Finite Group Databasebeta

← C3×#71950 #11798632

Subgroups — C3×#71950 #11798632

Order 1152.

670 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 90 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×H(3,𝔽4)C211yesmaximal, characteristic, complemented
2C3×#5061C211yesmaximal, characteristic, complemented
3C3×#5032C212yesmaximal, complemented
4C3×#4541C211yesmaximal, characteristic, complemented
5C3×#4536C212yesmaximal, complemented
6C3×#2199—31nomaximal
7#71950C311yesmaximal, characteristic, direct product factor
8C3×#1004C2211yescharacteristic
9C32×#124—21no—
10C32×#124—22no—
11C32×#124—21no—
12C32×#124C2211yescharacteristic
13C32×#120—22no—
14C32×#120C2212yes—
15C3×#953—22no—
16C3×#942—21no—
17C3×#834C2212yes—
18C3×#833C2211yescharacteristic
19C3×#806—22no—
20C3×#805—21no—
21C3×H(3,𝔽4)—21no—
22C3×H(3,𝔽4)D311yescharacteristic, complemented
23C3×H(3,𝔽4)C2×C311yescharacteristic, complemented
24C3×B(4,𝔽2)—32no—
25C3×#455—31no—
26#5061C2×C311yescharacteristic, complemented
27#5032C2×C312yescomplemented
28C3×#354—31no—
29C3×AΣL(1,𝔽16)—32no—
30#4541C2×C311yescharacteristic, complemented
31#4536C2×C312yescomplemented
32C24×C32D412yescomplemented
33C2×C32×Q8—21no—
34C2×C32×Q8D411yescharacteristic, complemented
35C2×C32×D4—21no—
36C2×C32×D4—21no—
37C2×C32×D4—22no—
38C2×C32×D4C2311yescharacteristic
39C2×C3×#67—22no—
40C2×C3×#67—42no—
41C3×#237—41no—
42C3×D3×D4—41no—
43C3×#235—21no—
44C32×C22⋊C4—22no—
45C32×C22⋊C4—22no—
46C32×C22⋊C4—22no—
47C32×C22⋊C4—22no—
48C32×C22⋊C4D412yescomplemented
49C32×C42—21no—
50C32×C42D411yescharacteristic, complemented
51C3×#217—22no—
52C3×#217—42no—
53C3×#216—41no—
54C3×#215—41no—
55C3×#208—21no—
56#2199—31noSylow
57C3×H(5,𝔽2)—31no—
58#1004C22×C311yescharacteristic
59C3×#137—61no—
60C3×#124—22no—
61C3×#124—24no—
62C3×#124—22no—
63C3×#124—21no—
64C3×#124—21no—
65C3×#124—22no—
66C3×#124—22no—
67C3×#124—21no—
68C3×#124—21no—
69C3×#124—21no—
70C3×#124C2×D311yescharacteristic
71C3×#124C22×C311yescharacteristic
72C3×#120—24no—
73C3×#120—22no—
74C3×#120—22no—
75C3×#120—22no—
76C3×#120—62no—
77C3×#120C2×D312yes—
78C3×#120C22×C312yes—
79#953—22no—
80#942—21no—
81C3×#104—61no—
82C3×#100—31no—
83C3×#99—32no—
84C3×#99—62no—
85#834C22×C312yes—
86#833C22×C311yescharacteristic
87#806—22no—
88#805—21no—
89C23×C32—22no—
90C23×C32—42no—
91C23×C32—82no—
92C23×C32C2×D412yes—
93C22×C3×D3—22no—
94C32×Q8—41no—
95C32×Q8—81no—
96C32×D4—41no—
97C32×D4—82no—
98C32×D4—81no—
99C32×D4—81no—
100C2×C32×C4—21no—
101C2×C32×C4—22no—
102C2×C32×C4—21no—
103C2×C32×C4C2×D411yescharacteristic
104C3×#67—42no—
105C3×#67—41no—
106C3×#67—84no—
107C2×C3×Dic3—22no—
108C2×C3×Dic3—42no—
109C3×D12—21no—
110C3×C4×D3—41no—
111C3×Dic6—21no—
112C3×C3⋊C8—41no—
113H(3,𝔽4)C3×D311yescharacteristic, complemented
114B(4,𝔽2)—32no—
115#455—31no—
116#354—31no—
117AΣL(1,𝔽16)—32no—
118C24×C3—22no—
119C24×C3#6712yescomplemented
120C24×C3C3×D412yescomplemented
121C3×#41—61no—
122C3×#41—121no—
123C2×C3×Q8—22no—
124C2×C3×Q8—21no—
125C2×C3×Q8—21no—
126C2×C3×Q8—21no—
127C2×C3×Q8#6711yescharacteristic, complemented
128C2×C3×Q8C3×D411yescharacteristic, complemented
129C2×C3×D4—22no—
130C2×C3×D4—22no—
131C2×C3×D4—21no—
132C2×C3×D4—21no—
133C2×C3×D4—21no—
134C2×C3×D4—21no—
135C2×C3×D4—24no—
136C2×C3×D4—22no—
137C2×C3×D4—22no—
138C2×C3×D4—21no—
139C2×C3×D4—62no—
140C2×C3×D4—121no—
141C2×C3×D4—122no—
142C2×C3×D4C22×D311yescharacteristic
143C2×C3×D4C23×C311yescharacteristic
144C2×#67—22no—
145C2×#67—42no—
146#237—41no—
147D3×D4—41no—
148#235—21no—
149C3×Q16—121no—
150C3×SD16—121no—
151C3×M16—61no—
152C3×C22⋊C4—24no—
153C3×C22⋊C4—24no—
154C3×C22⋊C4—22no—
155C3×C22⋊C4—22no—
156C3×C22⋊C4—22no—
157C3×C22⋊C4—22no—
158C3×C22⋊C4—24no—
159C3×C22⋊C4—24no—
160C3×C22⋊C4—22no—
161C3×C22⋊C4—22no—
162C3×C22⋊C4—22no—
163C3×C22⋊C4—22no—
164C3×C22⋊C4—22no—
165C3×C22⋊C4—62no—
166C3×C22⋊C4—122no—
167C3×C22⋊C4#6712yescomplemented
168C3×C22⋊C4C3×D412yescomplemented
169C3×C42—22no—
170C3×C42—21no—
171C3×C42—21no—
172C3×C42—21no—
173C3×C42#6711yescharacteristic, complemented
174C3×C42C3×D411yescharacteristic, complemented
175#217—22no—
176#217—42no—
177#216—41no—
178#215—41no—
179#208—21no—
180C22×C32—22no—
181C22×C32—42no—
182C22×C32—42no—
183C22×C32—42no—
184C22×C32—42no—
185C22×C32—44no—
186C22×C32—82no—
187C22×C32#12011yescharacteristic
188C2×C3×D3—41no—
189C2×C3×D3—82no—
190C32×C4—21no—
191C32×C4—42no—
192C32×C4—41no—
193C32×C4—41no—
194C3×Dic3—41no—
195C3×Dic3—82no—
196H(5,𝔽2)—31no—
197#137—61no—
198#124—21no—
199#124—22no—
200#124—21no—
201#124C2×C3×D311yescharacteristic
202#120—22no—
203#120—62no—
204#120C2×C3×D312yes—
205#104—61no—
206#100—31no—
207#99—32no—
208#99—62no—
209C23×C3—24no—
210C23×C3—22no—
211C23×C3—22no—
212C23×C3—22no—
213C23×C3—42no—
214C23×C3—42no—
215C23×C3—62no—
216C23×C3—82no—
217C23×C3—82no—
218C23×C3—82no—
219C23×C3—84no—
220C23×C3C2×#6712yes—
221C23×C3C2×C3×D412yes—
222C22×D3—22no—
223C3×Q8—41no—
224C3×Q8—41no—
225C3×Q8—61no—
226C3×Q8—81no—
227C3×Q8—82no—
228C3×Q8—81no—
229C3×Q8—81no—
230C3×D4—41no—
231C3×D4—41no—
232C3×D4—61no—
233C3×D4—81no—
234C3×D4—82no—
235C3×D4—82no—
236C3×D4—84no—
237C3×D4—82no—
238C3×D4—81no—
239C3×D4—81no—
240C3×D4—82no—
241C3×D4—81no—
242C3×D4—81no—
243C3×D4—122no—
244C3×D4—121no—
245C3×D4—244no—
246C2×C3×C4—22no—
247C2×C3×C4—24no—
248C2×C3×C4—22no—
249C2×C3×C4—21no—
250C2×C3×C4—22no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—22no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—62no—
257C2×C3×C4—121no—
258C2×C3×C4—122no—
259C2×C3×C4C2×#6711yescharacteristic
260C2×C3×C4C2×C3×D411yescharacteristic
261#67—42no—
262#67—41no—
263#67—84no—
264C2×Dic3—22no—
265C2×Dic3—42no—
266D12—21no—
267C4×D3—41no—
268Dic6—21no—
269C3×C8—121no—
270C3⋊C8—41no—
271C2×C32—21no—
272C2×C32—42no—
273C2×C32—82no—
274C2×C32B(4,𝔽2)11yescharacteristic
275C3×D3—81no—
276C24C3×#6712yescomplemented
277#41—61no—
278#41—121no—
279C2×Q8—21no—
280C2×Q8C3×#6711yescharacteristic, complemented
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—22no—
284C2×D4—62no—
285C2×D4—121no—
286C2×D4—122no—
287C2×D4C22×C3×D311yescharacteristic
288Q16—121no—
289SD16—121no—
290M16—61no—
291C22⋊C4—22no—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C22⋊C4—62no—
296C22⋊C4—122no—
297C22⋊C4C3×#6712yescomplemented
298C42—21no—
299C42C3×#6711yescharacteristic, complemented
300C22×C3—22no—
301C22×C3—22no—
302C22×C3—21no—
303C22×C3—42no—
304C22×C3—42no—
305C22×C3—42no—
306C22×C3—44no—
307C22×C3—44no—
308C22×C3—44no—
309C22×C3—42no—
310C22×C3—42no—
311C22×C3—42no—
312C22×C3—42no—
313C22×C3—44no—
314C22×C3—44no—
315C22×C3—44no—
316C22×C3—42no—
317C22×C3—42no—
318C22×C3—84no—
319C22×C3—82no—
320C22×C3—82no—
321C22×C3—84no—
322C22×C3—121no—
323C22×C3—242no—
324C22×C3#95311yescharacteristic
325C22×C3C3×#12011yescharacteristic
326C2×D3—41no—
327C2×D3—82no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—41no—
331C3×C4—44no—
332C3×C4—42no—
333C3×C4—42no—
334C3×C4—41no—
335C3×C4—42no—
336C3×C4—41no—
337C3×C4—42no—
338C3×C4—41no—
339C3×C4—41no—
340C3×C4—121no—
341C3×C4—242no—
342Dic3—41no—
343Dic3—82no—
344C32#219911yescharacteristic, complemented, Sylow
345C23—22no—
346C23—42no—
347C23—62no—
348C23—82no—
349C23C2×C3×#6712yes—
350Q8—41no—
351Q8—61no—
352Q8—81no—
353D4—41no—
354D4—61no—
355D4—82no—
356D4—81no—
357D4—81no—
358D4—122no—
359D4—121no—
360D4—244no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—62no—
365C2×C4—121no—
366C2×C4—122no—
367C2×C4C2×C3×#6711yescharacteristic
368C8—121no—
369C2×C3—22no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—42no—
374C2×C3—42no—
375C2×C3—82no—
376C2×C3—84no—
377C2×C3—82no—
378C2×C3—82no—
379C2×C3—241no—
380C2×C3#524411yescharacteristic
381C2×C3C3×B(4,𝔽2)11yescharacteristic
382D3—81no—
383C22—22no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—44no—
388C22—42no—
389C22—82no—
390C22—121no—
391C22—242no—
392C22C3×#95311yescharacteristic
393C4—21no—
394C4—42no—
395C4—41no—
396C4—41no—
397C4—121no—
398C4—242no—
399C3—21no—
400C3#7195011yescharacteristic, direct product factor
401C3C3×#219911yescharacteristic, complemented
402C2—21no—
403C2—42no—
404C2—82no—
405C2—241no—
406C2C3×#524411yescharacteristic
subgroupG/H
centerC2×C3#5244
commutator, cosocleC2×C3×D4C23×C3
FrattiniC2×D4C22×C3×D3
FittingC32×H(3,𝔽4)C2
socleC2×C32B(4,𝔽2)
hypercenterC3×H(3,𝔽4)D3
hypocenterC3C3×#2199
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4 ◁ C3×H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. C32×H(3,𝔽4)C2semidirect
2. C3×#5061C2semidirect
3. C3×#5032C2semidirect
4. C3×#4541C2semidirect
5. C3×#4536C2semidirect
7. #71950C3direct (N × H)
22. C3×H(3,𝔽4)D3semidirect
23. C3×H(3,𝔽4)C2×C3semidirect
26. #5061C2×C3semidirect
27. #5032C2×C3semidirect
30. #4541C2×C3semidirect
31. #4536C2×C3semidirect
32. C24×C32D4semidirect
34. C2×C32×Q8D4semidirect
48. C32×C22⋊C4D4semidirect
50. C32×C42D4semidirect
113. H(3,𝔽4)C3×D3semidirect
119. C24×C3#67semidirect
120. C24×C3C3×D4semidirect
127. C2×C3×Q8#67semidirect
128. C2×C3×Q8C3×D4semidirect
167. C3×C22⋊C4#67semidirect
168. C3×C22⋊C4C3×D4semidirect
173. C3×C42#67semidirect
174. C3×C42C3×D4semidirect
276. C24C3×#67semidirect
280. C2×Q8C3×#67semidirect
297. C22⋊C4C3×#67semidirect
299. C42C3×#67semidirect
344. C32#2199semidirect
400. C3#71950direct (N × H)
401. C3C3×#2199semidirect

Maximal subgroups

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

subgroupindex
C32×H(3,𝔽4)2
C3×#50612
C3×#50322
C3×#45412
C3×#45362
C3×#21993
#719503