Finite Group Databasebeta

← #13140553

Subgroups — #13140553

Order 1728.

512 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 95 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
1D4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
2C2×#3753⋊C3C211yesmaximal, characteristic
3#11721132C211yesmaximal, characteristic, complemented
4#11720578C211yesmaximal, characteristic, complemented
5#11720318C211yesmaximal, characteristic, complemented
6#11720312C211yesmaximal, characteristic, complemented
7#11720307C211yesmaximal, characteristic, complemented
8#10591436—31nomaximal
9#10591404—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3—21no—
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×((C3⋊D3)⋊C3)⋊C4—21no—
14C4⋊#1084C2211yescharacteristic
15C2×#6180C2211yescharacteristic
16#3757⋊C3—21no—
17#3753⋊C3C414yescomplemented
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88450—21no—
22D4×C3⋊Dic3D311yescharacteristic, complemented
23C2×#3753D311yescharacteristic
24C3×D4×Dic3—31no—
25C2×C3×#213—31no—
26#63898—31no—
27#63894—31no—
28C3×#832—31no—
29C3×#832—31no—
30C3×#832—61no—
31#63574—31no—
32#63573—31no—
33#63564—31no—
34#63563—31no—
35#63555—31no—
36#63554—31no—
37C23×H(3,𝔽3)—21no—
38C22×(C3⋊D3)⋊C3—21no—
39D4×H(3,𝔽3)—21no—
40D4×H(3,𝔽3)C2×C412yescomplemented
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—42no—
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46C2×(C3⋊Dic3)⋊C3D411yescharacteristic
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)C2×C412yescomplemented
49((C3⋊D3)⋊C3)⋊C4—42no—
50#6180—21no—
51(C3⋊(C3⋊C8))⋊C3C2×C412yescomplemented
52#5152—91no—
53#5014—91no—
54#4785—91no—
55C2×C32×D4—21no—
56C2×C32×D4C2×D311yescharacteristic
57C2×C32×D4C2×D311yescharacteristic
58C22×C3⋊Dic3—21no—
59C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
60C22×C3×Dic3—61no—
61C2×C3×D12—31no—
62C2×#624—61no—
63C2×D3×Dic3—61no—
64#3757—21no—
65#3753Dic314yescomplemented
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67C4×C3⋊Dic3C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
69C3×#217—61no—
70C3×#213—34no—
71C3×C4⋊Dic3—31no—
72C3×C4⋊Dic3—31no—
73C3×C4⋊Dic3—31no—
74C3×C4⋊Dic3—61no—
75C3×C4×Dic3—31no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—31no—
78C2×C3×C3⋊C8—31no—
79C2×C3×C3⋊C8—61no—
80#3722—61no—
81(C2×D3)⋊Dic3—61no—
82C22×H(3,𝔽3)—22no—
83C22×H(3,𝔽3)—41no—
84C22×H(3,𝔽3)—41no—
85C22×H(3,𝔽3)C2×D411yescharacteristic
86C2×(C3⋊D3)⋊C3—22no—
87C2×(C3⋊D3)⋊C3—41no—
88C2×(C3⋊D3)⋊C3—41no—
89C4×H(3,𝔽3)C22×C411yescharacteristic
90C4×H(3,𝔽3)#4111yescharacteristic
91#1084—41no—
92(C3⋊Dic3)⋊C3—21no—
93(C3⋊Dic3)⋊C3—21no—
94(C3⋊Dic3)⋊C3—41no—
95C2×C3×D8—91no—
96C3×C4×D4—91no—
97D4×Dic3—31no—
98D4×Dic3—91no—
99D4×Dic3—91no—
100C2×#213—31no—
101C2×#213—91no—
102#893—91no—
103C4×D12—91no—
104C3×#102—91no—
105#832—31no—
106#832—31no—
107#832—61no—
108#832—91no—
109#821—91no—
110#817—91no—
111#815—91no—
112#815—91no—
113#809—91no—
114#808—91no—
115#808—91no—
116#803—91no—
117C23×C32—21no—
118C23×C32—21no—
119C23×C32—41no—
120C22×C3⋊D3—21no—
121C22×C3×D3—61no—
122C32×D4—21no—
123C32×D4—21no—
124C32×D4—22no—
125C32×D4—41no—
126C32×D4C4×D312yescomplemented
127C32×D4C2×Dic312yescomplemented
128C2×C32×C4—21no—
129C2×C32×C4C22×D311yescharacteristic
130C2×C32×C4C22×D311yescharacteristic
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3—42no—
133C2×C3⋊Dic3#6711yescharacteristic
134C2×C3⋊Dic3#6711yescharacteristic
135C4⋊(C3⋊D3)—21no—
136C4⋊(C3⋊D3)C2×Dic312yescomplemented
137C2×C3×Dic3—31no—
138C2×C3×Dic3—31no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—121no—
143C2×C3×Dic3—122no—
144C3×D12—32no—
145C3×D12—61no—
146#624—122no—
147D3×Dic3—122no—
148C3⋊(C3⋊C8)C2×Dic312yescomplemented
149C3×C3⋊C8—32no—
150C3×C3⋊C8—61no—
151C3×C3⋊C8—61no—
152C3×C3⋊C8—121no—
153#442—271noSylow
154C2×H(3,𝔽3)—42no—
155C2×H(3,𝔽3)C4×D411yescharacteristic
156C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
157C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
158(C3⋊D3)⋊C3—42no—
159C2×C3×D4—31no—
160C2×C3×D4—31no—
161C2×C3×D4—61no—
162C2×C3×D4—91no—
163C2×C3×D4D3211yescharacteristic
164C22×C3×C4—181no—
165C22×Dic3—61no—
166C22×Dic3—181no—
167C22×Dic3—181no—
168C2×D12—31no—
169C2×D12—91no—
170C2×C4×D3—181no—
171C2×C4×D3—181no—
172C3×D8—94no—
173C2×C3×C8—91no—
174C2×C3×C8—91no—
175C3×C4⋊C4—91no—
176C3×C4⋊C4—91no—
177C3×C22⋊C4—181no—
178C3×C42—91no—
179#217—61no—
180#217—181no—
181#217—181no—
182#213—34no—
183#213—94no—
184(C2×D3)⋊C4—181no—
185(C2×D3)⋊C4—181no—
186C4⋊Dic3—31no—
187C4⋊Dic3—31no—
188C4⋊Dic3—31no—
189C4⋊Dic3—61no—
190C4⋊Dic3—91no—
191C4×Dic3—31no—
192C4×Dic3—91no—
193C2×C3⋊C8—31no—
194C2×C3⋊C8—31no—
195C2×C3⋊C8—31no—
196C2×C3⋊C8—61no—
197C2×C3⋊C8—91no—
198C22×C32—22no—
199C22×C32—22no—
200C22×C32—21no—
201C22×C32—42no—
202C22×C32—41no—
203C22×C32—41no—
204C22×C32—41no—
205C22×C32—41no—
206C22×C32—81no—
207C22×C32—81no—
208C22×C32D3×D411yescharacteristic
209C22×C32C2×#6711yescharacteristic
210C2×C3⋊D3—22no—
211C2×C3⋊D3—41no—
212C2×C3⋊D3—41no—
213C2×C3×D3—62no—
214C2×C3×D3—121no—
215C2×C3×D3—121no—
216C32×C4—21no—
217C32×C4—21no—
218C32×C4C2×C4×D311yescharacteristic
219C32×C4#23511yescharacteristic
220C32×C4#23711yescharacteristic
221C32×C4C22×Dic311yescharacteristic
222C3⋊Dic3—21no—
223C3⋊Dic3—21no—
224C3⋊Dic3—41no—
225C3×Dic3—61no—
226C3×Dic3—61no—
227C3×Dic3—121no—
228C3×Dic3—121no—
229C3×Dic3—121no—
230C3×Dic3—241no—
231C2×D8—91no—
232C4×D4—91no—
233C4×D4—271no—
234#106—271no—
235#102—91no—
236#102—271no—
237#97—271no—
238H(3,𝔽3)#44211yescharacteristic, complemented, Sylow
239C23×C3—21no—
240C23×C3—61no—
241C23×C3—61no—
242C23×C3—121no—
243C23×C3—181no—
244C22×D3—61no—
245C22×D3—181no—
246C3×D4—21no—
247C3×D4—32no—
248C3×D4—32no—
249C3×D4—61no—
250C3×D4—61no—
251C3×D4—62no—
252C3×D4—92no—
253C3×D4—121no—
254C3×D4—181no—
255C3×D4D3×Dic312yes—
256C2×C3×C4—31no—
257C2×C3×C4—31no—
258C2×C3×C4—61no—
259C2×C3×C4—91no—
260C2×C3×C4—91no—
261C2×C3×C4—181no—
262C2×C3×C4—181no—
263C2×C3×C4—362no—
264C2×C3×C4C2×D3211yescharacteristic
265C2×Dic3—31no—
266C2×Dic3—31no—
267C2×Dic3—61no—
268C2×Dic3—61no—
269C2×Dic3—61no—
270C2×Dic3—91no—
271C2×Dic3—91no—
272C2×Dic3—121no—
273C2×Dic3—122no—
274C2×Dic3—181no—
275C2×Dic3—362no—
276C2×Dic3—362no—
277D12—32no—
278D12—61no—
279D12—92no—
280D12—181no—
281C4×D3—362no—
282C4×D3—362no—
283C3×C8—92no—
284C3×C8—181no—
285C3⋊C8—32no—
286C3⋊C8—61no—
287C3⋊C8—61no—
288C3⋊C8—92no—
289C3⋊C8—121no—
290C2×C32—21no—
291C2×C32—21no—
292C2×C32—21no—
293C2×C32—42no—
294C2×C32—42no—
295C2×C32—82no—
296C2×C32#91111yescharacteristic
297C2×C32#91111yescharacteristic
298C2×C32C4×#6711yescharacteristic
299C2×C32#93211yescharacteristic
300C2×C32D4×Dic311yescharacteristic
301C2×C32#94911yescharacteristic
302C3⋊D3—42no—
303C3×D3—122no—
304C2×D4—91no—
305C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
306C22×C4—181no—
307C22×C4—541no—
308D8—94no—
309C2×C8—91no—
310C2×C8—91no—
311C4⋊C4—91no—
312C4⋊C4—91no—
313C22⋊C4—181no—
314C22⋊C4—541no—
315C42—91no—
316C22×C3—22no—
317C22×C3—31no—
318C22×C3—31no—
319C22×C3—41no—
320C22×C3—41no—
321C22×C3—62no—
322C22×C3—62no—
323C22×C3—61no—
324C22×C3—122no—
325C22×C3—121no—
326C22×C3—121no—
327C22×C3—121no—
328C22×C3—121no—
329C22×C3—182no—
330C22×C3—241no—
331C22×C3—241no—
332C22×C3—361no—
333C22×C3—361no—
334C22×C3D3×#6711yescharacteristic
335C2×D3—62no—
336C2×D3—121no—
337C2×D3—121no—
338C2×D3—182no—
339C2×D3—361no—
340C2×D3—361no—
341C3×C4—31no—
342C3×C4—31no—
343C3×C4—31no—
344C3×C4—31no—
345C3×C4—61no—
346C3×C4—61no—
347C3×C4—181no—
348C3×C4—181no—
349C3×C4—361no—
350C3×C4—361no—
351C3×C4C2×D3×Dic311yescharacteristic
352C3×C4#380411yescharacteristic
353Dic3—61no—
354Dic3—61no—
355Dic3—121no—
356Dic3—121no—
357Dic3—121no—
358Dic3—181no—
359Dic3—181no—
360Dic3—241no—
361Dic3—361no—
362C32—21no—
363C32#501411yescharacteristic, complemented
364C32#515211yescharacteristic, complemented
365C23—21no—
366C23—181no—
367D4—21no—
368D4—92no—
369D4—181no—
370D4((C3⋊D3)⋊C3)⋊C412yescomplemented
371C2×C4—91no—
372C2×C4—91no—
373C2×C4—181no—
374C2×C4—181no—
375C2×C4—362no—
376C2×C4—1082no—
377C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
378C8—92no—
379C8—181no—
380C2×C3—31no—
381C2×C3—31no—
382C2×C3—31no—
383C2×C3—31no—
384C2×C3—31no—
385C2×C3—31no—
386C2×C3—42no—
387C2×C3—61no—
388C2×C3—61no—
389C2×C3—61no—
390C2×C3—122no—
391C2×C3—122no—
392C2×C3—242no—
393C2×C3—362no—
394C2×C3#6392511yescharacteristic
395C2×C3#6393211yescharacteristic
396C2×C3Dic3×#6711yescharacteristic
397D3—122no—
398D3—362no—
399C22—22no—
400C22—41no—
401C22—41no—
402C22—182no—
403C22—361no—
404C22—361no—
405C22#8867611yescharacteristic
406C4—181no—
407C4—181no—
408C4—361no—
409C4—361no—
410C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
411C4#8867111yescharacteristic
412C3—31no—
413C3—31no—
414C3—61no—
415C3#1059143011yescharacteristic
416C2—42no—
417C2—362no—
418C2#1172119511yescharacteristic
419C2#1172120011yescharacteristic
420C2#1172131411yescharacteristic
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×#3753⋊C3C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#442
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
1. D4×(C3⋊Dic3)⋊C3C2semidirect
3. #11721132C2semidirect
4. #11720578C2semidirect
5. #11720318C2semidirect
6. #11720312C2semidirect
7. #11720307C2semidirect
17. #3753⋊C3C4semidirect
22. D4×C3⋊Dic3D3semidirect
40. D4×H(3,𝔽3)C2×C4semidirect
48. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
51. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
65. #3753Dic3semidirect
126. C32×D4C4×D3semidirect
127. C32×D4C2×Dic3semidirect
136. C4⋊(C3⋊D3)C2×Dic3semidirect
148. C3⋊(C3⋊C8)C2×Dic3semidirect
238. H(3,𝔽3)#442semidirect
363. C32#5014semidirect
364. C32#5152semidirect
370. D4((C3⋊D3)⋊C3)⋊C4semidirect

Maximal subgroups

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

subgroupindex
D4×(C3⋊Dic3)⋊C32
C2×#3753⋊C32
#117211322
#117205782
#117203182
#117203122
#117203072
#105914363
#105914043