Finite Group Databasebeta

← #15780468

Subgroups — #15780468

Order 3328.

676 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 111 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#13118476C211yesmaximal, characteristic
2C2×#11718070C212yesmaximal, complemented
3#13114760C211yesmaximal, characteristic
4#13112962C212yesmaximal, complemented
5C22×C13×#101C211yesmaximal, characteristic
6C23×Q8×C13C2211yescharacteristic
7C22×C13×M16—21no—
8C2×#88198C411yescharacteristic
9C2×#88198C2211yescharacteristic
10C2×#88178C411yescharacteristic
11C2×C13×#101—21no—
12C2×C13×#101—24no—
13C2×C13×#101C2212yes—
14#11718891C411yescharacteristic
15#11718891C2211yescharacteristic
16#11718890—21no—
17#11718890—21no—
18C2×#88076—21no—
19C2×#88076C2212yes—
20#11718871—21no—
21#11718871—21no—
22C4×C22⋊Dic13C411yescharacteristic
23#11718868—21no—
24#11718070—24no—
25C22×Q8×C13—22no—
26C22×Q8×C13—22no—
27C22×Q8×C13—41no—
28C22×Q8×C13—41no—
29C22×Q8×C13—42no—
30C22×Q8×C13C2×C411yescharacteristic
31C22×Q8×C13D411yescharacteristic
32C22×Q8×C13D411yescharacteristic
33C22×Q8×C13C2311yescharacteristic
34C23×C4×C13C2×C411yescharacteristic
35C23×C4×C13D411yescharacteristic
36C23×C4×C13D411yescharacteristic
37C22×Dic26C2×C411yescharacteristic
38C2×C13×M16—24no—
39C2×C13×M16—22no—
40C2×C13×M16—41no—
41C2×C13×M16—42no—
42C22×C8×C13—41no—
43C2×C22⋊Dic13—21no—
44C2×C22⋊Dic13—21no—
45#88198—22no—
46#88198—22no—
47#88198—42no—
48C2×C4⋊Dic13C2×C411yescharacteristic
49#88178—41no—
50#88178—41no—
51C2×#6101—21no—
52C2×#6101—21no—
53C2×#6101—21no—
54C2×#6101—21no—
55C2×#6101—21no—
56C2×C4×Dic13C2×C411yescharacteristic
57C2×C4×Dic13C2×C411yescharacteristic
58C2×#6099—22no—
59C13×#101—24no—
60C13×#101—22no—
61C13×#101D414yes—
62#88076—22no—
63#88076—22no—
64#88076—42no—
65#88071—21no—
66#88071—21no—
67#88071—21no—
68#88071—21no—
69#88071—21no—
70C22⋊#1055—41no—
71C22⋊#1055—41no—
72#13094—131nomaximal, Sylow
73C24×C13C22⋊C411yescharacteristic
74C2×Q8×C13—22no—
75C2×Q8×C13—22no—
76C2×Q8×C13—22no—
77C2×Q8×C13—21no—
78C2×Q8×C13—22no—
79C2×Q8×C13—21no—
80C2×Q8×C13—41no—
81C2×Q8×C13—41no—
82C2×Q8×C13—41no—
83C2×Q8×C13—41no—
84C2×Q8×C13—41no—
85C2×Q8×C13—41no—
86C2×Q8×C13—44no—
87C2×Q8×C13—44no—
88C2×Q8×C13—44no—
89C2×Q8×C13—44no—
90C2×Q8×C13C2×D412yes—
91C2×Q8×C13#4112yes—
92C22×C4×C13—21no—
93C22×C4×C13—21no—
94C22×C4×C13—22no—
95C22×C4×C13—21no—
96C22×C4×C13—21no—
97C22×C4×C13—21no—
98C22×C4×C13—21no—
99C22×C4×C13—21no—
100C22×C4×C13—21no—
101C22×C4×C13—22no—
102C22×C4×C13—22no—
103C22×C4×C13—41no—
104C22×C4×C13—41no—
105C22×C4×C13C22⋊C411yescharacteristic
106C22×C4×C13C22⋊C411yescharacteristic
107C22×C4×C13C22⋊C411yescharacteristic
108C22×C4×C13C22×C411yescharacteristic
109C22×C4×C13C2×D411yescharacteristic
110C22×C4×C13C2×D411yescharacteristic
111C22×Dic13—21no—
112C22×Dic13—21no—
113C22×Dic13—21no—
114C22×Dic13—21no—
115C2×Dic26—22no—
116C2×Dic26—41no—
117C2×Dic26—41no—
118C13×M16—24no—
119C13×M16—44no—
120C13×M16—42no—
121C2×C8×C13—44no—
122C2×C8×C13—42no—
123C22⋊Dic13—24no—
124C22⋊Dic13—41no—
125C22⋊Dic13—41no—
126C4⋊Dic13—41no—
127#6101—42no—
128#6101—42no—
129#6101—42no—
130#6101—41no—
131#6101—42no—
132#6101—41no—
133C4×Dic13—22no—
134C4×Dic13—41no—
135#6099—42no—
136#6099—42no—
137C2×#1055—42no—
138C22×#101D1311yescharacteristic
139#2434—131no—
140C2×#356—132no—
141#1784—131no—
142#1394—132no—
143C23×C13—22no—
144C23×C13—21no—
145C23×C13—21no—
146C23×C13—41no—
147C23×C13#9911yescharacteristic
148C23×C13#9911yescharacteristic
149C23×C13C2×C22⋊C411yescharacteristic
150Q8×C13—42no—
151Q8×C13—42no—
152Q8×C13—44no—
153Q8×C13—44no—
154Q8×C13—42no—
155Q8×C13—42no—
156C2×C4×C13—22no—
157C2×C4×C13—22no—
158C2×C4×C13—22no—
159C2×C4×C13—22no—
160C2×C4×C13—21no—
161C2×C4×C13—21no—
162C2×C4×C13—21no—
163C2×C4×C13—21no—
164C2×C4×C13—21no—
165C2×C4×C13—21no—
166C2×C4×C13—21no—
167C2×C4×C13—21no—
168C2×C4×C13—21no—
169C2×C4×C13—21no—
170C2×C4×C13—21no—
171C2×C4×C13—21no—
172C2×C4×C13—21no—
173C2×C4×C13—21no—
174C2×C4×C13—42no—
175C2×C4×C13—42no—
176C2×C4×C13—42no—
177C2×C4×C13—42no—
178C2×C4×C13#9912yes—
179C2×C4×C13C4×D412yes—
180C2×C4×C13#12011yescharacteristic
181C2×C4×C13#12112yes—
182C2×C4×C13#12311yescharacteristic
183C2×Dic13—22no—
184C2×Dic13—22no—
185C2×Dic13—41no—
186C2×Dic13—41no—
187C2×Dic13—41no—
188C2×Dic13—41no—
189C2×Dic13—42no—
190C2×Dic13—41no—
191C2×Dic13—41no—
192C2×Dic13—41no—
193C2×Dic13—41no—
194Dic26—82no—
195C8×C13—44no—
196#1055—82no—
197C23×Q8C2×D1311yescharacteristic
198C22×M16—21no—
199C2×C4⋊Q8—131no—
200C2×C4⋊Q8—131no—
201C2×C22⋊Q8—131no—
202C2×#101—21no—
203C2×#101—24no—
204C2×#101—132no—
205C2×#101—261no—
206C2×#101C2×D1312yes—
207#407—261no—
208#395—261no—
209#395—261no—
210#391—131no—
211#391—131no—
212#385—261no—
213#385—261no—
214C4×C22⋊C4—131no—
215#356—264no—
216C22×C13—21no—
217C22×C13—21no—
218C22×C13—21no—
219C22×C13—21no—
220C22×C13—21no—
221C22×C13—21no—
222C22×C13—21no—
223C22×C13—42no—
224C22×C13—42no—
225C22×C13#35612yescomplemented
226C22×C13#38611yescharacteristic
227C22×C13C2×#9911yescharacteristic
228C22×C13C2×#9911yescharacteristic
229C4×C13—22no—
230C4×C13—21no—
231C4×C13—22no—
232C4×C13—22no—
233C4×C13—21no—
234C4×C13—41no—
235C4×C13—41no—
236Dic13—42no—
237Dic13—42no—
238Dic13—81no—
239Dic13—81no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—41no—
243C22×Q8—41no—
244C22×Q8—42no—
245C22×Q8—131no—
246C22×Q8C4×D1311yescharacteristic
247C22×Q8D5211yescharacteristic
248C22×Q8C22⋊D1311yescharacteristic
249C22×Q8C22×D1311yescharacteristic
250C23×C4C4×D1311yescharacteristic
251C23×C4D5211yescharacteristic
252C23×C4C22⋊D1311yescharacteristic
253C2×M16—24no—
254C2×M16—22no—
255C2×M16—41no—
256C2×M16—42no—
257C2×M16—262no—
258C22×C8—41no—
259C4⋊Q8—262no—
260C4⋊Q8—262no—
261C4⋊Q8—522no—
262C22⋊Q8—521no—
263C22⋊Q8—521no—
264C2×C4⋊C4—131no—
265C2×C4⋊C4—261no—
266C2×C4⋊C4—261no—
267C2×C4⋊C4—261no—
268C2×C4⋊C4—261no—
269C2×C4⋊C4—261no—
270C2×C22⋊C4—261no—
271C2×C22⋊C4—261no—
272C2×C42—131no—
273C2×C42—131no—
274#101—24no—
275#101—22no—
276#101—262no—
277#101—262no—
278#101—522no—
279#101C22⋊D1314yes—
280C22⋊C8—521no—
281C22⋊C8—521no—
282#95—261no—
283#95—261no—
284#95—261no—
285#95—261no—
286#95—261no—
287C2×C13—21no—
288C2×C13—21no—
289C2×C13—22no—
290C2×C13—41no—
291C2×C13#189811yescharacteristic
292C2×C13C2×#35611yescharacteristic
293C2×C13#213311yescharacteristic
294C24#610311yescharacteristic
295C2×Q8—22no—
296C2×Q8—21no—
297C2×Q8—22no—
298C2×Q8—21no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—44no—
308C2×Q8—44no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—262no—
312C2×Q8—521no—
313C2×Q8—521no—
314C2×Q8#612712yes—
315C2×Q8C2×C22⋊D1312yes—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—261no—
330C22×C4—261no—
331C22×C4—261no—
332C22×C4—261no—
333C22×C4#610311yescharacteristic
334C22×C4#610311yescharacteristic
335C22×C4#610311yescharacteristic
336C22×C4C2×C4×D1311yescharacteristic
337C22×C4C2×D5211yescharacteristic
338C22×C4C2×C22⋊D1311yescharacteristic
339M16—24no—
340M16—44no—
341M16—42no—
342M16—522no—
343M16—522no—
344C2×C8—44no—
345C2×C8—42no—
346C2×C8—522no—
347C4⋊C4—521no—
348C4⋊C4—522no—
349C4⋊C4—522no—
350C4⋊C4—521no—
351C4⋊C4—522no—
352C4⋊C4—521no—
353C4⋊C4—522no—
354C22⋊C4—264no—
355C22⋊C4—521no—
356C22⋊C4—521no—
357C42—262no—
358C42—521no—
359C13#1309411yescharacteristic, complemented, Sylow
360C23—22no—
361C23—21no—
362C23—21no—
363C23—41no—
364C23#8804611yescharacteristic
365C23#8804611yescharacteristic
366C23C2×#610311yescharacteristic
367Q8—42no—
368Q8—42no—
369Q8—44no—
370Q8—44no—
371Q8—42no—
372Q8—42no—
373Q8—1042no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—262no—
397C2×C4—262no—
398C2×C4—521no—
399C2×C4—521no—
400C2×C4—521no—
401C2×C4—521no—
402C2×C4—522no—
403C2×C4—521no—
404C2×C4—521no—
405C2×C4—521no—
406C2×C4—521no—
407C2×C4#8804612yes—
408C2×C4C4×C22⋊D1312yes—
409C2×C4#8818311yescharacteristic
410C2×C4#8818412yes—
411C2×C4#8820711yescharacteristic
412C8—44no—
413C8—1042no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—42no—
422C22—42no—
423C22#1171807012yescomplemented
424C22C2×#8804611yescharacteristic
425C22C2×#8804611yescharacteristic
426C22#1171887411yescharacteristic
427C4—21no—
428C4—22no—
429C4—21no—
430C4—22no—
431C4—22no—
432C4—41no—
433C4—41no—
434C4—522no—
435C4—522no—
436C4—1041no—
437C4—1041no—
438C2—21no—
439C2—21no—
440C2—22no—
441C2—41no—
442C2#1311528911yescharacteristic
443C2C2×#1171807011yescharacteristic
444C2#1311652311yescharacteristic
subgroupG/H
centerC22C2×#88046
commutatorC22×C4×C13C22×C4
FrattiniC22×Q8C22×D13
FittingC22×C13×#101C2
socleC22×C13C2×#99
prime radicalC24×C13C22⋊C4
cosocleC22×Q8×C13C23
hypercenterC22×#101D13
hypocenterC13#13094
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ C22×#101

Complements

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

NH ≅ G/Ntype
2. C2×#11718070C2semidirect
4. #13112962C2semidirect
225. C22×C13#356semidirect
359. C13#13094semidirect
423. C22#11718070semidirect

Maximal subgroups

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

subgroupindex
#131184762
C2×#117180702
#131147602
#131129622
C22×C13×#1012
#1309413