Finite Group Databasebeta

← C22×#13049813 #72220310

Subgroups — C22×#13049813 #72220310

Order 5184.

9136 conjugacy classes of subgroups; only the 462 normal subgroups are shown.

normal subgroups only

This group has 9136 conjugacy classes of subgroups — too many to draw. Only its 462 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
1C22×#10620198C211yesmaximal, characteristic, complemented
2C22×#10620215C212yesmaximal, complemented
3C2×#13049813C2124yesmaximal, direct product factor
4C22×#10620206C214yesmaximal, complemented
5C22×C3⋊D32C2211yescharacteristic
6C22×C3⋊D32C2214yescomplemented
7C22×C3×#1113—24no—
8C22×#66472C2212yescomplemented
9C2×#10620215—212no—
10C2×#10620215C22124yescomplemented
11C2×#10620206C22148yescomplemented
12C2×#10620198C22112yescomplemented
13#13049813C22164yesdirect product factor
14C22×D33—64no—
15C22×C3⋊(C3⋊(C3⋊D3))C2311yescharacteristic
16C22×C32×C3⋊D3—24no—
17C22×C32×C3⋊D3D412yescomplemented
18C2×C3⋊D32—212no—
19C2×C3⋊D32D416yescomplemented
20C2×C3⋊D32C2316yes—
21C2×C3⋊D32C23124yescomplemented
22C2×C3×#1113—248no—
23C2×#66472C23112yescomplemented
24#10620215—248no—
25#10620215C23132yescomplemented
26#10620206C23164yescomplemented
27#10620198C23116yescomplemented
28C23×GO+(4,𝔽2)—92nomaximal
29C22×#1113—24no—
30C22×D3×C3⋊D3—68no—
31C22×D3×C3⋊D3—64no—
32C22×D3×C3⋊D3—124no—
33C22×C3×D32—68no—
34C22×C3×D32—64no—
35C2×D33—648no—
36C2×D33—648no—
37C22×C34C2×D411yescharacteristic, complemented
38C2×C3⋊(C3⋊(C3⋊D3))C2×D413yescomplemented
39C2×C3⋊(C3⋊(C3⋊D3))C2413yes—
40C2×C32×C3⋊D3—224no—
41C2×C32×C3⋊D3C2×D4112yescomplemented
42C3⋊D32—224no—
43C3⋊D32C2×D4112yescomplemented
44C3⋊D32C2414yes—
45C3⋊D32C24116yescomplemented
46C3×#1113—264no—
47#66472C2418yescomplemented
48C23×D32—94no—
49C23×D32—184no—
50C23×#157—92no—
51C22×GO+(4,𝔽2)—98no—
52C22×GO+(4,𝔽2)—948no—
53C22×GO+(4,𝔽2)—184no—
54C22×C3⋊(C3⋊D3)—64no—
55C22×C3⋊(C3⋊D3)—124no—
56C22×C3⋊(C3⋊D3)—242no—
57C22×C3×C3⋊D3—28no—
58C22×C3×C3⋊D3—24no—
59C22×C3×C3⋊D3—44no—
60C22×C32×D3—64no—
61C22×C32×D3—68no—
62C22×C32×D3—124no—
63C2×#1113—248no—
64C2×D3×C3⋊D3—648no—
65C2×D3×C3⋊D3—624no—
66C2×D3×C3⋊D3—624no—
67C2×D3×C3⋊D3—648no—
68C2×D3×C3⋊D3—1224no—
69C2×D3×C3⋊D3—1224no—
70C2×C3×D32—696no—
71C2×C3×D32—648no—
72D33—664no—
73D33—6192no—
74C2×C34C22×D413yescomplemented
75C3⋊(C3⋊(C3⋊D3))C22×D413yescomplemented
76C3⋊(C3⋊(C3⋊D3))C2511yescharacteristic
77C32×C3⋊D3—216no—
78C32×C3⋊D3C22×D418yescomplemented
79C23×C3⋊D3—92no—
80C23×C3⋊D3—184no—
81C23×C3×D3—184no—
82C23×C3×D3—188no—
83C22×D32—68no—
84C22×D32—64no—
85C22×D32—948no—
86C22×D32—98no—
87C22×D32—1824no—
88C22×D32—1848no—
89C22×D32—1848no—
90C22×D32—184no—
91C22×D32—184no—
92C22×D32—188no—
93C22×D32—362no—
94C22×D32—368no—
95C22×#157—94no—
96C22×#157—924no—
97C22×#157—184no—
98C22×#157—362no—
99C2×GO+(4,𝔽2)—996no—
100C2×GO+(4,𝔽2)—9128no—
101C2×GO+(4,𝔽2)—1848no—
102C22×C33—24no—
103C22×C33—44no—
104C22×C33—82no—
105C2×C3⋊(C3⋊D3)—612no—
106C2×C3⋊(C3⋊D3)—612no—
107C2×C3⋊(C3⋊D3)—1212no—
108C2×C3⋊(C3⋊D3)—1212no—
109C2×C3⋊(C3⋊D3)—246no—
110C2×C3⋊(C3⋊D3)—246no—
111C2×C3×C3⋊D3—248no—
112C2×C3×C3⋊D3—224no—
113C2×C3×C3⋊D3—424no—
114C2×C32×D3—624no—
115C2×C32×D3—648no—
116C2×C32×D3—1224no—
117#1113—264no—
118D3×C3⋊D3—632no—
119D3×C3⋊D3—648no—
120D3×C3⋊D3—616no—
121D3×C3⋊D3—696no—
122D3×C3⋊D3—1216no—
123D3×C3⋊D3—1248no—
124C3×D32—6128no—
125C3×D32—664no—
126C24×D3—544no—
127C34C23×D411yescharacteristic, complemented, Sylow
128C23×C32—92no—
129C23×C32—184no—
130C22×C3⋊D3—24no—
131C22×C3⋊D3—92no—
132C22×C3⋊D3—912no—
133C22×C3⋊D3—912no—
134C22×C3⋊D3—1824no—
135C22×C3⋊D3—1824no—
136C22×C3⋊D3—184no—
137C22×C3⋊D3—184no—
138C22×C3⋊D3—362no—
139C22×C3⋊D3—368no—
140C22×C3⋊D3—362no—
141C22×C3⋊D3—724no—
142C22×C3⋊D3—724no—
143C22×C3⋊D3GO+(4,𝔽2)12yescomplemented
144C22×C3×D3—68no—
145C22×C3×D3—68no—
146C22×C3×D3—68no—
147C22×C3×D3—124no—
148C22×C3×D3—128no—
149C22×C3×D3—128no—
150C22×C3×D3—128no—
151C22×C3×D3—1848no—
152C22×C3×D3—1896no—
153C2×D32—696no—
154C2×D32—648no—
155C2×D32—964no—
156C2×D32—948no—
157C2×D32—1896no—
158C2×D32—1864no—
159C2×D32—18192no—
160C2×D32—1824no—
161C2×D32—1824no—
162C2×D32—1824no—
163C2×D32—1824no—
164C2×D32—1824no—
165C2×D32—1848no—
166C2×D32—3612no—
167C2×D32—3612no—
168C2×D32—3648no—
169C2×D32—3648no—
170C2×D32—3624no—
171C2×#157—932no—
172C2×#157—924no—
173C2×#157—1824no—
174C2×#157—3612no—
175GO+(4,𝔽2)—9128no—
176GO+(4,𝔽2)—1864no—
177C23×D4—811noSylow
178C2×C33—212no—
179C2×C33—412no—
180C2×C33—86no—
181C3⋊(C3⋊D3)—612no—
182C3⋊(C3⋊D3)—64no—
183C3⋊(C3⋊D3)—1212no—
184C3⋊(C3⋊D3)—124no—
185C3⋊(C3⋊D3)—246no—
186C3⋊(C3⋊D3)—242no—
187C3×C3⋊D3—232no—
188C3×C3⋊D3—216no—
189C3×C3⋊D3—416no—
190C32×D3—616no—
191C32×D3—632no—
192C32×D3—1216no—
193C24×C3—544no—
194C23×D3—188no—
195C23×D3—184no—
196C23×D3—548no—
197C23×D3—544no—
198C23×D3—5448no—
199C23×D3—5448no—
200C23×D3—1084no—
201C22×C32—24no—
202C22×C32—24no—
203C22×C32—42no—
204C22×C32—48no—
205C22×C32—42no—
206C22×C32—84no—
207C22×C32—84no—
208C22×C32—912no—
209C22×C32—1824no—
210C22×C32C2×GO+(4,𝔽2)12yescomplemented
211C2×C3⋊D3—224no—
212C2×C3⋊D3—96no—
213C2×C3⋊D3—96no—
214C2×C3⋊D3—98no—
215C2×C3⋊D3—924no—
216C2×C3⋊D3—1816no—
217C2×C3⋊D3—1848no—
218C2×C3⋊D3—1812no—
219C2×C3⋊D3—1812no—
220C2×C3⋊D3—1812no—
221C2×C3⋊D3—1812no—
222C2×C3⋊D3—366no—
223C2×C3⋊D3—366no—
224C2×C3⋊D3—3624no—
225C2×C3⋊D3—3624no—
226C2×C3⋊D3—366no—
227C2×C3⋊D3—366no—
228C2×C3⋊D3—7212no—
229C2×C3⋊D3—7212no—
230C2×C3⋊D3—7212no—
231C2×C3⋊D3—7212no—
232C2×C3⋊D3C2×GO+(4,𝔽2)112yescomplemented
233C2×C3×D3—648no—
234C2×C3×D3—648no—
235C2×C3×D3—648no—
236C2×C3×D3—1224no—
237C2×C3×D3—1248no—
238C2×C3×D3—1248no—
239C2×C3×D3—1248no—
240C2×C3×D3—1864no—
241C2×C3×D3—18128no—
242D32—6128no—
243D32—664no—
244D32—932no—
245D32—1848no—
246D32—1816no—
247D32—1848no—
248D32—1816no—
249D32—1848no—
250D32—1832no—
251D32—3624no—
252D32—368no—
253D32—3632no—
254D32—3696no—
255D32—3648no—
256#157—916no—
257#157—1816no—
258#157—368no—
259C25—812no—
260C22×D4—8124no—
261C22×D4—814no—
262C23×C4—811no—
263C33—24no—
264C33—44no—
265C33—82no—
266C23×C3—184no—
267C23×C3—188no—
268C23×C3—364no—
269C23×C3—5448no—
270C22×D3—64no—
271C22×D3—68no—
272C22×D3—124no—
273C22×D3—1896no—
274C22×D3—1848no—
275C22×D3—5448no—
276C22×D3—5424no—
277C22×D3—5424no—
278C22×D3—5464no—
279C22×D3—544no—
280C22×D3—5448no—
281C22×D3—54192no—
282C22×D3—10824no—
283C22×D3—1084no—
284C22×D3—10824no—
285C22×D3—2162no—
286C2×C32—212no—
287C2×C32—212no—
288C2×C32—46no—
289C2×C32—424no—
290C2×C32—46no—
291C2×C32—812no—
292C2×C32—812no—
293C2×C32—98no—
294C2×C32—1816no—
295C2×C32C22×GO+(4,𝔽2)16yescomplemented
296C3⋊D3—216no—
297C3⋊D3—92no—
298C3⋊D3—96no—
299C3⋊D3—1812no—
300C3⋊D3—184no—
301C3⋊D3—184no—
302C3⋊D3—1812no—
303C3⋊D3—366no—
304C3⋊D3—362no—
305C3⋊D3—3624no—
306C3⋊D3—368no—
307C3⋊D3—362no—
308C3⋊D3—366no—
309C3⋊D3—7212no—
310C3⋊D3—724no—
311C3⋊D3—7212no—
312C3⋊D3—724no—
313C3⋊D3C22×GO+(4,𝔽2)18yescomplemented
314C3×D3—632no—
315C3×D3—632no—
316C3×D3—632no—
317C3×D3—1216no—
318C3×D3—1232no—
319C3×D3—1232no—
320C3×D3—1232no—
321C24—544no—
322C24—811no—
323C24—814no—
324C24—8124no—
325C24—16212no—
326C2×D4—8148no—
327C2×D4—8164no—
328C22×C4—812no—
329C22×C4—8112no—
330C22×C3—24no—
331C22×C3—44no—
332C22×C3—82no—
333C22×C3—1824no—
334C22×C3—1848no—
335C22×C3—3624no—
336C22×C3—5464no—
337C2×D3—624no—
338C2×D3—648no—
339C2×D3—1224no—
340C2×D3—18128no—
341C2×D3—1864no—
342C2×D3—5432no—
343C2×D3—5416no—
344C2×D3—5448no—
345C2×D3—5412no—
346C2×D3—5412no—
347C2×D3—5496no—
348C2×D3—10816no—
349C2×D3—10812no—
350C2×D3—10812no—
351C2×D3—10848no—
352C2×D3—2166no—
353C2×D3—2166no—
354C32—24no—
355C32—24no—
356C32—42no—
357C32—48no—
358C32—42no—
359C32—84no—
360C32—84no—
361C32C23×GO+(4,𝔽2)12yescomplemented
362C23—92no—
363C23—184no—
364C23—5448no—
365C23—816no—
366C23—816no—
367C23—8124no—
368C23—8132no—
369C23—811no—
370C23—16212no—
371C23—16248no—
372D4—8164no—
373C2×C4—8116no—
374C2×C4—8112no—
375C2×C3—212no—
376C2×C3—412no—
377C2×C3—86no—
378C2×C3—1816no—
379C2×C3—1832no—
380C2×C3—3616no—
381D3—616no—
382D3—632no—
383D3—1216no—
384D3—544no—
385D3—5412no—
386D3—1084no—
387D3—10812no—
388D3—2162no—
389D3—2166no—
390C22—912no—
391C22—1824no—
392C22—5464no—
393C22—814no—
394C22—8112no—
395C22—8116no—
396C22—813no—
397C22—813no—
398C22—16224no—
399C22#1304981311yescharacteristic, direct product factor
400C4—818no—
401C3—24no—
402C3—44no—
403C3—82no—
404C2—98no—
405C2—1816no—
406C2—811no—
407C2—813no—
408C2C2×#1304981313yesdirect product factor
subgroupG/H
center, hypercenterC22#13049813
commutator, cosocleC3⋊(C3⋊(C3⋊D3))C25
FrattiniC1G
Fitting, socleC22×C34C2×D4
hypocenterC34C23×D4
LCSG ▷ C3⋊(C3⋊(C3⋊D3)) ▷ C34
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×#10620198C2semidirect
2. C22×#10620215C2semidirect
3. C2×#13049813C2direct (N × H)
4. C22×#10620206C2semidirect
6. C22×C3⋊D32C22semidirect
8. C22×#66472C22semidirect
10. C2×#10620215C22semidirect
11. C2×#10620206C22semidirect
12. C2×#10620198C22semidirect
13. #13049813C22direct (N × H)
17. C22×C32×C3⋊D3D4semidirect
19. C2×C3⋊D32D4semidirect
21. C2×C3⋊D32C23semidirect
23. C2×#66472C23semidirect
25. #10620215C23semidirect
26. #10620206C23semidirect
27. #10620198C23semidirect
37. C22×C34C2×D4semidirect
38. C2×C3⋊(C3⋊(C3⋊D3))C2×D4semidirect
41. C2×C32×C3⋊D3C2×D4semidirect
43. C3⋊D32C2×D4semidirect
45. C3⋊D32C24semidirect
47. #66472C24semidirect
74. C2×C34C22×D4semidirect
75. C3⋊(C3⋊(C3⋊D3))C22×D4semidirect
78. C32×C3⋊D3C22×D4semidirect
127. C34C23×D4semidirect
143. C22×C3⋊D3GO+(4,𝔽2)semidirect
210. C22×C32C2×GO+(4,𝔽2)semidirect
232. C2×C3⋊D3C2×GO+(4,𝔽2)semidirect
295. C2×C32C22×GO+(4,𝔽2)semidirect
313. C3⋊D3C22×GO+(4,𝔽2)semidirect
361. C32C23×GO+(4,𝔽2)semidirect
399. C22#13049813direct (N × H)
408. C2C2×#13049813direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#106201982
C22×#106202152
C2×#130498132
C22×#106202062
C23×GO+(4,𝔽2)9