Finite Group Databasebeta

← #72220052

Subgroups — #72220052

Order 7776.

3194 conjugacy classes of subgroups; only the 349 normal subgroups are shown.

normal subgroups only

This group has 3194 conjugacy classes of subgroups — too many to draw. Only its 349 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#72220049—34nomaximal
2#72220048C311yesmaximal, characteristic, complemented
3#72220010C212yesmaximal, complemented
4C22⋊C4×#6900C211yesmaximal, characteristic, complemented
5C2×#14517678C211yesmaximal, characteristic, complemented
6C23×#92314C211yesmaximal, characteristic, complemented
7C2×C4⋊#92314C212yesmaximal, complemented
8C23×#6900C2211yescharacteristic
9C22×#92314—22no—
10C22×#92314—24no—
11C22×#92314C2212yes—
12C22×#92314C2211yescharacteristic, complemented
13C2×C4×#6900C2212yescomplemented
14#14517678—22no—
15C2×#11757618C2211yescharacteristic, complemented
16C4⋊#92314—24no—
17C23×C3⋊AGL(1,ℤ/9)—34no—
18C2×#10619980—34no—
19C2×C4⋊(C3⋊AGL(1,ℤ/9))—38no—
20C23×C9⋊D9C2×C311yescharacteristic, complemented
21C3×C22⋊C4×3−1+2D314yes—
22#13047949—38no—
23C2×#10619673C2×C311yescharacteristic, complemented
24C2×C4⋊(C9⋊D9)C2×C312yescomplemented
25C22⋊C4×C9⋊C9—24no—
26C92×C22⋊C4C2×C311yescharacteristic, complemented
27#13046768C2×C312yescomplemented
28C22×#6900—22no—
29C22×#6900D412yescomplemented
30C22×#6900C2311yescharacteristic
31C2×#92314—28no—
32C2×#92314—22no—
33C2×#92314—22no—
34C2×#92314—22no—
35C2×#92314—21no—
36C2×#92314D414yescomplemented
37C4×#6900—22no—
38#11757618—21no—
39C3×#64091—91no—
40#11721728—34no—
41#63444⋊C3—912no—
42C23×C3×3−1+2C2×D314yes—
43C22×C3⋊AGL(1,ℤ/9)—38no—
44C22×C3⋊AGL(1,ℤ/9)—34no—
45C22×C3⋊AGL(1,ℤ/9)—616no—
46C22×C3⋊AGL(1,ℤ/9)—68no—
47C2×C3×C4×3−1+2C2×D318yes—
48#10619980—68no—
49C2×C3⋊(3−1+2⋊C4)—34no—
50C4⋊(C3⋊AGL(1,ℤ/9))—616no—
51C23×C9⋊C9—24no—
52C23×C92C22×C311yescharacteristic
53C22×C9⋊D9—22no—
54C22×C9⋊D9—24no—
55C22×C9⋊D9C22×C312yes—
56C22×C9⋊D9C22×C311yescharacteristic, complemented
57C2×C4×C9⋊C9—28no—
58C2×C4×C92C22×C312yescomplemented
59#10619673—22no—
60C2×C9⋊Dic9C22×C311yescharacteristic, complemented
61C4⋊(C9⋊D9)—24no—
62C2×#6900—22no—
63C2×#6900C2×D411yescharacteristic
64C2×#6900C2×D412yes—
65#92314—24no—
66#92314—41no—
67C23×C3×C3⋊D3—91no—
68C2×C3×#638—91no—
69C2×C3×C4⋊(C3⋊D3)—92no—
70C23×C3⋊D9—34no—
71C23×AGL(1,ℤ/9)—912no—
72C33×C22⋊C4C3⋊D311yescharacteristic
73C3×#3752—92no—
74C2×#6233—34no—
75C2×C4⋊(C3⋊D9)—38no—
76C2×#611⋊C3—912no—
77C2×C4⋊AGL(1,ℤ/9)—924no—
78C22⋊C4×3−1+2—312no—
79C3×C9×C22⋊C4—24no—
80C3×C9×C22⋊C4C3×D314yes—
81#88536—38no—
82(C2×AGL(1,ℤ/9))⋊C4—924no—
83C22×C3×3−1+2—28no—
84C22×C3×3−1+2D1218yes—
85C22×C3×3−1+2C22×D314yes—
86C2×C3⋊AGL(1,ℤ/9)—316no—
87C2×C3⋊AGL(1,ℤ/9)—632no—
88C2×C3⋊AGL(1,ℤ/9)—68no—
89C2×C3⋊AGL(1,ℤ/9)—68no—
90C2×C3⋊AGL(1,ℤ/9)—64no—
91C2×C3⋊AGL(1,ℤ/9)—68no—
92C3×C4×3−1+2—28no—
93C3⋊(3−1+2⋊C4)—64no—
94C22×C9⋊C9—28no—
95C22×C9⋊C9—24no—
96C22×C9⋊C9—48no—
97C22×C92—22no—
98C22×C92C3×D412yescomplemented
99C22×C92C23×C311yescharacteristic
100C2×C9⋊D9—28no—
101C2×C9⋊D9—22no—
102C2×C9⋊D9—22no—
103C2×C9⋊D9—22no—
104C2×C9⋊D9—21no—
105C2×C9⋊D9C3×D414yescomplemented
106C4×C9⋊C9—48no—
107C4×C92—22no—
108C9⋊Dic9—21no—
109#64091—91no—
110C3×#882—274no—
111#63444—912no—
112#6900#12011yescharacteristic, complemented, Sylow
113C23×C33C2×C3⋊D311yescharacteristic
114C22×C3×C3⋊D3—92no—
115C22×C3×C3⋊D3—91no—
116C22×C3×C3⋊D3—184no—
117C22×C3×C3⋊D3—182no—
118C2×C33×C4C2×C3⋊D312yes—
119C3×#638—182no—
120C2×C3×C3⋊Dic3—91no—
121C3×C4⋊(C3⋊D3)—184no—
122C23×3−1+2—312no—
123C23×C3×C9—24no—
124C23×C3×C9C2×C3×D314yes—
125C22×C3⋊D9—38no—
126C22×C3⋊D9—34no—
127C22×C3⋊D9—616no—
128C22×C3⋊D9—68no—
129C22×AGL(1,ℤ/9)—924no—
130C22×AGL(1,ℤ/9)—912no—
131C22×AGL(1,ℤ/9)—1848no—
132C22×AGL(1,ℤ/9)—1824no—
133C2×C4×3−1+2—324no—
134C2×C3×C4×C9—28no—
135C2×C3×C4×C9C2×C3×D318yes—
136#6233—68no—
137C2×C3⋊Dic9—34no—
138C4⋊(C3⋊D9)—616no—
139#611⋊C3—1824no—
140C2×3−1+2⋊C4—912no—
141C4⋊AGL(1,ℤ/9)—1848no—
142C2×C3×3−1+2—28no—
143C2×C3×3−1+2C2×D1214yes—
144C2×C3×3−1+2D3×D418yes—
145C3⋊AGL(1,ℤ/9)—616no—
146C3⋊AGL(1,ℤ/9)—124no—
147C2×C9⋊C9—24no—
148C2×C9⋊C9—28no—
149C2×C9⋊C9—48no—
150C2×C92—22no—
151C2×C92C2×C3×D411yescharacteristic
152C2×C92C2×C3×D412yes—
153C9⋊D9—24no—
154C9⋊D9—41no—
155C23×C3⋊D3—91no—
156C23×C3×D3—274no—
157C2×#638—91no—
158C2×C4⋊(C3⋊D3)—92no—
159C2×C3×#67—274no—
160C2×C3×D12—278no—
161C23×D9—912no—
162C32×C22⋊C4—34no—
163C32×C22⋊C4C3×C3⋊D311yescharacteristic
164#3752—92no—
165C3×(C2×D3)⋊C4—278no—
166C2×#611—912no—
167C2×D36—924no—
168C9×C22⋊C4—64no—
169C9×C22⋊C4AGL(1,ℤ/9)112yescomplemented
170(C2×D9)⋊C4—924no—
171C22×C33—22no—
172C22×C33C4⋊(C3⋊D3)12yes—
173C22×C33C22×C3⋊D311yescharacteristic
174C2×C3×C3⋊D3—94no—
175C2×C3×C3⋊D3—188no—
176C2×C3×C3⋊D3—182no—
177C2×C3×C3⋊D3—182no—
178C2×C3×C3⋊D3—181no—
179C2×C3×C3⋊D3—182no—
180C33×C4—22no—
181C3×C3⋊Dic3—181no—
182C22×3−1+2—312no—
183C22×3−1+2—324no—
184C22×3−1+2—624no—
185C22×C3×C9—24no—
186C22×C3×C9—28no—
187C22×C3×C9—28no—
188C22×C3×C9—48no—
189C22×C3×C9C3×D1218yes—
190C22×C3×C9C22×C3×D314yes—
191C2×C3⋊D9—316no—
192C2×C3⋊D9—632no—
193C2×C3⋊D9—68no—
194C2×C3⋊D9—68no—
195C2×C3⋊D9—64no—
196C2×C3⋊D9—68no—
197C2×AGL(1,ℤ/9)—948no—
198C2×AGL(1,ℤ/9)—1824no—
199C2×AGL(1,ℤ/9)—1824no—
200C2×AGL(1,ℤ/9)—1896no—
201C2×AGL(1,ℤ/9)—1824no—
202C2×AGL(1,ℤ/9)—1812no—
203C4×3−1+2—624no—
204C3×C4×C9—28no—
205C3×C4×C9—48no—
206C3⋊Dic9—64no—
2073−1+2⋊C4—1812no—
208C3×#120—811no—
209#882—274no—
210C3×3−1+2#88214yes—
211C9⋊C9—24no—
212C92C3×#12011yescharacteristic, complemented
213C23×C32—34no—
214C23×C32C2×C3×C3⋊D311yescharacteristic
215C22×C3⋊D3—92no—
216C22×C3⋊D3—91no—
217C22×C3⋊D3—184no—
218C22×C3⋊D3—182no—
219C22×C3×D3—278no—
220C22×C3×D3—274no—
221C22×C3×D3—5416no—
222C22×C3×D3—548no—
223C2×C32×C4—38no—
224C2×C32×C4C2×C3×C3⋊D312yes—
225#638—182no—
226C2×C3⋊Dic3—91no—
227C4⋊(C3⋊D3)—184no—
228C3×#67—548no—
229C2×C3×Dic3—274no—
230C3×D12—5416no—
231C23×C9—64no—
232C23×C9C2×AGL(1,ℤ/9)112yes—
233C22×D9—924no—
234C22×D9—912no—
235C22×D9—1848no—
236C22×D9—1824no—
237C2×C4×C9—68no—
238C2×C4×C9C2×AGL(1,ℤ/9)124yescomplemented
239#611—1824no—
240C2×Dic9—912no—
241D36—1848no—
242C2×C33—22no—
243C2×C33C2×C4⋊(C3⋊D3)11yescharacteristic
244C2×C33D4×C3⋊D312yes—
245C3×C3⋊D3—184no—
246C3×C3⋊D3—361no—
247C2×3−1+2—312no—
248C2×3−1+2—324no—
249C2×3−1+2—624no—
250C2×C3×C9—28no—
251C2×C3×C9—24no—
252C2×C3×C9—28no—
253C2×C3×C9—48no—
254C2×C3×C9C2×C3×D1214yes—
255C2×C3×C9C3×D3×D418yes—
256C3⋊D9—616no—
257C3⋊D9—124no—
258AGL(1,ℤ/9)—1848no—
259AGL(1,ℤ/9)—3612no—
260C24×C3—811no—
261C23×D3—274no—
262C2×C3×D4—812no—
263C2×C3×D4—811no—
264C2×#67—274no—
265C2×D12—278no—
266C3×C22⋊C4—91no—
267C3×C22⋊C4—812no—
268C3×C22⋊C4C3⋊AGL(1,ℤ/9)14yes—
269(C2×D3)⋊C4—278no—
270C22×C32—22no—
271C22×C32—38no—
272C22×C32—34no—
273C22×C32—68no—
274C22×C32C3×C4⋊(C3⋊D3)12yes—
275C22×C32C22×C3×C3⋊D311yescharacteristic
276C2×C3⋊D3—94no—
277C2×C3⋊D3—188no—
278C2×C3⋊D3—182no—
279C2×C3⋊D3—182no—
280C2×C3⋊D3—181no—
281C2×C3⋊D3—182no—
282C2×C3×D3—2716no—
283C2×C3×D3—548no—
284C2×C3×D3—548no—
285C2×C3×D3—5432no—
286C2×C3×D3—548no—
287C2×C3×D3—544no—
288C32×C4—22no—
289C32×C4—68no—
290C3⋊Dic3—181no—
291C3×Dic3—544no—
292C22×C9—224no—
293C22×C9—68no—
294C22×C9—64no—
295C22×C9—128no—
296C22×C9C4⋊AGL(1,ℤ/9)124yescomplemented
297C22×C9C22×AGL(1,ℤ/9)112yes—
298C2×D9—948no—
299C2×D9—1896no—
300C2×D9—1824no—
301C2×D9—1824no—
302C2×D9—1812no—
303C2×D9—1824no—
304C4×C9—224no—
305C4×C9—128no—
306Dic9—1812no—
307#120—811noSylow
308C33#6409111yescharacteristic
3093−1+2—312no—
310C3×C9—24no—
311C3×C9C3×#88214yes—
312C23×C3—91no—
313C23×C3—812no—
314C23×C3—811no—
315C23×C3—1622no—
316C23×C3—1624no—
317C23×C3C2×C3⋊AGL(1,ℤ/9)14yes—
318C22×D3—278no—
319C22×D3—274no—
320C22×D3—5416no—
321C22×D3—548no—
322C3×D4—1624no—
323C3×D4—1622no—
324C2×C3×C4—92no—
325C2×C3×C4—811no—
326C2×C3×C4C2×C3⋊AGL(1,ℤ/9)18yes—
327#67—548no—
328C2×Dic3—274no—
329D12—5416no—
330C2×C32—22no—
331C2×C32—34no—
332C2×C32—38no—
333C2×C32—68no—
334C2×C32C2×C3×C4⋊(C3⋊D3)11yescharacteristic
335C2×C32C3×D4×C3⋊D312yes—
336C3⋊D3—184no—
337C3⋊D3—361no—
338C3×D3—5416no—
339C3×D3—1084no—
340C2×C9—224no—
341C2×C9—64no—
342C2×C9—68no—
343C2×C9—128no—
344C2×C9C2×C4⋊AGL(1,ℤ/9)112yes—
345C2×C9D4×AGL(1,ℤ/9)124yes—
346D9—1848no—
347D9—3612no—
348C24—811no—
349C2×D4—811no—
350C2×D4—812no—
351C22⋊C4—812no—
352C22⋊C4#9231411yescharacteristic, complemented
353C22×C3—28no—
354C22×C3—92no—
355C22×C3—91no—
356C22×C3—182no—
357C22×C3—814no—
358C22×C3—1622no—
359C22×C3—1621no—
360C22×C3—1622no—
361C22×C3—1622no—
362C22×C3—1628no—
363C22×C3C4⋊(C3⋊AGL(1,ℤ/9))18yes—
364C22×C3C22×C3⋊AGL(1,ℤ/9)14yes—
365C2×D3—2716no—
366C2×D3—5432no—
367C2×D3—548no—
368C2×D3—548no—
369C2×D3—544no—
370C2×D3—548no—
371C3×C4—28no—
372C3×C4—182no—
373C3×C4—1621no—
374Dic3—544no—
375C32—34no—
376C32C3×#6409111yescharacteristic
377C9—64no—
378C9#63444⋊C3112yescomplemented
379C23—812no—
380C23—811no—
381C23—1624no—
382C23—1622no—
383C23C2×#9231411yescharacteristic
384D4—1624no—
385D4—1622no—
386C2×C4—811no—
387C2×C4C2×#9231412yescomplemented
388C2×C3—28no—
389C2×C3—92no—
390C2×C3—91no—
391C2×C3—182no—
392C2×C3—3241no—
393C2×C3—1624no—
394C2×C3C2×C4⋊(C3⋊AGL(1,ℤ/9))14yes—
395C2×C3D4×C3⋊AGL(1,ℤ/9)18yes—
396D3—5416no—
397D3—1084no—
398C22—22no—
399C22—814no—
400C22—1628no—
401C22—1622no—
402C22—1621no—
403C22—1622no—
404C22—1622no—
405C22C4⋊#9231412yescomplemented
406C22C22×#9231411yescharacteristic
407C4—22no—
408C4—1621no—
409C3—91no—
410C3#7222004914yes—
411C2—22no—
412C2—3241no—
413C2—1624no—
414C2D4×#9231412yes—
415C2C2×C4⋊#9231411yescharacteristic
subgroupG/H
centerC22C22×#92314
commutator, cosocleC22×C92C23×C3
Frattini, socleC22×C32C22×C3×C3⋊D3
FittingC22⋊C4×#6900C2
hypercenterC22⋊C4#92314
hypocenterC92C3×#120
LCSG ▷ C22×C92 ▷ C92
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #72220048C3semidirect
3. #72220010C2semidirect
4. C22⋊C4×#6900C2semidirect
5. C2×#14517678C2semidirect
6. C23×#92314C2semidirect
7. C2×C4⋊#92314C2semidirect
12. C22×#92314C22semidirect
13. C2×C4×#6900C22semidirect
15. C2×#11757618C22semidirect
20. C23×C9⋊D9C2×C3semidirect
23. C2×#10619673C2×C3semidirect
24. C2×C4⋊(C9⋊D9)C2×C3semidirect
26. C92×C22⋊C4C2×C3semidirect
27. #13046768C2×C3semidirect
29. C22×#6900D4semidirect
36. C2×#92314D4semidirect
56. C22×C9⋊D9C22×C3semidirect
58. C2×C4×C92C22×C3semidirect
60. C2×C9⋊Dic9C22×C3semidirect
98. C22×C92C3×D4semidirect
105. C2×C9⋊D9C3×D4semidirect
112. #6900#120semidirect
169. C9×C22⋊C4AGL(1,ℤ/9)semidirect
212. C92C3×#120semidirect
238. C2×C4×C9C2×AGL(1,ℤ/9)semidirect
296. C22×C9C4⋊AGL(1,ℤ/9)semidirect
352. C22⋊C4#92314semidirect
378. C9#63444⋊C3semidirect
387. C2×C4C2×#92314semidirect
405. C22C4⋊#92314semidirect

Maximal subgroups

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

subgroupindex
#722200493
#722200483
#722200102
C22⋊C4×#69002
C2×#145176782
C23×#923142
C2×C4⋊#923142