Finite Group Databasebeta

← #13139945

Subgroups — #13139945

Order 1728.

659 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 659 conjugacy classes of subgroups — too many to draw. Only its 131 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×#3828⋊C3C211yesmaximal, characteristic
2C2×C4⋊#1084C211yesmaximal, characteristic, complemented
3C2×#3748⋊C3C211yesmaximal, characteristic, complemented
4#11720309C212yesmaximal, complemented
5#11720308C212yesmaximal, complemented
6#10591106—31nomaximal
7#10591087—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084—21no—
10C2×#638⋊C3—21no—
11#3828⋊C3—22no—
12#3828⋊C3C414yescomplemented
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C2×C4×(C3⋊D3)⋊C3—21no—
15C2×#634⋊C3C2211yescharacteristic
16C4⋊#1084—21no—
17C4⋊#1084C2212yes—
18#3748⋊C3—22no—
19C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
20C2×#3828D311yescharacteristic
21C2×C3×#235—31no—
22C2×#3748D311yescharacteristic, complemented
23C2×C3×C4⋊Dic3—31no—
24C2×C3×C4⋊Dic3—31no—
25C2×C3×C4⋊Dic3—61no—
26C2×C3×#208—31no—
27#63567—32no—
28#63565—32no—
29#63564—32no—
30#63562—32no—
31C23×H(3,𝔽3)D411yescharacteristic
32C22×(C3⋊D3)⋊C3—21no—
33C2×C4×H(3,𝔽3)C2×C412yes—
34C2×C4×H(3,𝔽3)D411yescharacteristic
35C2×C4×H(3,𝔽3)D412yes—
36C2×C4×H(3,𝔽3)C2311yescharacteristic
37C2×#1084—22no—
38C2×#1084—42no—
39#638⋊C3—24no—
40C2×(C3⋊Dic3)⋊C3—21no—
41C4⋊((C3⋊D3)⋊C3)—21no—
42C4⋊((C3⋊D3)⋊C3)C2×C412yescomplemented
43C4×(C3⋊D3)⋊C3—24no—
44#634⋊C3—21no—
45#634⋊C3C2×C412yescomplemented
46(C3⋊(C3⋊C8))⋊C3—22no—
47#5232—91no—
48#5133—91no—
49#4966—91no—
50C22×C32×C4—21no—
51C22×C32×C4C2×D311yescharacteristic
52C22×C32×C4C2×D311yescharacteristic
53C2×#638—21no—
54#3828—22no—
55#3828Dic314yescomplemented
56C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
57C2×C4×C3⋊D3—21no—
58C2×#634C2×D311yescharacteristic
59C2×C3×#67—61no—
60C22×C3×Dic3—62no—
61C22×C3×Dic3—61no—
62C22×C3×Dic3—61no—
63C3×#235—34no—
64C3×#235—62no—
65C2×C3×D12—31no—
66C2×C3×C4×D3—61no—
67C2×C3×Dic6—31no—
68#3748—22no—
69C2×C3⋊(C3⋊C8)C2×D312yes—
70C3×C4⋊Dic3—32no—
71C3×C4⋊Dic3—32no—
72C3×C4⋊Dic3—62no—
73C3×C4⋊Dic3—62no—
74C3×C4⋊Dic3—61no—
75C3×C4⋊Dic3—61no—
76C3×#208—62no—
77C2×C3×C3⋊C8—32no—
78C22×H(3,𝔽3)—21no—
79C22×H(3,𝔽3)—21no—
80C22×H(3,𝔽3)C22⋊C412yes—
81C22×H(3,𝔽3)C2×D411yescharacteristic
82C2×(C3⋊D3)⋊C3—22no—
83C2×(C3⋊D3)⋊C3—41no—
84C2×(C3⋊D3)⋊C3—41no—
85C4×H(3,𝔽3)C22⋊C412yes—
86C4×H(3,𝔽3)C22×C411yescharacteristic
87C4×H(3,𝔽3)C2×D411yescharacteristic
88#1084—42no—
89(C3⋊Dic3)⋊C3—22no—
90C2×C3×#41—91no—
91C2×#235—31no—
92C2×#235—91no—
93C2×C3×M16—91no—
94C2×C3×C4⋊C4—91no—
95C2×C4⋊Dic3—31no—
96C2×C4⋊Dic3—31no—
97C2×C4⋊Dic3—61no—
98C2×#208—31no—
99C2×#208—91no—
100#835—92no—
101#832—92no—
102#821—92no—
103#816—92no—
104#810—92no—
105#809—92no—
106C23×C32—21no—
107C23×C32D1211yescharacteristic
108C23×C32#6711yescharacteristic
109C22×C3⋊D3—21no—
110C22×C3×D3—61no—
111C2×C32×C4—22no—
112C2×C32×C4—22no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4C4×D312yes—
116C2×C32×C4D1211yescharacteristic
117C2×C32×C4C2×Dic312yes—
118C2×C32×C4#6712yes—
119C2×C32×C4#6712yes—
120C2×C32×C4#6711yescharacteristic
121C2×C32×C4C22×D311yescharacteristic
122C2×C32×C4C22×D311yescharacteristic
123#638—24no—
124C2×C3⋊Dic3—21no—
125C4⋊(C3⋊D3)—21no—
126C4⋊(C3⋊D3)C2×Dic312yescomplemented
127C4×C3⋊D3—24no—
128#634—21no—
129#634C2×Dic312yescomplemented
130C3×#67—64no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—64no—
133C2×C3×Dic3—62no—
134C2×C3×Dic3—62no—
135C2×C3×Dic3—124no—
136C2×C3×Dic3—122no—
137C2×C3×Dic3—122no—
138C3×D12—32no—
139C3×D12—61no—
140C3×C4×D3—64no—
141C3×Dic6—32no—
142C3×Dic6—61no—
143C3⋊(C3⋊C8)—22no—
144C3×C3⋊C8—62no—
145#417—271noSylow
146C2×H(3,𝔽3)—22no—
147C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
148C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
149C2×H(3,𝔽3)#13711yescharacteristic
150(C3⋊D3)⋊C3—42no—
151C3×#41—94no—
152C3×#41—182no—
153C2×C3×Q8—91no—
154C2×C3×D4—91no—
155C2×C3×D4—181no—
156C22×C3×C4—31no—
157C22×C3×C4—31no—
158C22×C3×C4—61no—
159C22×C3×C4—181no—
160C22×C3×C4—181no—
161C22×C3×C4D3211yescharacteristic
162C2×#67—61no—
163C2×#67—181no—
164C22×Dic3—61no—
165C22×Dic3—61no—
166C22×Dic3—62no—
167#235—34no—
168#235—62no—
169#235—94no—
170#235—182no—
171C2×D12—31no—
172C2×D12—91no—
173C2×C4×D3—61no—
174C2×C4×D3—181no—
175C2×Dic6—31no—
176C2×Dic6—91no—
177C3×M16—182no—
178C2×C3×C8—92no—
179C3×C4⋊C4—92no—
180C3×C4⋊C4—181no—
181C4⋊Dic3—32no—
182C4⋊Dic3—32no—
183C4⋊Dic3—62no—
184C4⋊Dic3—61no—
185C4⋊Dic3—61no—
186C4⋊Dic3—62no—
187#208—62no—
188#208—182no—
189C2×C3⋊C8—32no—
190C2×C3⋊C8—92no—
191C22×C32—22no—
192C22×C32—22no—
193C22×C32—21no—
194C22×C32—21no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32—22no—
198C22×C32—21no—
199C22×C32(C2×D3)⋊C412yes—
200C22×C32#21712yes—
201C22×C32C2×D1211yescharacteristic
202C22×C32C2×#6711yescharacteristic
203C2×C3⋊D3—22no—
204C2×C3⋊D3—41no—
205C2×C3⋊D3—41no—
206C2×C3×D3—62no—
207C2×C3×D3—121no—
208C2×C3×D3—121no—
209C32×C4—22no—
210C32×C4—21no—
211C32×C4—21no—
212C32×C4(C2×D3)⋊C412yes—
213C32×C4#21712yes—
214C32×C4C2×C4×D311yescharacteristic
215C32×C4C22×Dic311yescharacteristic
216C32×C4C2×#6711yescharacteristic
217C32×C4C2×#6711yescharacteristic
218C3⋊Dic3—22no—
219C3×Dic3—62no—
220C3×Dic3—124no—
221C3×Dic3—122no—
222C3×Dic3—122no—
223C2×#41—91no—
224C2×M16—91no—
225C2×C4⋊C4—91no—
226#103—272no—
227#102—272no—
228H(3,𝔽3)#41711yescharacteristic, complemented, Sylow
229C23×C3—31no—
230C23×C3—31no—
231C23×C3—61no—
232C23×C3—181no—
233C23×C3#62611yescharacteristic
234C22×D3—61no—
235C22×D3—181no—
236C3×Q8—92no—
237C3×Q8—181no—
238C3×D4—92no—
239C3×D4—181no—
240C3×D4—184no—
241C2×C3×C4—32no—
242C2×C3×C4—32no—
243C2×C3×C4—32no—
244C2×C3×C4—32no—
245C2×C3×C4—31no—
246C2×C3×C4—31no—
247C2×C3×C4—31no—
248C2×C3×C4—31no—
249C2×C3×C4—62no—
250C2×C3×C4—62no—
251C2×C3×C4—61no—
252C2×C3×C4—61no—
253C2×C3×C4—184no—
254C2×C3×C4—181no—
255C2×C3×C4—182no—
256C2×C3×C4—362no—
257C2×C3×C4D3×Dic312yes—
258C2×C3×C4#62512yes—
259C2×C3×C4#62611yescharacteristic
260C2×C3×C4C2×D3211yescharacteristic
261#67—64no—
262#67—184no—
263C2×Dic3—61no—
264C2×Dic3—62no—
265C2×Dic3—62no—
266C2×Dic3—64no—
267C2×Dic3—122no—
268C2×Dic3—122no—
269C2×Dic3—124no—
270C2×Dic3—181no—
271D12—32no—
272D12—61no—
273D12—92no—
274D12—181no—
275C4×D3—64no—
276C4×D3—184no—
277Dic6—32no—
278Dic6—61no—
279Dic6—92no—
280Dic6—181no—
281C3×C8—182no—
282C3⋊C8—62no—
283C3⋊C8—182no—
284C2×C32—24no—
285C2×C32—22no—
286C2×C32—22no—
287C2×C32—21no—
288C2×C32—21no—
289C2×C32—21no—
290C2×C32#90811yescharacteristic
291C2×C32#90911yescharacteristic
292C2×C32C2×(C2×D3)⋊C411yescharacteristic
293C2×C32#94911yescharacteristic
294C2×C32#95111yescharacteristic
295C2×C32C2×#21711yescharacteristic
296C3⋊D3—42no—
297C3×D3—122no—
298#41—94no—
299#41—182no—
300C2×Q8—91no—
301C2×D4—91no—
302C2×D4—181no—
303C22×C4—181no—
304C22×C4—181no—
305C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
306M16—182no—
307C2×C8—92no—
308C4⋊C4—92no—
309C4⋊C4—181no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—31no—
313C22×C3—31no—
314C22×C3—32no—
315C22×C3—32no—
316C22×C3—61no—
317C22×C3—61no—
318C22×C3—61no—
319C22×C3—61no—
320C22×C3—62no—
321C22×C3—62no—
322C22×C3—61no—
323C22×C3—62no—
324C22×C3—182no—
325C22×C3—361no—
326C22×C3—361no—
327C22×C3(C2×D3)⋊Dic312yes—
328C22×C3C2×#62611yescharacteristic
329C2×D3—62no—
330C2×D3—121no—
331C2×D3—121no—
332C2×D3—182no—
333C2×D3—361no—
334C2×D3—361no—
335C3×C4—31no—
336C3×C4—31no—
337C3×C4—31no—
338C3×C4—31no—
339C3×C4—32no—
340C3×C4—32no—
341C3×C4—61no—
342C3×C4—61no—
343C3×C4—62no—
344C3×C4—182no—
345C3×C4—362no—
346C3×C4(C2×D3)⋊Dic312yes—
347C3×C4C2×D3×Dic311yescharacteristic
348C3×C4C2×#62511yescharacteristic
349Dic3—62no—
350Dic3—122no—
351Dic3—122no—
352Dic3—124no—
353Dic3—182no—
354C32—21no—
355C32#513311yescharacteristic, complemented
356C32#523211yescharacteristic, complemented
357C23—181no—
358C23#620011yescharacteristic
359Q8—92no—
360Q8—181no—
361D4—92no—
362D4—184no—
363D4—181no—
364C2×C4—181no—
365C2×C4—184no—
366C2×C4—182no—
367C2×C4—362no—
368C2×C4((C3⋊D3)⋊C3)⋊C412yes—
369C2×C4#619812yes—
370C2×C4#620011yescharacteristic
371C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
372C8—182no—
373C2×C3—22no—
374C2×C3—31no—
375C2×C3—31no—
376C2×C3—31no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3—31no—
380C2×C3—62no—
381C2×C3—62no—
382C2×C3—64no—
383C2×C3—61no—
384C2×C3—61no—
385C2×C3—61no—
386C2×C3—362no—
387C2×C3#6382511yescharacteristic
388C2×C3#6383111yescharacteristic
389C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
390D3—122no—
391D3—362no—
392C22—21no—
393C22—21no—
394C22—182no—
395C22—361no—
396C22—361no—
397C22#8845012yes—
398C22C2×#620011yescharacteristic
399C4—182no—
400C4—362no—
401C4#8845012yes—
402C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
403C4C2×#619811yescharacteristic
404C3—31no—
405C3—31no—
406C3—61no—
407C3#1059108811yescharacteristic
408C2—22no—
409C2—362no—
410C2#1172106811yescharacteristic
411C2#1172107411yescharacteristic
412C2C2×#8845011yescharacteristic
subgroupG/H
centerC22C2×#6200
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC2×#3828⋊C3C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#417
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊#1084C2semidirect
3. C2×#3748⋊C3C2semidirect
4. #11720309C2semidirect
5. #11720308C2semidirect
12. #3828⋊C3C4semidirect
22. C2×#3748D3semidirect
42. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
45. #634⋊C3C2×C4semidirect
55. #3828Dic3semidirect
126. C4⋊(C3⋊D3)C2×Dic3semidirect
129. #634C2×Dic3semidirect
228. H(3,𝔽3)#417semidirect
355. C32#5133semidirect
356. C32#5232semidirect

Maximal subgroups

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

subgroupindex
C2×#3828⋊C32
C2×C4⋊#10842
C2×#3748⋊C32
#117203092
#117203082
#105911063
#105910873