Finite Group Databasebeta

← #72249174

Subgroups — #72249174

Order 3402.

362 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 58 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#13133671C211yesmaximal, characteristic, complemented, Hall
2C7×C3⋊AGL(1,ℤ/9)C311yesmaximal, characteristic, complemented
3#11765745C311yesmaximal, characteristic, complemented
4#11765745C311yesmaximal, characteristic, complemented
5#11765745C311yesmaximal, characteristic, complemented
6C3×(C3⋊D3)⋊ΓL(1,𝔽8)—31nomaximal
7ΓL(1,𝔽8)×AGL(1,ℤ/9)—31nomaximal
8ΓL(1,𝔽8)×AGL(1,ℤ/9)—31nomaximal
9ΓL(1,𝔽8)×AGL(1,ℤ/9)—31nomaximal
10C3×C7×3−1+2C2×C311yescharacteristic, complemented
11C3×C7⋊H(3,𝔽3)D311yescharacteristic
12C3×#4430—21no—
13#10588065—21no—
14#10588065—21no—
15#10588065—21no—
16#10588065—21no—
17#10588065—21no—
18#10588065—21no—
19#10588065—21no—
20#10588065—21no—
21#10588065—21no—
22ΓL(1,𝔽8)×3−1+2—21no—
23ΓL(1,𝔽8)×3−1+2—21no—
24ΓL(1,𝔽8)×3−1+2—21no—
25ΓL(1,𝔽8)×3−1+2D311yescharacteristic, complemented
26ΓL(1,𝔽8)×3−1+2D311yescharacteristic, complemented
27ΓL(1,𝔽8)×3−1+2D311yescharacteristic, complemented
28#10588061—21no—
29#10588061—21no—
30#10588061—21no—
31#10588061C2×C311yescharacteristic, complemented
32#10588061C2×C311yescharacteristic, complemented
33#10588061C2×C311yescharacteristic, complemented
34#4207⋊C3—71nomaximal, Hall
35C3×C7×C3⋊D3—31no—
36C3×D3×ΓL(1,𝔽8)—91no—
37C7×C3⋊D9C3211yescharacteristic, complemented
38C7×AGL(1,ℤ/9)—31no—
39C7×AGL(1,ℤ/9)—31no—
40C7×AGL(1,ℤ/9)—31no—
41(C3⋊D3)⋊ΓL(1,𝔽8)—31no—
42(C3⋊D3)⋊ΓL(1,𝔽8)—31no—
43(C3⋊D3)⋊ΓL(1,𝔽8)—31no—
44D9×ΓL(1,𝔽8)—31no—
45D9×ΓL(1,𝔽8)—31no—
46D9×ΓL(1,𝔽8)—31no—
47#67361—31no—
48#67361—31no—
49#67361—31no—
50ΓL(1,𝔽64)—31no—
51ΓL(1,𝔽64)—31no—
52ΓL(1,𝔽64)—31no—
533−1+4—71noSylow
54C33×C7C3×D311yescharacteristic
55C32×ΓL(1,𝔽8)—21no—
56C32×ΓL(1,𝔽8)C3⋊D311yescharacteristic
57C7×3−1+2—21no—
58C7×3−1+2—21no—
59C7×3−1+2—21no—
60C7×3−1+2C3×D311yescharacteristic
61C7×3−1+2C3×D311yescharacteristic
62C7×3−1+2C3×D311yescharacteristic
63C3×C7×C9—21no—
64C3×C7×C9C2×C3211yescharacteristic, complemented
65C7⋊H(3,𝔽3)—21no—
66C7⋊H(3,𝔽3)—21no—
67C7⋊H(3,𝔽3)—21no—
68C7⋊H(3,𝔽3)C3×D311yescharacteristic
69C7⋊H(3,𝔽3)C3×D311yescharacteristic
70C7⋊H(3,𝔽3)C3×D311yescharacteristic
71#4430—21no—
72#4430—21no—
73#4430—21no—
74#4430—21no—
75#4430—21no—
76#4430—21no—
77#4430—21no—
78#4430—21no—
79#4430—21no—
80C3×C7⋊C9—21no—
81C3×C7⋊C9—21no—
82C3×C7⋊C9—21no—
83#4428—21no—
84#4428—21no—
85#4428—21no—
86#4428—21no—
87#4428—21no—
88#4428—21no—
89#4428—21no—
90#4428—21no—
91#4428—21no—
92#4428—21no—
93#4428—21no—
94#4428—21no—
95#4428C3×D311yescharacteristic, complemented
96#4428C3×D311yescharacteristic, complemented
97#4428C3×D311yescharacteristic, complemented
98#4427—21no—
99#4427—21no—
100#4427—21no—
101#4427—21no—
102#4427—21no—
103#4427—21no—
104#4427—21no—
105#4427—21no—
106#4427—21no—
107#4427—21no—
108#4427—21no—
109#4427—21no—
110#4427C3×D311yescharacteristic, complemented
111#4427C3×D311yescharacteristic, complemented
112#4427C3×D311yescharacteristic, complemented
113C9×ΓL(1,𝔽8)—21no—
114C9×ΓL(1,𝔽8)—21no—
115C9×ΓL(1,𝔽8)—21no—
116C9×ΓL(1,𝔽8)—21no—
117C9×ΓL(1,𝔽8)—21no—
118C9×ΓL(1,𝔽8)—21no—
119C9×ΓL(1,𝔽8)—21no—
120C9×ΓL(1,𝔽8)—21no—
121C9×ΓL(1,𝔽8)—21no—
122C9×ΓL(1,𝔽8)—21no—
123C9×ΓL(1,𝔽8)—21no—
124C9×ΓL(1,𝔽8)—21no—
125C9×ΓL(1,𝔽8)C3×D311yescharacteristic, complemented
126C9×ΓL(1,𝔽8)C3×D311yescharacteristic, complemented
127C9×ΓL(1,𝔽8)C3×D311yescharacteristic, complemented
128#4207—71no—
129#4207—71no—
130#4207—71no—
131C3⋊AGL(1,ℤ/9)ΓL(1,𝔽8)11yescharacteristic, complemented
132C3×AGL(1,ℤ/9)—211no—
133C3×AGL(1,ℤ/9)—211no—
134C3×AGL(1,ℤ/9)—211no—
135C3×(C3⋊D3)⋊C3—211no—
136C7×C3⋊D3—31no—
137C3×D3×C7—91no—
138C3×D3×C7—271no—
139C2×C3×ΓL(1,𝔽8)—271no—
140D3×ΓL(1,𝔽8)—91no—
141D3×ΓL(1,𝔽8)—91no—
142D3×ΓL(1,𝔽8)—91no—
143C7×D9—31no—
144C7×D9—31no—
145C7×D9—31no—
146#737—71no—
147#737—71no—
148#737—71no—
149#737—141no—
150#737—141no—
151#737—141no—
152#737—141no—
153#737—141no—
154#737—141no—
155#737—141no—
156#737—141no—
157#737—141no—
158#737—141no—
159#737—141no—
160#737—141no—
161C3×3−1+2—71no—
162C3×3−1+2—71no—
163C3×3−1+2—71no—
164C3×3−1+2—141no—
165C3×3−1+2—141no—
166C3×3−1+2—141no—
167C3×3−1+2—141no—
168C3×3−1+2C2×ΓL(1,𝔽8)11yescharacteristic, complemented
169C3×H(3,𝔽3)—71no—
170C32×C7—21no—
171C32×C7—181no—
172C32×C7C32×D311yescharacteristic
173C32×C7C3×C3⋊D311yescharacteristic
174C3×ΓL(1,𝔽8)—21no—
175C3×ΓL(1,𝔽8)—21no—
176C3×ΓL(1,𝔽8)—21no—
177C3×ΓL(1,𝔽8)—181no—
178C3×ΓL(1,𝔽8)—181no—
179C3×ΓL(1,𝔽8)C3×C3⋊D311yescharacteristic
180C3×ΓL(1,𝔽8)C3×C3⋊D311yescharacteristic
181C3×ΓL(1,𝔽8)C3×C3⋊D311yescharacteristic
182C7×C9—21no—
183C7×C9—21no—
184C7×C9—21no—
185C7×C9C32×D311yescharacteristic
186C7×C9C32×D311yescharacteristic
187C7×C9C32×D311yescharacteristic
188C7⋊C9—21no—
189C7⋊C9—21no—
190C7⋊C9—21no—
191C7⋊C9—21no—
192C7⋊C9—21no—
193C7⋊C9—21no—
194C7⋊C9—21no—
195C7⋊C9—21no—
196C7⋊C9—21no—
197C3×C3⋊D3—31no—
198C32×D3—631no—
199C3⋊D9C3×ΓL(1,𝔽8)11yescharacteristic, complemented
200AGL(1,ℤ/9)—31no—
201AGL(1,ℤ/9)—31no—
202AGL(1,ℤ/9)—31no—
203AGL(1,ℤ/9)—211no—
204AGL(1,ℤ/9)—211no—
205AGL(1,ℤ/9)—211no—
206AGL(1,ℤ/9)—211no—
207AGL(1,ℤ/9)—211no—
208AGL(1,ℤ/9)—211no—
209(C3⋊D3)⋊C3—211no—
210(C3⋊D3)⋊C3—211no—
211(C3⋊D3)⋊C3—211no—
212C3×D9—211no—
213C3×D9—211no—
214C3×D9—211no—
215C2×C3×C7—271no—
216D3×C7—91no—
217D3×C7—271no—
218C2×ΓL(1,𝔽8)—271no—
219C2×ΓL(1,𝔽8)—271no—
220C2×ΓL(1,𝔽8)—271no—
221C33—71no—
222C33—141no—
223C33D3×ΓL(1,𝔽8)11yescharacteristic
2243−1+2—21no—
2253−1+2—21no—
2263−1+2—21no—
2273−1+2—71no—
2283−1+2—71no—
2293−1+2—71no—
2303−1+2—71no—
2313−1+2—71no—
2323−1+2—71no—
2333−1+2—141no—
2343−1+2—141no—
2353−1+2—141no—
2363−1+2—141no—
2373−1+2—141no—
2383−1+2—141no—
2393−1+2—141no—
2403−1+2—141no—
2413−1+2—141no—
2423−1+2—141no—
2433−1+2—141no—
2443−1+2—141no—
2453−1+2—141no—
2463−1+2—141no—
2473−1+2—141no—
2483−1+2—141no—
2493−1+2—141no—
2503−1+2—141no—
2513−1+2—141no—
2523−1+2—141no—
2533−1+2—141no—
2543−1+2—141no—
2553−1+2—141no—
2563−1+2—141no—
2573−1+2—141no—
2583−1+2—141no—
2593−1+2—141no—
2603−1+2—141no—
2613−1+2—141no—
2623−1+2—141no—
2633−1+2—141no—
2643−1+2—141no—
2653−1+2—141no—
2663−1+2D3×ΓL(1,𝔽8)11yescharacteristic
2673−1+2D3×ΓL(1,𝔽8)11yescharacteristic
2683−1+2D3×ΓL(1,𝔽8)11yescharacteristic
269H(3,𝔽3)—71no—
270H(3,𝔽3)—71no—
271H(3,𝔽3)—71no—
272H(3,𝔽3)—141no—
273H(3,𝔽3)—141no—
274H(3,𝔽3)—141no—
275C3×C9—21no—
276C3×C9—71no—
277C3×C9—71no—
278C3×C9—71no—
279C3×C9—141no—
280C3×C9—141no—
281C3×C9—141no—
282C3×C9—141no—
283C3×C9—141no—
284C3×C9—141no—
285C3×C9—141no—
286C3×C9—141no—
287C3×C9—141no—
288C3×C9—141no—
289C3×C9—141no—
290C3×C9—141no—
291C3×C9—141no—
292C3×C9—141no—
293C3×C9—141no—
294C3×C9C2×C3×ΓL(1,𝔽8)11yescharacteristic, complemented
295C3×C7—31no—
296C3×C7—61no—
297C3×C7—61no—
298C3×C7C32×C3⋊D311yescharacteristic
299ΓL(1,𝔽8)—31no—
300ΓL(1,𝔽8)—31no—
301ΓL(1,𝔽8)—61no—
302ΓL(1,𝔽8)—61no—
303ΓL(1,𝔽8)—61no—
304ΓL(1,𝔽8)—61no—
305C2×C32—1891no—
306C3⋊D3—31no—
307C3×D3—91no—
308C3×D3—271no—
309C3×D3—631no—
310C3×D3—631no—
311C3×D3—631no—
312D9—31no—
313D9—31no—
314D9—31no—
315C2×C7—271noHall
316C32—21no—
317C32—71no—
318C32—71no—
319C32—71no—
320C32—91no—
321C32—141no—
322C32—141no—
323C32—141no—
324C32—1261no—
325C32—1261no—
326C32C3×D3×ΓL(1,𝔽8)11yescharacteristic
327C32C3⋊D3×ΓL(1,𝔽8)11yescharacteristic
328C9—21no—
329C9—21no—
330C9—21no—
331C9—141no—
332C9—141no—
333C9—141no—
334C9—141no—
335C9—141no—
336C9—141no—
337C9—141no—
338C9—141no—
339C9—141no—
340C9C3×D3×ΓL(1,𝔽8)11yescharacteristic
341C9C3×D3×ΓL(1,𝔽8)11yescharacteristic
342C9C3×D3×ΓL(1,𝔽8)11yescharacteristic
343C7#4207⋊C311yescharacteristic, complemented, Sylow
344C2×C3—271no—
345C2×C3—1891no—
346C2×C3—1891no—
347C2×C3—1891no—
348D3—91no—
349D3—271no—
350C3—31no—
351C3—61no—
352C3—61no—
353C3—211no—
354C3—421no—
355C3—421no—
356C3—421no—
357C3—421no—
358C3—421no—
359C3C3×C3⋊D3×ΓL(1,𝔽8)11yescharacteristic
360C2—271noSylow
subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC3×C7×C9C2×C32
FrattiniC3C3×C3⋊D3×ΓL(1,𝔽8)
FittingC3×C7×3−1+2C2×C3
socleC3×C7C32×C3⋊D3
LCSG ▷ C3×C7×C9
UCSC1

Complements

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

NH ≅ G/Ntype
1. #13133671C2semidirect
2. C7×C3⋊AGL(1,ℤ/9)C3semidirect
3. #11765745C3semidirect
4. #11765745C3semidirect
5. #11765745C3semidirect
10. C3×C7×3−1+2C2×C3semidirect
25. ΓL(1,𝔽8)×3−1+2D3semidirect
26. ΓL(1,𝔽8)×3−1+2D3semidirect
27. ΓL(1,𝔽8)×3−1+2D3semidirect
31. #10588061C2×C3semidirect
32. #10588061C2×C3semidirect
33. #10588061C2×C3semidirect
37. C7×C3⋊D9C32semidirect
64. C3×C7×C9C2×C32semidirect
95. #4428C3×D3semidirect
96. #4428C3×D3semidirect
97. #4428C3×D3semidirect
110. #4427C3×D3semidirect
111. #4427C3×D3semidirect
112. #4427C3×D3semidirect
125. C9×ΓL(1,𝔽8)C3×D3semidirect
126. C9×ΓL(1,𝔽8)C3×D3semidirect
127. C9×ΓL(1,𝔽8)C3×D3semidirect
131. C3⋊AGL(1,ℤ/9)ΓL(1,𝔽8)semidirect
168. C3×3−1+2C2×ΓL(1,𝔽8)semidirect
199. C3⋊D9C3×ΓL(1,𝔽8)semidirect
294. C3×C9C2×C3×ΓL(1,𝔽8)semidirect
343. C7#4207⋊C3semidirect

Maximal subgroups

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

subgroupindex
#131336712
C7×C3⋊AGL(1,ℤ/9)3
#117657453
C3×(C3⋊D3)⋊ΓL(1,𝔽8)3
ΓL(1,𝔽8)×AGL(1,ℤ/9)3
#4207⋊C37