Finite Group Databasebeta

← #29579730

Subgroups — #29579730

Order 15616.

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

normal subgroups only

This group has 510 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
1C2×#16839989C211yesmaximal, characteristic
2#21445855C211yesmaximal, characteristic, complemented
3#21445598C211yesmaximal, characteristic, complemented
4#21444340C211yesmaximal, characteristic, complemented
5#21444335C211yesmaximal, characteristic
6#21444182C211yesmaximal, characteristic, complemented
7C61×#1471C211yesmaximal, characteristic
8C22×#11758200C411yescharacteristic, complemented
9C23×#92418C2211yescharacteristic
10C2×#14519756C2211yescharacteristic
11#16839989—21no—
12#16839989—21no—
13#16839989—21no—
14#16839989—21no—
15C2×#14519712—21no—
16C2×#14519708C411yescharacteristic, complemented
17C2×#14519700C411yescharacteristic, complemented
18C2×#14519698C411yescharacteristic, complemented
19#16839726C2211yescharacteristic
20C2×#14519687—21no—
21C4×C22⋊Dic61C2211yescharacteristic
22#16839526—21no—
23#16839516—21no—
24#16839500—22no—
25#16839496—22no—
26#16839487—22no—
27C2×#121×C61C2211yescharacteristic
28C61×#407C2211yescharacteristic
29C2×C22⋊C8×C61C2211yescharacteristic
30C61×#323—22no—
31C22×Dic122C2×C411yescharacteristic
32C2×#11758200—21no—
33C2×#11758200—21no—
34C2×#11758200—42no—
35C2×#11758194—21no—
36C22×#92418—42no—
37C22×#92418C2×C411yescharacteristic
38C22×#92418C2×C411yescharacteristic
39C22×#92418D411yescharacteristic
40C22×#92418D411yescharacteristic
41C22×#92418D411yescharacteristic
42C22×#92418D411yescharacteristic
43#14519756—21no—
44#14519756—21no—
45#14519756—21no—
46#14519756—21no—
47C2×C4⋊Dic61C2×C411yescharacteristic
48C2×#11758196—21no—
49C2×C22⋊Dic61—21no—
50C2×C22⋊Dic61—21no—
51C2×C4×Dic61—21no—
52#14519712—42no—
53#14519708—21no—
54#14519708—21no—
55#14519700—21no—
56#14519700—21no—
57#14519698—21no—
58#14519698—21no—
59C22⋊#92418—21no—
60C22⋊#92418—21no—
61#14519687—42no—
62#14519684D412yes—
63C22×D4×C61—21no—
64C22×D4×C61C2×C411yescharacteristic
65C23×C4×C61C2311yescharacteristic
66C2×M16×C61—21no—
67C22×C8×C61—21no—
68#121×C61—21no—
69#121×C61—21no—
70#121×C61—21no—
71#121×C61—21no—
72C2×C4⋊C4×C61C2×C411yescharacteristic
73C2×C22⋊C4×C61—21no—
74C22⋊C8×C61—21no—
75C22⋊C8×C61—21no—
76C22⋊C8×C61—21no—
77C22⋊C8×C61D412yes—
78C24×C61—21no—
79C24×C61#4111yescharacteristic
80C2×D4×C61—21no—
81C2×D4×C61—21no—
82C2×D4×C61—41no—
83C2×D4×C61—41no—
84C2×D4×C61—41no—
85C2×D4×C61—41no—
86C2×D4×C61—82no—
87C22×C4×C61—21no—
88C22×C4×C61—41no—
89C22×C4×C61—41no—
90C22×C4×C61C22×C411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61#4111yescharacteristic
96C22×Dic61—21no—
97C22×Dic61—22no—
98C22×Dic61—21no—
99C2×Dic122—21no—
100C2×Dic122—21no—
101C2×Dic122—41no—
102C2×Dic122—41no—
103M16×C61—41no—
104M16×C61—41no—
105C2×C8×C61—22no—
106C2×C8×C61—41no—
107C2×C8×C61—41no—
108C2×C8×C61—41no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—21no—
111C22⋊C4×C61—41no—
112C22⋊C4×C61—41no—
113C22⋊Dic61—42no—
114C22⋊Dic61—41no—
115C22⋊Dic61—41no—
116#11758200—42no—
117#11758200—42no—
118C4⋊Dic61—21no—
119C4⋊Dic61—21no—
120#11758196—41no—
121#11758196—41no—
122C4×Dic61—41no—
123C4×Dic61—41no—
124#11758194—41no—
125#11758194—41no—
126C2×#92418—22no—
127C2×#92418—22no—
128C2×#92418—22no—
129C2×#92418—41no—
130C2×#92418—41no—
131C2×#92418—41no—
132C2×#92418—41no—
133C2×#92418C22⋊C411yescharacteristic
134C2×#92418C22⋊C411yescharacteristic
135C2×#92418C22⋊C411yescharacteristic
136C2×#92418C22⋊C411yescharacteristic
137C23×C61—22no—
138C23×C61—22no—
139C23×C61—41no—
140C23×C61—41no—
141C23×C61—42no—
142C23×C61—81no—
143C23×C61—81no—
144C23×C61C4×D411yescharacteristic
145C23×C61#12111yescharacteristic
146C23×C61#12411yescharacteristic
147D4×C61—41no—
148D4×C61—41no—
149D4×C61—81no—
150D4×C61—82no—
151D4×C61—82no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—81no—
163C2×C4×C61C2×C22⋊C411yescharacteristic
164C2×C4×C61C4×D411yescharacteristic
165C2×C4×C61#12111yescharacteristic
166C2×C4×C61#12711yescharacteristic
167C2×Dic61—42no—
168C2×Dic61—42no—
169C2×Dic61—41no—
170C2×Dic61—41no—
171C2×Dic61—42no—
172C2×Dic61—42no—
173C2×Dic61—81no—
174Dic122—41no—
175Dic122—41no—
176Dic122—81no—
177C8×C61—42no—
178C8×C61—81no—
179#92418—21no—
180#92418—21no—
181#92418—21no—
182#92418—81no—
183#92418#9912yes—
184#9816—611nomaximal, Sylow
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—44no—
193C22×C61—81no—
194C22×C61—81no—
195C22×C61—81no—
196C22×C61—162no—
197C22×C61#39011yescharacteristic
198C22×C61#44412yes—
199C22×C61#47112yes—
200C4×C61—21no—
201C4×C61—21no—
202C4×C61—21no—
203C4×C61—81no—
204C4×C61C2×#9911yescharacteristic
205C4×C61#41011yescharacteristic
206Dic61—82no—
207Dic61—81no—
208Dic61—81no—
209C2×#465—611no—
210#1810—611no—
211#1752—611no—
212#1508—611no—
213#1507—611no—
214#1472—611no—
215#1471D6111yescharacteristic
216C2×C61—22no—
217C2×C61—22no—
218C2×C61—41no—
219C2×C61—161no—
220C2×C61#196411yescharacteristic
221C2×C61#197511yescharacteristic
222C2×C61#197711yescharacteristic
223C22×SD16—611no—
224C23×C8—611no—
225C2×C22⋊Q8—611no—
226C2×#121C2×D6111yescharacteristic
227#465—1221no—
228#465—1221no—
229#465—1221no—
230#465—1221no—
231C2×#108—1221no—
232C2×#106—611no—
233C2×#103—611no—
234C2×#102—611no—
235#407—611no—
236#407C2×D6111yescharacteristic
237C2×C22⋊C8C2×D6111yescharacteristic
238C2×#97—1221no—
239C4×C22⋊C4—611no—
240#341—1221no—
241#336—1221no—
242#328—1222no—
243#327—1222no—
244#324—1222no—
245#323—22no—
246C61#981611yescharacteristic, complemented, Sylow
247C22×Q8—611no—
248C22×D4—21no—
249C22×D4C4×D6111yescharacteristic
250C23×C4C22×D6111yescharacteristic
251C2×SD16—1221no—
252C2×SD16—1221no—
253C2×SD16—2442no—
254C2×M16—21no—
255C2×M16—1221no—
256C22×C8—21no—
257C22×C8—611no—
258C22×C8—611no—
259C22×C8—611no—
260C22×C8—611no—
261C22×C8—611no—
262C22×C8—611no—
263C22×C8—2442no—
264C22⋊Q8—1221no—
265C22⋊Q8—1221no—
266C22⋊Q8—1221no—
267C22⋊Q8—1221no—
268#121—21no—
269#121—21no—
270#121—21no—
271#121—21no—
272C2×C4⋊C4—611no—
273C2×C4⋊C4—1221no—
274C2×C4⋊C4C4×D6111yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—1221no—
277C2×C22⋊C4—1221no—
278C2×C42—1221no—
279#108—2442no—
280#106—1221no—
281#106—1221no—
282#103—1221no—
283#103—1221no—
284#102—1221no—
285#102—1221no—
286C22⋊C8—21no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—1221no—
290C22⋊C8—1221no—
291C22⋊C8C22⋊D6112yes—
292#97—2442no—
293#95—612no—
294C24—21no—
295C24#1175822211yescharacteristic
296C2×Q8—1221no—
297C2×Q8—1221no—
298C2×Q8—2441no—
299C2×Q8—2441no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—82no—
307C22×C4—21no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—1221no—
311C22×C4—1221no—
312C22×C4—1222no—
313C22×C4C2×C4×D6111yescharacteristic
314C22×C4D4×D6111yescharacteristic
315C22×C4D4×D6111yescharacteristic
316C22×C4D4×D6111yescharacteristic
317C22×C4#1175822411yescharacteristic
318C22×C4C2×C22⋊D6111yescharacteristic
319SD16—2442no—
320SD16—2442no—
321M16—41no—
322M16—41no—
323M16—2441no—
324M16—2441no—
325C2×C8—22no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—611no—
330C2×C8—611no—
331C2×C8—611no—
332C2×C8—611no—
333C2×C8—1222no—
334C2×C8—1222no—
335C2×C8—1222no—
336C2×C8—2441no—
337C2×C8—2441no—
338C2×C8—2441no—
339C2×C8—2441no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—1221no—
343C4⋊C4—1221no—
344C4⋊C4—2441no—
345C4⋊C4—2441no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—2442no—
349C22⋊C4—2441no—
350C22⋊C4—2441no—
351C42—2441no—
352C42—2441no—
353C23—22no—
354C23—22no—
355C23—41no—
356C23—41no—
357C23—42no—
358C23—81no—
359C23—81no—
360C23C4×C22⋊D6111yescharacteristic
361C23#1451975011yescharacteristic
362C23#1451976311yescharacteristic
363Q8—4881no—
364Q8—2441no—
365Q8—2441no—
366D4—41no—
367D4—41no—
368D4—81no—
369D4—82no—
370D4—82no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—81no—
382C2×C4—4881no—
383C2×C4—2442no—
384C2×C4—2442no—
385C2×C4—2442no—
386C2×C4—2442no—
387C2×C4—2441no—
388C2×C4—2441no—
389C2×C4C22⋊C4×D6111yescharacteristic
390C2×C4#1451973711yescharacteristic
391C2×C4#1451974911yescharacteristic
392C2×C4#1451977311yescharacteristic
393C8—42no—
394C8—81no—
395C8—612no—
396C8—1221no—
397C8—1221no—
398C8—1221no—
399C8—4881no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—44no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—162no—
412C22#1683966211yescharacteristic
413C22#1683988212yes—
414C22#1684002412yes—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—81no—
419C4—4882no—
420C4—4881no—
421C4—4881no—
422C4#99×D6111yescharacteristic
423C4#1683973411yescharacteristic
424C2—22no—
425C2—22no—
426C2—41no—
427C2—161no—
428C2#2144665311yescharacteristic
429C2#2144671111yescharacteristic
430C2#2144671911yescharacteristic
subgroupG/H
centerC22#16839662
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1471C2
socleC22×C61#390
prime radicalC2×#121×C61C22
cosocleC23×C4×C61C23
hypercenter#1471D61
hypocenterC61#9816
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1471

Complements

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

NH ≅ G/Ntype
2. #21445855C2semidirect
3. #21445598C2semidirect
4. #21444340C2semidirect
6. #21444182C2semidirect
8. C22×#11758200C4semidirect
16. C2×#14519708C4semidirect
17. C2×#14519700C4semidirect
18. C2×#14519698C4semidirect
246. C61#9816semidirect

Maximal subgroups

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

subgroupindex
C2×#168399892
#214458552
#214455982
#214443402
#214443352
#214441822
C61×#14712
#981661