Finite Group Databasebeta

← #13140597

Subgroups — #13140597

Order 1728.

545 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 545 conjugacy classes of subgroups — too many to draw. Only its 91 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×#88545C211yesmaximal, characteristic
2D4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
3#11721149C211yesmaximal, characteristic, complemented
4#63659⋊C3C211yesmaximal, characteristic, complemented
5#11720324C211yesmaximal, characteristic, complemented
6#11720315C211yesmaximal, characteristic, complemented
7#11720310C211yesmaximal, characteristic, complemented
8#10591435—31nomaximal
9Dic3×#213—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12C22×(C3⋊Dic3)⋊C3—21no—
13C2×#6199—21no—
14#88545C414yescomplemented
15C2×#6180C2211yescharacteristic
16#3757⋊C3—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88451—21no—
22D4×C3⋊Dic3D311yescharacteristic, complemented
23C3×D4×Dic3—31no—
24C2×C3×#213—31no—
25C2×C3×#213—31no—
26C2×C3×#213—61no—
27#63898—31no—
28Dic3×D12—31no—
29#63659D311yescharacteristic, complemented
30C3×#832—31no—
31#63575—31no—
32#63575—31no—
33#63563—31no—
34#63562—31no—
35#63553—31no—
36Dic3×C3⋊C8—31no—
37C23×H(3,𝔽3)—21no—
38C22×#272—21no—
39D4×H(3,𝔽3)—21no—
40D4×H(3,𝔽3)C2×C412yescomplemented
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C4⋊#272—21no—
43C4⋊#272C2×C412yescomplemented
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—42no—
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48C2×(C3⋊Dic3)⋊C3D411yescharacteristic
49#6199—42no—
50#6180C2×C412yescomplemented
51(C3⋊(C3⋊C8))⋊C3—21no—
52Dic3×D8—91no—
53C4×#213—91no—
54#4756—91no—
55C2×C32×D4—21no—
56C2×C32×D4C2×D311yescharacteristic
57C2×C32×D4C2×D311yescharacteristic
58C22×C3⋊Dic3—21no—
59C22×C3×Dic3—61no—
60C2×C3×D12—31no—
61C2×C3×D12—31no—
62C2×C3×D12—61no—
63C2×D3×Dic3—61no—
64C2×D3×Dic3—61no—
65#3757—21no—
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67C4⋊(C3⋊Dic3)C2×D311yescharacteristic
68C4×C3⋊Dic3C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C3×#217—61no—
71C3×#213—34no—
72C3×#213—34no—
73C3×#213—64no—
74C3×C4⋊Dic3—31no—
75C3×C4⋊Dic3—31no—
76C3×C4×Dic3—31no—
77C2×C3×C3⋊C8—31no—
78C2×C3×C3⋊C8—31no—
79C2×C3×C3⋊C8—31no—
80C2×C3×C3⋊C8—61no—
81(C2×D3)⋊Dic3—61no—
82(C2×D3)⋊Dic3—61no—
83C22×H(3,𝔽3)—22no—
84C22×H(3,𝔽3)—41no—
85C22×H(3,𝔽3)—41no—
86C22×H(3,𝔽3)C2×D411yescharacteristic
87C2×#272—22no—
88C2×#272—41no—
89C2×#272—41no—
90C4×H(3,𝔽3)C22×C411yescharacteristic
91C4×H(3,𝔽3)#4111yescharacteristic
92(C3⋊Dic3)⋊C3—21no—
93(C3⋊Dic3)⋊C3—41no—
94(C3⋊Dic3)⋊C3—41no—
95(C3⋊Dic3)⋊C3D812yescomplemented
96C2×C3×D8—91no—
97C3×C4×D4—91no—
98D4×Dic3—31no—
99D4×Dic3—91no—
100D4×Dic3—91no—
101C2×#213—31no—
102C2×#213—31no—
103C2×#213—61no—
104#893—91no—
105C4×D12—91no—
106C3×#102—91no—
107#832—31no—
108#832—91no—
109#832—91no—
110#821—91no—
111#818—91no—
112C8×Dic3—91no—
113C8×Dic3—91no—
114#809—91no—
115#807—91no—
116#807—91no—
117C4×C3⋊C8—91no—
118C23×C32—21no—
119C23×C32—21no—
120C23×C32—41no—
121C22×C3×D3—61no—
122C22×C3×D3—61no—
123C22×C3×D3—121no—
124C32×D4—21no—
125C32×D4—21no—
126C32×D4—22no—
127C32×D4—41no—
128C32×D4C4×D312yescomplemented
129C32×D4C4×D312yescomplemented
130C2×C32×C4—21no—
131C2×C32×C4C22×D311yescharacteristic
132C2×C32×C4C22×D311yescharacteristic
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3—21no—
135C2×C3⋊Dic3—42no—
136C2×C3⋊Dic3#6711yescharacteristic
137C2×C3⋊Dic3#6711yescharacteristic
138C2×C3×Dic3—31no—
139C2×C3×Dic3—31no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—122no—
143C3×D12—32no—
144C3×D12—32no—
145C3×D12—62no—
146C3×D12—61no—
147C3×D12—61no—
148C3×D12—121no—
149D3×Dic3—122no—
150D3×Dic3—122no—
151C3⋊(C3⋊C8)—21no—
152C3×C3⋊C8—32no—
153C3×C3⋊C8—32no—
154C3×C3⋊C8—62no—
155C3×C3⋊C8—61no—
156C4×D8—271noSylow
157C2×H(3,𝔽3)—42no—
158C2×H(3,𝔽3)C4×D411yescharacteristic
159C2×H(3,𝔽3)C2×D811yescharacteristic
160C2×H(3,𝔽3)#13511yescharacteristic
161#272—42no—
162C2×C3×D4—31no—
163C2×C3×D4—31no—
164C2×C3×D4—61no—
165C2×C3×D4—91no—
166C2×C3×D4D3211yescharacteristic
167C22×C3×C4—181no—
168C22×Dic3—61no—
169C22×Dic3—181no—
170C22×Dic3—181no—
171C2×D12—31no—
172C2×D12—31no—
173C2×D12—61no—
174C2×C4×D3—181no—
175C2×C4×D3—181no—
176C3×D8—94no—
177C2×C3×C8—91no—
178C2×C3×C8—91no—
179C3×C4⋊C4—91no—
180C3×C4⋊C4—91no—
181C3×C22⋊C4—181no—
182C3×C42—91no—
183#217—61no—
184#217—181no—
185#217—181no—
186#213—34no—
187#213—34no—
188#213—64no—
189(C2×D3)⋊C4—181no—
190(C2×D3)⋊C4—181no—
191C4⋊Dic3—31no—
192C4⋊Dic3—31no—
193C4⋊Dic3—91no—
194C4⋊Dic3—91no—
195C4×Dic3—31no—
196C4×Dic3—91no—
197C2×C3⋊C8—31no—
198C2×C3⋊C8—31no—
199C2×C3⋊C8—31no—
200C2×C3⋊C8—61no—
201C2×C3⋊C8—91no—
202C22×C32—22no—
203C22×C32—22no—
204C22×C32—21no—
205C22×C32—42no—
206C22×C32—41no—
207C22×C32—41no—
208C22×C32—41no—
209C22×C32—41no—
210C22×C32—81no—
211C22×C32—81no—
212C22×C32D3×D411yescharacteristic
213C22×C32C2×#6711yescharacteristic
214C2×C3×D3—62no—
215C2×C3×D3—62no—
216C2×C3×D3—122no—
217C2×C3×D3—121no—
218C2×C3×D3—121no—
219C2×C3×D3—121no—
220C2×C3×D3—121no—
221C2×C3×D3—241no—
222C2×C3×D3—241no—
223C32×C4—21no—
224C32×C4—21no—
225C32×C4C2×C4×D311yescharacteristic
226C32×C4C2×C4×D311yescharacteristic
227C32×C4#23511yescharacteristic
228C32×C4#23711yescharacteristic
229C3⋊Dic3—21no—
230C3⋊Dic3—41no—
231C3⋊Dic3—41no—
232C3⋊Dic3#21312yescomplemented
233C3×Dic3—32no—
234C3×Dic3—61no—
235C3×Dic3—121no—
236C3×Dic3—121no—
237C2×D8—91no—
238C4×D4—91no—
239C4×D4—271no—
240#107—271no—
241#102—91no—
242#102—271no—
243C4×C8—271no—
244H(3,𝔽3)C4×D811yescharacteristic, complemented, Sylow
245C23×C3—21no—
246C23×C3—61no—
247C23×C3—61no—
248C23×C3—121no—
249C23×C3—181no—
250C22×D3—61no—
251C22×D3—61no—
252C22×D3—121no—
253C3×D4—21no—
254C3×D4—32no—
255C3×D4—32no—
256C3×D4—61no—
257C3×D4—61no—
258C3×D4—62no—
259C3×D4—92no—
260C3×D4—121no—
261C3×D4—181no—
262C3×D4#62412yes—
263C2×C3×C4—31no—
264C2×C3×C4—31no—
265C2×C3×C4—61no—
266C2×C3×C4—91no—
267C2×C3×C4—91no—
268C2×C3×C4—181no—
269C2×C3×C4—181no—
270C2×C3×C4—362no—
271C2×C3×C4C2×D3211yescharacteristic
272C2×Dic3—31no—
273C2×Dic3—31no—
274C2×Dic3—61no—
275C2×Dic3—61no—
276C2×Dic3—91no—
277C2×Dic3—91no—
278C2×Dic3—122no—
279C2×Dic3—181no—
280C2×Dic3—181no—
281C2×Dic3—362no—
282C2×Dic3—362no—
283D12—32no—
284D12—32no—
285D12—62no—
286D12—61no—
287D12—61no—
288D12—121no—
289C4×D3—362no—
290C4×D3—362no—
291C3×C8—92no—
292C3×C8—181no—
293C3⋊C8—32no—
294C3⋊C8—32no—
295C3⋊C8—62no—
296C3⋊C8—61no—
297C3⋊C8—181no—
298C2×C32—21no—
299C2×C32—21no—
300C2×C32—21no—
301C2×C32—42no—
302C2×C32—42no—
303C2×C32—82no—
304C2×C32#67⋊C411yescharacteristic
305C2×C32D3×D811yescharacteristic
306C2×C32#91611yescharacteristic
307C2×C32C4×#6711yescharacteristic
308C2×C32C2×#21311yescharacteristic
309C2×C32#95011yescharacteristic
310C3×D3—122no—
311C3×D3—122no—
312C3×D3—242no—
313C2×D4—91no—
314C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
315C22×C4—181no—
316C22×C4—541no—
317D8—94no—
318C2×C8—91no—
319C2×C8—91no—
320C4⋊C4—91no—
321C4⋊C4—91no—
322C22⋊C4—181no—
323C22⋊C4—541no—
324C42—91no—
325C22×C3—22no—
326C22×C3—31no—
327C22×C3—31no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—62no—
331C22×C3—62no—
332C22×C3—61no—
333C22×C3—122no—
334C22×C3—121no—
335C22×C3—121no—
336C22×C3—121no—
337C22×C3—121no—
338C22×C3—182no—
339C22×C3—241no—
340C22×C3—241no—
341C22×C3—361no—
342C22×C3—361no—
343C22×C3D3×#6711yescharacteristic
344C2×D3—62no—
345C2×D3—62no—
346C2×D3—122no—
347C2×D3—121no—
348C2×D3—121no—
349C2×D3—121no—
350C2×D3—121no—
351C2×D3—241no—
352C2×D3—241no—
353C3×C4—31no—
354C3×C4—31no—
355C3×C4—31no—
356C3×C4—31no—
357C3×C4—61no—
358C3×C4—61no—
359C3×C4—92no—
360C3×C4—181no—
361C3×C4—361no—
362C3×C4—361no—
363C3×C4#380411yescharacteristic
364C3×C4C2×#62411yescharacteristic
365Dic3—32no—
366Dic3—61no—
367Dic3—92no—
368Dic3—121no—
369Dic3—121no—
370Dic3—181no—
371Dic3—361no—
372Dic3—361no—
373C32—21no—
374C32#475611yescharacteristic, complemented
375C32C4×#21311yescharacteristic, complemented
376C23—21no—
377C23—181no—
378D4—21no—
379D4—92no—
380D4—181no—
381D4#619912yescomplemented
382C2×C4—91no—
383C2×C4—91no—
384C2×C4—181no—
385C2×C4—181no—
386C2×C4—362no—
387C2×C4—1082no—
388C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
389C8—92no—
390C8—181no—
391C2×C3—31no—
392C2×C3—31no—
393C2×C3—31no—
394C2×C3—31no—
395C2×C3—31no—
396C2×C3—31no—
397C2×C3—42no—
398C2×C3—61no—
399C2×C3—61no—
400C2×C3—61no—
401C2×C3—122no—
402C2×C3—122no—
403C2×C3—242no—
404C2×C3—362no—
405C2×C3D3×#21311yescharacteristic
406C2×C3#6393511yescharacteristic
407C2×C3#6395211yescharacteristic
408D3—122no—
409D3—122no—
410D3—242no—
411C22—22no—
412C22—41no—
413C22—41no—
414C22—182no—
415C22—361no—
416C22—361no—
417C22#8867611yescharacteristic
418C4—92no—
419C4—181no—
420C4—361no—
421C4—361no—
422C4#8867111yescharacteristic
423C4C2×#619911yescharacteristic
424C3—31no—
425C3—31no—
426C3—61no—
427C3#1059140011yescharacteristic
428C2—42no—
429C2—362no—
430C2#1172119411yescharacteristic
431C2#1172119711yescharacteristic
432C2#1172129311yescharacteristic
subgroupG/H
centerC22#88676
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC2×#88545C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C4×D8
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. D4×(C3⋊Dic3)⋊C3C2semidirect
3. #11721149C2semidirect
4. #63659⋊C3C2semidirect
5. #11720324C2semidirect
6. #11720315C2semidirect
7. #11720310C2semidirect
14. #88545C4semidirect
22. D4×C3⋊Dic3D3semidirect
29. #63659D3semidirect
40. D4×H(3,𝔽3)C2×C4semidirect
43. C4⋊#272C2×C4semidirect
50. #6180C2×C4semidirect
95. (C3⋊Dic3)⋊C3D8semidirect
128. C32×D4C4×D3semidirect
129. C32×D4C4×D3semidirect
232. C3⋊Dic3#213semidirect
244. H(3,𝔽3)C4×D8semidirect
374. C32#4756semidirect
375. C32C4×#213semidirect
381. D4#6199semidirect

Maximal subgroups

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

subgroupindex
C2×#885452
D4×(C3⋊Dic3)⋊C32
#117211492
#63659⋊C32
#117203242
#117203152
#117203102
#105914353
Dic3×#2133