Finite Group Databasebeta

← C3×#85517 #11922801

Subgroups — C3×#85517 #11922801

Order 1152.

657 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 657 conjugacy classes of subgroups — too many to draw. Only its 80 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×C3×#41⋊C3C211yesmaximal, characteristic
2C2×C3×#984C211yesmaximal, characteristic, complemented
3C3×#5426C211yesmaximal, characteristic, complemented
4C3×#5425C212yesmaximal, complemented
5C3×#5419C212yesmaximal, complemented
6#85517C311yesmaximal, characteristic, direct product factor
7C3×#3007—31nomaximal
8C2×C3×#41⋊C3—22no—
9C2×C3×#41⋊C3C2211yescharacteristic
10C2×C3×#41⋊C3C2211yescharacteristic
11C22×C3×SL(2,𝔽3)C2211yescharacteristic
12C3×#984—22no—
13C2×C3×#226C2212yescomplemented
14C3×#924—41nomaximal
15C3×#859C2212yescomplemented
16C22×C3×#41D311yescharacteristic
17C22×#41⋊C3—21no—
18C22×#41⋊C3C2×C311yescharacteristic
19C2×#984C2×C311yescharacteristic, complemented
20C2×C3×#137—31no—
21C2×C3×#137—31no—
22C2×C3×C22⋊Q8—31no—
23#5426C2×C311yescharacteristic, complemented
24#5425C2×C312yescomplemented
25#5419C2×C312yescomplemented
26C3×#452—32no—
27C3×#452—32no—
28C3×#451—32no—
29C3×#450—32no—
30C3×#417—31no—
31C3×#417—31no—
32C3×#407—31no—
33C22×C32×C4—41no—
34C2×C3×Dic6—41no—
35C3×#41⋊C3—21no—
36C3×#41⋊C3D412yes—
37C2×C3×SL(2,𝔽3)—21no—
38C2×C3×SL(2,𝔽3)C2311yescharacteristic
39C3×#226—22no—
40C3×#217—42no—
41C3×C4⋊Dic3—41no—
42C3×Dic3⋊C4—42no—
43#3007—31noSylow
44C2×C3×#41—22no—
45C2×C3×#41—31no—
46C2×C3×#41—31no—
47C2×C3×#41—62no—
48C2×C3×#41C2×D311yescharacteristic
49C2×C3×#41C2×D311yescharacteristic
50C22×C3×Q8—31no—
51C22×C3×Q8C2×D311yescharacteristic
52C22×C3×D4—31no—
53C23×C3×C4—31no—
54C2×#41⋊C3—24no—
55C2×#41⋊C3—22no—
56C2×#41⋊C3—21no—
57C2×#41⋊C3—21no—
58C2×#41⋊C3C22×C311yescharacteristic
59C2×#41⋊C3C22×C311yescharacteristic
60C22×SL(2,𝔽3)—21no—
61C22×SL(2,𝔽3)C22×C311yescharacteristic
62#984—22no—
63C2×#226C22×C312yescomplemented
64C3×#137—62no—
65C3×#137—64no—
66C3×#137—62no—
67C2×C3×Q16—32no—
68C2×C3×Q16—32no—
69C2×C3×SD16—32no—
70C2×C3×SD16—32no—
71C2×C3×M16—31no—
72C2×C3×M16—31no—
73C3×C22⋊Q8—32no—
74C3×C22⋊Q8—32no—
75C3×C22⋊Q8—61no—
76C3×C22⋊Q8—61no—
77C2×C3×C4⋊C4—31no—
78C2×C3×C4⋊C4—61no—
79C2×C3×C22⋊C4—61no—
80#924—41no—
81#859C22×C312yescomplemented
82C3×#103—32no—
83C3×#103—32no—
84C3×#102—32no—
85C3×#102—32no—
86C3×C22⋊C8—32no—
87C3×C22⋊C8—32no—
88C23×C32—41no—
89C2×C32×C4—41no—
90C2×C32×C4—41no—
91C2×C32×C4—82no—
92C2×C3×Dic3—42no—
93C2×C3×Dic3—42no—
94C3×Dic6—82no—
95C3×SL(2,𝔽3)C2×D411yescharacteristic
96C22×#41C3×D311yescharacteristic
97C2×#137—31no—
98C2×#137—31no—
99C2×C22⋊Q8—31no—
100#452—32no—
101#452—32no—
102#451—32no—
103#450—32no—
104#417—31no—
105#417—31no—
106#407—31no—
107C24×C3—31no—
108C3×#41—21no—
109C3×#41—32no—
110C3×#41—34no—
111C3×#41—62no—
112C3×#41—62no—
113C3×#41—61no—
114C3×#41—62no—
115C3×#41#6712yes—
116C2×C3×Q8—21no—
117C2×C3×Q8—31no—
118C2×C3×Q8—32no—
119C2×C3×Q8—62no—
120C2×C3×Q8—62no—
121C2×C3×Q8—61no—
122C2×C3×Q8—61no—
123C2×C3×Q8C22×D311yescharacteristic
124C2×C3×D4—31no—
125C2×C3×D4—31no—
126C2×C3×D4—61no—
127C2×C3×D4—62no—
128C2×C3×D4—62no—
129C22×C3×C4—31no—
130C22×C3×C4—31no—
131C22×C3×C4—31no—
132C22×C3×C4—31no—
133C22×C3×C4—31no—
134C22×C3×C4—41no—
135C22×C3×C4—62no—
136C22×C3×C4—62no—
137C22×C3×C4—61no—
138C22×C3×C4—61no—
139C22×C3×C4—81no—
140C22×C3×C4S411yescharacteristic
141C2×Dic6—41no—
142#41⋊C3—22no—
143#41⋊C3—21no—
144#41⋊C3—22no—
145#41⋊C3C3×D412yes—
146C2×SL(2,𝔽3)—22no—
147C2×SL(2,𝔽3)—21no—
148C2×SL(2,𝔽3)—21no—
149C2×SL(2,𝔽3)C23×C311yescharacteristic
150#226—22no—
151C3×Q16—62no—
152C3×Q16—64no—
153C3×Q16—62no—
154C3×SD16—64no—
155C3×SD16—64no—
156C3×M16—62no—
157C3×M16—62no—
158C2×C3×C8—32no—
159C2×C3×C8—32no—
160C3×C4⋊C4—32no—
161C3×C4⋊C4—62no—
162C3×C4⋊C4—62no—
163C3×C4⋊C4—61no—
164C3×C22⋊C4—62no—
165C3×C22⋊C4—62no—
166#217—42no—
167C4⋊Dic3—41no—
168Dic3⋊C4—42no—
169C22×C32—42no—
170C22×C32—41no—
171C22×C32—81no—
172C22×C32—81no—
173C32×C4—42no—
174C32×C4—81no—
175C3×Dic3—82no—
176C3×Dic3—82no—
177C2×#41—22no—
178C2×#41—31no—
179C2×#41—31no—
180C2×#41—62no—
181C2×#41C2×C3×D311yescharacteristic
182C2×#41C2×C3×D311yescharacteristic
183C22×Q8—31no—
184C22×Q8C2×C3×D311yescharacteristic
185C22×D4—31no—
186C23×C4—31no—
187#137—62no—
188#137—64no—
189#137—62no—
190C2×Q16—32no—
191C2×Q16—32no—
192C2×SD16—32no—
193C2×SD16—32no—
194C2×M16—31no—
195C2×M16—31no—
196C22⋊Q8—32no—
197C22⋊Q8—32no—
198C22⋊Q8—61no—
199C22⋊Q8—61no—
200C2×C4⋊C4—31no—
201C2×C4⋊C4—61no—
202C2×C22⋊C4—61no—
203#103—32no—
204#103—32no—
205#102—32no—
206#102—32no—
207C22⋊C8—32no—
208C22⋊C8—32no—
209C23×C3—31no—
210C23×C3—31no—
211C23×C3—41no—
212C23×C3—62no—
213C23×C3—62no—
214C23×C3—81no—
215C23×C3—121no—
216C23×C3C2×S411yescharacteristic
217C3×Q8—31no—
218C3×Q8—32no—
219C3×Q8—61no—
220C3×Q8—62no—
221C3×Q8—62no—
222C3×Q8—122no—
223C3×Q8C2×#6711yescharacteristic
224C3×D4—32no—
225C3×D4—32no—
226C3×D4—61no—
227C3×D4—61no—
228C3×D4—64no—
229C2×C3×C4—22no—
230C2×C3×C4—31no—
231C2×C3×C4—32no—
232C2×C3×C4—32no—
233C2×C3×C4—31no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—61no—
237C2×C3×C4—61no—
238C2×C3×C4—62no—
239C2×C3×C4—61no—
240C2×C3×C4—61no—
241C2×C3×C4—62no—
242C2×C3×C4—62no—
243C2×C3×C4—62no—
244C2×C3×C4—82no—
245C2×C3×C4—84no—
246C2×C3×C4—81no—
247C2×C3×C4—81no—
248C2×C3×C4—122no—
249C2×C3×C4—122no—
250C2×C3×C4C2×S411yescharacteristic
251C2×C3×C4C2×S411yescharacteristic
252C2×Dic3—42no—
253C2×Dic3—42no—
254Dic6—82no—
255SL(2,𝔽3)—21no—
256SL(2,𝔽3)C2×C3×D411yescharacteristic
257C3×C8—62no—
258C3×C8—62no—
259C2×C32—41no—
260C2×C32—41no—
261C2×C32—41no—
262C2×C32—82no—
263C24—31no—
264#41—21no—
265#41—32no—
266#41—34no—
267#41—62no—
268#41—62no—
269#41—61no—
270#41—62no—
271#41C3×#6712yes—
272C2×Q8—21no—
273C2×Q8—31no—
274C2×Q8—32no—
275C2×Q8—62no—
276C2×Q8—62no—
277C2×Q8—61no—
278C2×Q8—61no—
279C2×Q8C22×C3×D311yescharacteristic
280C2×D4—31no—
281C2×D4—31no—
282C2×D4—61no—
283C2×D4—62no—
284C2×D4—62no—
285C22×C4—31no—
286C22×C4—31no—
287C22×C4—31no—
288C22×C4—31no—
289C22×C4—31no—
290C22×C4—62no—
291C22×C4—62no—
292C22×C4—61no—
293C22×C4—61no—
294C22×C4C3×S411yescharacteristic
295Q16—62no—
296Q16—64no—
297Q16—62no—
298SD16—64no—
299SD16—64no—
300M16—62no—
301M16—62no—
302C2×C8—32no—
303C2×C8—32no—
304C4⋊C4—32no—
305C4⋊C4—62no—
306C4⋊C4—61no—
307C4⋊C4—62no—
308C22⋊C4—62no—
309C22⋊C4—62no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—32no—
313C22×C3—41no—
314C22×C3—42no—
315C22×C3—61no—
316C22×C3—61no—
317C22×C3—61no—
318C22×C3—61no—
319C22×C3—61no—
320C22×C3—81no—
321C22×C3—81no—
322C22×C3—82no—
323C22×C3—82no—
324C22×C3—81no—
325C22×C3—82no—
326C22×C3—124no—
327C22×C3#98412yescomplemented
328C22×C3C22×S411yescharacteristic
329C3×C4—21no—
330C3×C4—31no—
331C3×C4—31no—
332C3×C4—42no—
333C3×C4—61no—
334C3×C4—81no—
335C3×C4—82no—
336C3×C4—82no—
337C3×C4—122no—
338C3×C4—122no—
339C3×C4GL(2,ℤ/4)12yes—
340Dic3—82no—
341Dic3—82no—
342C32—41noSylow
343C23—31no—
344C23—31no—
345C23—62no—
346C23—62no—
347C23—121no—
348C23C2×C3×S411yescharacteristic
349Q8—31no—
350Q8—32no—
351Q8—61no—
352Q8—62no—
353Q8—62no—
354Q8—122no—
355Q8C2×C3×#6711yescharacteristic
356D4—32no—
357D4—32no—
358D4—61no—
359D4—61no—
360D4—64no—
361C2×C4—22no—
362C2×C4—31no—
363C2×C4—32no—
364C2×C4—32no—
365C2×C4—31no—
366C2×C4—61no—
367C2×C4—61no—
368C2×C4—62no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—62no—
372C2×C4—62no—
373C2×C4—62no—
374C2×C4—122no—
375C2×C4—122no—
376C2×C4C2×C3×S411yescharacteristic
377C2×C4C2×C3×S411yescharacteristic
378C8—62no—
379C8—62no—
380C2×C3—22no—
381C2×C3—41no—
382C2×C3—41no—
383C2×C3—41no—
384C2×C3—62no—
385C2×C3—82no—
386C2×C3—84no—
387C2×C3—81no—
388C2×C3—81no—
389C2×C3—81no—
390C2×C3—121no—
391C2×C3C2×#98411yescharacteristic
392C2×C3#592311yescharacteristic
393C2×C3C2×GL(2,ℤ/4)11yescharacteristic
394C22—21no—
395C22—21no—
396C22—32no—
397C22—61no—
398C22—61no—
399C22—61no—
400C22—61no—
401C22—61no—
402C22—124no—
403C22C3×#98412yescomplemented
404C22C22×C3×S411yescharacteristic
405C4—21no—
406C4—31no—
407C4—31no—
408C4—61no—
409C4—122no—
410C4—122no—
411C4C3×GL(2,ℤ/4)12yes—
412C3—41no—
413C3—81no—
414C3#8551711yescharacteristic, direct product factor
415C2—22no—
416C2—62no—
417C2—121no—
418C2C2×C3×#98411yescharacteristic
419C2C3×#592311yescharacteristic
420C2C2×C3×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, socleC22×C3C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C3
FrattiniC22C22×C3×S4
FittingC22×C3×#41D3
prime radicalC22×C3×#41⋊C3C2
hypercenterC22×C3×C4S4
hypocenterSL(2,𝔽3)C2×C3×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#984C2semidirect
3. C3×#5426C2semidirect
4. C3×#5425C2semidirect
5. C3×#5419C2semidirect
6. #85517C3direct (N × H)
13. C2×C3×#226C22semidirect
15. C3×#859C22semidirect
19. C2×#984C2×C3semidirect
23. #5426C2×C3semidirect
24. #5425C2×C3semidirect
25. #5419C2×C3semidirect
63. C2×#226C22×C3semidirect
81. #859C22×C3semidirect
327. C22×C3#984semidirect
403. C22C3×#984semidirect
414. C3#85517direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#41⋊C32
C2×C3×#9842
C3×#54262
C3×#54252
C3×#54192
#855173
C3×#30073
C3×#9244