Finite Group Databasebeta

← #13140543

Subgroups — #13140543

Order 1728.

435 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 435 conjugacy classes of subgroups — too many to draw. Only its 69 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#64149⋊C3C211yesmaximal, characteristic
2#11721093C211yesmaximal, characteristic, complemented
3#11720578C211yesmaximal, characteristic, complemented
4#63659⋊C3C211yesmaximal, characteristic, complemented
5#11720318C211yesmaximal, characteristic, complemented
6#11720312C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic, complemented
8#10591480—31nomaximal
9#10591477—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3—21no—
12C2×#638⋊C3—21no—
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C4⋊#1084C2211yescharacteristic
15C2×#6180C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88449—21no—
20#88448—21no—
21#64149D311yescharacteristic
22C3×#940—31no—
23#63872—31no—
24#63869—31no—
25#63659D311yescharacteristic, complemented
26C3×#832—31no—
27C3×#832—31no—
28C3×#832—31no—
29C3×#832—61no—
30#63564—31no—
31#63564—31no—
32#63563—31no—
33#63562—31no—
34#63555—31no—
35#63554—31no—
36C23×H(3,𝔽3)—21no—
37C22×(C3⋊D3)⋊C3—21no—
38D4×H(3,𝔽3)—21no—
39D4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084—21no—
42#638⋊C3—42no—
43#638⋊C3—42no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46C2×(C3⋊Dic3)⋊C3D411yescharacteristic
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)—21no—
49#6180—21no—
50(C3⋊(C3⋊C8))⋊C3—21no—
51#5071—91no—
52#4764—91no—
53#4764—91no—
54C2×C32×D4—21no—
55C2×C32×D4C2×D311yescharacteristic
56C2×C32×D4C2×D311yescharacteristic
57C2×#638—21no—
58C2×#638—21no—
59C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
60C2×C3×#67—61no—
61C2×C3×#67—61no—
62C2×C3×D12—31no—
63C4⋊(C3⋊Dic3)C2×D311yescharacteristic
64C4×C3⋊Dic3C2×D311yescharacteristic
65C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—61no—
70C3×C4×Dic3—31no—
71C2×C3×C3⋊C8—31no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—61no—
75#3724—61no—
76Dic3⋊Dic3—61no—
77Dic3⋊Dic3—61no—
78Dic3⋊Dic3—61no—
79C22×H(3,𝔽3)—41no—
80C22×H(3,𝔽3)—41no—
81C22×H(3,𝔽3)—41no—
82C22×H(3,𝔽3)C2×D411yescharacteristic
83C2×(C3⋊D3)⋊C3—41no—
84C2×(C3⋊D3)⋊C3—41no—
85C2×(C3⋊D3)⋊C3—41no—
86C4×H(3,𝔽3)#4111yescharacteristic
87C4×H(3,𝔽3)#4111yescharacteristic
88#1084—41no—
89(C3⋊Dic3)⋊C3—21no—
90(C3⋊Dic3)⋊C3—21no—
91(C3⋊Dic3)⋊C3—41no—
92C3×#127—91no—
93#940—31no—
94#940—91no—
95#890—91no—
96#890—91no—
97#870—91no—
98C3×#102—91no—
99C3×#102—91no—
100#832—31no—
101#832—31no—
102#832—31no—
103#832—61no—
104#832—91no—
105#832—91no—
106#832—91no—
107#821—91no—
108#821—91no—
109#815—91no—
110#815—91no—
111#809—91no—
112#809—91no—
113#803—91no—
114C23×C32—21no—
115C23×C32—21no—
116C23×C32—41no—
117C22×C3⋊D3—21no—
118C22×C3×D3—61no—
119C32×D4—22no—
120C32×D4—21no—
121C32×D4—21no—
122C32×D4—21no—
123C32×D4—21no—
124C32×D4—41no—
125C2×C32×C4—21no—
126C2×C32×C4C22×D311yescharacteristic
127C2×C32×C4C22×D311yescharacteristic
128#638—42no—
129#638—42no—
130C2×C3⋊Dic3—21no—
131C2×C3⋊Dic3#6711yescharacteristic
132C2×C3⋊Dic3#6711yescharacteristic
133C4⋊(C3⋊D3)—21no—
134C4⋊(C3⋊D3)—21no—
135C3×#67—122no—
136C3×#67—122no—
137C2×C3×Dic3—31no—
138C2×C3×Dic3—31no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—121no—
143C3×D12—61no—
144C3×D12—61no—
145C3⋊(C3⋊C8)—21no—
146C3×C3⋊C8—61no—
147C3×C3⋊C8—61no—
148C3×C3⋊C8—61no—
149C3×C3⋊C8—121no—
150#490—271noSylow
151C2×H(3,𝔽3)—81no—
152C2×H(3,𝔽3)#12411yescharacteristic
153C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
154C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
155(C3⋊D3)⋊C3—81no—
156C2×C3×D4—31no—
157C2×C3×D4—31no—
158C2×C3×D4—61no—
159C2×C3×D4—91no—
160C2×C3×D4—181no—
161C2×C3×D4—181no—
162C2×C3×D4D3211yescharacteristic
163C2×#67—61no—
164C2×#67—61no—
165C2×#67—181no—
166C2×#67—181no—
167C2×D12—31no—
168C2×D12—91no—
169C2×C3×C8—91no—
170C2×C3×C8—91no—
171C3×C4⋊C4—91no—
172C3×C4⋊C4—91no—
173C3×C42—91no—
174C4⋊Dic3—31no—
175C4⋊Dic3—31no—
176C4⋊Dic3—31no—
177C4⋊Dic3—61no—
178C4⋊Dic3—91no—
179C4⋊Dic3—181no—
180C4⋊Dic3—181no—
181Dic3⋊C4—181no—
182Dic3⋊C4—181no—
183Dic3⋊C4—181no—
184Dic3⋊C4—181no—
185C4×Dic3—31no—
186C4×Dic3—91no—
187C2×C3⋊C8—31no—
188C2×C3⋊C8—31no—
189C2×C3⋊C8—31no—
190C2×C3⋊C8—61no—
191C2×C3⋊C8—91no—
192C22×C32—21no—
193C22×C32—42no—
194C22×C32—41no—
195C22×C32—41no—
196C22×C32—41no—
197C22×C32—41no—
198C22×C32—41no—
199C22×C32—41no—
200C22×C32—81no—
201C22×C32—81no—
202C22×C32D3×D411yescharacteristic
203C22×C32C2×#6711yescharacteristic
204C2×C3⋊D3—41no—
205C2×C3⋊D3—41no—
206C2×C3⋊D3—41no—
207C2×C3×D3—121no—
208C2×C3×D3—121no—
209C2×C3×D3—121no—
210C32×C4—21no—
211C32×C4—21no—
212C32×C4#23511yescharacteristic
213C32×C4#23711yescharacteristic
214C32×C4#23711yescharacteristic
215C32×C4#23711yescharacteristic
216C3⋊Dic3—21no—
217C3⋊Dic3—21no—
218C3⋊Dic3—41no—
219C3×Dic3—61no—
220C3×Dic3—61no—
221C3×Dic3—121no—
222C3×Dic3—121no—
223C3×Dic3—121no—
224C3×Dic3—241no—
225#127—91no—
226#125—271no—
227#102—91no—
228#102—91no—
229#102—271no—
230#102—271no—
231#97—271no—
232H(3,𝔽3)#49011yescharacteristic, complemented, Sylow
233C23×C3—21no—
234C23×C3—61no—
235C23×C3—61no—
236C23×C3—121no—
237C23×C3—181no—
238C22×D3—61no—
239C22×D3—181no—
240C3×D4—21no—
241C3×D4—21no—
242C3×D4—61no—
243C3×D4—61no—
244C3×D4—61no—
245C3×D4—61no—
246C3×D4—62no—
247C3×D4—121no—
248C3×D4—181no—
249C3×D4—181no—
250C3×D4—362no—
251C3×D4—362no—
252C2×C3×C4—31no—
253C2×C3×C4—31no—
254C2×C3×C4—61no—
255C2×C3×C4—91no—
256C2×C3×C4—91no—
257C2×C3×C4—181no—
258C2×C3×C4—181no—
259C2×C3×C4C2×D3211yescharacteristic
260#67—122no—
261#67—122no—
262#67—362no—
263#67—362no—
264C2×Dic3—31no—
265C2×Dic3—31no—
266C2×Dic3—61no—
267C2×Dic3—61no—
268C2×Dic3—61no—
269C2×Dic3—91no—
270C2×Dic3—91no—
271C2×Dic3—121no—
272C2×Dic3—181no—
273D12—61no—
274D12—61no—
275D12—181no—
276D12—181no—
277C3×C8—181no—
278C3×C8—181no—
279C3⋊C8—61no—
280C3⋊C8—61no—
281C3⋊C8—61no—
282C3⋊C8—121no—
283C3⋊C8—181no—
284C2×C32—21no—
285C2×C32—21no—
286C2×C32—21no—
287C2×C32—82no—
288C2×C32—81no—
289C2×C32—81no—
290C2×C32#88511yescharacteristic
291C2×C32#91111yescharacteristic
292C2×C32#91411yescharacteristic
293C2×C32#93211yescharacteristic
294C2×C32#93211yescharacteristic
295C2×C32#93611yescharacteristic
296C3⋊D3—81no—
297C3×D3—241no—
298C2×D4—91no—
299C2×D4—181no—
300C2×D4—181no—
301C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
302C2×C8—91no—
303C2×C8—91no—
304C4⋊C4—91no—
305C4⋊C4—91no—
306C4⋊C4—541no—
307C4⋊C4—541no—
308C42—91no—
309C22×C3—31no—
310C22×C3—31no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3—41no—
314C22×C3—61no—
315C22×C3—121no—
316C22×C3—121no—
317C22×C3—121no—
318C22×C3—121no—
319C22×C3—121no—
320C22×C3—121no—
321C22×C3—122no—
322C22×C3—241no—
323C22×C3—241no—
324C22×C3—361no—
325C22×C3—361no—
326C22×C3—361no—
327C22×C3D3×#6711yescharacteristic
328C2×D3—121no—
329C2×D3—121no—
330C2×D3—121no—
331C2×D3—361no—
332C2×D3—361no—
333C2×D3—361no—
334C3×C4—31no—
335C3×C4—31no—
336C3×C4—31no—
337C3×C4—31no—
338C3×C4—61no—
339C3×C4—61no—
340C3×C4—181no—
341C3×C4—181no—
342C3×C4—361no—
343C3×C4—361no—
344C3×C4#380411yescharacteristic
345C3×C4#380511yescharacteristic
346Dic3—61no—
347Dic3—61no—
348Dic3—121no—
349Dic3—121no—
350Dic3—121no—
351Dic3—181no—
352Dic3—181no—
353Dic3—241no—
354Dic3—361no—
355C32—21no—
356C32#476411yescharacteristic, complemented
357C32#507111yescharacteristic, complemented
358C23—21no—
359C23—181no—
360D4—21no—
361D4—21no—
362D4—181no—
363D4—181no—
364D4—362no—
365D4—362no—
366C2×C4—91no—
367C2×C4—91no—
368C2×C4—181no—
369C2×C4—181no—
370C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
371C8—181no—
372C8—181no—
373C2×C3—31no—
374C2×C3—31no—
375C2×C3—31no—
376C2×C3—31no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3—61no—
380C2×C3—61no—
381C2×C3—61no—
382C2×C3—81no—
383C2×C3—241no—
384C2×C3—241no—
385C2×C3—242no—
386C2×C3—721no—
387C2×C3#6393211yescharacteristic
388C2×C3#6393411yescharacteristic
389C2×C3#6395911yescharacteristic
390D3—241no—
391D3—721no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—361no—
396C22—361no—
397C22—361no—
398C22#8867611yescharacteristic
399C4—181no—
400C4—181no—
401C4—361no—
402C4—361no—
403C4#8867111yescharacteristic
404C4#8867211yescharacteristic
405C3—31no—
406C3—31no—
407C3—61no—
408C3#1059147711yescharacteristic
409C2—81no—
410C2—721no—
411C2#1172119611yescharacteristic
412C2#1172120011yescharacteristic
413C2#1172130411yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radical#64149⋊C3C2
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#490
LCSG ▷ C2×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. #11721093C2semidirect
3. #11720578C2semidirect
4. #63659⋊C3C2semidirect
5. #11720318C2semidirect
6. #11720312C2semidirect
7. #11720308C2semidirect
25. #63659D3semidirect
232. H(3,𝔽3)#490semidirect
356. C32#4764semidirect
357. C32#5071semidirect

Maximal subgroups

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

subgroupindex
#64149⋊C32
#117210932
#117205782
#63659⋊C32
#117203182
#117203122
#117203082
#105914803
#105914773