
87C22×C4×C61—42no—
88C22×C4×C61—42no—
89C22×C4×C61C2×D412yes—
90C22×C4×C61C2×D412yes—
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61#4114yes—
93C22×C4×C61C2411yescharacteristic
94C2×C22⋊D61—24no—
95C2×C22⋊D61—42no—
96C2×C22⋊D61—42no—
97C22×Dic61—21no—
98C22×Dic61—21no—
99C22×Dic61—21no—
100#11758222—48no—
101#11758222—48no—
102C2×D244—22no—
103C2×D244—22no—
104C2×D244—21no—
105C2×D244#4112yes—
106C2×C4×D61—24no—
107C2×C4×D61—42no—
108C2×C4×D61—42no—
109C2×Dic122—22no—
110C2×Dic122—22no—
111C2×Dic122—21no—
112C2×Dic122#4112yes—
113C4⋊C4×C61—22no—
114C4⋊C4×C61—22no—
115C4⋊C4×C61—22no—
116C4⋊C4×C61—22no—
117C4⋊C4×C61—44no—
118C4⋊C4×C61—44no—
119C22⋊C4×C61—44no—
120C42×C61—22no—
121C42×C61—22no—
122C42×C61—42no—
123C22⋊Dic61—42no—
124#11758198—44no—
125#11758198—44no—
126C4⋊Dic61—22no—
127C4⋊Dic61—22no—
128C4×Dic61—42no—
129C2×#92418—22no—
130C2×#92418—22no—
131C2×#92418—22no—
132C2×#92418—22no—
133C2×#92418—22no—
134C2×#92418—22no—
135C23×C61—21no—
136C23×C61—21no—
137C23×C61—22no—
138C23×C61—22no—
139C23×C61#12412yes—
140C23×C61C22×D411yescharacteristic
141C22×D61—22no—
142C22×D61—41no—
143C22×D61—41no—
144C22×D61—42no—
145C22×D61—42no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—24no—
149C2×C4×C61—24no—
150C2×C4×C61—22no—
151C2×C4×C61—22no—
152C2×C4×C61—22no—
153C2×C4×C61—22no—
154C2×C4×C61—22no—
155C2×C4×C61—22no—
156C2×C4×C61—48no—
157C2×C4×C61—42no—
158C2×C4×C61—42no—
159C2×C4×C61—42no—
160C2×C4×C61—42no—
161C2×C4×C61—42no—
162C2×C4×C61—42no—
163C2×C4×C61#12412yes—
164C2×C4×C61C2×#4111yescharacteristic
165C2×C4×C61C2×#4111yescharacteristic
166C2×C4×C61C2×#4112yes—
167C2×C4×C61H(5,𝔽2)12yes—
168C22⋊D61—48no—
169C2×Dic61—22no—
170C2×Dic61—41no—
171C2×Dic61—41no—
172C2×Dic61—41no—
173C2×Dic61—41no—
174C2×Dic61—42no—
175C2×Dic61—42no—
176D244—24no—
177D244—42no—
178C4×D61—48no—
179Dic122—24no—
180Dic122—42no—
181#92418—42no—
182#92418—42no—
183#29114—611nomaximal, Sylow
184C22×C61—22no—
185C22×C61—22no—
186C22×C61—22no—
187C22×C61—42no—
188C22×C61—42no—
189C22×C61#52511yescharacteristic
190C22×C61C2×#12411yescharacteristic
191C22×C61#53411yescharacteristic
192C22×C61#57512yes—
193C22×C61#57512yes—
194C2×D61—42no—
195C2×D61—44no—
196C2×D61—81no—
197C2×D61—81no—
198C2×D61—82no—
199C4×C61—22no—
200C4×C61—42no—
201C4×C61—42no—
202C4×C61—82no—
203C4×C61—82no—
204C4×C61#55112yes—
205C4×C61#55112yes—
206Dic61—42no—
207Dic61—81no—
208Dic61—81no—
209#2851D6111yescharacteristic, complemented
210#2297—611no—
211#2074—611no—
212#2073—611no—
213#2026—611no—
214#2025—611no—
215#1936—612no—
216#1935—612no—
217#1825—612no—
218#1788—612no—
219#1749—611no—
220C2×C61—42no—
221C2×C61#256111yescharacteristic
222C2×C61#311112yes—
223C2×C61#313511yescharacteristic
224C2×C61#314211yescharacteristic
225C2×C61#319712yes—
226D61—82no—
227C22×#41—611no—
228C2×#125C2×D6111yescharacteristic
229C2×#117—611no—
230C22×C42C2×D6111yescharacteristic
231C2×#103—612no—
232C2×#103—612no—
233C2×#102—612no—
234C2×#102—612no—
235C2×C22⋊C8—612no—
236C2×#97—612no—
237#391—611no—
238#391—611no—
239#390—611no—
240#390—611no—
241#389C2×D6112yes—
242#387—22no—
243#386—1222no—
244#385—22no—
245#382C2×D6111yescharacteristic
246#380C2×D6112yes—
247C4×C22⋊C4—1222no—
248#340—612no—
249#340—612no—
250#340—612no—
251#340—612no—
252#336—611no—
253#336—611no—
254C61#2911411yescharacteristic, complemented, Sylow
255C2×#41—1222no—
256C2×#41—1222no—
257C2×#41—1224no—
258C2×#41—2442no—
259C22×Q8—611no—
260C22×D4—611no—
261C22×D4—1221no—
262C23×C4—1221no—
263C23×C4C22⋊D6112yes—
264C23×C4C22×D6111yescharacteristic
265C22×C8—612no—
266C22×C8—612no—
267#125—42no—
268#117—1224no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—611no—
272C2×C4⋊C4—611no—
273C2×C4⋊C4C22×D6112yes—
274C2×C4⋊C4C22×D6112yes—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—1221no—
277C2×C22⋊C4—1222no—
278C2×C22⋊C4—1222no—
279C2×C42—42no—
280C2×C42—1221no—
281C2×C42C22⋊D6112yes—
282C2×C42C22×D6111yescharacteristic
283C2×C42C22×D6111yescharacteristic
284#103—1224no—
285#103—2442no—
286#102—1224no—
287#102—2442no—
288C22⋊C8—1224no—
289#97—2442no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—1222no—
294#95—1222no—
295C24—1221no—
296C24C2×C22⋊D6111yescharacteristic
297#41—2448no—
298#41—2448no—
299C2×Q8—612no—
300C2×Q8—1221no—
301C2×Q8—1222no—
302C2×Q8—1222no—
303C2×D4—612no—
304C2×D4—1224no—
305C2×D4—1221no—
306C2×D4—1222no—
307C2×D4—1222no—
308C2×D4—2442no—
309C2×D4—2442no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—1224no—
319C22×C4—1221no—
320C22×C4—1221no—
321C22×C4—1221no—
322C22×C4—2442no—
323C22×C4—2442no—
324C22×C4#1175822614yes—
325C22×C4C2×C22⋊D6112yes—
326C22×C4C2×C22⋊D6112yes—
327C22×C4C2×C22⋊D6111yescharacteristic
328C22×C4C23×D6111yescharacteristic
329C2×C8—1222no—
330C2×C8—1222no—
331C2×C8—1222no—
332C2×C8—1222no—
333C2×C8—1222no—
334C2×C8—1222no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—44no—
340C4⋊C4—44no—
341C4⋊C4—1222no—
342C4⋊C4—1222no—
343C22⋊C4—44no—
344C22⋊C4—2444no—
345C22⋊C4—2444no—
346C22⋊C4—2442no—
347C42—22no—
348C42—22no—
349C42—42no—
350C42—2442no—
351C23—21no—
352C23—22no—
353C23—22no—
354C23—21no—
355C23—1222no—
356C23—2441no—
357C23—2442no—
358C23—2442no—
359C23—2441no—
360C23#1451976612yes—
361C23C22×C22⋊D6111yescharacteristic
362Q8—1224no—
363Q8—2442no—
364D4—1224no—
365D4—2448no—
366D4—2442no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—48no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—1222no—
385C2×C4—2448no—
386C2×C4—2441no—
387C2×C4—2441no—
388C2×C4—2442no—
389C2×C4—2442no—
390C2×C4—2441no—
391C2×C4—2441no—
392C2×C4#1451976612yes—
393C2×C4#41×D6111yescharacteristic
394C2×C4#41×D6111yescharacteristic
395C2×C4C2×#1175822612yes—
396C2×C4#1451982212yes—
397C8—2442no—
398C8—2442no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—42no—
403C22—42no—
404C22—4882no—
405C22—4881no—
406C22—4881no—
407C22—2444no—
408C22—2442no—
409C22#1684026911yescharacteristic
410C22C2×#1451976611yescharacteristic
411C22#1684032111yescharacteristic
412C22#1684056012yes—
413C22#1684056012yes—
414C4—22no—
415C4—42no—
416C4—42no—
417C4—82no—
418C4—82no—
419C4—4881no—
420C4—4881no—
421C4—2442no—
422C4#1684043912yes—
423C4#1684043912yes—
424C2—42no—
425C2—4882no—
426C2#2145097811yescharacteristic
427C2#2145592212yes—
428C2#2145616711yescharacteristic
429C2#2145622211yescharacteristic
430C2#2145665612yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2851C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
hypercenter#2851D61
hypocenterC61#29114
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2851

Complements

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

NH ≅ G/Ntype
2. #21447238C2semidirect
3. #21447233C2semidirect
4. #21446967C2semidirect
5. #21446962C2semidirect
6. #21446515C2semidirect
7. #21446508C2semidirect
8. #21445911C2semidirect
10. #21445581C2semidirect
11. C61×#2851C2semidirect
209. #2851D61semidirect
254. C61#29114semidirect

Maximal subgroups

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

subgroupindex
#214483092
#214472382
#214472332
#214469672
#214469622
#214465152
#214465082
#214459112
#214457572
#214455812
C61×#28512
#2911461