Finite Group Databasebeta

← #13137953

Subgroups — #13137953

Order 1728.

498 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 83 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
1#64102⋊C3C211yesmaximal, characteristic, complemented
2#11721132C211yesmaximal, characteristic, complemented
3C2×#88432C211yesmaximal, characteristic
4#11720530C211yesmaximal, characteristic, complemented
5#63636⋊C3C211yesmaximal, characteristic, complemented
6#11720312C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic, complemented
8#10590589—31nomaximal
9#10590462—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×C4×(C3⋊D3)⋊C3—21no—
13C2×((C3⋊D3)⋊C3)⋊C4—21no—
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15C4⋊#1084C2211yescharacteristic
16C2×#6180C2211yescharacteristic
17#3752⋊C3—21no—
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88450—21no—
21#88432C414yescomplemented
22#64102D311yescharacteristic, complemented
23C3×#893—31no—
24#63898—31no—
25#63894—31no—
26C2×#3714—31no—
27C2×#3714—31no—
28#63636D311yescharacteristic, complemented
29C3×#809—31no—
30C3×#808—31no—
31C3×#808—31no—
32C3×#808—61no—
33#63564—31no—
34#63562—31no—
35#63555—31no—
36#63554—31no—
37C22×(C3⋊D3)⋊C3—21no—
38C22×(C3⋊D3)⋊C3—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084—21no—
42C2×(C3⋊Dic3)⋊C3D411yescharacteristic
43C2×(C3⋊Dic3)⋊C3D411yescharacteristic
44C4⋊((C3⋊D3)⋊C3)—21no—
45C4⋊((C3⋊D3)⋊C3)—21no—
46C4⋊((C3⋊D3)⋊C3)C2×C412yescomplemented
47C4⋊((C3⋊D3)⋊C3)C2×C412yescomplemented
48C4×(C3⋊D3)⋊C3—42no—
49((C3⋊D3)⋊C3)⋊C4—42no—
50#6180C2×C412yescomplemented
51(C3⋊(C3⋊C8))⋊C3—21no—
52#5014—91no—
53#4869—91no—
54#4785—91no—
55C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
56C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
57C2×C4×C3⋊D3—21no—
58C2×C3×D12—31no—
59C2×C3×D12—31no—
60C2×C3×C4×D3—61no—
61C2×#624—61no—
62C2×D3×Dic3—61no—
63C32×C4⋊C4—21no—
64C32×C4⋊C4C2×D311yescharacteristic
65C32×C4⋊C4C2×D311yescharacteristic
66#3752—21no—
67C4×C3⋊Dic3C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
69C3×(C2×D3)⋊C4—61no—
70C3×C4⋊Dic3—31no—
71C3×C4⋊Dic3—31no—
72C3×C4⋊Dic3—61no—
73C3×C4×Dic3—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—61no—
78#3722—61no—
79(C2×D3)⋊Dic3—61no—
80#3714—34no—
81#3714—34no—
82C22×H(3,𝔽3)C2×D411yescharacteristic
83C2×(C3⋊D3)⋊C3—22no—
84C2×(C3⋊D3)⋊C3—22no—
85C2×(C3⋊D3)⋊C3—41no—
86C2×(C3⋊D3)⋊C3—41no—
87C2×(C3⋊D3)⋊C3—41no—
88C2×(C3⋊D3)⋊C3—41no—
89C4×H(3,𝔽3)—41no—
90C4×H(3,𝔽3)C22×C411yescharacteristic
91C4×H(3,𝔽3)#4111yescharacteristic
92#1084—41no—
93(C3⋊Dic3)⋊C3—21no—
94(C3⋊Dic3)⋊C3—21no—
95C3×C4×D4—91no—
96D4×Dic3—91no—
97C2×#213—91no—
98C2×#213—91no—
99C2×D24—91no—
100#893—31no—
101#893—91no—
102#893—91no—
103C4×D12—91no—
104C3×#106—91no—
105C3×#102—91no—
106#832—91no—
107#821—91no—
108#815—91no—
109#815—91no—
110#809—31no—
111#809—91no—
112#809—91no—
113#808—31no—
114#808—31no—
115#808—61no—
116#803—91no—
117C22×C3⋊D3—21no—
118C22×C3⋊D3—21no—
119C22×C3×D3—61no—
120C22×C3×D3—61no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—41no—
125C2×C32×C4C22×D311yescharacteristic
126C2×C32×C4C22×D311yescharacteristic
127C2×C3⋊Dic3#6711yescharacteristic
128C2×C3⋊Dic3#6711yescharacteristic
129C4⋊(C3⋊D3)—21no—
130C4⋊(C3⋊D3)—21no—
131C4⋊(C3⋊D3)C4×D312yescomplemented
132C4⋊(C3⋊D3)C4×D312yescomplemented
133C4×C3⋊D3—42no—
134C2×C3×Dic3—31no—
135C2×C3×Dic3—31no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—121no—
139C3×D12—32no—
140C3×D12—32no—
141C3×D12—61no—
142C3×D12—61no—
143C3×C4×D3—122no—
144#624—122no—
145D3×Dic3—122no—
146C3⋊(C3⋊C8)—21no—
147C3×C3⋊C8—32no—
148C3×C3⋊C8—32no—
149C3×C3⋊C8—62no—
150C3×C3⋊C8—61no—
151#442—271noSylow
152C2×H(3,𝔽3)C4×D411yescharacteristic
153C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
154C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
155(C3⋊D3)⋊C3—42no—
156(C3⋊D3)⋊C3—42no—
157C2×C3×D4—91no—
158C2×C3×D4—91no—
159C22×C3×C4—181no—
160C22×Dic3—181no—
161C2×D12—31no—
162C2×D12—31no—
163C2×D12—91no—
164C2×D12—91no—
165C2×C4×D3—61no—
166C2×C4×D3—181no—
167C2×C4×D3—181no—
168C2×C4×D3—181no—
169C2×C3×C8—91no—
170C2×C3×C8—91no—
171C3×C4⋊C4—31no—
172C3×C4⋊C4—31no—
173C3×C4⋊C4—61no—
174C3×C4⋊C4—91no—
175C3×C4⋊C4D3211yescharacteristic
176C3×C22⋊C4—181no—
177C3×C42—91no—
178#217—181no—
179#213—94no—
180#213—94no—
181(C2×D3)⋊C4—61no—
182(C2×D3)⋊C4—181no—
183(C2×D3)⋊C4—181no—
184(C2×D3)⋊C4—181no—
185C4⋊Dic3—31no—
186C4⋊Dic3—31no—
187C4⋊Dic3—61no—
188C4×Dic3—31no—
189C4×Dic3—91no—
190C2×C3⋊C8—31no—
191C2×C3⋊C8—31no—
192C2×C3⋊C8—31no—
193C2×C3⋊C8—61no—
194C2×C3⋊C8—91no—
195D24—94no—
196C22×C32—21no—
197C22×C32D3×D411yescharacteristic
198C22×C32C2×#6711yescharacteristic
199C2×C3⋊D3—22no—
200C2×C3⋊D3—22no—
201C2×C3⋊D3—41no—
202C2×C3⋊D3—41no—
203C2×C3⋊D3—41no—
204C2×C3⋊D3—41no—
205C2×C3×D3—62no—
206C2×C3×D3—62no—
207C2×C3×D3—121no—
208C2×C3×D3—121no—
209C2×C3×D3—121no—
210C2×C3×D3—121no—
211C32×C4—21no—
212C32×C4—21no—
213C32×C4—41no—
214C32×C4—41no—
215C32×C4—81no—
216C32×C4C2×C4×D311yescharacteristic
217C32×C4C2×C4×D311yescharacteristic
218C32×C4#23511yescharacteristic
219C32×C4#23711yescharacteristic
220C3⋊Dic3—21no—
221C3⋊Dic3—21no—
222C3×Dic3—61no—
223C3×Dic3—61no—
224C3×Dic3—121no—
225C3×Dic3—121no—
226C3×Dic3—241no—
227C2×D8—271no—
228C4×D4—91no—
229C4×D4—271no—
230#106—91no—
231#102—91no—
232#102—271no—
233#97—271no—
234H(3,𝔽3)#44211yescharacteristic, complemented, Sylow
235C23×C3—181no—
236C23×C3—181no—
237C22×D3—61no—
238C22×D3—61no—
239C22×D3—181no—
240C22×D3—181no—
241C3×D4—92no—
242C3×D4—92no—
243C3×D4—181no—
244C3×D4—181no—
245C2×C3×C4—21no—
246C2×C3×C4—31no—
247C2×C3×C4—31no—
248C2×C3×C4—61no—
249C2×C3×C4—61no—
250C2×C3×C4—61no—
251C2×C3×C4—91no—
252C2×C3×C4—91no—
253C2×C3×C4—121no—
254C2×C3×C4—181no—
255C2×C3×C4—362no—
256C2×C3×C4C2×D3211yescharacteristic
257C2×Dic3—31no—
258C2×Dic3—31no—
259C2×Dic3—61no—
260C2×Dic3—61no—
261C2×Dic3—91no—
262C2×Dic3—91no—
263C2×Dic3—121no—
264C2×Dic3—362no—
265D12—32no—
266D12—32no—
267D12—61no—
268D12—61no—
269D12—92no—
270D12—92no—
271D12—181no—
272D12—181no—
273C4×D3—122no—
274C4×D3—362no—
275C4×D3—362no—
276C4×D3—362no—
277C3×C8—92no—
278C3×C8—181no—
279C3⋊C8—32no—
280C3⋊C8—32no—
281C3⋊C8—62no—
282C3⋊C8—61no—
283C3⋊C8—181no—
284C2×C32—21no—
285C2×C32—21no—
286C2×C32—21no—
287C2×C32#67⋊C411yescharacteristic
288C2×C32#91111yescharacteristic
289C2×C32#91411yescharacteristic
290C2×C32C4×#6711yescharacteristic
291C2×C32#93211yescharacteristic
292C2×C32#94911yescharacteristic
293C3⋊D3—42no—
294C3⋊D3—42no—
295C3×D3—122no—
296C3×D3—122no—
297C2×D4—91no—
298C2×D4—91no—
299C22×C4—181no—
300C22×C4—541no—
301D8—274no—
302C2×C8—91no—
303C2×C8—91no—
304C4⋊C4—91no—
305C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
306C22⋊C4—181no—
307C22⋊C4—541no—
308C42—91no—
309C22×C3—31no—
310C22×C3—31no—
311C22×C3—61no—
312C22×C3—182no—
313C22×C3—182no—
314C22×C3—361no—
315C22×C3—361no—
316C22×C3—361no—
317C22×C3—361no—
318C22×C3D3×#6711yescharacteristic
319C2×D3—62no—
320C2×D3—62no—
321C2×D3—121no—
322C2×D3—121no—
323C2×D3—121no—
324C2×D3—121no—
325C2×D3—182no—
326C2×D3—182no—
327C2×D3—361no—
328C2×D3—361no—
329C2×D3—361no—
330C2×D3—361no—
331C3×C4—31no—
332C3×C4—31no—
333C3×C4—31no—
334C3×C4—31no—
335C3×C4—41no—
336C3×C4—61no—
337C3×C4—61no—
338C3×C4—121no—
339C3×C4—121no—
340C3×C4—181no—
341C3×C4—181no—
342C3×C4—241no—
343C3×C4—361no—
344C3×C4#379511yescharacteristic
345C3×C4C4×D3211yescharacteristic
346Dic3—61no—
347Dic3—61no—
348Dic3—121no—
349Dic3—121no—
350Dic3—181no—
351Dic3—181no—
352Dic3—241no—
353C32—21no—
354C32#478511yescharacteristic, complemented
355C32#501411yescharacteristic, complemented
356C23—181no—
357C23—181no—
358D4—92no—
359D4—92no—
360D4—181no—
361D4—181no—
362C2×C4—21no—
363C2×C4—91no—
364C2×C4—91no—
365C2×C4—181no—
366C2×C4—362no—
367C2×C4—1082no—
368C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
369C8—92no—
370C8—181no—
371C2×C3—31no—
372C2×C3—31no—
373C2×C3—31no—
374C2×C3—31no—
375C2×C3—31no—
376C2×C3—31no—
377C2×C3—61no—
378C2×C3—61no—
379C2×C3—61no—
380C2×C3—362no—
381C2×C3—362no—
382C2×C3#6387911yescharacteristic
383C2×C3#6393211yescharacteristic
384C2×C3#6393911yescharacteristic
385D3—122no—
386D3—122no—
387D3—362no—
388D3—362no—
389C22—182no—
390C22—182no—
391C22—361no—
392C22—361no—
393C22—361no—
394C22—361no—
395C22#8867611yescharacteristic
396C4—41no—
397C4—181no—
398C4—181no—
399C4—361no—
400C4#8864811yescharacteristic
401C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
402C3—31no—
403C3—31no—
404C3—61no—
405C3#1059045511yescharacteristic
406C2—362no—
407C2—362no—
408C2#1172112211yescharacteristic
409C2#1172120011yescharacteristic
410C2#1172123311yescharacteristic
subgroupG/H
centerC22#88676
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC2×#88432C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#442
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64102⋊C3C2semidirect
2. #11721132C2semidirect
4. #11720530C2semidirect
5. #63636⋊C3C2semidirect
6. #11720312C2semidirect
7. #11720308C2semidirect
21. #88432C4semidirect
22. #64102D3semidirect
28. #63636D3semidirect
46. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
47. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
50. #6180C2×C4semidirect
131. C4⋊(C3⋊D3)C4×D3semidirect
132. C4⋊(C3⋊D3)C4×D3semidirect
234. H(3,𝔽3)#442semidirect
354. C32#4785semidirect
355. C32#5014semidirect

Maximal subgroups

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

subgroupindex
#64102⋊C32
#117211322
C2×#884322
#117205302
#63636⋊C32
#117203122
#117203082
#105905893
#105904623