Finite Group Databasebeta

← C3×#85538 #11922824

Subgroups — C3×#85538 #11922824

Order 1152.

667 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 667 conjugacy classes of subgroups — too many to draw. Only its 76 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
1C2×C3×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2C2×C3×#985C211yesmaximal, characteristic, complemented
3C3×#5426C211yesmaximal, characteristic, complemented
4C3×#5421C212yesmaximal, complemented
5C3×#5419C212yesmaximal, complemented
6#85538C311yesmaximal, characteristic, direct product factor
7C3×#3013—31nomaximal
8C3×2−1+4⋊C3—22no—
9C2×C3×#41⋊C3C2211yescharacteristic
10C22×C3×SL(2,𝔽3)C2212yes—
11C3×#985—22no—
12C2×C3×GL(2,𝔽3)C2211yescharacteristic
13C2×C3×#226C2211yescharacteristic
14C3×#938—41nomaximal
15C3×#859C2212yes—
16C2×C3×2−1+4D311yescharacteristic, complemented
17C2×2−1+4⋊C3—21no—
18C2×2−1+4⋊C3C2×C311yescharacteristic, complemented
19C2×#985C2×C311yescharacteristic, complemented
20C2×C3×#137—31no—
21C2×C3×#135—31no—
22C3×#537—31no—
23#5426C2×C311yescharacteristic, complemented
24#5421C2×C312yescomplemented
25#5419C2×C312yescomplemented
26C3×#452—32no—
27C3×#451—32no—
28C3×#449—32no—
29C3×#448—32no—
30C3×#417—31no—
31C2×C3×#103—31no—
32C3×#408—31no—
33C2×C32×D4—41no—
34C2×C3×#67—42no—
35C2×C3×C4×D3—41no—
36C3×#41⋊C3D412yescomplemented
37C2×C3×SL(2,𝔽3)—22no—
38C2×C3×SL(2,𝔽3)C2311yescharacteristic
39C3×GL(2,𝔽3)—21no—
40C3×#226—21no—
41C3×#217—42no—
42C3×C4⋊Dic3—41no—
43#3013—31noSylow
44C3×2−1+4—22no—
45C3×2−1+4—62no—
46C2×C3×#41—31no—
47C2×C3×#41—32no—
48C2×C3×#41—31no—
49C2×C3×#41C2×D311yescharacteristic
50C22×C3×Q8—31no—
51C22×C3×Q8C2×D312yes—
522−1+4⋊C3—24no—
532−1+4⋊C3—22no—
54C2×#41⋊C3—21no—
55C2×#41⋊C3C22×C311yescharacteristic
56C22×SL(2,𝔽3)—22no—
57C22×SL(2,𝔽3)C22×C312yes—
58#985—22no—
59C2×GL(2,𝔽3)C22×C311yescharacteristic
60C2×#226C22×C311yescharacteristic
61C3×#137—64no—
62C3×#135—62no—
63C3×#135—62no—
64C2×C3×Q16—32no—
65C2×C3×Q16—31no—
66C2×C3×SD16—31no—
67C2×C3×SD16—32no—
68C2×C3×SD16—31no—
69C2×C3×D8—31no—
70C2×C3×M16—31no—
71C22×C3×C8—31no—
72C3×C22⋊Q8—32no—
73C3×C22⋊Q8—61no—
74C3×#121—32no—
75C3×#121—62no—
76C3×#121—61no—
77C2×C3×C4⋊C4—31no—
78#938—41no—
79#859C22×C312yes—
80C3×#103—32no—
81C3×#103—32no—
82C3×#103—32no—
83C3×#102—32no—
84C3×C22⋊C8—32no—
85C3×C22⋊C8—32no—
86C23×C32—42no—
87C22×C3×D3—41no—
88C32×D4—82no—
89C2×C32×C4—41no—
90C3×#67—84no—
91C2×C3×Dic3—42no—
92C2×C3×Dic3—41no—
93C3×C4×D3—82no—
94C3×SL(2,𝔽3)C2×D411yescharacteristic
95C2×2−1+4C3×D311yescharacteristic, complemented
96C2×#137—31no—
97C2×#135—31no—
98#537—31no—
99#452—32no—
100#451—32no—
101#449—32no—
102#448—32no—
103#417—31no—
104C2×#103—31no—
105#408—31no—
106C3×#41—32no—
107C3×#41—62no—
108C3×#41—62no—
109C3×#41—62no—
110C3×#41—62no—
111C3×#41—64no—
112C3×#41—64no—
113C3×#41#6712yescomplemented
114C2×C3×Q8—22no—
115C2×C3×Q8—31no—
116C2×C3×Q8—32no—
117C2×C3×Q8—34no—
118C2×C3×Q8—31no—
119C2×C3×Q8—62no—
120C2×C3×Q8—62no—
121C2×C3×Q8C22×D311yescharacteristic
122C2×C3×D4—31no—
123C2×C3×D4—32no—
124C2×C3×D4—31no—
125C2×C3×D4—41no—
126C2×C3×D4—61no—
127C2×C3×D4—62no—
128C2×C3×D4—81no—
129C2×C3×D4S411yescharacteristic
130C22×C3×C4—31no—
131C22×C3×C4—32no—
132C22×C3×C4—61no—
133C22×C3×C4—61no—
134C22×C3×C4—61no—
135C2×#67—42no—
136C2×C4×D3—41no—
137#41⋊C3—22no—
138#41⋊C3C3×D412yescomplemented
139C2×SL(2,𝔽3)—24no—
140C2×SL(2,𝔽3)—22no—
141C2×SL(2,𝔽3)—21no—
142C2×SL(2,𝔽3)C23×C311yescharacteristic
143GL(2,𝔽3)—21no—
144#226—21no—
145C3×Q16—61no—
146C3×Q16—64no—
147C3×Q16—61no—
148C3×SD16—62no—
149C3×SD16—61no—
150C3×SD16—64no—
151C3×SD16—61no—
152C3×D8—62no—
153C3×M16—62no—
154C2×C3×C8—32no—
155C2×C3×C8—31no—
156C2×C3×C8—31no—
157C2×C3×C8—62no—
158C3×C4⋊C4—32no—
159C3×C4⋊C4—62no—
160C3×C4⋊C4—61no—
161C3×C22⋊C4—62no—
162C3×C22⋊C4—62no—
163#217—42no—
164C4⋊Dic3—41no—
165C22×C32—41no—
166C22×C32—82no—
167C22×C32—82no—
168C22×C32—82no—
169C2×C3×D3—42no—
170C2×C3×D3—81no—
171C2×C3×D3—81no—
172C32×C4—42no—
173C3×Dic3—81no—
174C3×Dic3—82no—
1752−1+4—22no—
1762−1+4—62no—
177C2×#41—31no—
178C2×#41—32no—
179C2×#41—31no—
180C2×#41C2×C3×D311yescharacteristic
181C22×Q8—31no—
182C22×Q8C2×C3×D312yes—
183#137—64no—
184#135—62no—
185#135—62no—
186C2×Q16—32no—
187C2×Q16—31no—
188C2×SD16—31no—
189C2×SD16—32no—
190C2×SD16—31no—
191C2×D8—31no—
192C2×M16—31no—
193C22×C8—31no—
194C22⋊Q8—32no—
195C22⋊Q8—61no—
196#121—32no—
197#121—62no—
198#121—61no—
199C2×C4⋊C4—31no—
200#103—32no—
201#103—32no—
202#103—32no—
203#102—32no—
204C22⋊C8—32no—
205C22⋊C8—32no—
206C23×C3—31no—
207C23×C3—42no—
208C23×C3—61no—
209C23×C3—82no—
210C23×C3C2×S412yes—
211C22×D3—41no—
212C3×Q8—31no—
213C3×Q8—34no—
214C3×Q8—62no—
215C3×Q8—61no—
216C3×Q8—61no—
217C3×Q8—62no—
218C3×Q8C2×#6711yescharacteristic
219C3×D4—22no—
220C3×D4—32no—
221C3×D4—61no—
222C3×D4—61no—
223C3×D4—61no—
224C3×D4—64no—
225C3×D4—82no—
226C3×D4—84no—
227C3×D4—122no—
228C3×D4—242no—
229C2×C3×C4—31no—
230C2×C3×C4—32no—
231C2×C3×C4—32no—
232C2×C3×C4—32no—
233C2×C3×C4—41no—
234C2×C3×C4—62no—
235C2×C3×C4—62no—
236C2×C3×C4—64no—
237C2×C3×C4—62no—
238C2×C3×C4—61no—
239C2×C3×C4—81no—
240C2×C3×C4—122no—
241C2×C3×C4—241no—
242C2×C3×C4C2×S411yescharacteristic
243#67—84no—
244C2×Dic3—42no—
245C2×Dic3—41no—
246C4×D3—82no—
247SL(2,𝔽3)—21no—
248SL(2,𝔽3)C2×C3×D411yescharacteristic
249C3×C8—61no—
250C3×C8—61no—
251C3×C8—62no—
252C2×C32—41no—
253C2×C32—41no—
254C2×C32—41no—
255C2×C32—162no—
256C3×D3—82no—
257#41—32no—
258#41—62no—
259#41—62no—
260#41—62no—
261#41—62no—
262#41—64no—
263#41—64no—
264#41C3×#6712yescomplemented
265C2×Q8—22no—
266C2×Q8—31no—
267C2×Q8—32no—
268C2×Q8—34no—
269C2×Q8—31no—
270C2×Q8—62no—
271C2×Q8—62no—
272C2×Q8C22×C3×D311yescharacteristic
273C2×D4—31no—
274C2×D4—32no—
275C2×D4—31no—
276C2×D4—61no—
277C2×D4—62no—
278C2×D4C3×S411yescharacteristic
279C22×C4—31no—
280C22×C4—32no—
281C22×C4—61no—
282C22×C4—61no—
283C22×C4—61no—
284Q16—61no—
285Q16—64no—
286Q16—61no—
287SD16—62no—
288SD16—61no—
289SD16—64no—
290SD16—61no—
291D8—62no—
292M16—62no—
293C2×C8—32no—
294C2×C8—31no—
295C2×C8—31no—
296C2×C8—62no—
297C4⋊C4—32no—
298C4⋊C4—62no—
299C4⋊C4—61no—
300C22⋊C4—62no—
301C22⋊C4—62no—
302C22×C3—22no—
303C22×C3—22no—
304C22×C3—22no—
305C22×C3—32no—
306C22×C3—41no—
307C22×C3—61no—
308C22×C3—61no—
309C22×C3—82no—
310C22×C3—82no—
311C22×C3—82no—
312C22×C3—84no—
313C22×C3—81no—
314C22×C3—121no—
315C22×C3—121no—
316C22×C3—121no—
317C22×C3—162no—
318C22×C3—162no—
319C22×C3C22×S411yescharacteristic
320C2×D3—42no—
321C2×D3—81no—
322C2×D3—81no—
323C3×C4—31no—
324C3×C4—31no—
325C3×C4—42no—
326C3×C4—62no—
327C3×C4—82no—
328C3×C4—122no—
329C3×C4—121no—
330C3×C4GL(2,ℤ/4)12yes—
331Dic3—81no—
332Dic3—82no—
333C32—41noSylow
334C23—31no—
335C23—61no—
336C23C2×C3×S412yes—
337Q8—31no—
338Q8—34no—
339Q8—62no—
340Q8—61no—
341Q8—61no—
342Q8—62no—
343Q8C2×C3×#6711yescharacteristic
344D4—22no—
345D4—32no—
346D4—61no—
347D4—61no—
348D4—61no—
349D4—64no—
350D4—122no—
351D4—242no—
352C2×C4—31no—
353C2×C4—32no—
354C2×C4—32no—
355C2×C4—32no—
356C2×C4—62no—
357C2×C4—62no—
358C2×C4—64no—
359C2×C4—62no—
360C2×C4—61no—
361C2×C4—122no—
362C2×C4—241no—
363C2×C4C2×C3×S411yescharacteristic
364C8—61no—
365C8—61no—
366C8—62no—
367C2×C3—42no—
368C2×C3—41no—
369C2×C3—41no—
370C2×C3—41no—
371C2×C3—62no—
372C2×C3—81no—
373C2×C3—81no—
374C2×C3—81no—
375C2×C3—162no—
376C2×C3—164no—
377C2×C3—241no—
378C2×C3#592611yescharacteristic
379C2×C3#592711yescharacteristic
380C2×C3C2×GL(2,ℤ/4)11yescharacteristic
381D3—82no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—32no—
386C22—61no—
387C22—61no—
388C22—121no—
389C22—121no—
390C22—121no—
391C22C22×C3×S411yescharacteristic
392C4—31no—
393C4—31no—
394C4—62no—
395C4—122no—
396C4—121no—
397C4C3×GL(2,ℤ/4)12yes—
398C3—41no—
399C3—81no—
400C3#8553811yescharacteristic, direct product factor
401C2—42no—
402C2—62no—
403C2—241no—
404C2C3×#592611yescharacteristic
405C2C3×#592711yescharacteristic
406C2C2×C3×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, socleC22×C3C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C3
FrattiniC22C22×C3×S4
FittingC2×C3×2−1+4D3
hypercenterC2×C3×D4S4
hypocenterSL(2,𝔽3)C2×C3×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×2−1+4⋊C3C2semidirect
2. C2×C3×#985C2semidirect
3. C3×#5426C2semidirect
4. C3×#5421C2semidirect
5. C3×#5419C2semidirect
6. #85538C3direct (N × H)
16. C2×C3×2−1+4D3semidirect
18. C2×2−1+4⋊C3C2×C3semidirect
19. C2×#985C2×C3semidirect
23. #5426C2×C3semidirect
24. #5421C2×C3semidirect
25. #5419C2×C3semidirect
36. C3×#41⋊C3D4semidirect
95. C2×2−1+4C3×D3semidirect
113. C3×#41#67semidirect
138. #41⋊C3C3×D4semidirect
264. #41C3×#67semidirect
400. C3#85538direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×2−1+4⋊C32
C2×C3×#9852
C3×#54262
C3×#54212
C3×#54192
#855383
C3×#30133
C3×#9384