Finite Group Databasebeta

← #15778541

Subgroups — #15778541

Order 3328.

708 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 127 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#13116482C212yesmaximal
2C2×#11718131C212yesmaximal, complemented
3#13114947C211yesmaximal, characteristic, complemented
4#13114944C211yesmaximal, characteristic, complemented
5C13×#3491C211yesmaximal, characteristic
6C13×#552—22no—
7C13×#552—22no—
8C13×#551—22no—
9C13×#551—22no—
10C13×#543C411yescharacteristic, complemented
11C13×#539—22no—
12C13×#534C2211yescharacteristic
13C13×#529—22no—
14C2×C13×#123C411yescharacteristic, complemented
15C2×C13×#123C2211yescharacteristic
16C13×#518C411yescharacteristic, complemented
17C13×#518C2211yescharacteristic
18C2×C13×#117C411yescharacteristic, complemented
19C2×#88074C2212yes—
20C2×#88073C2212yes—
21#11718131—22no—
22#11718131—22no—
23#11718131—22no—
24#11718128—22no—
25#11718125—22no—
26#11718119—22no—
27C22×D4×C13C2×C411yescharacteristic
28C22×D4×C13C2311yescharacteristic
29C23×C4×C13C2×C411yescharacteristic
30C13×#126—22no—
31C13×#126—22no—
32C13×#126—22no—
33C13×#126—22no—
34C13×#125—22no—
35C13×#125—22no—
36C13×#125—22no—
37C13×#124—22no—
38C13×#123—22no—
39C13×#123—22no—
40C13×#123—22no—
41C13×#123—21no—
42C13×#123—21no—
43C13×#123—21no—
44C13×#123—22no—
45C13×#123D412yes—
46C13×#123D412yes—
47C13×#123D412yes—
48C13×C22⋊Q8—22no—
49C13×C22⋊Q8—22no—
50C13×C22⋊Q8—22no—
51C13×C22⋊Q8—22no—
52C13×C22⋊Q8—22no—
53C13×#121—22no—
54C13×#120—21no—
55C13×#120D412yes—
56C4×Q8×C13—22no—
57C4×D4×C13—22no—
58C4×D4×C13—22no—
59C4×D4×C13—22no—
60C4×D4×C13—22no—
61C4×D4×C13—22no—
62C13×#117—22no—
63C13×#117—22no—
64C13×#117—21no—
65C13×#117—22no—
66C13×#117—22no—
67C13×#117D412yes—
68C2×C13×C4⋊C4—21no—
69C2×C13×C4⋊C4—21no—
70C2×C13×C4⋊C4—21no—
71C2×C13×C22⋊C4—21no—
72C2×C13×C22⋊C4—21no—
73C2×C13×C22⋊C4—21no—
74C2×C13×C22⋊C4C2×C411yescharacteristic
75C2×C13×C22⋊C4C2×C411yescharacteristic
76C2×C13×C22⋊C4C2×C411yescharacteristic
77C2×C13×C22⋊C4C2×C411yescharacteristic
78C2×C13×C22⋊C4D412yes—
79C2×C42×C13—21no—
80C2×C22⋊Dic13—22no—
81C2×#6099—22no—
82#88074—22no—
83#88074—22no—
84#88073—22no—
85#88073—22no—
86#12726—131nomaximal, Sylow
87C24×C13C22⋊C411yescharacteristic
88C24×C13C22×C411yescharacteristic
89C24×C13C2×D411yescharacteristic
90C2×Q8×C13—22no—
91C2×D4×C13—21no—
92C2×D4×C13—21no—
93C2×D4×C13—22no—
94C2×D4×C13—42no—
95C2×D4×C13—42no—
96C2×D4×C13C2×D412yes—
97C2×D4×C13C2×D412yes—
98C22×C4×C13—21no—
99C22×C4×C13—21no—
100C22×C4×C13—22no—
101C22×C4×C13—21no—
102C22×C4×C13—21no—
103C22×C4×C13—21no—
104C22×C4×C13—21no—
105C22×C4×C13—22no—
106C22×C4×C13—42no—
107C22×C4×C13C22⋊C411yescharacteristic
108C22×C4×C13C22⋊C411yescharacteristic
109C22×C4×C13C22⋊C411yescharacteristic
110C22×C4×C13C2×D411yescharacteristic
111C22×Dic13—42no—
112C13×C4⋊C4—22no—
113C13×C4⋊C4—22no—
114C13×C4⋊C4—22no—
115C13×C4⋊C4—22no—
116C13×C4⋊C4—22no—
117C13×C4⋊C4—22no—
118C13×C4⋊C4—22no—
119C13×C4⋊C4—22no—
120C13×C4⋊C4—22no—
121C13×C4⋊C4—22no—
122C13×C4⋊C4—22no—
123C13×C4⋊C4—22no—
124C13×C22⋊C4—22no—
125C13×C22⋊C4—22no—
126C13×C22⋊C4—22no—
127C13×C22⋊C4—21no—
128C13×C22⋊C4—21no—
129C13×C22⋊C4—21no—
130C13×C22⋊C4—21no—
131C13×C22⋊C4—22no—
132C13×C22⋊C4—22no—
133C13×C22⋊C4—22no—
134C13×C22⋊C4—22no—
135C13×C22⋊C4—22no—
136C13×C22⋊C4C22⋊C412yes—
137C13×C22⋊C4C22⋊C412yes—
138C13×C22⋊C4C22⋊C412yes—
139C13×C22⋊C4C22⋊C412yes—
140C42×C13—22no—
141C42×C13—22no—
142C42×C13—22no—
143C42×C13—22no—
144C22⋊Dic13—42no—
145C22⋊Dic13—42no—
146#6099—42no—
147#6099—42no—
148C2×#1055—42no—
149#3491D1311yescharacteristic
150#2121—132no—
151C2×#352—132no—
152#1829—131no—
153#1828—131no—
154C23×C13—22no—
155C23×C13—22no—
156C23×C13—21no—
157C23×C13—41no—
158C23×C13—81no—
159C23×C13—81no—
160C23×C13—81no—
161C23×C13C2×C22⋊C411yescharacteristic
162C23×C13C2×C22⋊C411yescharacteristic
163C23×C13C2×C22⋊C411yescharacteristic
164C23×C13#12012yes—
165Q8×C13—82no—
166D4×C13—42no—
167D4×C13—42no—
168D4×C13—42no—
169D4×C13—42no—
170D4×C13—82no—
171C2×C4×C13—22no—
172C2×C4×C13—21no—
173C2×C4×C13—22no—
174C2×C4×C13—22no—
175C2×C4×C13—21no—
176C2×C4×C13—22no—
177C2×C4×C13—22no—
178C2×C4×C13—21no—
179C2×C4×C13—42no—
180C2×C4×C13—42no—
181C2×C4×C13—41no—
182C2×C4×C13—41no—
183C2×C4×C13—41no—
184C2×C4×C13—81no—
185C2×C4×C13—82no—
186C2×C4×C13—82no—
187C2×C4×C13—81no—
188C2×C4×C13—81no—
189C2×C4×C13—81no—
190C2×C4×C13#12012yes—
191C2×Dic13—82no—
192C2×Dic13—162no—
193#1055—82no—
194#552—22no—
195#552—22no—
196#551—22no—
197#551—22no—
198#543Dic1311yescharacteristic, complemented
199#539—22no—
200#534C2×D1311yescharacteristic
201#529—22no—
202C2×#123Dic1311yescharacteristic, complemented
203C2×#123C2×D1311yescharacteristic
204#518Dic1311yescharacteristic, complemented
205#518C2×D1311yescharacteristic
206C2×#117Dic1311yescharacteristic, complemented
207C2×#100—132no—
208C2×#99—132no—
209#352—262no—
210#352—262no—
211#352—262no—
212AΣL(1,𝔽16)—262no—
213#344—262no—
214#342—262no—
215C22×C13—22no—
216C22×C13—21no—
217C22×C13—21no—
218C22×C13—41no—
219C22×C13—41no—
220C22×C13—42no—
221C22×C13—41no—
222C22×C13—42no—
223C22×C13—41no—
224C22×C13—82no—
225C22×C13—82no—
226C22×C13—161no—
227C22×C13#37911yescharacteristic
228C22×C13#41012yes—
229C4×C13—22no—
230C4×C13—42no—
231C4×C13—42no—
232C4×C13—42no—
233C4×C13—42no—
234C4×C13—41no—
235C4×C13—41no—
236C4×C13—41no—
237Dic13—162no—
238C22×D4C2×Dic1311yescharacteristic
239C22×D4C22×D1311yescharacteristic
240C23×C4C2×Dic1311yescharacteristic
241C2×M16—262no—
242#126—22no—
243#126—22no—
244#126—22no—
245#126—22no—
246#125—22no—
247#125—22no—
248#125—22no—
249#124—22no—
250#123—22no—
251#123—22no—
252#123—22no—
253#123—21no—
254#123—21no—
255#123—21no—
256#123—22no—
257#123C22⋊D1312yes—
258#123C22⋊D1312yes—
259#123C22⋊D1312yes—
260C22⋊Q8—22no—
261C22⋊Q8—22no—
262C22⋊Q8—22no—
263C22⋊Q8—22no—
264C22⋊Q8—22no—
265#121—22no—
266#120—21no—
267#120C22⋊D1312yes—
268C4×Q8—22no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—22no—
273C4×D4—22no—
274#117—22no—
275#117—22no—
276#117—21no—
277#117—22no—
278#117—22no—
279#117C22⋊D1312yes—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—262no—
287C2×C22⋊C4C2×Dic1311yescharacteristic
288C2×C22⋊C4C2×Dic1311yescharacteristic
289C2×C22⋊C4C2×Dic1311yescharacteristic
290C2×C22⋊C4C2×Dic1311yescharacteristic
291C2×C22⋊C4C22⋊D1312yes—
292C2×C42—21no—
293#100—262no—
294#100—262no—
295#99—262no—
296#99—262no—
297C2×C13—21no—
298C2×C13—22no—
299C2×C13—42no—
300C2×C13—41no—
301C2×C13—41no—
302C2×C13—81no—
303C2×C13#178211yescharacteristic
304C24C22⋊Dic1311yescharacteristic
305C24C22×Dic1311yescharacteristic
306C24C2×C22⋊D1311yescharacteristic
307C2×Q8—22no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—22no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4C2×C22⋊D1312yes—
314C2×D4C2×C22⋊D1312yes—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—42no—
324C22×C4—522no—
325C22×C4C22⋊Dic1311yescharacteristic
326C22×C4C22⋊Dic1311yescharacteristic
327C22×C4C22⋊Dic1311yescharacteristic
328C22×C4C2×C22⋊D1311yescharacteristic
329M16—522no—
330M16—522no—
331C2×C8—522no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—22no—
347C22⋊C4—21no—
348C22⋊C4—21no—
349C22⋊C4—21no—
350C22⋊C4—21no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—522no—
357C22⋊C4—522no—
358C22⋊C4C22⋊Dic1312yes—
359C22⋊C4C22⋊Dic1312yes—
360C22⋊C4C22⋊Dic1312yes—
361C22⋊C4C22⋊Dic1312yes—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—22no—
366C13#1272611yescharacteristic, complemented, Sylow
367C23—22no—
368C23—22no—
369C23—21no—
370C23—41no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23C2×C22⋊Dic1311yescharacteristic
375C23C2×C22⋊Dic1311yescharacteristic
376C23C2×C22⋊Dic1311yescharacteristic
377C23#8820712yes—
378Q8—82no—
379D4—42no—
380D4—42no—
381D4—42no—
382D4—42no—
383D4—82no—
384C2×C4—22no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—81no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—1042no—
404C2×C4—2082no—
405C2×C4#8820712yes—
406C8—1042no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—42no—
413C22—41no—
414C22—42no—
415C22—41no—
416C22—82no—
417C22—82no—
418C22—161no—
419C22#1171890112yes—
420C22#1171890811yescharacteristic
421C4—22no—
422C4—42no—
423C4—42no—
424C4—42no—
425C4—42no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—2082no—
430C2—21no—
431C2—22no—
432C2—42no—
433C2—41no—
434C2—41no—
435C2—81no—
436C2#1311475411yescharacteristic
subgroupG/H
centerC2#13114754
commutatorC24×C13C22×C4
FrattiniC22×D4C22×D13
FittingC13×#3491C2
socleC2×C13#1782
prime radicalC13×#534C22
cosocleC22×D4×C13C23
hypercenter#3491D13
hypocenterC13#12726
LCSG ▷ C24×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C2 ◁ C23 ◁ #518 ◁ #3491

Complements

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

NH ≅ G/Ntype
2. C2×#11718131C2semidirect
3. #13114947C2semidirect
4. #13114944C2semidirect
10. C13×#543C4semidirect
14. C2×C13×#123C4semidirect
16. C13×#518C4semidirect
18. C2×C13×#117C4semidirect
198. #543Dic13semidirect
202. C2×#123Dic13semidirect
204. #518Dic13semidirect
206. C2×#117Dic13semidirect
366. C13#12726semidirect

Maximal subgroups

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

subgroupindex
#131164822
C2×#117181312
#131149472
#131149442
C13×#34912
#1272613