Finite Group Databasebeta

← #30600917

Subgroups — #30600917

Order 15616.

472 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 84 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#21462745C211yesmaximal, characteristic
2#21462011C211yesmaximal, characteristic, complemented
3#21461332C211yesmaximal, characteristic, complemented
4#21461154C211yesmaximal, characteristic
5#21460727C211yesmaximal, characteristic, complemented
6#21460703C211yesmaximal, characteristic, complemented
7#21455304C211yesmaximal, characteristic
8#16840813—21no—
9#16840793—22no—
10D4×#92420—21no—
11#16840782—21no—
12#16840779C2211yescharacteristic
13#16840765C2211yescharacteristic
14#16840765C2211yescharacteristic
15C2×#14519844—21no—
16C4×#11758217C2211yescharacteristic
17#16840642—22no—
18#16840622—21no—
19#16840622—21no—
20Q8⋊#92420—21no—
21D4⋊#92420—21no—
22C2×#14519807C411yescharacteristic
23C2×#14519775C2211yescharacteristic
24#41×Dic61C2211yescharacteristic
25#16839970—21no—
26#16839970—21no—
27#16839965—21no—
28#16839965—21no—
29#16839785C2211yescharacteristic
30#16839772C411yescharacteristic
31#16839771C411yescharacteristic
32C2×#137×C61C411yescharacteristic
33C2×#11758217—21no—
34C2×#11758217—21no—
35C2×#11758217D412yes—
36C22×#92420—22no—
37#14519844—22no—
38#14519844—21no—
39#14519844—21no—
40#14519844D411yescharacteristic
41#14519844D411yescharacteristic
42C4⋊#92420—21no—
43C4⋊#92420—21no—
44C22⋊#92420—22no—
45C22⋊#92420—21no—
46#14519832—22no—
47#14519832—21no—
48C4×#92420—21no—
49C4×#92420—21no—
50C22×Dic122C2×C411yescharacteristic
51#14519807—42no—
52C2×#11758202—21no—
53C2×#11758200—21no—
54C2×#11758194C2×C411yescharacteristic
55#14519775—21no—
56#14519775—21no—
57#14519775—21no—
58#14519775—21no—
59Q8×Dic61D411yescharacteristic
60Q8×Dic61D411yescharacteristic
61D4×Dic61—21no—
62D4×Dic61—21no—
63D4×Dic61D411yescharacteristic
64D4×Dic61D411yescharacteristic
65#14519736—21no—
66#14519736C2×C411yescharacteristic
67C2×#11758196—21no—
68C2×#11758196—21no—
69C2×C4×Dic61—21no—
70C2×C4×Dic61C2311yescharacteristic
71#14519706—21no—
72#14519706—21no—
73#14519702—21no—
74#14519702—21no—
75#14519701—21no—
76#14519701—21no—
77C2×#41×C61C2×C411yescharacteristic
78C22×Q8×C61C2×C411yescharacteristic
79#137×C61—42no—
80C2×Q16×C61—21no—
81C2×SD16×C61—21no—
82C2×M16×C61C2×C411yescharacteristic
83#41×C61—22no—
84#41×C61—41no—
85C2×Q8×C61—21no—
86C2×Q8×C61—21no—
87C2×Q8×C61—41no—
88C2×Q8×C61—41no—
89C2×Q8×C61C22⋊C411yescharacteristic
90C2×D4×C61—21no—
91C2×D4×C61C22⋊C411yescharacteristic
92C22×C4×C61—21no—
93C22×C4×C61—21no—
94C22×C4×C61C22×C411yescharacteristic
95C22×Dic61—21no—
96C22×Dic61—21no—
97C22×Dic61—21no—
98C22×Dic61C2×D411yescharacteristic
99C22×Dic61#4111yescharacteristic
100C2×Dic122—21no—
101C2×Dic122—21no—
102C2×Dic122—41no—
103C2×Dic122—41no—
104#11758217—22no—
105#11758217—41no—
106#11758217—42no—
107#11758217—81no—
108C2×#92420—21no—
109C2×#92420—21no—
110C2×#92420—22no—
111C2×#92420—41no—
112C2×#92420—41no—
113C2×#92420—82no—
114Q16×C61—42no—
115SD16×C61—42no—
116M16×C61—41no—
117C2×C8×C61—21no—
118C22⋊Dic61—21no—
119C22⋊Dic61—21no—
120C22⋊Dic61—21no—
121#11758202—42no—
122#11758200—42no—
123C4⋊Dic61—21no—
124C4⋊Dic61—21no—
125C4⋊Dic61C22⋊C411yescharacteristic
126C4⋊Dic61C22⋊C411yescharacteristic
127#11758196—41no—
128#11758196—41no—
129#11758196—41no—
130#11758196—41no—
131C4×Dic61—21no—
132C4×Dic61—21no—
133C4×Dic61—21no—
134C4×Dic61C2×D411yescharacteristic
135C4×Dic61C2×D411yescharacteristic
136C4×Dic61C2×D411yescharacteristic
137C4×Dic61#4111yescharacteristic
138#11758194—41no—
139C2×#92418—21no—
140C23×C61—21no—
141C23×C61C4×D411yescharacteristic
142Q8×C61—21no—
143Q8×C61—21no—
144Q8×C61—41no—
145Q8×C61—41no—
146Q8×C61—81no—
147D4×C61—21no—
148D4×C61—42no—
149D4×C61#10012yes—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—42no—
154C2×C4×C61—41no—
155C2×C4×C61—81no—
156C2×C4×C61C2×C22⋊C411yescharacteristic
157C2×C4×C61C4×D411yescharacteristic
158C2×Dic61—21no—
159C2×Dic61—21no—
160C2×Dic61—21no—
161C2×Dic61—21no—
162C2×Dic61—21no—
163C2×Dic61—21no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—81no—
170C2×Dic61#12011yescharacteristic
171C2×Dic61#12111yescharacteristic
172C2×Dic61#12111yescharacteristic
173C2×Dic61#12311yescharacteristic
174Dic122—41no—
175Dic122—41no—
176Dic122—81no—
177#92420—42no—
178#92420—41no—
179#92420—42no—
180#92420—41no—
181#92420—81no—
182C8×C61—41no—
183#92418—41no—
184#9353—611nomaximal, Sylow
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—22no—
188C22×C61—21no—
189C22×C61—41no—
190C22×C61—41no—
191C22×C61#38611yescharacteristic
192C4×C61—21no—
193C4×C61—41no—
194C4×C61—81no—
195C4×C61C2×#10011yescharacteristic
196C4×C61#41311yescharacteristic
197Dic61—21no—
198Dic61—21no—
199Dic61—41no—
200Dic61—41no—
201Dic61—81no—
202Dic61#45511yescharacteristic
203Dic61#45611yescharacteristic
204Dic61#47111yescharacteristic
205Dic61#47311yescharacteristic
206#3039—611no—
207#2990—611no—
208#1997—611no—
209#1548—611no—
210#1489—611no—
211#1309—611no—
212#1299—611no—
213C2×C61—41no—
214C2×C61—42no—
215C2×C61#188211yescharacteristic
216C2×C61#190011yescharacteristic
217C2×C61#190811yescharacteristic
218C2×#137—611no—
219C2×#137#692311yescharacteristic
220C2×C4⋊Q8—611no—
221C4×#41—611no—
222#462—1221no—
223#462—1221no—
224#461—1221no—
225#461—1221no—
226#446—1221no—
227#435—1222no—
228C8×D4—1221no—
229#432—1221no—
230#431—611no—
231#423—611no—
232#423—611no—
233#423—611no—
234C2×C4⋊C8—1221no—
235#419—611no—
236#419—611no—
237C4×M16—611no—
238#343—1222no—
239#333—1221no—
240#333—1221no—
241Q8⋊C8—1221no—
242D4⋊C8—1221no—
243C61#935311yescharacteristic, complemented, Sylow
244C2×#41C2×#692311yescharacteristic
245C22×Q8—611no—
246C22×Q8C2×#692311yescharacteristic
247#137—42no—
248#137—2442no—
249C2×Q16—21no—
250C2×Q16—1221no—
251C2×SD16—21no—
252C2×SD16—1221no—
253C2×M16—612no—
254C2×M16—611no—
255C2×M16—1221no—
256C2×M16—1221no—
257C2×M16C2×#692311yescharacteristic
258C22×C8—1222no—
259C4⋊Q8—1221no—
260C4⋊Q8—1221no—
261C4⋊Q8—1221no—
262C4⋊Q8—1221no—
263C4×Q8—611no—
264C4×Q8—611no—
265C4×D4—611no—
266C4×D4—611no—
267C4×D4—1221no—
268C4×D4—1221no—
269#117—611no—
270#117—1221no—
271C2×C4⋊C4—1221no—
272C2×C4⋊C4—1221no—
273C2×C42—611no—
274C2×C42—1221no—
275C4⋊C8—611no—
276C4⋊C8—611no—
277C4⋊C8—1221no—
278C4⋊C8—1221no—
279C4⋊C8—1221no—
280C4⋊C8—1221no—
281C4⋊C8—1221no—
282C4⋊C8—1221no—
283C4⋊C8—1222no—
284#103—1221no—
285#103—1221no—
286#103—1221no—
287#103—1221no—
288C22⋊C8—1221no—
289C22⋊C8—1222no—
290#97—1221no—
291#97—1222no—
292C4×C8—1221no—
293C4×C8—1221no—
294#41—22no—
295#41—41no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—1221no—
301C2×Q8—1221no—
302C2×Q8—2441no—
303C2×Q8—2441no—
304C2×Q8C22⋊#692311yescharacteristic
305C2×D4—21no—
306C2×D4C22⋊#692311yescharacteristic
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—611no—
310C22×C4—611no—
311C22×C4—1221no—
312C22×C4—1221no—
313C22×C4—1221no—
314C22×C4C22×#692311yescharacteristic
315Q16—42no—
316Q16—2442no—
317SD16—42no—
318SD16—2442no—
319M16—41no—
320M16—1222no—
321M16—4881no—
322M16—2441no—
323M16—2441no—
324M16—2442no—
325C2×C8—21no—
326C2×C8—1221no—
327C2×C8—1221no—
328C2×C8—1222no—
329C2×C8—1221no—
330C2×C8—4882no—
331C2×C8—2441no—
332C2×C8—2441no—
333C4⋊C4—611no—
334C4⋊C4—611no—
335C4⋊C4—1221no—
336C4⋊C4—1221no—
337C4⋊C4—2441no—
338C4⋊C4—2441no—
339C4⋊C4—2441no—
340C4⋊C4—2441no—
341C22⋊C4—1221no—
342C22⋊C4—1221no—
343C22⋊C4—1221no—
344C42—611no—
345C42—611no—
346C42—611no—
347C42—611no—
348C42—1221no—
349C42—1221no—
350C42—1221no—
351C23—21no—
352C23D4×#692311yescharacteristic
353Q8—21no—
354Q8—21no—
355Q8—41no—
356Q8—41no—
357Q8—81no—
358Q8—4881no—
359Q8—2441no—
360Q8—2441no—
361D4—21no—
362D4—42no—
363D4#1451983712yes—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—81no—
370C2×C4—611no—
371C2×C4—611no—
372C2×C4—611no—
373C2×C4—611no—
374C2×C4—1221no—
375C2×C4—1221no—
376C2×C4—1221no—
377C2×C4—1221no—
378C2×C4—1221no—
379C2×C4—1221no—
380C2×C4—4881no—
381C2×C4—2441no—
382C2×C4—2441no—
383C2×C4—2441no—
384C2×C4—2441no—
385C2×C4—2441no—
386C2×C4C2×C22⋊#692311yescharacteristic
387C2×C4D4×#692311yescharacteristic
388C8—41no—
389C8—4881no—
390C8—2441no—
391C8—2441no—
392C8—2442no—
393C8—2442no—
394C8—2441no—
395C22—21no—
396C22—21no—
397C22—22no—
398C22—21no—
399C22—41no—
400C22—41no—
401C22#1684071011yescharacteristic
402C4—21no—
403C4—41no—
404C4—81no—
405C4—611no—
406C4—611no—
407C4—611no—
408C4—611no—
409C4—1221no—
410C4—1221no—
411C4—4881no—
412C4—2441no—
413C4—2441no—
414C4C2×#1451983711yescharacteristic
415C4#1684074911yescharacteristic
416C2—41no—
417C2—42no—
418C2#2146181811yescharacteristic
419C2#2146184411yescharacteristic
420C2#2146185911yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#137×C61C4
socleC22×C61#386
prime radicalC2×#41×C61C2×C4
cosocleC2×C4×Dic61C23
hypercenterC2×#137#6923
hypocenterC61#9353
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. #21462011C2semidirect
3. #21461332C2semidirect
5. #21460727C2semidirect
6. #21460703C2semidirect
243. C61#9353semidirect

Maximal subgroups

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

subgroupindex
#214627452
#214620112
#214613322
#214611542
#214607272
#214607032
#214553042
#935361