Finite Group Databasebeta

← C3×#70991 #11798629

Subgroups — C3×#70991 #11798629

Order 1152.

616 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 82 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
1C3×#5683C211yesmaximal, characteristic, complemented
2C32×#455C211yesmaximal, characteristic, complemented
3C3×#5244C212yesmaximal, complemented
4C3×#4541C211yesmaximal, characteristic, complemented
5C3×#4475C212yesmaximal, complemented
6C3×#2199—31nomaximal
7#70991C311yesmaximal, characteristic, direct product factor
8C32×H(5,𝔽2)C2211yescharacteristic
9C32×#137—21no—
10C32×#124C2211yescharacteristic
11C3×#947—21no—
12C3×#937—22no—
13C3×#937C2212yes—
14C3×#885—22no—
15C3×#882—22no—
16C3×#875—21no—
17C32×#104—21no—
18C32×#100C2211yescharacteristic
19C3×#834—22no—
20C3×#834C2212yes—
21C3×H(3,𝔽4)—31no—
22#5683C2×C311yescharacteristic, complemented
23C3×B(4,𝔽2)—32no—
24C3×#455—21no—
25C3×#455D311yescharacteristic, complemented
26C3×#455C2×C311yescharacteristic, complemented
27#5244C2×C312yescomplemented
28C3×#354—31no—
29C3×AΣL(1,𝔽16)—32no—
30#4541C2×C311yescharacteristic, complemented
31#4475C2×C312yescomplemented
32C23×C3×D3D412yescomplemented
33C32×#41—21no—
34C32×#41—41no—
35C2×C32×Q8D411yescharacteristic, complemented
36C2×C32×D4—22no—
37C2×C32×D4—41no—
38C2×C32×D4C2311yescharacteristic
39C2×C3×#67—22no—
40C2×C3×#67—42no—
41C2×C3×D12—21no—
42C2×C3×D12—21no—
43C2×C3×Dic6—21no—
44C32×Q16—41no—
45C32×SD16—41no—
46C32×M16—21no—
47C32×C22⋊C4—22no—
48C32×C42D411yescharacteristic, complemented
49C3×#217—22no—
50C3×#217D412yescomplemented
51C3×(C2×D3)⋊C4—22no—
52C3×(C2×D3)⋊C4—22no—
53C3×(C2×D3)⋊C4—22no—
54C3×(C2×D3)⋊C4—42no—
55C3×C4×Dic3—21no—
56#2199—31noSylow
57C3×H(5,𝔽2)—21no—
58C3×H(5,𝔽2)C2×D311yescharacteristic
59C3×H(5,𝔽2)C22×C311yescharacteristic
60C3×#137—22no—
61C3×#137—21no—
62C3×#137—21no—
63C3×#124—21no—
64C3×#124—62no—
65C3×#124—61no—
66C3×#124—61no—
67C3×#124C2×D311yescharacteristic
68C3×#124C22×C311yescharacteristic
69C3×#120—32no—
70C3×#120—62no—
71C3×#120—62no—
72#947—21no—
73#937—22no—
74#937C22×C312yes—
75#885—22no—
76#882—22no—
77#875—21no—
78C3×#104—22no—
79C3×#104—21no—
80C3×#104—21no—
81C3×#100—21no—
82C3×#100C2×D311yescharacteristic
83C3×#100C22×C311yescharacteristic
84C3×#99—32no—
85C3×#99—62no—
86#834—22no—
87#834C22×C312yes—
88C23×C32—22no—
89C23×C32C2×D412yes—
90C22×C3×D3—22no—
91C22×C3×D3—42no—
92C22×C3×D3—82no—
93C32×Q8—21no—
94C32×Q8—41no—
95C32×D4—21no—
96C32×D4—41no—
97C32×D4—42no—
98C32×D4—41no—
99C2×C32×C4—22no—
100C2×C32×C4—21no—
101C2×C32×C4—21no—
102C2×C32×C4—41no—
103C2×C32×C4C2×D411yescharacteristic
104C3×#67—82no—
105C3×#67—84no—
106C2×C3×Dic3—22no—
107C2×C3×Dic3—42no—
108C3×D12—81no—
109C3×D12—81no—
110C3×Dic6—81no—
111C32×C8—41no—
112H(3,𝔽4)—31no—
113B(4,𝔽2)—32no—
114#455C3×D311yescharacteristic, complemented
115#354—31no—
116AΣL(1,𝔽16)—32no—
117C24×C3—32no—
118C23×D3C3×D412yescomplemented
119C3×#41—22no—
120C3×#41—21no—
121C3×#41—21no—
122C3×#41—41no—
123C3×#41—41no—
124C3×#41—42no—
125C2×C3×Q8—21no—
126C2×C3×Q8—61no—
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—42no—
133C2×C3×D4—41no—
134C2×C3×D4—41no—
135C2×C3×D4—42no—
136C2×C3×D4—61no—
137C2×C3×D4—61no—
138C2×C3×D4—62no—
139C2×C3×D4—122no—
140C2×C3×D4C22×D311yescharacteristic
141C2×C3×D4C23×C311yescharacteristic
142C2×#67—22no—
143C2×#67—42no—
144C2×D12—21no—
145C2×D12—21no—
146C2×Dic6—21no—
147C3×Q16—41no—
148C3×Q16—41no—
149C3×Q16—42no—
150C3×SD16—41no—
151C3×SD16—41no—
152C3×SD16—42no—
153C3×M16—22no—
154C3×M16—21no—
155C3×M16—21no—
156C3×C22⋊C4—22no—
157C3×C22⋊C4—22no—
158C3×C22⋊C4—32no—
159C3×C22⋊C4—42no—
160C3×C22⋊C4—62no—
161C3×C22⋊C4—62no—
162C3×C22⋊C4—62no—
163C3×C22⋊C4—62no—
164C3×C22⋊C4—122no—
165C3×C42—21no—
166C3×C42—61no—
167C3×C42#6711yescharacteristic, complemented
168C3×C42C3×D411yescharacteristic, complemented
169#217—22no—
170#217C3×D412yescomplemented
171(C2×D3)⋊C4—22no—
172(C2×D3)⋊C4—22no—
173(C2×D3)⋊C4—22no—
174(C2×D3)⋊C4—42no—
175C4×Dic3—21no—
176C22×C32—22no—
177C22×C32—41no—
178C22×C32—42no—
179C22×C32—82no—
180C22×C32#12011yescharacteristic
181C2×C3×D3—42no—
182C2×C3×D3—42no—
183C2×C3×D3—44no—
184C2×C3×D3—42no—
185C2×C3×D3—82no—
186C32×C4—21no—
187C32×C4—41no—
188C32×C4—41no—
189C32×C4—41no—
190C3×Dic3—42no—
191C3×Dic3—82no—
192H(5,𝔽2)C2×C3×D311yescharacteristic
193#137—21no—
194#124—62no—
195#124—61no—
196#124—61no—
197#124C2×C3×D311yescharacteristic
198#120—32no—
199#120—62no—
200#120—62no—
201#104—21no—
202#100C2×C3×D311yescharacteristic
203#99—32no—
204#99—62no—
205C23×C3—22no—
206C23×C3—22no—
207C23×C3—22no—
208C23×C3—42no—
209C23×C3—62no—
210C23×C3—122no—
211C23×C3—242no—
212C23×C3D3×D412yes—
213C23×C3C2×C3×D412yes—
214C22×D3—22no—
215C22×D3—42no—
216C22×D3—82no—
217C3×Q8—22no—
218C3×Q8—21no—
219C3×Q8—21no—
220C3×Q8—41no—
221C3×Q8—41no—
222C3×Q8—42no—
223C3×Q8—241no—
224C3×D4—22no—
225C3×D4—21no—
226C3×D4—21no—
227C3×D4—41no—
228C3×D4—41no—
229C3×D4—42no—
230C3×D4—42no—
231C3×D4—42no—
232C3×D4—42no—
233C3×D4—44no—
234C3×D4—41no—
235C3×D4—41no—
236C3×D4—241no—
237C3×D4—241no—
238C3×D4—242no—
239C3×D4—244no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—42no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—42no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—62no—
254C2×C3×C4—122no—
255C2×C3×C4C2×#6711yescharacteristic
256C2×C3×C4C2×C3×D411yescharacteristic
257#67—82no—
258#67—84no—
259C2×Dic3—22no—
260C2×Dic3—42no—
261D12—81no—
262D12—81no—
263Dic6—81no—
264C3×C8—42no—
265C3×C8—41no—
266C3×C8—41no—
267C2×C32—21no—
268C2×C32—42no—
269C2×C32—81no—
270C2×C32B(4,𝔽2)11yescharacteristic
271C3×D3—82no—
272C24—32no—
273#41—21no—
274#41—41no—
275C2×Q8—61no—
276C2×Q8C3×#6711yescharacteristic, complemented
277C2×D4—22no—
278C2×D4—41no—
279C2×D4—61no—
280C2×D4—61no—
281C2×D4—62no—
282C2×D4—122no—
283C2×D4C22×C3×D311yescharacteristic
284Q16—41no—
285SD16—41no—
286M16—21no—
287C22⋊C4—22no—
288C22⋊C4—32no—
289C22⋊C4—62no—
290C22⋊C4—62no—
291C22⋊C4—62no—
292C22⋊C4—62no—
293C22⋊C4—122no—
294C42—61no—
295C42C3×#6711yescharacteristic, complemented
296C22×C3—22no—
297C22×C3—22no—
298C22×C3—21no—
299C22×C3—42no—
300C22×C3—42no—
301C22×C3—42no—
302C22×C3—42no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—82no—
306C22×C3—82no—
307C22×C3—82no—
308C22×C3—122no—
309C22×C3—122no—
310C22×C3—124no—
311C22×C3—122no—
312C22×C3—162no—
313C22×C3—242no—
314C22×C3#93711yescharacteristic
315C22×C3C3×#12011yescharacteristic
316C2×D3—42no—
317C2×D3—42no—
318C2×D3—44no—
319C2×D3—42no—
320C2×D3—82no—
321C3×C4—22no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—41no—
325C3×C4—41no—
326C3×C4—42no—
327C3×C4—41no—
328C3×C4—41no—
329C3×C4—41no—
330C3×C4—41no—
331C3×C4—81no—
332C3×C4—81no—
333C3×C4—122no—
334C3×C4—242no—
335Dic3—42no—
336Dic3—82no—
337C32#219911yescharacteristic, complemented, Sylow
338C23—22no—
339C23—62no—
340C23—122no—
341C23—242no—
342C23C3×D3×D412yes—
343Q8—21no—
344Q8—41no—
345Q8—241no—
346D4—21no—
347D4—41no—
348D4—42no—
349D4—41no—
350D4—241no—
351D4—241no—
352D4—242no—
353D4—244no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—41no—
358C2×C4—62no—
359C2×C4—122no—
360C2×C4C2×C3×#6711yescharacteristic
361C8—41no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3—21no—
365C2×C3—41no—
366C2×C3—42no—
367C2×C3—42no—
368C2×C3—82no—
369C2×C3—82no—
370C2×C3—81no—
371C2×C3—81no—
372C2×C3—242no—
373C2×C3#503211yescharacteristic
374C2×C3C3×B(4,𝔽2)11yescharacteristic
375D3—82no—
376C22—22no—
377C22—42no—
378C22—41no—
379C22—82no—
380C22—122no—
381C22—122no—
382C22—124no—
383C22—122no—
384C22—242no—
385C22C3×#93711yescharacteristic
386C4—21no—
387C4—41no—
388C4—41no—
389C4—41no—
390C4—122no—
391C4—242no—
392C3—21no—
393C3#7099111yescharacteristic, direct product factor
394C3C3×#219911yescharacteristic, complemented
395C2—21no—
396C2—42no—
397C2—81no—
398C2—242no—
399C2C3×#503211yescharacteristic
subgroupG/H
centerC2×C3#5032
commutator, cosocleC2×C3×D4C23×C3
FrattiniC2×D4C22×C3×D3
FittingC32×#455C2
socleC2×C32B(4,𝔽2)
hypercenterC3×#455D3
hypocenterC3C3×#2199
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4 ◁ C3×#455

Complements

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

NH ≅ G/Ntype
1. C3×#5683C2semidirect
2. C32×#455C2semidirect
3. C3×#5244C2semidirect
4. C3×#4541C2semidirect
5. C3×#4475C2semidirect
7. #70991C3direct (N × H)
22. #5683C2×C3semidirect
25. C3×#455D3semidirect
26. C3×#455C2×C3semidirect
27. #5244C2×C3semidirect
30. #4541C2×C3semidirect
31. #4475C2×C3semidirect
32. C23×C3×D3D4semidirect
35. C2×C32×Q8D4semidirect
48. C32×C42D4semidirect
50. C3×#217D4semidirect
114. #455C3×D3semidirect
118. C23×D3C3×D4semidirect
127. C2×C3×Q8#67semidirect
128. C2×C3×Q8C3×D4semidirect
167. C3×C42#67semidirect
168. C3×C42C3×D4semidirect
170. #217C3×D4semidirect
276. C2×Q8C3×#67semidirect
295. C42C3×#67semidirect
337. C32#2199semidirect
393. C3#70991direct (N × H)
394. C3C3×#2199semidirect

Maximal subgroups

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

subgroupindex
C3×#56832
C32×#4552
C3×#52442
C3×#45412
C3×#44752
C3×#21993
#709913