Finite Group Databasebeta

← #29598316

Subgroups — #29598316

Order 15616.

628 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 99 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#21450705C211yesmaximal, characteristic
2C2×#16839553C212yesmaximal, complemented
3C2×#16839550C212yesmaximal
4C4×#14519692C211yesmaximal, characteristic, complemented
5C61×#1800C211yesmaximal, characteristic, complemented
6C2×#14519758C2211yescharacteristic
7C2×#14519745C2211yescharacteristic
8C22×#11758198C411yescharacteristic, complemented
9C2×#14519692C2212yes—
10#16839684—21no—
11#16839678—21no—
12#16839653—21no—
13#16839641—21no—
14#16839623C411yescharacteristic, complemented
15#16839622C411yescharacteristic, complemented
16#16839617C411yescharacteristic, complemented
17C4×C22⋊Dic61—21no—
18#16839553—24no—
19#16839550—24no—
20#16839529—21no—
21C2×C4×D4×C61C2211yescharacteristic
22C61×#423—21no—
23C2×#100×C61C2212yes—
24C61×#343—21no—
25C24×D61C2×C411yescharacteristic
26C22×C22⋊D61—21no—
27C23×Dic61C2×C411yescharacteristic
28#14519758—22no—
29#14519758—22no—
30#14519745—22no—
31#14519745—22no—
32C2×C4⋊Dic61—21no—
33C2×#11758196—21no—
34C2×C22⋊Dic61—22no—
35C2×C22⋊Dic61—21no—
36C2×C22⋊Dic61C2×C411yescharacteristic
37C2×C22⋊Dic61C2×C411yescharacteristic
38C2×#11758198—21no—
39C2×#11758198—21no—
40C2×#11758198—21no—
41C2×#11758198—21no—
42C2×#11758198—21no—
43C2×#11758198—41no—
44C2×#11758198—41no—
45C2×C4×Dic61—41no—
46#14519692—22no—
47#14519692D414yes—
48#14519684—21no—
49#14519684—21no—
50#14519684—21no—
51#14519684—41no—
52C22×D4×C61C2311yescharacteristic
53C23×C4×C61C2×C411yescharacteristic
54C23×C4×C61D411yescharacteristic
55C2×M16×C61—22no—
56C4×D4×C61—41no—
57C4×D4×C61—41no—
58C2×C4⋊C4×C61D411yescharacteristic
59C2×C22⋊C4×C61C2×C411yescharacteristic
60C2×C22⋊C4×C61D411yescharacteristic
61C2×C42×C61D411yescharacteristic
62C4⋊C8×C61—41no—
63C4⋊C8×C61—41no—
64#100×C61—22no—
65#100×C61—22no—
66C24×C61C22×C411yescharacteristic
67C24×C61C2×D411yescharacteristic
68C23×D61—21no—
69C23×D61—22no—
70C23×D61—42no—
71C23×D61—41no—
72C23×D61—41no—
73C23×D61—81no—
74C2×D4×C61—41no—
75C2×D4×C61—41no—
76C2×D4×C61C2×D412yes—
77C2×D4×C61#4112yes—
78C22×C4×C61—21no—
79C22×C4×C61—21no—
80C22×C4×C61—21no—
81C22×C4×C61—21no—
82C22×C4×C61—81no—
83C22×C4×C61—81no—
84C22×C4×C61C22⋊C411yescharacteristic
85C22×C4×C61C22⋊C411yescharacteristic
86C22×C4×C61C22⋊C411yescharacteristic
87C22×C4×C61C22⋊C411yescharacteristic
88C22×C4×C61C2×D411yescharacteristic
89C2×C22⋊D61—42no—
90C2×C22⋊D61—42no—
91C2×C22⋊D61—81no—
92C22×Dic61—21no—
93C22×Dic61—21no—
94C22×Dic61—21no—
95C22×Dic61—22no—
96C22×Dic61—42no—
97C22×Dic61—41no—
98C22×Dic61—41no—
99M16×C61—42no—
100M16×C61—42no—
101C2×C8×C61—42no—
102C4⋊C4×C61—41no—
103C22⋊C4×C61—41no—
104C22⋊C4×C61—41no—
105C42×C61—21no—
106C42×C61—21no—
107C22⋊Dic61—22no—
108C22⋊Dic61—22no—
109C22⋊Dic61—24no—
110C22⋊Dic61—42no—
111C22⋊Dic61—42no—
112#11758198—42no—
113#11758198—42no—
114#11758198—41no—
115#11758198—41no—
116#11758198—41no—
117#11758198—41no—
118#11758198—81no—
119C4⋊Dic61—81no—
120#11758196—41no—
121#11758196—41no—
122C4×Dic61—82no—
123C23×C61—21no—
124C23×C61—21no—
125C23×C61—21no—
126C23×C61—21no—
127C23×C61—41no—
128C23×C61—41no—
129C23×C61—41no—
130C23×C61—41no—
131C23×C61C2×C22⋊C411yescharacteristic
132C23×C61C4×D412yes—
133C23×C61#12112yes—
134C22×D61—22no—
135C22×D61—41no—
136C22×D61—41no—
137C22×D61—41no—
138C22×D61—42no—
139C22×D61—41no—
140C22×D61—41no—
141C22×D61—41no—
142C22×D61—42no—
143C22×D61—84no—
144C22×D61—84no—
145C22×D61—82no—
146C22×D61—82no—
147D4×C61—42no—
148D4×C61—42no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—41no—
155C2×C4×C61—42no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—81no—
160C2×C4×C61—81no—
161C2×C4×C61—81no—
162C2×C4×C61#9911yescharacteristic
163C2×C4×C61#9911yescharacteristic
164C2×C4×C61#12411yescharacteristic
165C2×C4×C61#12711yescharacteristic
166C22⋊D61—84no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—44no—
171C2×Dic61—41no—
172C2×Dic61—41no—
173C2×Dic61—41no—
174C2×Dic61—41no—
175C2×Dic61—41no—
176C2×Dic61—41no—
177C2×Dic61—82no—
178C2×Dic61—82no—
179C2×Dic61—82no—
180C8×C61—82no—
181#13222—611nomaximal, Sylow
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—22no—
185C22×C61—22no—
186C22×C61—42no—
187C22×C61—42no—
188C22×C61—42no—
189C22×C61—41no—
190C22×C61—41no—
191C22×C61—41no—
192C22×C61—41no—
193C22×C61—81no—
194C22×C61#39011yescharacteristic
195C22×C61C2×#9911yescharacteristic
196C22×C61#41011yescharacteristic
197C2×D61—42no—
198C2×D61—42no—
199C2×D61—42no—
200C2×D61—81no—
201C2×D61—81no—
202C2×D61—84no—
203C2×D61—81no—
204C2×D61—81no—
205C2×D61—84no—
206C4×C61—21no—
207C4×C61—21no—
208C4×C61—21no—
209C4×C61—21no—
210C4×C61—41no—
211C4×C61—41no—
212C4×C61—41no—
213Dic61—81no—
214Dic61—81no—
215Dic61—84no—
216Dic61—81no—
217#2534—611no—
218C2×#352—612no—
219C2×AΣL(1,𝔽16)—612no—
220#1800D6111yescharacteristic, complemented
221C4×#99—611no—
222C2×C61—21no—
223C2×C61—21no—
224C2×C61—42no—
225C2×C61—42no—
226C2×C61#196411yescharacteristic
227C2×C61#212011yescharacteristic
228C2×C61#212111yescharacteristic
229D61—82no—
230C2×#123—611no—
231C2×#120—611no—
232C2×C4×D4C2×D6111yescharacteristic
233C22×C22⋊C4—611no—
234#423—21no—
235C2×#100C2×D6112yes—
236C2×#99—612no—
237#396—1221no—
238#394—1221no—
239#388—1221no—
240#386—1221no—
241#381—611no—
242#380—611no—
243#379—611no—
244C4×C22⋊C4—1221no—
245#352—1224no—
246AΣL(1,𝔽16)—1224no—
247#343—21no—
248#342—1221no—
249C61#1322211yescharacteristic, complemented, Sylow
250C25—611no—
251C22×D4—1221no—
252C22×D4C22×D6111yescharacteristic
253C23×C4—611no—
254C23×C4C4×D6111yescharacteristic
255C23×C4D24411yescharacteristic
256C2×M16—22no—
257#123—1222no—
258#123—1222no—
259#120—1222no—
260#120—1222no—
261C4×D4—41no—
262C4×D4—41no—
263C2×C4⋊C4—1221no—
264C2×C4⋊C4—1221no—
265C2×C4⋊C4C22⋊D6111yescharacteristic
266C2×C22⋊C4—611no—
267C2×C22⋊C4—611no—
268C2×C22⋊C4—1222no—
269C2×C22⋊C4—1221no—
270C2×C22⋊C4—1221no—
271C2×C22⋊C4—1221no—
272C2×C22⋊C4—1221no—
273C2×C22⋊C4—1221no—
274C2×C22⋊C4—1221no—
275C2×C22⋊C4—2441no—
276C2×C22⋊C4—2441no—
277C2×C22⋊C4C4×D6111yescharacteristic
278C2×C22⋊C4D24411yescharacteristic
279C2×C42—2441no—
280C2×C42C22⋊D6111yescharacteristic
281C4⋊C8—41no—
282C4⋊C8—41no—
283#100—22no—
284#100—22no—
285#99—614no—
286#99—1222no—
287#95—1221no—
288#95—1221no—
289#95—1221no—
290#95—2441no—
291C24—1221no—
292C24—1222no—
293C24—4881no—
294C24—2441no—
295C24—2441no—
296C24—2442no—
297C24C2×C4×D6111yescharacteristic
298C24C2×D24411yescharacteristic
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—4881no—
302C2×D4—2442no—
303C2×D4—2442no—
304C2×D4D4×D6112yes—
305C2×D4#1175822612yes—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—81no—
311C22×C4—81no—
312C22×C4—1222no—
313C22×C4—1221no—
314C22×C4—1221no—
315C22×C4—1221no—
316C22×C4—2442no—
317C22×C4—2441no—
318C22×C4—2441no—
319C22×C4#1175819811yescharacteristic
320C22×C4#1175819811yescharacteristic
321C22×C4#1175819811yescharacteristic
322C22×C4#1175819811yescharacteristic
323C22×C4C2×C22⋊D6111yescharacteristic
324M16—42no—
325M16—42no—
326C2×C8—42no—
327C4⋊C4—41no—
328C4⋊C4—4881no—
329C4⋊C4—2441no—
330C4⋊C4—2441no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—1222no—
334C22⋊C4—1222no—
335C22⋊C4—1224no—
336C22⋊C4—4881no—
337C22⋊C4—2442no—
338C22⋊C4—2442no—
339C22⋊C4—2442no—
340C22⋊C4—2442no—
341C22⋊C4—2441no—
342C22⋊C4—2441no—
343C22⋊C4—2441no—
344C22⋊C4—2441no—
345C42—21no—
346C42—21no—
347C42—4882no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—1222no—
357C23—4882no—
358C23—4882no—
359C23—4884no—
360C23—4884no—
361C23—2441no—
362C23—2441no—
363C23—2441no—
364C23—2442no—
365C23—2442no—
366C23—2441no—
367C23—2441no—
368C23—2441no—
369C23C2×#1175819811yescharacteristic
370C23#1451973912yes—
371C23#1451975212yes—
372D4—42no—
373D4—42no—
374D4—4884no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—4882no—
389C2×C4—4882no—
390C2×C4—4882no—
391C2×C4—2444no—
392C2×C4—2441no—
393C2×C4—2441no—
394C2×C4—2441no—
395C2×C4—2441no—
396C2×C4—2441no—
397C2×C4—2441no—
398C2×C4—2441no—
399C2×C4—2441no—
400C2×C4—2441no—
401C2×C4#1451969111yescharacteristic
402C2×C4#1451969111yescharacteristic
403C2×C4#1451976611yescharacteristic
404C2×C4#1451977311yescharacteristic
405C8—82no—
406C22—21no—
407C22—21no—
408C22—22no—
409C22—22no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—81no—
418C22—4881no—
419C22—4881no—
420C22—4884no—
421C22—4884no—
422C22—4881no—
423C22—4881no—
424C22—2442no—
425C22—2442no—
426C22—2442no—
427C22#1683966311yescharacteristic
428C22C2×#1451969111yescharacteristic
429C22#1683973511yescharacteristic
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—4881no—
438C4—4881no—
439C4—4884no—
440C4—4881no—
441C2—21no—
442C2—21no—
443C2—42no—
444C2—42no—
445C2—4882no—
446C2#2144665411yescharacteristic
447C2#2144746111yescharacteristic
448C2#2144746611yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC24×C61C22×C4
FrattiniC22×D4C22×D61
FittingC61×#1800C2
socleC22×C61C2×#99
prime radicalC2×#14519745C22
cosocleC22×D4×C61C23
hypercenter#1800D61
hypocenterC61#13222
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1800

Complements

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

NH ≅ G/Ntype
2. C2×#16839553C2semidirect
4. C4×#14519692C2semidirect
5. C61×#1800C2semidirect
8. C22×#11758198C4semidirect
14. #16839623C4semidirect
15. #16839622C4semidirect
16. #16839617C4semidirect
220. #1800D61semidirect
249. C61#13222semidirect

Maximal subgroups

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

subgroupindex
#214507052
C2×#168395532
C2×#168395502
C4×#145196922
C61×#18002
#1322261