Finite Group Databasebeta

← C32×#2200 #11769249

Subgroups — C32×#2200 #11769249

Order 1152.

780 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 168 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
1C32×#560C211yesmaximal, characteristic, complemented
2C32×#458C211yesmaximal, characteristic, complemented
3C32×B(4,𝔽2)C211yesmaximal, characteristic, complemented
4C32×#453C211yesmaximal, characteristic, complemented
5C32×#353C211yesmaximal, characteristic, complemented
6C32×#352C211yesmaximal, characteristic, complemented
7C32×AΣL(1,𝔽16)C211yesmaximal, characteristic, complemented
8C3×#2200C314yesmaximal, direct product factor
9C32×H(5,𝔽2)C2211yescharacteristic
10C32×AGL(1,ℤ/8)—21no—
11C32×#127C2211yescharacteristic
12C32×#126—21no—
13C32×#123—21no—
14C32×#123—21no—
15C32×#123C2211yescharacteristic
16C32×#121—21no—
17C32×#121—21no—
18C32×#121—21no—
19C32×#120—21no—
20C32×#120—21no—
21C32×#120C2211yescharacteristic
22C32×#104—21no—
23C32×#100C2211yescharacteristic
24C32×#99—21no—
25C32×#99—21no—
26C32×#99C2211yescharacteristic
27C32×#99C2211yescharacteristic
28C3×#560C2×C314yescomplemented
29C3×#458C2×C314yescomplemented
30C3×B(4,𝔽2)C2×C314yescomplemented
31C3×#453C2×C314yescomplemented
32C3×#353C2×C314yescomplemented
33C3×#352C2×C314yescomplemented
34C3×AΣL(1,𝔽16)C2×C314yescomplemented
35C24×C32D411yescharacteristic
36C32×#41—21no—
37C32×#41—41no—
38C2×C32×D4—21no—
39C2×C32×D4—21no—
40C2×C32×D4—21no—
41C2×C32×D4—21no—
42C2×C32×D4—21no—
43C2×C32×D4—21no—
44C2×C32×D4—21no—
45C2×C32×D4—41no—
46C2×C32×D4—41no—
47C2×C32×D4D411yescharacteristic, complemented
48C2×C32×D4C2311yescharacteristic
49C22×C32×C4—21no—
50C22×C32×C4D411yescharacteristic, complemented
51C32×SD16—41no—
52C32×D8—41no—
53C32×M16—21no—
54C32×C4⋊C4—21no—
55C32×C4⋊C4—21no—
56C32×C4⋊C4—21no—
57C32×C4⋊C4—41no—
58C32×C22⋊C4—21no—
59C32×C22⋊C4—21no—
60C32×C22⋊C4—21no—
61C32×C22⋊C4—21no—
62C32×C22⋊C4—21no—
63C32×C22⋊C4—21no—
64C32×C22⋊C4—21no—
65C32×C22⋊C4—41no—
66C32×C22⋊C4—41no—
67C32×C22⋊C4D411yescharacteristic
68C32×C22⋊C4D411yescharacteristic, complemented
69C32×C42D411yescharacteristic, complemented
70#2200C3211yescharacteristic, direct product factor, Sylow
71C3×H(5,𝔽2)C22×C314yes—
72C3×AGL(1,ℤ/8)—24no—
73C3×#127C22×C314yes—
74C3×#126—24no—
75C3×#123—24no—
76C3×#123—24no—
77C3×#123C22×C314yes—
78C3×#121—24no—
79C3×#121—24no—
80C3×#121—24no—
81C3×#120—24no—
82C3×#120—24no—
83C3×#120C22×C314yes—
84C3×#104—24no—
85C3×#100C22×C314yes—
86C3×#99—24no—
87C3×#99—24no—
88C3×#99C22×C314yes—
89C3×#99C22×C314yes—
90C23×C32—21no—
91C23×C32—21no—
92C23×C32—21no—
93C23×C32—21no—
94C23×C32—41no—
95C23×C32—81no—
96C23×C32C2×D411yescharacteristic
97C23×C32C2×D411yescharacteristic
98C32×Q8—21no—
99C32×D4—21no—
100C32×D4—41no—
101C32×D4—41no—
102C32×D4—41no—
103C32×D4—41no—
104C32×D4—41no—
105C32×D4—81no—
106C32×D4—81no—
107C32×D4—81no—
108C32×D4—81no—
109C32×D4—81no—
110C32×D4—82no—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—41no—
118C2×C32×C4—41no—
119C2×C32×C4—41no—
120C2×C32×C4—41no—
121C2×C32×C4—81no—
122C2×C32×C4C2×D411yescharacteristic
123C32×C8—41no—
124#560C2×C3211yescharacteristic, complemented
125#458C2×C3211yescharacteristic, complemented
126B(4,𝔽2)C2×C3211yescharacteristic, complemented
127#453C2×C3211yescharacteristic, complemented
128#353C2×C3211yescharacteristic, complemented
129#352C2×C3211yescharacteristic, complemented
130AΣL(1,𝔽16)C2×C3211yescharacteristic, complemented
131C24×C3C3×D414yes—
132C3×#41—24no—
133C3×#41—44no—
134C2×C3×D4—24no—
135C2×C3×D4—24no—
136C2×C3×D4—24no—
137C2×C3×D4—24no—
138C2×C3×D4—24no—
139C2×C3×D4—24no—
140C2×C3×D4—24no—
141C2×C3×D4—44no—
142C2×C3×D4—44no—
143C2×C3×D4C3×D414yescomplemented
144C2×C3×D4C23×C314yes—
145C22×C3×C4—24no—
146C22×C3×C4C3×D414yescomplemented
147C3×SD16—44no—
148C3×D8—44no—
149C3×M16—24no—
150C3×C4⋊C4—24no—
151C3×C4⋊C4—24no—
152C3×C4⋊C4—24no—
153C3×C4⋊C4—44no—
154C3×C22⋊C4—24no—
155C3×C22⋊C4—24no—
156C3×C22⋊C4—24no—
157C3×C22⋊C4—24no—
158C3×C22⋊C4—24no—
159C3×C22⋊C4—24no—
160C3×C22⋊C4—24no—
161C3×C22⋊C4—44no—
162C3×C22⋊C4—44no—
163C3×C22⋊C4C3×D414yes—
164C3×C22⋊C4C3×D414yescomplemented
165C3×C42C3×D414yescomplemented
166C22×C32—21no—
167C22×C32—21no—
168C22×C32—41no—
169C22×C32—41no—
170C22×C32—41no—
171C22×C32—41no—
172C22×C32—41no—
173C22×C32—41no—
174C22×C32—42no—
175C22×C32—41no—
176C22×C32—41no—
177C22×C32—41no—
178C22×C32—81no—
179C22×C32—81no—
180C22×C32—81no—
181C22×C32#12011yescharacteristic
182C32×C4—21no—
183C32×C4—41no—
184C32×C4—41no—
185C32×C4—41no—
186C32×C4—41no—
187C32×C4—41no—
188C32×C4—81no—
189C32×C4—81no—
190H(5,𝔽2)C22×C3211yescharacteristic
191AGL(1,ℤ/8)—21no—
192#127C22×C3211yescharacteristic
193#126—21no—
194#123—21no—
195#123—21no—
196#123C22×C3211yescharacteristic
197#121—21no—
198#121—21no—
199#121—21no—
200#120—21no—
201#120—21no—
202#120C22×C3211yescharacteristic
203#104—21no—
204#100C22×C3211yescharacteristic
205#99—21no—
206#99—21no—
207#99C22×C3211yescharacteristic
208#99C22×C3211yescharacteristic
209C23×C3—24no—
210C23×C3—24no—
211C23×C3—24no—
212C23×C3—24no—
213C23×C3—44no—
214C23×C3—84no—
215C23×C3C2×C3×D414yes—
216C23×C3C2×C3×D414yes—
217C3×Q8—24no—
218C3×D4—24no—
219C3×D4—44no—
220C3×D4—44no—
221C3×D4—44no—
222C3×D4—44no—
223C3×D4—44no—
224C3×D4—84no—
225C3×D4—84no—
226C3×D4—84no—
227C3×D4—84no—
228C3×D4—84no—
229C3×D4—88no—
230C2×C3×C4—24no—
231C2×C3×C4—24no—
232C2×C3×C4—24no—
233C2×C3×C4—24no—
234C2×C3×C4—24no—
235C2×C3×C4—24no—
236C2×C3×C4—44no—
237C2×C3×C4—44no—
238C2×C3×C4—44no—
239C2×C3×C4—44no—
240C2×C3×C4—84no—
241C2×C3×C4C2×C3×D414yes—
242C3×C8—44no—
243C2×C32—21no—
244C2×C32—41no—
245C2×C32—41no—
246C2×C32—81no—
247C2×C32—81no—
248C2×C32—81no—
249C2×C32B(4,𝔽2)11yescharacteristic
250C24C32×D411yescharacteristic
251#41—21no—
252#41—41no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4C32×D411yescharacteristic, complemented
263C2×D4C23×C3211yescharacteristic
264C22×C4—21no—
265C22×C4C32×D411yescharacteristic, complemented
266SD16—41no—
267D8—41no—
268M16—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—41no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4C32×D411yescharacteristic
283C22⋊C4C32×D411yescharacteristic, complemented
284C42C32×D411yescharacteristic, complemented
285C22×C3—24no—
286C22×C3—24no—
287C22×C3—44no—
288C22×C3—44no—
289C22×C3—44no—
290C22×C3—44no—
291C22×C3—44no—
292C22×C3—44no—
293C22×C3—48no—
294C22×C3—44no—
295C22×C3—44no—
296C22×C3—44no—
297C22×C3—84no—
298C22×C3—84no—
299C22×C3—84no—
300C22×C3C3×#12014yes—
301C3×C4—24no—
302C3×C4—44no—
303C3×C4—44no—
304C3×C4—44no—
305C3×C4—44no—
306C3×C4—44no—
307C3×C4—84no—
308C3×C4—84no—
309C32#220011yescharacteristic, direct product factor, Sylow
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—41no—
315C23—81no—
316C23C2×C32×D411yescharacteristic
317C23C2×C32×D411yescharacteristic
318Q8—21no—
319D4—21no—
320D4—41no—
321D4—41no—
322D4—41no—
323D4—41no—
324D4—41no—
325D4—81no—
326D4—81no—
327D4—81no—
328D4—81no—
329D4—81no—
330D4—82no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—81no—
342C2×C4C2×C32×D411yescharacteristic
343C8—41no—
344C2×C3—24no—
345C2×C3—44no—
346C2×C3—44no—
347C2×C3—84no—
348C2×C3—84no—
349C2×C3—84no—
350C2×C3C3×B(4,𝔽2)14yes—
351C22—21no—
352C22—21no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—42no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22C32×#12011yescharacteristic
367C4—21no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—81no—
374C4—81no—
375C3C3×#220014yesdirect product factor
376C2—21no—
377C2—41no—
378C2—41no—
379C2—81no—
380C2—81no—
381C2—81no—
382C2C32×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2×C32B(4,𝔽2)
commutator, Frattini, cosocleC2×D4C23×C32
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C32 ◁ C22×C32 ◁ C2×C32×D4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C32×#560C2semidirect
2. C32×#458C2semidirect
3. C32×B(4,𝔽2)C2semidirect
4. C32×#453C2semidirect
5. C32×#353C2semidirect
6. C32×#352C2semidirect
7. C32×AΣL(1,𝔽16)C2semidirect
8. C3×#2200C3direct (N × H)
28. C3×#560C2×C3semidirect
29. C3×#458C2×C3semidirect
30. C3×B(4,𝔽2)C2×C3semidirect
31. C3×#453C2×C3semidirect
32. C3×#353C2×C3semidirect
33. C3×#352C2×C3semidirect
34. C3×AΣL(1,𝔽16)C2×C3semidirect
47. C2×C32×D4D4semidirect
50. C22×C32×C4D4semidirect
68. C32×C22⋊C4D4semidirect
69. C32×C42D4semidirect
70. #2200C32direct (N × H)
124. #560C2×C32semidirect
125. #458C2×C32semidirect
126. B(4,𝔽2)C2×C32semidirect
127. #453C2×C32semidirect
128. #353C2×C32semidirect
129. #352C2×C32semidirect
130. AΣL(1,𝔽16)C2×C32semidirect
143. C2×C3×D4C3×D4semidirect
146. C22×C3×C4C3×D4semidirect
164. C3×C22⋊C4C3×D4semidirect
165. C3×C42C3×D4semidirect
262. C2×D4C32×D4semidirect
265. C22×C4C32×D4semidirect
283. C22⋊C4C32×D4semidirect
284. C42C32×D4semidirect
309. C32#2200direct (N × H)
375. C3C3×#2200direct (N × H)

Maximal subgroups

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

subgroupindex
C32×#5602
C32×#4582
C32×B(4,𝔽2)2
C32×#4532
C32×#3532
C32×#3522
C32×AΣL(1,𝔽16)2
C3×#22003