Finite Group Databasebeta

← #85874⋊C3 #11923217

Subgroups — #85874⋊C3 #11923217

Order 1152.

757 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 757 conjugacy classes of subgroups — too many to draw. Only its 87 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×#1007⋊C3C211yesmaximal, characteristic, complemented
2C22×C3⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
3C3×#519⋊C3C211yesmaximal, characteristic, complemented
4#5576⋊C3C212yesmaximal, complemented
5C4⋊(C3⋊(#41⋊C3))C212yesmaximal, complemented
6#85874C311yesmaximal, characteristic, complemented
7#3487⋊C3—31nomaximal
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9#1007⋊C3—22no—
10C2×D3×SL(2,𝔽3)C2211yescharacteristic, complemented
11C2×C3⋊(#41⋊C3)—22no—
12C2×C3⋊(#41⋊C3)C2212yes—
13C2×C3⋊(#41⋊C3)C2211yescharacteristic, complemented
14C3×#886—41nomaximal
15C3×C4×SL(2,𝔽3)C2212yescomplemented
16C2×#1007C2×C311yescharacteristic
17C22×#239C2×C311yescharacteristic, complemented
18C2×2−1+4⋊C3—31no—
19C22×#41⋊C3—31no—
20C3×#519C2×C311yescharacteristic, complemented
21#5577—32no—
22#5577C2×C312yescomplemented
23#5576—32no—
24#5576C2×C312yescomplemented
25#5525—31no—
26#5525—31no—
27#5524—31no—
28#5524—31no—
29C4⋊(#41⋊C3)—32no—
30#548⋊C3—32no—
31#519⋊C3—21no—
32#519⋊C3D311yescharacteristic, complemented
33C2×C3×#67—41no—
34C22×C3×Dic3—41no—
35C2×C3×SL(2,𝔽3)D412yescomplemented
36C2×C3×SL(2,𝔽3)C2311yescharacteristic
37D3×SL(2,𝔽3)—21no—
38C3⋊(#41⋊C3)—22no—
39C3⋊(#41⋊C3)—21no—
40C32×C22⋊C4—41no—
41C3×(C2×D3)⋊C4—42no—
42C3×C4⋊Dic3—42no—
43#3487—31noSylow
44C22×C3×Q8C22×C311yescharacteristic
45#1007—22no—
46#1007—62no—
47C2×#239—22no—
48C2×#239—31no—
49C2×#239—32no—
50C2×#239—62no—
51C2×#239C22×C312yes—
52C2×#239C22×C311yescharacteristic
53C2×D3×Q8—31no—
54C2×D3×Q8C22×C311yescharacteristic
55C2×#235—31no—
56C2×#235—31no—
57C22×D12—31no—
58C22×C4×D3—31no—
592−1+4⋊C3—62no—
60C2×#41⋊C3—32no—
61C2×#41⋊C3—31no—
62C2×#41⋊C3—62no—
63C22×SL(2,𝔽3)—21no—
64C22×SL(2,𝔽3)—31no—
65C22×SL(2,𝔽3)C2×D311yescharacteristic
66C3×#123—121no—
67C3×C4×Q8—32no—
68C3×C4×Q8C22×C312yes—
69C3×#117—31no—
70C3×#117—31no—
71#897—32no—
72#897—32no—
73#895—32no—
74#895—34no—
75#895—32no—
76#895—32no—
77#895—32no—
78#886—31no—
79#886A411yescharacteristic
80#882—32no—
81#882—31no—
82#882—31no—
83#875—32no—
84#875—32no—
85#874—32no—
86#874—32no—
87C4×D12—34no—
88C4×D12—32no—
89C4×D12—32no—
90C4×SL(2,𝔽3)—22no—
91C4×SL(2,𝔽3)C2×D312yescomplemented
92C23×C32—41no—
93C22×C3×D3—41no—
94C2×C32×C4—42no—
95C3×#67—82no—
96C2×C3×Dic3—42no—
97C2×C3×Dic3—41no—
98C2×C3×Dic3—82no—
99C3×SL(2,𝔽3)C2×D411yescharacteristic
100C2×2−1+4—31no—
101C22×#41—31no—
102#550—32no—
103#550—92no—
104#548—32no—
105#548—92no—
106#535—91no—
107#535—91no—
108#525—91no—
109#525—91no—
110#519C3×D311yescharacteristic, complemented
111C23×D3—31no—
112C2×C3×Q8—31no—
113C2×C3×Q8—32no—
114C2×C3×Q8C3×D412yes—
115C2×C3×Q8C23×C311yescharacteristic
116C2×C3×D4—121no—
117C22×C3×C4—31no—
118C22×C3×C4—121no—
119C2×#67—31no—
120C2×#67C2×A411yescharacteristic
121C22×Dic3C2×A411yescharacteristic
122#239—22no—
123#239—21no—
124#239—64no—
125#239—62no—
126#239—62no—
127#239—64no—
128#239—62no—
129#239—61no—
130D3×Q8—21no—
131D3×Q8—62no—
132D3×Q8—62no—
133D3×Q8—61no—
134#235—62no—
135#235—62no—
136#235—62no—
137#235—62no—
138C2×D12—31no—
139C2×D12—32no—
140C2×D12—32no—
141C2×D12—31no—
142C2×D12—34no—
143C2×D12—32no—
144C2×D12—32no—
145C2×D12—62no—
146C2×D12—62no—
147C2×C4×D3—32no—
148C2×C4×D3—31no—
149C2×C4×D3—31no—
150C2×C4×D3—32no—
151C2×C4×D3—32no—
152C2×C4×D3—62no—
153C2×C4×D3—62no—
154C2×Dic6—31no—
155C2×Dic6—31no—
156#41⋊C3—62no—
157#41⋊C3—61no—
158C2×SL(2,𝔽3)—22no—
159C2×SL(2,𝔽3)—21no—
160C2×SL(2,𝔽3)—61no—
161C2×SL(2,𝔽3)D1212yescomplemented
162C2×SL(2,𝔽3)C22×D311yescharacteristic
163C3×C4⋊C4—32no—
164C3×C4⋊C4—32no—
165C3×C4⋊C4—122no—
166C3×C22⋊C4—31no—
167C3×C22⋊C4—41no—
168C3×C22⋊C4—81no—
169C3×C22⋊C4—122no—
170C3×C22⋊C4C2×A411yescharacteristic
171C3×C42—32no—
172C3×C42—32no—
173(C2×D3)⋊C4—34no—
174(C2×D3)⋊C4—32no—
175(C2×D3)⋊C4—32no—
176(C2×D3)⋊C4—32no—
177(C2×D3)⋊C4C2×A412yes—
178C4⋊Dic3—32no—
179C4⋊Dic3C2×A412yes—
180C22×C32—42no—
181C22×C32—41no—
182C22×C32—82no—
183C2×C3×D3—81no—
184C2×C3×D3—82no—
185C32×C4—82no—
186C3×Dic3—82no—
187C3×Dic3—81no—
1882−1+4—62no—
1892−1+4—182no—
190C2×#41—32no—
191C2×#41—31no—
192C2×#41—62no—
193C2×#41—91no—
194C2×#41—91no—
195C2×#41—91no—
196C2×#41—92no—
197C2×#41—182no—
198C22×Q8—31no—
199C22×Q8—91no—
200C22×Q8C2×C3×D311yescharacteristic
201C22×D4—91no—
202C23×C4—91no—
203#127—92no—
204#127—92no—
205#124—92no—
206#124—92no—
207#123—31no—
208#123—91no—
209C22⋊Q8—92no—
210C22⋊Q8—92no—
211#121—92no—
212#121—92no—
213#121—94no—
214#121—92no—
215#121—92no—
216#120—91no—
217#120—92no—
218#120—91no—
219C4×Q8—32no—
220C4×Q8C2×C3×D312yes—
221C4×D4—92no—
222C4×D4—92no—
223C4×D4—94no—
224#117—31no—
225#117—31no—
226C23×C3—41no—
227C23×C3—81no—
228C23×C3—121no—
229C23×C3C22×A411yescharacteristic
230C22×D3—32no—
231C22×D3—31no—
232C22×D3—62no—
233C22×D3—122no—
234C22×D3C22×A411yescharacteristic
235C3×Q8—32no—
236C3×Q8—31no—
237C3×Q8—32no—
238C3×Q8C2×C3×D411yescharacteristic
239C3×D4—242no—
240C2×C3×C4—32no—
241C2×C3×C4—32no—
242C2×C3×C4—31no—
243C2×C3×C4—31no—
244C2×C3×C4—32no—
245C2×C3×C4—42no—
246C2×C3×C4—82no—
247C2×C3×C4—122no—
248C2×C3×C4—121no—
249C2×C3×C4—242no—
250C2×C3×C4C22×A412yes—
251#67—22no—
252#67—62no—
253C2×Dic3—22no—
254C2×Dic3—31no—
255C2×Dic3C22×A412yes—
256C2×Dic3C22×A411yescharacteristic
257D12—62no—
258D12—62no—
259D12—62no—
260D12—61no—
261D12—61no—
262D12—64no—
263D12—122no—
264D12—122no—
265D12—124no—
266C4×D3—62no—
267C4×D3—64no—
268C4×D3—62no—
269C4×D3—61no—
270C4×D3—61no—
271C4×D3—61no—
272C4×D3—61no—
273C4×D3—64no—
274Dic6—62no—
275Dic6—61no—
276Dic6—61no—
277SL(2,𝔽3)—21no—
278SL(2,𝔽3)C2×D1211yescharacteristic
279C2×C32—41no—
280C2×C32—42no—
281C2×C32—82no—
282C3×D3—161no—
283C24—91no—
284#41—62no—
285#41—61no—
286#41—182no—
287#41—182no—
288#41—182no—
289#41—182no—
290#41—184no—
291#41—182no—
292#41—182no—
293#41—184no—
294#41—182no—
295#41—181no—
296C2×Q8—31no—
297C2×Q8—32no—
298C2×Q8—61no—
299C2×Q8—91no—
300C2×Q8—91no—
301C2×Q8—182no—
302C2×Q8—182no—
303C2×Q8—181no—
304C2×Q8C3×D1212yes—
305C2×Q8C22×C3×D311yescharacteristic
306C2×D4—31no—
307C2×D4—91no—
308C2×D4—92no—
309C2×D4—91no—
310C2×D4—92no—
311C2×D4—92no—
312C2×D4—94no—
313C2×D4—91no—
314C2×D4—92no—
315C2×D4—182no—
316C2×D4—182no—
317C22×C4—31no—
318C22×C4—31no—
319C22×C4—91no—
320C22×C4—92no—
321C22×C4—92no—
322C22×C4—91no—
323C22×C4—92no—
324C22×C4—182no—
325C22×C4—182no—
326C4⋊C4—32no—
327C4⋊C4—32no—
328C4⋊C4—32no—
329C4⋊C4—92no—
330C22⋊C4—31no—
331C22⋊C4—32no—
332C22⋊C4—92no—
333C22⋊C4—92no—
334C22⋊C4—94no—
335C22⋊C4—92no—
336C22⋊C4D3×A411yescharacteristic
337C42—32no—
338C42—32no—
339C22×C3—22no—
340C22×C3—42no—
341C22×C3—41no—
342C22×C3—82no—
343C22×C3—82no—
344C22×C3—81no—
345C22×C3—162no—
346C22×C3—241no—
347C22×C3—242no—
348C22×C3D4×A412yes—
349C22×C3C23×A411yescharacteristic
350C2×D3—21no—
351C2×D3—22no—
352C2×D3—61no—
353C2×D3—62no—
354C2×D3—62no—
355C2×D3—62no—
356C2×D3—62no—
357C2×D3—124no—
358C3×C4—22no—
359C3×C4—32no—
360C3×C4—31no—
361C3×C4—31no—
362C3×C4—62no—
363C3×C4—82no—
364C3×C4—162no—
365C3×C4—242no—
366C3×C4—241no—
367Dic3—21no—
368Dic3—22no—
369Dic3—61no—
370C32—41noSylow
371C23—31no—
372C23—92no—
373C23—91no—
374C23—182no—
375C23—362no—
376C23C2×D3×A411yescharacteristic
377Q8—32no—
378Q8—31no—
379Q8—32no—
380Q8—182no—
381Q8—181no—
382Q8—181no—
383Q8C2×C3×D1211yescharacteristic
384D4—62no—
385D4—182no—
386D4—182no—
387D4—182no—
388D4—181no—
389D4—181no—
390D4—184no—
391D4—182no—
392D4—362no—
393D4—362no—
394D4—364no—
395C2×C4—32no—
396C2×C4—32no—
397C2×C4—31no—
398C2×C4—31no—
399C2×C4—32no—
400C2×C4—32no—
401C2×C4—31no—
402C2×C4—62no—
403C2×C4—91no—
404C2×C4—182no—
405C2×C4—182no—
406C2×C4—184no—
407C2×C4—181no—
408C2×C4—181no—
409C2×C4—181no—
410C2×C4—181no—
411C2×C4—184no—
412C2×C4C2×D3×A412yes—
413C2×C3—22no—
414C2×C3—41no—
415C2×C3—42no—
416C2×C3—82no—
417C2×C3—81no—
418C2×C3—82no—
419C2×C3—162no—
420C2×C3—481no—
421C2×C3C2×D4×A411yescharacteristic
422C2×C3#594512yes—
423D3—41no—
424D3—121no—
425D3—122no—
426C22—22no—
427C22—61no—
428C22—62no—
429C22—181no—
430C22—182no—
431C22—182no—
432C22—182no—
433C22—182no—
434C22—364no—
435C22A4×D1212yes—
436C22C22×D3×A411yescharacteristic
437C4—22no—
438C4—31no—
439C4—32no—
440C4—31no—
441C4—62no—
442C4—61no—
443C4—62no—
444C4—181no—
445C3—41no—
446C3—81no—
447C3#3487⋊C311yescharacteristic, complemented
448C2—22no—
449C2—121no—
450C2—362no—
451C2—361no—
452C2C2×A4×D1211yescharacteristic
453C2#5968⋊C312yes—
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×#519C2×C3
socleC22×C3C23×A4
hypercenterC22⋊C4D3×A4
hypocenterC3×Q8C2×C3×D4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#1007⋊C3C2semidirect
2. C22×C3⋊(#41⋊C3)C2semidirect
3. C3×#519⋊C3C2semidirect
4. #5576⋊C3C2semidirect
5. C4⋊(C3⋊(#41⋊C3))C2semidirect
6. #85874C3semidirect
10. C2×D3×SL(2,𝔽3)C22semidirect
13. C2×C3⋊(#41⋊C3)C22semidirect
15. C3×C4×SL(2,𝔽3)C22semidirect
17. C22×#239C2×C3semidirect
20. C3×#519C2×C3semidirect
22. #5577C2×C3semidirect
24. #5576C2×C3semidirect
32. #519⋊C3D3semidirect
35. C2×C3×SL(2,𝔽3)D4semidirect
91. C4×SL(2,𝔽3)C2×D3semidirect
110. #519C3×D3semidirect
161. C2×SL(2,𝔽3)D12semidirect
447. C3#3487⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#1007⋊C32
C22×C3⋊(#41⋊C3)2
C3×#519⋊C32
#5576⋊C32
C4⋊(C3⋊(#41⋊C3))2
#858743
#3487⋊C33
C3×#8864