Finite Group Databasebeta

← #16808817

Subgroups — #16808817

Order 3328.

1702 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13131749C214yesmaximal, complemented
2#13131747C214yesmaximal, complemented
3#13131740C218yesmaximal, complemented
4#13131638C214yesmaximal, complemented
5Q8×#6118C214yesmaximal, complemented
6#13131517C214yesmaximal
7#13131513C212yesmaximal
8D13×#533C211yesmaximal, characteristic
9C13×#533C411yescharacteristic
10#11719433C2212yes—
11D13×C4⋊Q8C412yes—
12#11719393C2212yes—
13D13×#125C412yes—
14D13×C22⋊Q8C414yes—
15#11719327C2214yes—
16#11719283C414yes—
17C4×Q8×D13C2214yes—
18D13×#117C2212yes—
19#11719222C412yes—
20C2×C42×D13C2211yescharacteristic
21#11719175C411yescharacteristic
22#11719158C2218yescomplemented
23Q8×#1057C2218yescomplemented
24#11719130C2214yes—
25#11719127C2214yes—
26#11719119C2214yes—
27#11719080C2218yescomplemented
28#11719080C22116yescomplemented
29#11719078C22116yescomplemented
30#11719077C22116yes—
31#11719074C2218yescomplemented
32#11719068C22116yes—
33#11719054C2214yes—
34#11719053C2214yes—
35#11719052C2212yes—
36#11719052C2214yes—
37#11719052C2212yes—
38#11719051C2214yes—
39C4×#6118C2214yes—
40C4×#6118C2218yes—
41C2×Q8×D13C2×C414yes—
42C22×C4×D13C2×C412yes—
43C22×C4×D13C2311yescharacteristic
44C2×#6118C2314yes—
45C2×#6118C2314yes—
46C13×C4⋊Q8C2×C412yes—
47C13×#125C2×C412yes—
48C13×C22⋊Q8C2×C414yes—
49C4×Q8×C13C2×C414yes—
50C13×#117C2×C412yes—
51C2×C42×C13C2×C411yescharacteristic
52#88200C2×C414yes—
53Q8×Dic13C2314yes—
54#88198C2×C414yes—
55#88180C2×C412yes—
56#88178C2×C414yes—
57C2×C4×Dic13C2×C411yescharacteristic
58C2×C4×Dic13C2312yes—
59#88151C2×C414yes—
60#88150C2×C418yes—
61#88146C2314yes—
62C4⋊C4×D13C2×C414yes—
63C4⋊C4×D13C2×C414yes—
64C4⋊C4×D13C2×C418yes—
65#88144C2×C414yes—
66#88143C2×C418yes—
67#88142C2×C414yes—
68#88142C2×C414yes—
69#88141C2314yes—
70#88141C2318yes—
71C22⋊C4×D13C2×C414yes—
72#88132C2×C418yes—
73#88131C2314yes—
74#88126C2×C414yes—
75C42×D13C2×C411yescharacteristic
76C42×D13C2×C412yes—
77C42×D13C2×C411yescharacteristic
78C42×D13C2311yescharacteristic
79C42×D13C2312yes—
80C42×D13C2311yescharacteristic
81C42×D13C2314yes—
82#88124C2×C412yes—
83#88123C2×C412yes—
84C4×Dic26C2×C414yes—
85C4×Dic26C2×C418yes—
86C22⋊#1057C2318yes—
87#88112C23116yes—
88#88111C2318yes—
89#88110C2318yes—
90#88110C23116yes—
91C4⋊#1057C2314yes—
92C4⋊#1057C2314yes—
93C4⋊#1057C2314yes—
94C4⋊#1057C2314yes—
95C4⋊#1057C23116yes—
96C4×#1057C2318yes—
97C4×#1057C23116yes—
98#60249—131nomaximal, Sylow
99C23×D13C22×C411yescharacteristic
100C2×Q8×C13C22×C414yes—
101C22×C4×C13C22×C411yescharacteristic
102C22×C4×C13C22×C412yes—
103C22×Dic13C22×C412yes—
104C22×Dic13C2411yescharacteristic
105Q8×D13—48no—
106C2×C4×D13—22no—
107C2×C4×D13—22no—
108C2×C4×D13—44no—
109C2×C4×D13C22×C411yescharacteristic
110C2×C4×D13C22×C411yescharacteristic
111C2×C4×D13C22×C412yes—
112C2×C4×D13C22×C412yes—
113C2×C4×D13C22×C412yes—
114C2×C4×D13C22×C412yes—
115C2×C4×D13C22×C418yes—
116C2×C4×D13C2411yescharacteristic
117C2×C4×D13C2411yescharacteristic
118C2×Dic26C22×C414yes—
119C2×Dic26C22×C418yes—
120C2×#1057C2418yes—
121C2×#1057C2418yes—
122#6118—28no—
123#6118—28no—
124C13×C4⋊C4C22×C414yes—
125C13×C4⋊C4C22×C418yes—
126C13×C4⋊C4C22×C414yes—
127C13×C22⋊C4C22×C414yes—
128C42×C13C22×C412yes—
129C42×C13C22×C414yes—
130C42×C13C22×C411yescharacteristic
131C42×C13C22×C411yescharacteristic
132C22⋊Dic13C22×C414yes—
133#6103C22×C418yes—
134C4⋊Dic13C22×C414yes—
135C4⋊Dic13C22×C418yes—
136C4⋊Dic13C22×C414yes—
137#6101C22×C418yes—
138#6101C22×C418yes—
139#6101C22×C418yes—
140#6101C22×C418yes—
141C4×Dic13C22×C412yes—
142C4×Dic13C22×C414yes—
143C4×Dic13C22×C412yes—
144C4×Dic13C22×C412yes—
145C4×Dic13C22×C411yescharacteristic
146C4×Dic13C22×C411yescharacteristic
147C4×Dic13C2412yes—
148C4×Dic13C2418yes—
149C4×Dic13C2412yes—
150C2×#533—131no—
151#2987—134no—
152#2986—134no—
153#2985—138no—
154Q8×M16—134no—
155Q8×M16—134no—
156#2921—134no—
157#2920—132no—
158C23×C13C23×C411yescharacteristic
159C22×D13—22no—
160C22×D13—42no—
161C22×D13C23×C411yescharacteristic
162C22×D13C23×C411yescharacteristic
163Q8×C13—28no—
164C2×C4×C13—22no—
165C2×C4×C13—24no—
166C2×C4×C13C23×C411yescharacteristic
167C2×C4×C13C23×C412yes—
168C2×C4×C13C23×C418yes—
169C2×C4×C13C23×C411yescharacteristic
170C2×C4×C13C23×C412yes—
171C2×Dic13—21no—
172C2×Dic13—21no—
173C2×Dic13—42no—
174C2×Dic13C23×C411yescharacteristic
175C2×Dic13C23×C412yes—
176C2×Dic13C23×C412yes—
177C2×Dic13C23×C418yes—
178C2×Dic13C2511yescharacteristic
179C4×D13—22no—
180C4×D13—22no—
181C4×D13—24no—
182C4×D13—24no—
183C4×D13—21no—
184C4×D13—21no—
185C4×D13—42no—
186C4×D13—42no—
187C4×D13—48no—
188C4×D132−1+412yes—
189C4×D132−1+412yes—
190Dic26—28no—
191Dic26—48no—
192#1057—28no—
193#1057—28no—
194#533—132no—
195#533—132no—
196#533—134no—
197#533—134no—
198#533—132no—
199#533—131no—
200#533#26111yescharacteristic
201C2×C4⋊Q8—132no—
202C2×#125—132no—
203C2×C22⋊Q8—134no—
204C2×C4×Q8—134no—
205C2×#117—132no—
206C22×C42—131no—
207#446—1316no—
208#446—138no—
209#446—1316no—
210#446—138no—
211C8×Q8—138no—
212C8×Q8—138no—
213#433—1316no—
214#433—1316no—
215#432—134no—
216#432—134no—
217#431—134no—
218#431—134no—
219#423—132no—
220#423—132no—
221#423—134no—
222#423—134no—
223#407—134no—
224C4×M16—134no—
225C4×M16—138no—
226C22×C13—21no—
227C22×C13—22no—
228C22×C13C24×C411yescharacteristic
229C2×D13—22no—
230C2×D13—21no—
231C2×D13—22no—
232C2×D13—44no—
233C2×D13#51912yes—
234C4×C13—21no—
235C4×C13—28no—
236C4×C13—22no—
237C4×C13#51912yes—
238C4×C13#56814yes—
239Dic13—21no—
240Dic13—28no—
241Dic13—22no—
242Dic13—22no—
243Dic13C2×2−1+411yescharacteristic
244Dic13C2×2−1+411yescharacteristic
245C22×Q8—134no—
246C23×C4—131no—
247C23×C4—132no—
248C2×M16—134no—
249C2×M16—134no—
250C4⋊Q8—134no—
251C4⋊Q8—134no—
252C4⋊Q8—134no—
253C4⋊Q8—132no—
254C4⋊Q8C2×#26112yes—
255#125—138no—
256#125—134no—
257#125—132no—
258#125C2×#26112yes—
259C22⋊Q8—134no—
260C22⋊Q8—134no—
261C22⋊Q8—138no—
262C22⋊Q8—134no—
263C22⋊Q8—138no—
264C22⋊Q8C2×#26114yes—
265C4×Q8—134no—
266C4×Q8—134no—
267C4×Q8—138no—
268C4×Q8—138no—
269C4×Q8—134no—
270C4×Q8C2×#26114yes—
271#117—132no—
272#117—134no—
273#117—134no—
274#117—134no—
275#117C2×#26112yes—
276C2×C4⋊C4—134no—
277C2×C4⋊C4—138no—
278C2×C4⋊C4—134no—
279C2×C22⋊C4—134no—
280C2×C42—131no—
281C2×C42—132no—
282C2×C42—134no—
283C2×C42—132no—
284C2×C42—131no—
285C2×C42—131no—
286C2×C42—132no—
287C2×C42—131no—
288C2×C42—131no—
289C2×C42C2×#26111yescharacteristic
290C4⋊C8—134no—
291C4⋊C8—134no—
292C4⋊C8—134no—
293C4⋊C8—134no—
294C4⋊C8—1316no—
295C4⋊C8—1316no—
296C22⋊C8—138no—
297C22⋊C8—138no—
298#97—138no—
299#97—1316no—
300C4×C8—138no—
301C4×C8—1316no—
302C2×C13—41no—
303C2×C13C2×#51911yescharacteristic
304C2×C13C2×#56812yes—
305D13—22no—
306D13—41no—
307C24—131no—
308C2×Q8—134no—
309C2×Q8—138no—
310C2×Q8—528no—
311C2×Q8C22×#26114yes—
312C22×C4—131no—
313C22×C4—131no—
314C22×C4—132no—
315C22×C4—131no—
316C22×C4—131no—
317C22×C4—132no—
318C22×C4—131no—
319C22×C4—132no—
320C22×C4—132no—
321C22×C4—132no—
322C22×C4—138no—
323C22×C4—262no—
324C22×C4—262no—
325C22×C4—524no—
326C22×C4C22×#26111yescharacteristic
327C22×C4C22×#26112yes—
328M16—268no—
329M16—268no—
330C2×C8—138no—
331C2×C8—138no—
332C4⋊C4—138no—
333C4⋊C4—134no—
334C4⋊C4—138no—
335C4⋊C4—138no—
336C4⋊C4—138no—
337C4⋊C4—138no—
338C4⋊C4—134no—
339C4⋊C4C22×#26114yes—
340C4⋊C4C22×#26118yes—
341C4⋊C4C22×#26114yes—
342C22⋊C4—134no—
343C22⋊C4—138no—
344C22⋊C4C22×#26114yes—
345C42—132no—
346C42—138no—
347C42—132no—
348C42—132no—
349C42—131no—
350C42—132no—
351C42—132no—
352C42—131no—
353C42—134no—
354C42C22×#26112yes—
355C42C22×#26111yescharacteristic
356C42C22×#26111yescharacteristic
357C42C22×#26114yes—
358C13#6024911yescharacteristic, complemented, Sylow
359C23—131no—
360C23—131no—
361C23—262no—
362C23—522no—
363C23C23×#26111yescharacteristic
364Q8—28no—
365Q8—268no—
366Q8—528no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—131no—
370C2×C4—132no—
371C2×C4—132no—
372C2×C4—131no—
373C2×C4—132no—
374C2×C4—132no—
375C2×C4—138no—
376C2×C4—261no—
377C2×C4—261no—
378C2×C4—262no—
379C2×C4—261no—
380C2×C4—261no—
381C2×C4—262no—
382C2×C4—264no—
383C2×C4—264no—
384C2×C4—522no—
385C2×C4—522no—
386C2×C4—528no—
387C2×C4—522no—
388C2×C4C23×#26111yescharacteristic
389C2×C4C23×#26112yes—
390C2×C4C23×#26118yes—
391C2×C4C23×#26111yescharacteristic
392C2×C4C23×#26112yes—
393C8—268no—
394C8—268no—
395C22—21no—
396C22—22no—
397C22—132no—
398C22—262no—
399C22—261no—
400C22—262no—
401C22—524no—
402C22C24×#26111yescharacteristic
403C4—21no—
404C4—22no—
405C4—28no—
406C4—131no—
407C4—131no—
408C4—261no—
409C4—262no—
410C4—262no—
411C4—268no—
412C4#1171962712yes—
413C4#1171962914yes—
414C2—41no—
415C2—262no—
416C2—521no—
417C2C2×#1171962711yescharacteristic
418C2C2×#1171962912yes—
subgroupG/H
center, FrattiniC22C24×#261
commutator, socleC22×C13C24×C4
FittingC13×#533C4
prime radicalC23×D13C22×C4
cosocleC2×Dic13C25
hypercenter#533#261
hypocenterC13#60249
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ #533

Complements

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

NH ≅ G/Ntype
1. #13131749C2semidirect
2. #13131747C2semidirect
3. #13131740C2semidirect
4. #13131638C2semidirect
5. Q8×#6118C2semidirect
22. #11719158C22semidirect
23. Q8×#1057C22semidirect
27. #11719080C22semidirect
28. #11719080C22semidirect
29. #11719078C22semidirect
31. #11719074C22semidirect
358. C13#60249semidirect

Maximal subgroups

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

subgroupindex
#131317492
#131317472
#131317402
#131316382
Q8×#61182
#131315172
#131315132
D13×#5332
#6024913