Finite Group Databasebeta

← #10588507⋊C3 #13141995

Subgroups — #10588507⋊C3 #13141995

Order 1728.

460 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 90 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#11722642C211yesmaximal, characteristic, complemented
2#63512⋊C3C211yesmaximal, characteristic, complemented
3#63470⋊C3C211yesmaximal, characteristic, complemented
4#63437⋊C3C211yesmaximal, characteristic, complemented
53−1+2×#104C211yesmaximal, characteristic, complemented
6#63382⋊C3C211yesmaximal, characteristic, complemented
7#63364⋊C3C211yesmaximal, characteristic, complemented
8C3×#4827—31nomaximal
9#10588507C311yesmaximal, characteristic, complemented
10#41×3−1+2C2211yescharacteristic
11Q8×AGL(1,ℤ/9)—21no—
12#3699⋊C3—21no—
13#3699⋊C3—22no—
14#3697⋊C3—21no—
15#3697⋊C3C2211yescharacteristic
16C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
17C2×3−1+2⋊Q8—21no—
18C2×3−1+2⋊Q8C2211yescharacteristic
19M16×3−1+2C2211yescharacteristic
20C42×3−1+2C2211yescharacteristic
21#3675⋊C3—21no—
22#3673⋊C3—21no—
23(C2×AGL(1,ℤ/9))⋊C4—22no—
24#3667⋊C3C2211yescharacteristic
25D72⋊C3—21no—
26#3664⋊C3—21no—
27C3×#1010—31no—
28C3×#949—31no—
29C3×#908—31no—
30C3×#875—31no—
31#63716C2×C311yescharacteristic, complemented
32C32×#104D311yescharacteristic
33C3×#824—31no—
34C3×#805—31no—
35#63512C2×C311yescharacteristic, complemented
36#63470C2×C311yescharacteristic, complemented
37#63437C2×C311yescharacteristic, complemented
38C9×#104—21no—
39C9×#104C2×C311yescharacteristic, complemented
40#63382C2×C311yescharacteristic, complemented
41#63364C2×C311yescharacteristic, complemented
42C22×AGL(1,ℤ/9)—21no—
43Q8×3−1+2D411yescharacteristic
44D4×3−1+2—21no—
45D4×3−1+2D411yescharacteristic
46C2×C4×3−1+2—21no—
47C2×C4×3−1+2—21no—
48C2×C4×3−1+2C2311yescharacteristic
49#611⋊C3—22no—
50#611⋊C3—21no—
51C2×3−1+2⋊C4—22no—
52C2×3−1+2⋊C4D412yes—
53C4⋊AGL(1,ℤ/9)—21no—
54C4⋊AGL(1,ℤ/9)—21no—
55C4⋊AGL(1,ℤ/9)D411yescharacteristic
56C4×AGL(1,ℤ/9)—22no—
57C4×AGL(1,ℤ/9)—21no—
583−1+2⋊Q8—22no—
593−1+2⋊Q8—21no—
603−1+2⋊Q8—21no—
613−1+2⋊Q8D411yescharacteristic
62C8×3−1+2—21no—
633−1+2⋊C8—21no—
64C3×#454—91no—
65#4827—31no—
66C32×#41C2×D311yescharacteristic
67C3×D3×Q8—61no—
68C3×#237—62no—
69C3×#237—61no—
70C3×#235—31no—
71C3×#235—61no—
72C2×C3×D12—31no—
73C2×C3×Dic6—31no—
74C2×C3×Dic6—61no—
75C32×M16C2×D311yescharacteristic
76C32×C42C2×D311yescharacteristic
77C3×#215—61no—
78C3×#213—61no—
79C3×(C2×D3)⋊C4—62no—
80C3×#208—31no—
81C3×D24—61no—
82C3×#204—61no—
83C9×#41—21no—
84C9×#41C22×C311yescharacteristic
85Q8×D9—21no—
86#3699—21no—
87#3699—22no—
88#3697—21no—
89#3697C22×C311yescharacteristic
90C2×D36C22×C311yescharacteristic
91C2×Dic18—21no—
92C2×Dic18C22×C311yescharacteristic
93C9×M16—21no—
94C9×M16C22×C311yescharacteristic
95C42×C9—21no—
96C42×C9C22×C311yescharacteristic
97#3675—21no—
98#3673—21no—
99(C2×D9)⋊C4—22no—
100#3667C22×C311yescharacteristic
101D72—21no—
102#3664—21no—
103C22×3−1+2—21no—
104C22×3−1+2C2×D411yescharacteristic
105C2×AGL(1,ℤ/9)—21no—
106C2×AGL(1,ℤ/9)—42no—
107C2×AGL(1,ℤ/9)—41no—
108C4×3−1+2—21no—
109C4×3−1+2—41no—
110C4×3−1+2C2×D411yescharacteristic
111C4×3−1+2C2×D411yescharacteristic
1123−1+2⋊C4—22no—
1133−1+2⋊C4—21no—
114C3×2−1+4—91no—
115#1010—31no—
116C3×AGL(1,ℤ/8)—91no—
117C3×AGL(1,ℤ/8)—91no—
118C3×#124—91no—
119#949—31no—
120#908—31no—
121#875—31no—
122C3×#104—31no—
123C3×#104—91no—
124C3×#104C3×D311yescharacteristic
125C3×#101—91no—
126#824—31no—
127#805—31no—
128C22×C3×D3—61no—
129C32×Q8D1211yescharacteristic
130C32×D4—21no—
131C32×D4D1211yescharacteristic
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4C22×D311yescharacteristic
135C3×#67—62no—
136C3×#67—61no—
137C2×C3×Dic3—32no—
138C2×C3×Dic3—62no—
139C3×D12—31no—
140C3×D12—61no—
141C3×D12—61no—
142C3×C4×D3—62no—
143C3×C4×D3—61no—
144C3×Dic6—31no—
145C3×Dic6—62no—
146C3×Dic6—61no—
147C3×Dic6—61no—
148C22×D9—21no—
149C32×C8—21no—
150C3×C3⋊C8—61no—
151Q8×C9—21no—
152Q8×C9C3×D411yescharacteristic
153D4×C9—21no—
154D4×C9—21no—
155D4×C9—41no—
156D4×C9C3×D411yescharacteristic
157C2×C4×C9—21no—
158C2×C4×C9—21no—
159C2×C4×C9—21no—
160C2×C4×C9—41no—
161C2×C4×C9—41no—
162C2×C4×C9C23×C311yescharacteristic
163#611—22no—
164#611—21no—
165C2×Dic9—22no—
166C2×Dic9C3×D412yes—
167D36—21no—
168D36—21no—
169D36C3×D411yescharacteristic
170C4×D9—22no—
171C4×D9—21no—
172Dic18—22no—
173Dic18—21no—
174Dic18—21no—
175Dic18C3×D411yescharacteristic
176C8×C9—21no—
177C8×C9—41no—
178C9⋊C8—21no—
179#454—91noSylow
180C2×3−1+2—21no—
181C2×3−1+2—41no—
182C2×3−1+2#12011yescharacteristic
183AGL(1,ℤ/9)—41no—
184AGL(1,ℤ/9)—81no—
185C3×#41—31no—
186C3×#41—91no—
187C3×#41—181no—
188C3×#41—182no—
189C3×#41—181no—
190C3×#41C2×C3×D311yescharacteristic
191C2×C3×Q8—91no—
192C2×C3×Q8—181no—
193C2×C3×Q8—181no—
194C2×C3×D4—91no—
195D3×Q8—61no—
196#237—62no—
197#237—61no—
198#235—31no—
199#235—61no—
200C2×D12—31no—
201C2×Dic6—31no—
202C2×Dic6—61no—
203C3×SD16—181no—
204C3×SD16—181no—
205C3×D8—181no—
206C3×D8—181no—
207C3×M16—31no—
208C3×M16—91no—
209C3×M16C2×C3×D311yescharacteristic
210C3×C22⋊C4—182no—
211C3×C42—31no—
212C3×C42C2×C3×D311yescharacteristic
213#215—61no—
214#213—61no—
215(C2×D3)⋊C4—62no—
216#208—31no—
217D24—61no—
218#204—61no—
219C22×C32—21no—
220C22×C32D3×D411yescharacteristic
221C2×C3×D3—61no—
222C2×C3×D3—122no—
223C2×C3×D3—121no—
224C32×C4—21no—
225C32×C4—41no—
226C32×C4C2×D1211yescharacteristic
227C32×C4D3×D411yescharacteristic
228C3×Dic3—61no—
229C3×Dic3—62no—
230C22×C9—21no—
231C22×C9—21no—
232C22×C9—41no—
233C22×C9C2×C3×D411yescharacteristic
234C2×D9—21no—
235C2×D9—42no—
236C2×D9—41no—
237C4×C9—21no—
238C4×C9—21no—
239C4×C9—21no—
240C4×C9—41no—
241C4×C9—41no—
242C4×C9—42no—
243C4×C9C2×C3×D411yescharacteristic
244C4×C9C2×C3×D411yescharacteristic
245Dic9—22no—
246Dic9—21no—
2472−1+4—91no—
248AGL(1,ℤ/8)—91no—
249AGL(1,ℤ/8)—91no—
250#124—91no—
251#104—91no—
252#104AGL(1,ℤ/9)11yescharacteristic, complemented
253#101—91no—
2543−1+2#45411yescharacteristic, complemented, Sylow
255C23×C3—181no—
256C22×D3—61no—
257C3×Q8—31no—
258C3×Q8—91no—
259C3×Q8—181no—
260C3×Q8—181no—
261C3×Q8—182no—
262C3×Q8C3×D1211yescharacteristic
263C3×D4—21no—
264C3×D4—31no—
265C3×D4—61no—
266C3×D4—91no—
267C3×D4—181no—
268C3×D4—181no—
269C3×D4—181no—
270C3×D4—182no—
271C3×D4C3×D1211yescharacteristic
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—31no—
275C2×C3×C4—61no—
276C2×C3×C4—61no—
277C2×C3×C4—92no—
278C2×C3×C4—181no—
279C2×C3×C4—182no—
280C2×C3×C4—182no—
281C2×C3×C4C22×C3×D311yescharacteristic
282#67—62no—
283#67—61no—
284C2×Dic3—32no—
285C2×Dic3—62no—
286D12—31no—
287D12—61no—
288D12—61no—
289C4×D3—62no—
290C4×D3—61no—
291Dic6—31no—
292Dic6—62no—
293Dic6—61no—
294Dic6—61no—
295C3×C8—21no—
296C3×C8—61no—
297C3×C8—181no—
298C3⋊C8—61no—
299C2×C32—21no—
300C2×C32—41no—
301C2×C32#88211yescharacteristic
302C3×D3—121no—
303C3×D3—241no—
304C2×C9—21no—
305C2×C9—21no—
306C2×C9—41no—
307C2×C9—41no—
308C2×C9—81no—
309C2×C9C3×#12011yescharacteristic
310D9—41no—
311D9—81no—
312#41—91no—
313#41—181no—
314#41—182no—
315#41—181no—
316#41C2×AGL(1,ℤ/9)11yescharacteristic
317C2×Q8—91no—
318C2×Q8—181no—
319C2×Q8—181no—
320C2×D4—91no—
321SD16—181no—
322SD16—181no—
323D8—181no—
324D8—181no—
325M16—91no—
326M16C2×AGL(1,ℤ/9)11yescharacteristic
327C22⋊C4—182no—
328C42C2×AGL(1,ℤ/9)11yescharacteristic
329C22×C3—21no—
330C22×C3—31no—
331C22×C3—61no—
332C22×C3—181no—
333C22×C3—361no—
334C22×C3—362no—
335C22×C3C3×D3×D411yescharacteristic
336C2×D3—61no—
337C2×D3—122no—
338C2×D3—121no—
339C3×C4—21no—
340C3×C4—31no—
341C3×C4—31no—
342C3×C4—41no—
343C3×C4—61no—
344C3×C4—121no—
345C3×C4—182no—
346C3×C4—181no—
347C3×C4C2×C3×D1211yescharacteristic
348C3×C4C3×D3×D411yescharacteristic
349Dic3—61no—
350Dic3—62no—
351C32#482711yescharacteristic
352C9—21no—
353C9C3×#45411yescharacteristic, complemented
354C23—181no—
355Q8—91no—
356Q8—182no—
357Q8—181no—
358Q8—181no—
359Q8C4⋊AGL(1,ℤ/9)11yescharacteristic
360D4—21no—
361D4—91no—
362D4—182no—
363D4—181no—
364D4—181no—
365D4—181no—
366D4C4⋊AGL(1,ℤ/9)11yescharacteristic
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—92no—
370C2×C4—182no—
371C2×C4—182no—
372C2×C4—181no—
373C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
374C8—21no—
375C8—181no—
376C2×C3—21no—
377C2×C3—31no—
378C2×C3—41no—
379C2×C3—61no—
380C2×C3—121no—
381C2×C3—361no—
382C2×C3—721no—
383C2×C3C3×#88211yescharacteristic
384D3—121no—
385D3—241no—
386C22—21no—
387C22—181no—
388C22—362no—
389C22—361no—
390C22D4×AGL(1,ℤ/9)11yescharacteristic
391C4—21no—
392C4—41no—
393C4—182no—
394C4—181no—
395C4C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
396C4D4×AGL(1,ℤ/9)11yescharacteristic
397C3—31no—
398C3C3×#482711yescharacteristic
399C2—21no—
400C2—41no—
401C2—361no—
402C2—721no—
403C2#63444⋊C311yescharacteristic
subgroupG/H
centerC2#63444⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#104C2
socleC2×C3C3×#882
hypercenter#104AGL(1,ℤ/9)
hypocenterC9C3×#454
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ #104

Complements

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

NH ≅ G/Ntype
1. #11722642C2semidirect
2. #63512⋊C3C2semidirect
3. #63470⋊C3C2semidirect
4. #63437⋊C3C2semidirect
5. 3−1+2×#104C2semidirect
6. #63382⋊C3C2semidirect
7. #63364⋊C3C2semidirect
9. #10588507C3semidirect
31. #63716C2×C3semidirect
35. #63512C2×C3semidirect
36. #63470C2×C3semidirect
37. #63437C2×C3semidirect
39. C9×#104C2×C3semidirect
40. #63382C2×C3semidirect
41. #63364C2×C3semidirect
252. #104AGL(1,ℤ/9)semidirect
254. 3−1+2#454semidirect
353. C9C3×#454semidirect

Maximal subgroups

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

subgroupindex
#117226422
#63512⋊C32
#63470⋊C32
#63437⋊C32
3−1+2×#1042
#63382⋊C32
#63364⋊C32
C3×#48273
#105885073