Finite Group Databasebeta

← #29661333

Subgroups — #29661333

Order 15616.

820 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 820 conjugacy classes of subgroups — too many to draw. Only its 171 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#21457046C212yesmaximal, complemented
2#21456201C211yesmaximal, characteristic, complemented
3#21445153C214yesmaximal, complemented
4#21445147C212yesmaximal, complemented
5#21445142C212yesmaximal, complemented
6#21444457C211yesmaximal, characteristic, complemented
7#21444426C212yesmaximal, complemented
8C61×#3531C211yesmaximal, characteristic
9C2×#14519777C2211yescharacteristic
10#16840172C2214yes—
11#16840132C2212yes—
12#16840132C2212yes—
13#16840121C2214yes—
14#16840114—22no—
15#16840095—22no—
16#16840078—22no—
17#16840047—22no—
18#16840042—22no—
19#16839819C2212yes—
20#16839770—22no—
21C4×#11758194C2211yescharacteristic
22#16839504C2212yes—
23#16839504C2214yes—
24#16839504C2212yes—
25#16839502C2214yes—
26C61×#564C2211yescharacteristic
27C61×#563—22no—
28C61×#563—22no—
29C61×#563C2212yes—
30C61×#562—22no—
31C61×#562C2211yescharacteristic
32C61×#533—22no—
33C61×#533C2211yescharacteristic
34C61×#529—22no—
35C61×#529—22no—
36C61×#529C2212yes—
37C22×Dic122—21no—
38C2×#11758194—21no—
39#14519777—22no—
40#14519777—21no—
41#14519777C2312yes—
42C2×C4⋊Dic61—22no—
43#14519710—24no—
44#14519708—24no—
45C4⋊#92418—22no—
46#14519701—24no—
47#14519700—24no—
48#14519700—24no—
49#14519698—24no—
50C22⋊#92418—22no—
51#14519688C2312yes—
52#14519688C2314yes—
53C4×#92418—21no—
54C4⋊Q8×C61—22no—
55C4⋊Q8×C61—22no—
56C4⋊Q8×C61D412yes—
57#126×C61—22no—
58#126×C61—22no—
59#126×C61—22no—
60#126×C61—22no—
61#126×C61—22no—
62#125×C61—22no—
63#125×C61—22no—
64#125×C61—22no—
65#125×C61—22no—
66#125×C61C2311yescharacteristic
67#125×C61C2311yescharacteristic
68#125×C61C2312yes—
69#124×C61—22no—
70#124×C61—22no—
71#124×C61C2312yes—
72#123×C61—22no—
73#123×C61—22no—
74#123×C61—22no—
75#123×C61—22no—
76#123×C61—22no—
77C22⋊Q8×C61—22no—
78C22⋊Q8×C61—22no—
79C22⋊Q8×C61—22no—
80C22⋊Q8×C61—22no—
81C22⋊Q8×C61—22no—
82C22⋊Q8×C61—22no—
83C22⋊Q8×C61—22no—
84C22⋊Q8×C61D412yes—
85#121×C61—22no—
86#121×C61—22no—
87#121×C61D412yes—
88C4×Q8×C61—22no—
89C4×Q8×C61—22no—
90C4×Q8×C61D412yes—
91C4×D4×C61—22no—
92C4×D4×C61—22no—
93C4×D4×C61D412yes—
94#117×C61—22no—
95#117×C61—22no—
96#117×C61D412yes—
97C2×C42×C61—22no—
98C2×C42×C61C2311yescharacteristic
99C2×Q8×C61—22no—
100C2×Q8×C61—22no—
101C2×Q8×C61C2×D412yes—
102C2×D4×C61—22no—
103C2×D4×C61—22no—
104C2×D4×C61C2×D412yes—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D412yes—
110C22×Dic61—41no—
111C2×Dic122—22no—
112C2×Dic122—44no—
113C2×Dic122—41no—
114C4⋊C4×C61—22no—
115C4⋊C4×C61—22no—
116C4⋊C4×C61—22no—
117C4⋊C4×C61—22no—
118C4⋊C4×C61—22no—
119C4⋊C4×C61—22no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61—22no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—22no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61C2×D412yes—
128C4⋊C4×C61C2×D412yes—
129C4⋊C4×C61C2×D412yes—
130C4⋊C4×C61C2×D412yes—
131C22⋊C4×C61—22no—
132C22⋊C4×C61—22no—
133C22⋊C4×C61—22no—
134C22⋊C4×C61—22no—
135C22⋊C4×C61—22no—
136C22⋊C4×C61—22no—
137C22⋊C4×C61—22no—
138C22⋊C4×C61—22no—
139C22⋊C4×C61—22no—
140C22⋊C4×C61—22no—
141C42×C61—22no—
142C42×C61—22no—
143C42×C61—22no—
144C42×C61—22no—
145C42×C61C2×D412yes—
146C42×C61C2×D411yescharacteristic
147C42×C61C2411yescharacteristic
148C4⋊Dic61—24no—
149C4⋊Dic61—42no—
150#11758194—42no—
151C2×#92418—22no—
152C2×#92418—24no—
153C23×C61—22no—
154C23×C61#12011yescharacteristic
155Q8×C61—42no—
156Q8×C61—82no—
157Q8×C61—82no—
158D4×C61—42no—
159D4×C61—82no—
160D4×C61—82no—
161C2×C4×C61—22no—
162C2×C4×C61—22no—
163C2×C4×C61—22no—
164C2×C4×C61—22no—
165C2×C4×C61—22no—
166C2×C4×C61—22no—
167C2×C4×C61—42no—
168C2×C4×C61—44no—
169C2×C4×C61—82no—
170C2×C4×C61—82no—
171C2×C4×C61#12012yes—
172C2×C4×C61#12011yescharacteristic
173C2×C4×C61C22×D412yes—
174C2×C4×C61C22×D411yescharacteristic
175C2×Dic61—42no—
176C2×Dic61—82no—
177Dic122—44no—
178Dic122—82no—
179#92418—44no—
180#92418—42no—
181#19005—611nomaximal, Sylow
182C22×C61—21no—
183C22×C61—22no—
184C22×C61—42no—
185C22×C61—42no—
186C22×C61—42no—
187C22×C61C2×#12011yescharacteristic
188C4×C61—21no—
189C4×C61—22no—
190C4×C61—22no—
191C4×C61—42no—
192C4×C61—42no—
193C4×C61—42no—
194C4×C61—42no—
195C4×C61—42no—
196C4×C61#45412yes—
197Dic61—82no—
198#3531D6111yescharacteristic
199#3239—612no—
200#3139—611no—
201#1664—614no—
202#1663—612no—
203#1662—612no—
204#1537—611no—
205#1529—612no—
206C2×C61—41no—
207C2×C61—82no—
208C2×C61C2×#45411yescharacteristic
209C2×C61#302212yes—
210#564C2×D6111yescharacteristic
211#563—22no—
212#563—22no—
213#563C2×D6112yes—
214#562—22no—
215#562C2×D6111yescharacteristic
216#533—22no—
217#533C2×D6111yescharacteristic
218C2×C4⋊Q8—611no—
219#529—22no—
220#529—22no—
221#529C2×D6112yes—
222#501—614no—
223#491—612no—
224#491—612no—
225#489—614no—
226#487—1222no—
227#484—1222no—
228#481—1222no—
229#475—1222no—
230#474—1222no—
231#432—612no—
232#419—1222no—
233C4×M16—611no—
234#330—612no—
235#330—612no—
236#330—614no—
237#329—614no—
238C61#1900511yescharacteristic, complemented, Sylow
239C22×Q8—1221no—
240C2×M16—1221no—
241C4⋊Q8—22no—
242C4⋊Q8—22no—
243C4⋊Q8—612no—
244C4⋊Q8—1222no—
245C4⋊Q8—1221no—
246C4⋊Q8C22⋊D6112yes—
247#126—22no—
248#126—22no—
249#126—22no—
250#126—22no—
251#126—22no—
252#125—22no—
253#125—22no—
254#125—22no—
255#125—22no—
256#125C22×D6112yes—
257#125C22×D6111yescharacteristic
258#125C22×D6111yescharacteristic
259#124—22no—
260#124—22no—
261#124C22×D6112yes—
262#123—22no—
263#123—22no—
264#123—22no—
265#123—22no—
266#123—22no—
267C22⋊Q8—22no—
268C22⋊Q8—22no—
269C22⋊Q8—22no—
270C22⋊Q8—22no—
271C22⋊Q8—22no—
272C22⋊Q8—22no—
273C22⋊Q8—22no—
274C22⋊Q8C22⋊D6112yes—
275#121—22no—
276#121—22no—
277#121C22⋊D6112yes—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×Q8C22⋊D6112yes—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4C22⋊D6112yes—
284#117—22no—
285#117—22no—
286#117C22⋊D6112yes—
287C2×C4⋊C4—1222no—
288C2×C42—22no—
289C2×C42C22×D6111yescharacteristic
290#107—1224no—
291#106—1224no—
292C4⋊C8—1222no—
293#103—1224no—
294#103—1224no—
295#103—1224no—
296#102—1224no—
297C22⋊C8—1222no—
298#97—612no—
299#97—614no—
300C4×C8—1221no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—1222no—
304C2×Q8—2444no—
305C2×Q8—2441no—
306C2×Q8C2×C22⋊D6112yes—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4C2×C22⋊D6112yes—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—2441no—
314C22×C4C2×C22⋊D6111yescharacteristic
315C22×C4C2×C22⋊D6112yes—
316M16—2442no—
317C2×C8—1222no—
318C2×C8—1224no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—1224no—
333C4⋊C4—2442no—
334C4⋊C4C2×C22⋊D6112yes—
335C4⋊C4C2×C22⋊D6112yes—
336C4⋊C4C2×C22⋊D6112yes—
337C4⋊C4C2×C22⋊D6112yes—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—22no—
342C22⋊C4—22no—
343C22⋊C4—22no—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—22no—
347C22⋊C4—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42C2×C22⋊D6111yescharacteristic
353C42C2×C22⋊D6112yes—
354C42C23×D6111yescharacteristic
355C23—22no—
356C23#1451974611yescharacteristic
357Q8—42no—
358Q8—82no—
359Q8—82no—
360Q8—4882no—
361Q8—2444no—
362D4—42no—
363D4—82no—
364D4—82no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—4882no—
376C2×C4—2442no—
377C2×C4#1451974612yes—
378C2×C4#1451974611yescharacteristic
379C2×C4C22×C22⋊D6111yescharacteristic
380C2×C4C22×C22⋊D6112yes—
381C8—2444no—
382C8—2442no—
383C22—21no—
384C22—22no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22C2×#1451974611yescharacteristic
389C4—21no—
390C4—22no—
391C4—22no—
392C4—42no—
393C4—42no—
394C4—42no—
395C4—42no—
396C4—42no—
397C4—4882no—
398C4#1683993512yes—
399C2—41no—
400C2—82no—
401C2C2×#1683993511yescharacteristic
402C2#2145519812yes—
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3531C2
socleC22×C61C2×#120
hypercenter#3531D61
hypocenterC61#19005
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3531

Complements

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

NH ≅ G/Ntype
1. #21457046C2semidirect
2. #21456201C2semidirect
3. #21445153C2semidirect
4. #21445147C2semidirect
5. #21445142C2semidirect
6. #21444457C2semidirect
7. #21444426C2semidirect
238. C61#19005semidirect

Maximal subgroups

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

subgroupindex
#214570462
#214562012
#214451532
#214451472
#214451422
#214444572
#214444262
C61×#35312
#1900561