Finite Group Databasebeta

← #15754445

Subgroups — #15754445

Order 3328.

720 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 135 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#13114602C211yesmaximal, characteristic, complemented
2#13113456C212yesmaximal, complemented
3#13113296C212yesmaximal, complemented
4#13113170C211yesmaximal, characteristic, complemented
5C13×#3493C211yesmaximal, characteristic
6C13×#556—22no—
7C13×#555—22no—
8C13×#553—22no—
9C13×#552—22no—
10C13×#544C411yescharacteristic
11C13×#538—21no—
12C13×#538C412yes—
13C13×#537C411yescharacteristic
14C13×#529—21no—
15C13×#529C2212yes—
16C13×#525—21no—
17C13×#520—21no—
18C2×C13×#117C2211yescharacteristic
19C2×C22⋊#1055—21no—
20#11718587C2212yes—
21C2×#88043C2211yescharacteristic
22C2×C4×#1055C2211yescharacteristic
23#11718135—22no—
24#11718132—22no—
25Q8⋊#1055—22no—
26D4⋊#1055—22no—
27#11718120—21no—
28#11718117—21no—
29#11718116—21no—
30C2×C13×#41—21no—
31C23×C4×C13D411yescharacteristic
32C13×C4⋊Q8—22no—
33C13×#126—22no—
34C13×#126—22no—
35C13×#126—22no—
36C13×#126—22no—
37C13×#125—21no—
38C13×#125—22no—
39C13×#125D412yes—
40C13×#124—21no—
41C13×#124—22no—
42C13×#124—22no—
43C13×#124D412yes—
44C13×#123—22no—
45C13×#123—22no—
46C13×#123—22no—
47C13×C22⋊Q8—21no—
48C13×C22⋊Q8—21no—
49C13×C22⋊Q8—21no—
50C13×C22⋊Q8—22no—
51C13×C22⋊Q8C2×C412yes—
52C13×#121—21no—
53C13×#121—21no—
54C13×#121—22no—
55C13×#121—22no—
56C13×#121—22no—
57C13×#121—21no—
58C13×#121C2×C412yes—
59C13×#120—22no—
60C4×Q8×C13—22no—
61C4×Q8×C13—21no—
62C4×Q8×C13D412yes—
63C4×D4×C13—22no—
64C4×D4×C13—22no—
65C4×D4×C13—22no—
66C4×D4×C13—21no—
67C4×D4×C13—22no—
68C4×D4×C13—22no—
69C4×D4×C13D412yes—
70C13×#117—22no—
71C13×#117—22no—
72C13×#117—21no—
73C13×#117—22no—
74C13×#117—22no—
75C13×#117—22no—
76C2×C13×C4⋊C4—22no—
77C2×C13×C4⋊C4C2×C411yescharacteristic
78C2×C13×C4⋊C4D411yescharacteristic
79C2×C13×C22⋊C4C2×C411yescharacteristic
80C2×C13×C22⋊C4D411yescharacteristic
81C2×C42×C13D411yescharacteristic
82C2×C42×C13C2311yescharacteristic
83C22×#1055—21no—
84C22×#1055—21no—
85C22⋊#1055—24no—
86C22⋊#1055—22no—
87C4⋊#1055—22no—
88#88043—22no—
89#88043—22no—
90C4×#1055—22no—
91#8349—131nomaximal, Sylow
92C24×C13C22⋊C411yescharacteristic
93C13×#41—42no—
94C13×#41—42no—
95C2×Q8×C13—21no—
96C2×Q8×C13—21no—
97C2×Q8×C13C22⋊C412yes—
98C2×D4×C13—21no—
99C2×D4×C13—22no—
100C2×D4×C13—22no—
101C2×D4×C13—22no—
102C2×D4×C13—21no—
103C2×D4×C13C22⋊C412yes—
104C22×C4×C13—22no—
105C22×C4×C13—22no—
106C22×C4×C13—22no—
107C22×C4×C13—21no—
108C22×C4×C13—21no—
109C22×C4×C13—22no—
110C22×C4×C13—41no—
111C22×C4×C13C22⋊C411yescharacteristic
112C22×C4×C13C22⋊C411yescharacteristic
113C22×C4×C13C22⋊C411yescharacteristic
114C22×C4×C13C22×C411yescharacteristic
115C22×C4×C13C2×D411yescharacteristic
116C22×C4×C13C2×D411yescharacteristic
117C13×C4⋊C4—21no—
118C13×C4⋊C4—22no—
119C13×C4⋊C4—21no—
120C13×C4⋊C4—22no—
121C13×C4⋊C4—21no—
122C13×C4⋊C4—21no—
123C13×C4⋊C4—22no—
124C13×C4⋊C4—22no—
125C13×C4⋊C4—22no—
126C13×C4⋊C4—22no—
127C13×C4⋊C4—22no—
128C13×C4⋊C4—22no—
129C13×C4⋊C4C22⋊C412yes—
130C13×C4⋊C4C22⋊C412yes—
131C13×C22⋊C4—22no—
132C13×C22⋊C4—22no—
133C13×C22⋊C4—22no—
134C13×C22⋊C4—22no—
135C13×C22⋊C4—22no—
136C13×C22⋊C4—22no—
137C13×C22⋊C4—22no—
138C13×C22⋊C4—22no—
139C13×C22⋊C4—22no—
140C13×C22⋊C4—22no—
141C13×C22⋊C4—22no—
142C42×C13—22no—
143C42×C13—22no—
144C42×C13—21no—
145C42×C13—21no—
146C42×C13—22no—
147C42×C13C2×D412yes—
148C42×C13C2×D412yes—
149C2×#1055—22no—
150C2×#1055—22no—
151C2×#1055—41no—
152C2×#1055—41no—
153C2×#1055—41no—
154C2×#1055—41no—
155C2×#1055—42no—
156#3493D1311yescharacteristic
157#1750—131no—
158#1532—132no—
159#1494—132no—
160#1464—131no—
161C23×C13—21no—
162C23×C13—22no—
163C23×C13—21no—
164C23×C13—21no—
165C23×C13—41no—
166C23×C13—41no—
167C23×C13C2×C22⋊C411yescharacteristic
168Q8×C13—42no—
169Q8×C13—42no—
170Q8×C13—81no—
171D4×C13—42no—
172D4×C13—42no—
173D4×C13—42no—
174D4×C13—42no—
175D4×C13—82no—
176D4×C13—81no—
177D4×C13—82no—
178C2×C4×C13—21no—
179C2×C4×C13—21no—
180C2×C4×C13—21no—
181C2×C4×C13—21no—
182C2×C4×C13—21no—
183C2×C4×C13—22no—
184C2×C4×C13—21no—
185C2×C4×C13—22no—
186C2×C4×C13—22no—
187C2×C4×C13—41no—
188C2×C4×C13—41no—
189C2×C4×C13—42no—
190C2×C4×C13—42no—
191C2×C4×C13—44no—
192C2×C4×C13—41no—
193C2×C4×C13—41no—
194C2×C4×C13—82no—
195C2×C4×C13—82no—
196C2×C4×C13#10412yes—
197C2×C4×C13C2×C22⋊C412yes—
198C2×C4×C13#12012yes—
199C2×C4×C13#12011yescharacteristic
200C2×C4×C13#12011yescharacteristic
201#1055—42no—
202#1055—42no—
203#1055—82no—
204#556—22no—
205#555—22no—
206#553—22no—
207#552—22no—
208#544Dic1311yescharacteristic
209#538—21no—
210#538Dic1312yes—
211#537Dic1311yescharacteristic
212#529—21no—
213#529C2×D1312yes—
214#525—21no—
215#520—21no—
216C2×#117C2×D1311yescharacteristic
217#432—132no—
218C2×C22⋊C8—261no—
219C2×#97—131no—
220C2×C4×C8—131no—
221#336—261no—
222#330—262no—
223#329—262no—
224Q8⋊C8—262no—
225D4⋊C8—262no—
226#324—261no—
227#323—261no—
228C22×C13—21no—
229C22×C13—21no—
230C22×C13—42no—
231C22×C13—42no—
232C22×C13—41no—
233C22×C13—41no—
234C22×C13—42no—
235C22×C13—42no—
236C22×C13—41no—
237C22×C13—81no—
238C22×C13#37911yescharacteristic
239C22×C13C2×#10411yescharacteristic
240C22×C13#42111yescharacteristic
241C4×C13—22no—
242C4×C13—21no—
243C4×C13—21no—
244C4×C13—22no—
245C4×C13—22no—
246C4×C13—41no—
247C4×C13—42no—
248C4×C13—42no—
249C4×C13—41no—
250C4×C13—42no—
251C2×#41—21no—
252C23×C4C22⋊D1311yescharacteristic
253C22×C8—261no—
254C22×C8—261no—
255C4⋊Q8—22no—
256#126—22no—
257#126—22no—
258#126—22no—
259#126—22no—
260#125—21no—
261#125—22no—
262#125C22⋊D1312yes—
263#124—21no—
264#124—22no—
265#124—22no—
266#124C22⋊D1312yes—
267#123—22no—
268#123—22no—
269#123—22no—
270C22⋊Q8—21no—
271C22⋊Q8—21no—
272C22⋊Q8—21no—
273C22⋊Q8—22no—
274C22⋊Q8C2×Dic1312yes—
275#121—21no—
276#121—21no—
277#121—22no—
278#121—22no—
279#121—22no—
280#121—21no—
281#121C2×Dic1312yes—
282#120—22no—
283C4×Q8—22no—
284C4×Q8—21no—
285C4×Q8C22⋊D1312yes—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—22no—
289C4×D4—21no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4C22⋊D1312yes—
293#117—22no—
294#117—22no—
295#117—21no—
296#117—22no—
297#117—22no—
298#117—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4C2×Dic1311yescharacteristic
301C2×C4⋊C4C22⋊D1311yescharacteristic
302C2×C22⋊C4C2×Dic1311yescharacteristic
303C2×C22⋊C4C22⋊D1311yescharacteristic
304C2×C42C22⋊D1311yescharacteristic
305C2×C42C22×D1311yescharacteristic
306C4⋊C8—262no—
307C22⋊C8—264no—
308C22⋊C8—262no—
309#97—262no—
310#97—262no—
311C4×C8—262no—
312C2×C13—21no—
313C2×C13—21no—
314C2×C13—82no—
315C2×C13—81no—
316C2×C13#178911yescharacteristic
317C2×C13#179111yescharacteristic
318C2×C13#179411yescharacteristic
319C24C22⋊Dic1311yescharacteristic
320#41—42no—
321#41—42no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8C22⋊Dic1312yes—
325C2×D4—21no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—21no—
330C2×D4C22⋊Dic1312yes—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—41no—
338C22×C4C22⋊Dic1311yescharacteristic
339C22×C4C22⋊Dic1311yescharacteristic
340C22×C4C22⋊Dic1311yescharacteristic
341C22×C4C22×Dic1311yescharacteristic
342C22×C4C2×C22⋊D1311yescharacteristic
343C22×C4C2×C22⋊D1311yescharacteristic
344C2×C8—262no—
345C2×C8—262no—
346C2×C8—521no—
347C2×C8—521no—
348C2×C8—521no—
349C2×C8—521no—
350C2×C8—522no—
351C4⋊C4—21no—
352C4⋊C4—22no—
353C4⋊C4—21no—
354C4⋊C4—22no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4C22⋊Dic1312yes—
364C4⋊C4C22⋊Dic1312yes—
365C22⋊C4—22no—
366C22⋊C4—22no—
367C22⋊C4—22no—
368C22⋊C4—22no—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C22⋊C4—22no—
372C22⋊C4—22no—
373C22⋊C4—22no—
374C22⋊C4—22no—
375C22⋊C4—22no—
376C42—22no—
377C42—22no—
378C42—21no—
379C42—21no—
380C42—22no—
381C42C2×C22⋊D1312yes—
382C42C2×C22⋊D1312yes—
383C13#834911yescharacteristic, complemented, Sylow
384C23—21no—
385C23—21no—
386C23—22no—
387C23—21no—
388C23—41no—
389C23—41no—
390C23C2×C22⋊Dic1311yescharacteristic
391Q8—42no—
392Q8—42no—
393Q8—81no—
394D4—42no—
395D4—42no—
396D4—42no—
397D4—42no—
398D4—81no—
399D4—82no—
400D4—82no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4#8807712yes—
420C2×C4C2×C22⋊Dic1312yes—
421C2×C4#8820711yescharacteristic
422C2×C4#8820711yescharacteristic
423C2×C4#8820712yes—
424C8—522no—
425C8—522no—
426C8—1042no—
427C22—21no—
428C22—21no—
429C22—42no—
430C22—41no—
431C22—42no—
432C22—42no—
433C22—41no—
434C22—42no—
435C22—41no—
436C22—81no—
437C22C2×#8807711yescharacteristic
438C22#1171889711yescharacteristic
439C22#1171890811yescharacteristic
440C4—21no—
441C4—21no—
442C4—22no—
443C4—22no—
444C4—22no—
445C4—42no—
446C4—41no—
447C4—42no—
448C4—41no—
449C4—42no—
450C2—21no—
451C2—21no—
452C2—81no—
453C2—82no—
454C2#1311478011yescharacteristic
455C2#1311478711yescharacteristic
456C2#1311480011yescharacteristic
subgroupG/H
centerC22#11718908
commutatorC22×C4×C13C22×C4
FrattiniC2×C42C22×D13
Fitting, prime radicalC13×#3493C2
socleC22×C13#379
cosocleC2×C42×C13C23
hypercenter#3493D13
hypocenterC13#8349
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3493

Complements

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

NH ≅ G/Ntype
1. #13114602C2semidirect
2. #13113456C2semidirect
3. #13113296C2semidirect
4. #13113170C2semidirect
383. C13#8349semidirect

Maximal subgroups

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

subgroupindex
#131146022
#131134562
#131132962
#131131702
C13×#34932
#834913