Finite Group Databasebeta

← C3×#85533 #11922817

Subgroups — C3×#85533 #11922817

Order 1152.

627 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 627 conjugacy classes of subgroups — too many to draw. Only its 72 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
2C3×#5923C211yesmaximal, characteristic, complemented
3C3×#5431C212yesmaximal, complemented
4C3×#5430C212yesmaximal, complemented
5C3×#5423C211yesmaximal, characteristic, complemented
6#85533C311yesmaximal, characteristic, direct product factor
7C3×#3020—31nomaximal
8C3×2−1+4⋊C3—21no—
9C3×2−1+4⋊C3C2212yes—
10C2×C3×#41⋊C3C2211yescharacteristic
11C22×C3×SL(2,𝔽3)—21no—
12C3×#986C2211yescharacteristic, complemented
13C3×#985—21no—
14C3×#984C2211yescharacteristic, complemented
15C3×#983—21no—
16C3×#932—41nomaximal
17C3×GU(2,𝔽3)C2212yescomplemented
18C2×C3×2−1+4D311yescharacteristic, complemented
19C2×2−1+4⋊C3—21no—
20C2×2−1+4⋊C3C2×C311yescharacteristic, complemented
21#5923C2×C311yescharacteristic, complemented
22C3×#575—31no—
23C3×#575—31no—
24C3×#536—31no—
25#5431C2×C312yescomplemented
26#5430C2×C312yescomplemented
27#5423C2×C311yescharacteristic, complemented
28C3×#456—32no—
29C3×#456—32no—
30C3×#454—32no—
31C3×#454—32no—
32C3×#421—31no—
33C3×#421—31no—
34C2×C3×#101—31no—
35C2×C32×D4—41no—
36C3×#235—41no—
37C3×#41⋊C3D411yescharacteristic
38C3×#41⋊C3C2311yescharacteristic
39C2×C3×SL(2,𝔽3)—22no—
40C2×C3×SL(2,𝔽3)D411yescharacteristic
41C3×GL(2,𝔽3)—21no—
42C3×#226—21no—
43C3×#214—42no—
44C3×#213—42no—
45C3×#208—41no—
46#3020—31noSylow
47C3×2−1+4—21no—
48C3×2−1+4—32no—
49C3×2−1+4—61no—
50C3×2−1+4C2×D312yes—
51C2×C3×#41—31no—
52C2×C3×#41—31no—
53C2×C3×#41—61no—
54C2×C3×#41C2×D311yescharacteristic
55C22×C3×Q8—21no—
56C22×C3×Q8—31no—
572−1+4⋊C3—22no—
582−1+4⋊C3—21no—
592−1+4⋊C3—22no—
602−1+4⋊C3C22×C312yes—
61C2×#41⋊C3—21no—
62C2×#41⋊C3C22×C311yescharacteristic
63C22×SL(2,𝔽3)—22no—
64C22×SL(2,𝔽3)—21no—
65#986C22×C311yescharacteristic, complemented
66#985—21no—
67#984C22×C311yescharacteristic, complemented
68#983—21no—
69C3×#137—31no—
70C3×#137—32no—
71C3×#137—31no—
72C3×#137—61no—
73C3×#137—61no—
74C3×AGL(1,ℤ/8)—31no—
75C3×AGL(1,ℤ/8)—32no—
76C3×AGL(1,ℤ/8)—31no—
77C3×AGL(1,ℤ/8)—61no—
78C3×AGL(1,ℤ/8)—61no—
79C3×#135—61no—
80C3×#135—62no—
81C3×#135—61no—
82C2×C3×M16—31no—
83C2×C3×M16—31no—
84C3×C4⋊Q8—32no—
85C3×#124—32no—
86C3×#123—62no—
87C3×C22⋊Q8—62no—
88C3×#117—31no—
89#932—41no—
90GU(2,𝔽3)C22×C312yescomplemented
91C3×#104—32no—
92C3×#104—32no—
93C3×#104—32no—
94C3×#104—32no—
95C3×#101—32no—
96C3×#101—32no—
97C23×C32—81no—
98C32×D4—42no—
99C32×D4—81no—
100C2×C32×C4—41no—
101C3×#67—81no—
102C3×D12—41no—
103C3×C4×D3—81no—
104C3×Dic6—41no—
105C3×SL(2,𝔽3)C2×D411yescharacteristic
106C3×C3⋊C8—42no—
107C2×2−1+4C3×D311yescharacteristic, complemented
108#575—31no—
109#575—31no—
110#536—31no—
111#456—32no—
112#456—32no—
113#454—32no—
114#454—32no—
115#421—31no—
116#421—31no—
117C2×#101—31no—
118C3×#41—31no—
119C3×#41—31no—
120C3×#41—32no—
121C3×#41—32no—
122C3×#41—62no—
123C3×#41—61no—
124C3×#41—61no—
125C3×#41—64no—
126C3×#41—61no—
127C3×#41—61no—
128C3×#41—61no—
129C3×#41—62no—
130C3×#41—61no—
131C3×#41#6711yescharacteristic
132C3×#41C22×D311yescharacteristic
133C2×C3×Q8—22no—
134C2×C3×Q8—32no—
135C2×C3×Q8—32no—
136C2×C3×Q8—31no—
137C2×C3×Q8—31no—
138C2×C3×Q8—61no—
139C2×C3×Q8—61no—
140C2×C3×Q8—61no—
141C2×C3×Q8—62no—
142C2×C3×Q8#6711yescharacteristic
143C2×C3×D4—31no—
144C2×C3×D4—31no—
145C2×C3×D4—41no—
146C2×C3×D4—61no—
147C2×C3×D4—61no—
148C2×C3×D4—81no—
149C2×C3×D4S411yescharacteristic
150C22×C3×C4—31no—
151C22×C3×C4—32no—
152C22×C3×C4—61no—
153C22×C3×C4—61no—
154#235—41no—
155#41⋊C3—21no—
156#41⋊C3—21no—
157#41⋊C3C3×D411yescharacteristic
158#41⋊C3C23×C311yescharacteristic
159C2×SL(2,𝔽3)—24no—
160C2×SL(2,𝔽3)—22no—
161C2×SL(2,𝔽3)—21no—
162C2×SL(2,𝔽3)C3×D411yescharacteristic
163GL(2,𝔽3)—21no—
164#226—21no—
165C3×Q16—61no—
166C3×Q16—62no—
167C3×Q16—61no—
168C3×SD16—61no—
169C3×SD16—61no—
170C3×SD16—61no—
171C3×SD16—62no—
172C3×SD16—61no—
173C3×SD16—62no—
174C3×D8—61no—
175C3×D8—61no—
176C3×D8—62no—
177C3×M16—31no—
178C3×M16—31no—
179C3×M16—32no—
180C3×M16—61no—
181C3×M16—61no—
182C2×C3×C8—61no—
183C2×C3×C8—61no—
184C3×C4⋊C4—61no—
185C3×C4⋊C4—64no—
186C3×C22⋊C4—64no—
187C3×C22⋊C4—61no—
188C3×C42—32no—
189#214—42no—
190#213—42no—
191#208—41no—
192C22×C32—41no—
193C22×C32—82no—
194C22×C32—162no—
195C2×C3×D3—81no—
196C32×C4—41no—
197C32×C4—41no—
198C3×Dic3—81no—
1992−1+4—21no—
2002−1+4—32no—
2012−1+4—61no—
2022−1+4C2×C3×D312yes—
203C2×#41—31no—
204C2×#41—31no—
205C2×#41—61no—
206C2×#41C2×C3×D311yescharacteristic
207C22×Q8—21no—
208C22×Q8—31no—
209#137—31no—
210#137—32no—
211#137—31no—
212#137—61no—
213#137—61no—
214AGL(1,ℤ/8)—31no—
215AGL(1,ℤ/8)—32no—
216AGL(1,ℤ/8)—31no—
217AGL(1,ℤ/8)—61no—
218AGL(1,ℤ/8)—61no—
219#135—61no—
220#135—62no—
221#135—61no—
222C2×M16—31no—
223C2×M16—31no—
224C4⋊Q8—32no—
225#124—32no—
226#123—62no—
227C22⋊Q8—62no—
228#117—31no—
229#104—32no—
230#104—32no—
231#104—32no—
232#104—32no—
233#101—32no—
234#101—32no—
235C23×C3—21no—
236C23×C3—31no—
237C23×C3—61no—
238C23×C3—81no—
239C23×C3—82no—
240C3×Q8—31no—
241C3×Q8—32no—
242C3×Q8—62no—
243C3×Q8—62no—
244C3×Q8—61no—
245C3×Q8—61no—
246C3×Q8—61no—
247C3×Q8C2×#6711yescharacteristic
248C3×D4—21no—
249C3×D4—31no—
250C3×D4—31no—
251C3×D4—42no—
252C3×D4—61no—
253C3×D4—61no—
254C3×D4—62no—
255C3×D4—62no—
256C3×D4—61no—
257C3×D4—81no—
258C3×D4—82no—
259C3×D4—82no—
260C3×D4—121no—
261C3×D4—121no—
262C3×D4C2×S412yes—
263C2×C3×C4—31no—
264C2×C3×C4—31no—
265C2×C3×C4—31no—
266C2×C3×C4—31no—
267C2×C3×C4—31no—
268C2×C3×C4—34no—
269C2×C3×C4—41no—
270C2×C3×C4—62no—
271C2×C3×C4—62no—
272C2×C3×C4—62no—
273C2×C3×C4—61no—
274C2×C3×C4—62no—
275C2×C3×C4—61no—
276C2×C3×C4—81no—
277C2×C3×C4—121no—
278C2×C3×C4—121no—
279C2×C3×C4C2×S411yescharacteristic
280#67—81no—
281D12—41no—
282C4×D3—81no—
283Dic6—41no—
284SL(2,𝔽3)—21no—
285SL(2,𝔽3)C2×C3×D411yescharacteristic
286C3×C8—61no—
287C3×C8—61no—
288C3×C8—62no—
289C3⋊C8—42no—
290C2×C32—41no—
291C2×C32—81no—
292C2×C32—162no—
293C3×D3—161no—
294#41—31no—
295#41—31no—
296#41—32no—
297#41—32no—
298#41—62no—
299#41—61no—
300#41—61no—
301#41—64no—
302#41—61no—
303#41—61no—
304#41—61no—
305#41—62no—
306#41—61no—
307#41C3×#6711yescharacteristic
308#41C22×C3×D311yescharacteristic
309C2×Q8—22no—
310C2×Q8—32no—
311C2×Q8—32no—
312C2×Q8—31no—
313C2×Q8—31no—
314C2×Q8—61no—
315C2×Q8—61no—
316C2×Q8—61no—
317C2×Q8—62no—
318C2×Q8C3×#6711yescharacteristic
319C2×D4—31no—
320C2×D4—31no—
321C2×D4—61no—
322C2×D4—61no—
323C2×D4C3×S411yescharacteristic
324C22×C4—31no—
325C22×C4—32no—
326C22×C4—61no—
327C22×C4—61no—
328Q16—61no—
329Q16—62no—
330Q16—61no—
331SD16—61no—
332SD16—61no—
333SD16—61no—
334SD16—62no—
335SD16—61no—
336SD16—62no—
337D8—61no—
338D8—61no—
339D8—62no—
340M16—31no—
341M16—31no—
342M16—32no—
343M16—61no—
344M16—61no—
345C2×C8—61no—
346C2×C8—61no—
347C4⋊C4—61no—
348C4⋊C4—64no—
349C22⋊C4—64no—
350C22⋊C4—61no—
351C42—32no—
352C22×C3—22no—
353C22×C3—31no—
354C22×C3—31no—
355C22×C3—42no—
356C22×C3—41no—
357C22×C3—82no—
358C22×C3—84no—
359C22×C3—81no—
360C22×C3—121no—
361C22×C3—122no—
362C22×C3—121no—
363C22×C3—162no—
364C22×C3—164no—
365C22×C3GL(2,ℤ/4)11yescharacteristic
366C2×D3—81no—
367C3×C4—31no—
368C3×C4—31no—
369C3×C4—41no—
370C3×C4—41no—
371C3×C4—62no—
372C3×C4—81no—
373C3×C4—81no—
374C3×C4—122no—
375C3×C4—121no—
376C3×C4GL(2,ℤ/4)11yescharacteristic
377C3×C4C22×S411yescharacteristic
378Dic3—81no—
379C32—41noSylow
380C23—21no—
381C23—31no—
382C23—61no—
383Q8—31no—
384Q8—32no—
385Q8—62no—
386Q8—62no—
387Q8—61no—
388Q8—61no—
389Q8—61no—
390Q8C2×C3×#6711yescharacteristic
391D4—21no—
392D4—31no—
393D4—31no—
394D4—61no—
395D4—61no—
396D4—62no—
397D4—62no—
398D4—61no—
399D4—121no—
400D4—121no—
401D4C2×C3×S412yes—
402C2×C4—31no—
403C2×C4—31no—
404C2×C4—31no—
405C2×C4—31no—
406C2×C4—31no—
407C2×C4—34no—
408C2×C4—62no—
409C2×C4—62no—
410C2×C4—62no—
411C2×C4—61no—
412C2×C4—62no—
413C2×C4—61no—
414C2×C4—121no—
415C2×C4—121no—
416C2×C4C2×C3×S411yescharacteristic
417C8—61no—
418C8—61no—
419C8—62no—
420C2×C3—21no—
421C2×C3—42no—
422C2×C3—41no—
423C2×C3—61no—
424C2×C3—61no—
425C2×C3—81no—
426C2×C3—82no—
427C2×C3—81no—
428C2×C3—162no—
429C2×C3—164no—
430C2×C3—241no—
431C2×C3C2×GL(2,ℤ/4)11yescharacteristic
432D3—161no—
433C22—22no—
434C22—31no—
435C22—31no—
436C22—42no—
437C22—121no—
438C22—122no—
439C22—121no—
440C22C3×GL(2,ℤ/4)11yescharacteristic
441C4—31no—
442C4—31no—
443C4—62no—
444C4—122no—
445C4—121no—
446C4C3×GL(2,ℤ/4)11yescharacteristic
447C4C22×C3×S411yescharacteristic
448C3—41no—
449C3—81no—
450C3#8553311yescharacteristic, direct product factor
451C2—21no—
452C2—42no—
453C2—61no—
454C2—61no—
455C2—241no—
456C2C2×C3×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, socleC2×C3C2×GL(2,ℤ/4)
commutator, cosocle#41⋊C3C23×C3
FrattiniC4C22×C3×S4
FittingC2×C3×2−1+4D3
hypercenterC2×C3×D4S4
hypocenterSL(2,𝔽3)C2×C3×D4
LCSG ▷ #41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ 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. C3×#5923C2semidirect
3. C3×#5431C2semidirect
4. C3×#5430C2semidirect
5. C3×#5423C2semidirect
6. #85533C3direct (N × H)
12. C3×#986C22semidirect
14. C3×#984C22semidirect
17. C3×GU(2,𝔽3)C22semidirect
18. C2×C3×2−1+4D3semidirect
20. C2×2−1+4⋊C3C2×C3semidirect
21. #5923C2×C3semidirect
25. #5431C2×C3semidirect
26. #5430C2×C3semidirect
27. #5423C2×C3semidirect
65. #986C22×C3semidirect
67. #984C22×C3semidirect
90. GU(2,𝔽3)C22×C3semidirect
107. C2×2−1+4C3×D3semidirect
450. C3#85533direct (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
C3×#59232
C3×#54312
C3×#54302
C3×#54232
#855333
C3×#30203
C3×#9324