Finite Group Databasebeta

← C3×3−1+2×(C3⋊D3)⋊C3 #72243492

Subgroups — C3×3−1+2×(C3⋊D3)⋊C3 #72243492

Order 4374.

5356 conjugacy classes of subgroups; only the 734 normal subgroups are shown.

normal subgroups only

This group has 5356 conjugacy classes of subgroups — too many to draw. Only its 734 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
1C3×H(3,𝔽3)×3−1+2C211yesmaximal, characteristic, complemented, Sylow
2C3×C3⋊D3×3−1+2C311yesmaximal, characteristic, complemented
3C32×D3×3−1+2—31nomaximal
4C33×(C3⋊D3)⋊C3C311yesmaximal, characteristic
53−1+2×(C3⋊D3)⋊C3C3127yesmaximal, direct product factor
6C3×#92262C316yesmaximal, complemented
7C3×#92259C312yesmaximal, complemented
8C3×C9×(C3⋊D3)⋊C3C313yesmaximal, complemented
9C33×3−1+2—21no—
10C33×3−1+2D311yescharacteristic, complemented
11C33×3−1+2C2×C311yescharacteristic, complemented
12C33×H(3,𝔽3)C2×C311yescharacteristic
13H(3,𝔽3)×3−1+2—227no—
14H(3,𝔽3)×3−1+2C2×C3127yescomplemented
15C3×C9⋊H(3,𝔽3)—26no—
16C3×C9⋊H(3,𝔽3)—23no—
17C3×C9⋊H(3,𝔽3)C2×C316yescomplemented
18C3×#6891—21no—
19C3×#6891—22no—
20C3×#6891C2×C312yescomplemented
21C3×C9×H(3,𝔽3)C2×C313yescomplemented
22C33×C3⋊D3C3211yescharacteristic
23C34×D3—31no—
24C2×C32×3−1+2—91no—
25C3⋊D3×3−1+2C3219yescomplemented
26C3×C9×C3⋊D3C3213yescomplemented
27C3×D3×3−1+2—32no—
28C3×D3×3−1+2—31no—
29C3×D3×3−1+2—36no—
30C3×D3×3−1+2—327no—
31C3×D3×3−1+2—91no—
32C32×(C3⋊D3)⋊C3—39no—
33C32×(C3⋊D3)⋊C3C3212yes—
34C32×(C3⋊D3)⋊C3C3219yes—
35C32×(C3⋊D3)⋊C3C3211yescharacteristic
36C32×D3×C9—33no—
37#92262C32154yescomplemented
38#92259C32118yescomplemented
39C9×(C3⋊D3)⋊C3C32127yescomplemented
40C3×(C3⋊D3)⋊C9C3216yescomplemented
41C35—21no—
42C35C3×D311yescharacteristic
43C35C2×C3211yescharacteristic
44C32×3−1+2—21no—
45C32×3−1+2—22no—
46C32×3−1+2—29no—
47C32×3−1+2—227no—
48C32×3−1+2—23no—
49C32×3−1+2—26no—
50C32×3−1+2—31no—
51C32×3−1+2—31no—
52C32×3−1+2—63no—
53C32×3−1+2—63no—
54C32×3−1+2—69no—
55C32×3−1+2—61no—
56C32×3−1+2—69no—
57C32×3−1+2—62no—
58C32×3−1+2—618no—
59C32×3−1+2—66no—
60C32×3−1+2—61no—
61C32×3−1+2—61no—
62C32×3−1+2C3×D311yescharacteristic
63C32×3−1+2C3×D312yes—
64C32×3−1+2C3×D316yescomplemented
65C32×3−1+2C3×D3127yescomplemented
66C32×3−1+2C2×C3219yescomplemented
67C32×H(3,𝔽3)—22no—
68C32×H(3,𝔽3)—21no—
69C32×H(3,𝔽3)—29no—
70C32×H(3,𝔽3)—39no—
71C32×H(3,𝔽3)—69no—
72C32×H(3,𝔽3)C2×C3211yescharacteristic
73C32×H(3,𝔽3)C2×C3212yes—
74C32×H(3,𝔽3)C2×C3219yes—
75C33×C9—23no—
76C33×C9C3×D313yescomplemented
77C33×C9C2×C3213yescomplemented
78C9⋊H(3,𝔽3)—2162no—
79C9⋊H(3,𝔽3)—254no—
80C9⋊H(3,𝔽3)—281no—
81C9⋊H(3,𝔽3)C2×C32154yescomplemented
82#6891—227no—
83#6891—254no—
84#6891—218no—
85#6891C2×C32118yescomplemented
86C9×H(3,𝔽3)—227no—
87C9×H(3,𝔽3)C2×C32127yescomplemented
88C3×C32⋊C9—23no—
89C3×C32⋊C9—26no—
90C3×C32⋊C9—26no—
91C3×C32⋊C9—212no—
92C3×C32⋊C9—26no—
93C3×C32⋊C9C2×C3216yescomplemented
94C2×C34—91no—
95C32×C3⋊D3—33no—
96C32×C3⋊D3C3313yes—
97C32×C3⋊D3C3311yescharacteristic
98C33×D3—31no—
99C33×D3—39no—
100C33×D3—32no—
101C33×D3—31no—
102C33×D3—99no—
103C33×D3—91no—
104C2×C3×3−1+2—92no—
105C2×C3×3−1+2—927no—
106C2×C3×3−1+2—96no—
107C2×C3×3−1+2—91no—
108C2×C32×C9—93no—
109C9×C3⋊D3C3319yescomplemented
110D3×3−1+2—318no—
111D3×3−1+2—39no—
112D3×3−1+2—354no—
113D3×3−1+2—99no—
114C3×(C3⋊D3)⋊C3—36no—
115C3×(C3⋊D3)⋊C3—327no—
116C3×(C3⋊D3)⋊C33−1+219yesdirect product factor
117C3×(C3⋊D3)⋊C3C3316yes—
118C3×(C3⋊D3)⋊C3C3313yes—
119C3×D3×C9—36no—
120C3×D3×C9—33no—
121C3×D3×C9—327no—
122C3×D3×C9—93no—
123(C3⋊D3)⋊C9C33118yescomplemented
124C34—21no—
125C34—21no—
126C34—22no—
127C34—23no—
128C34—29no—
129C34—31no—
130C34—39no—
131C34—31no—
132C34—33no—
133C34—63no—
134C34—61no—
135C34—69no—
136C34—63no—
137C34—63no—
138C34—61no—
139C34—66no—
140C34—62no—
141C34—61no—
142C34—183no—
143C34—183no—
144C34—186no—
145C34C32×D311yescharacteristic
146C34C32×D312yes—
147C34C32×D319yes—
148C34C32×D311yescharacteristic
149C34C2×C3311yescharacteristic
150C34C2×C3313yes—
151C3×3−1+2—21no—
152C3×3−1+2—29no—
153C3×3−1+2—218no—
154C3×3−1+2—29no—
155C3×3−1+2—23no—
156C3×3−1+2—254no—
157C3×3−1+2—227no—
158C3×3−1+2—32no—
159C3×3−1+2—327no—
160C3×3−1+2—39no—
161C3×3−1+2—36no—
162C3×3−1+2—66no—
163C3×3−1+2—618no—
164C3×3−1+2—66no—
165C3×3−1+2—612no—
166C3×3−1+2—618no—
167C3×3−1+2—636no—
168C3×3−1+2—62no—
169C3×3−1+2—681no—
170C3×3−1+2—627no—
171C3×3−1+2—627no—
172C3×3−1+2—618no—
173C3×3−1+2—654no—
174C3×3−1+2—654no—
175C3×3−1+2—6162no—
176C3×3−1+2—69no—
177C3×3−1+2—66no—
178C3×3−1+2—66no—
179C3×3−1+2—612no—
180C3×3−1+2—62no—
181C3×3−1+2—64no—
182C3×3−1+2—61no—
183C3×3−1+2—62no—
184C3×3−1+2—69no—
185C3×3−1+2—627no—
186C3×3−1+2—63no—
187C3×3−1+2—66no—
188C3×3−1+2(C3⋊D3)⋊C311yescharacteristic, direct product factor
189C3×3−1+2C32×D319yes—
190C3×3−1+2C32×D3118yes—
191C3×3−1+2C32×D3154yescomplemented
192C3×H(3,𝔽3)—29no—
193C3×H(3,𝔽3)—23no—
194C3×H(3,𝔽3)—26no—
195C3×H(3,𝔽3)—218no—
196C3×H(3,𝔽3)—29no—
197C3×H(3,𝔽3)—36no—
198C3×H(3,𝔽3)—327no—
199C3×H(3,𝔽3)—681no—
200C3×H(3,𝔽3)—627no—
201C3×H(3,𝔽3)—66no—
202C3×H(3,𝔽3)—618no—
203C3×H(3,𝔽3)—69no—
204C3×H(3,𝔽3)C2×3−1+219yescomplemented
205C3×H(3,𝔽3)C2×C3313yes—
206C3×H(3,𝔽3)C2×C3316yes—
207C32×C9—26no—
208C32×C9—23no—
209C32×C9—227no—
210C32×C9—29no—
211C32×C9—33no—
212C32×C9—33no—
213C32×C9—69no—
214C32×C9—63no—
215C32×C9—69no—
216C32×C9—66no—
217C32×C9—618no—
218C32×C9—63no—
219C32×C9—63no—
220C32×C9C32×D313yes—
221C32×C9C32×D316yes—
222C32×C9C32×D3127yescomplemented
223C32×C9C2×C3319yescomplemented
224C32⋊C9—227no—
225C32⋊C9—254no—
226C32⋊C9—254no—
227C32⋊C9—2108no—
228C32⋊C9—254no—
229C32⋊C9—218no—
230C32⋊C9C2×C33118yescomplemented
231C2×C33—91no—
232C2×C33—99no—
233C2×C33—92no—
234C2×C33—91no—
235C2×C33—279no—
236C3×C3⋊D3—33no—
237C3×C3⋊D3C3×3−1+213yescomplemented
238C3×C3⋊D3C3411yescharacteristic
239C32×D3—39no—
240C32×D3—33no—
241C32×D3—31no—
242C32×D3—36no—
243C32×D3—33no—
244C32×D3—96no—
245C32×D3—927no—
246C32×D3—93no—
247C32×D3—91no—
248C32×D3—93no—
249C32×D3—273no—
250C2×3−1+2—918no—
251C2×3−1+2—954no—
252C2×3−1+2—99no—
253C2×C3×C9—96no—
254C2×C3×C9—927no—
255C2×C3×C9—93no—
256(C3⋊D3)⋊C3—36no—
257(C3⋊D3)⋊C3C3×3−1+219yesdirect product factor
258D3×C9—318no—
259D3×C9—39no—
260D3×C9—99no—
261C33—23no—
262C33—21no—
263C33—23no—
264C33—29no—
265C33—21no—
266C33—23no—
267C33—23no—
268C33—26no—
269C33—31no—
270C33—31no—
271C33—33no—
272C33—39no—
273C33—32no—
274C33—36no—
275C33—327no—
276C33—33no—
277C33—33no—
278C33—61no—
279C33—63no—
280C33—62no—
281C33—69no—
282C33—61no—
283C33—63no—
284C33—63no—
285C33—627no—
286C33—69no—
287C33—63no—
288C33—63no—
289C33—62no—
290C33—66no—
291C33—66no—
292C33—62no—
293C33—64no—
294C33—66no—
295C33—618no—
296C33—66no—
297C33—612no—
298C33—61no—
299C33—63no—
300C33—69no—
301C33—61no—
302C33—62no—
303C33—66no—
304C33—63no—
305C33—99no—
306C33—93no—
307C33—189no—
308C33—183no—
309C33—183no—
310C33—189no—
311C33—189no—
312C33—183no—
313C33—186no—
314C33—1818no—
315C33—1827no—
316C33—189no—
317C33—186no—
318C33—1818no—
319C33—1854no—
320C33—1818no—
321C33C3×(C3⋊D3)⋊C311yescharacteristic
322C33D3×3−1+219yescomplemented
323C33C2×C3×3−1+213yescomplemented
324C33C33×D311yescharacteristic
325C33C33×D313yes—
326C33C33×D316yes—
327C33C33×D313yes—
328C33C2×C3411yescharacteristic
3293−1+2—29no—
3303−1+2—227no—
3313−1+2—318no—
3323−1+2—354no—
3333−1+2—654no—
3343−1+2—654no—
3353−1+2—6108no—
3363−1+2—618no—
3373−1+2—618no—
3383−1+2—636no—
3393−1+2—69no—
3403−1+2—618no—
3413−1+2—627no—
3423−1+2—654no—
3433−1+2C3×(C3⋊D3)⋊C319yesdirect product factor
344H(3,𝔽3)—227no—
345H(3,𝔽3)—29no—
346H(3,𝔽3)—36no—
347H(3,𝔽3)—66no—
348H(3,𝔽3)—618no—
349H(3,𝔽3)—654no—
350H(3,𝔽3)—627no—
351H(3,𝔽3)C2×C3×3−1+219yescomplemented
352C3×C9—23no—
353C3×C9—29no—
354C3×C9—29no—
355C3×C9—218no—
356C3×C9—36no—
357C3×C9—327no—
358C3×C9—39no—
359C3×C9—627no—
360C3×C9—618no—
361C3×C9—654no—
362C3×C9—69no—
363C3×C9—66no—
364C3×C9—66no—
365C3×C9—612no—
366C3×C9—63no—
367C3×C9—66no—
368C3×C9—69no—
369C3×C9—627no—
370C3×C9C3×(C3⋊D3)⋊C313yescomplemented
371C3×C9C33×D319yes—
372C3×C9C33×D3118yes—
373C2×C32—93no—
374C2×C32—99no—
375C2×C32—91no—
376C2×C32—96no—
377C2×C32—93no—
378C2×C32—273no—
379C2×C32—2727no—
380C2×C32—276no—
381C3⋊D3C32×3−1+211yescharacteristic, complemented
382C3×D3—33no—
383C3×D3—39no—
384C3×D3—31no—
385C3×D3—96no—
386C3×D3—93no—
387C3×D3—91no—
388C3×D3—93no—
389C3×D3—273no—
390C2×C9—918no—
391C2×C9—99no—
392C32—21no—
393C32—23no—
394C32—23no—
395C32—23no—
396C32—21no—
397C32—29no—
398C32—33no—
399C32—39no—
400C32—33no—
401C32—31no—
402C32—33no—
403C32—33no—
404C32—36no—
405C32—36no—
406C32—69no—
407C32—63no—
408C32—63no—
409C32—63no—
410C32—61no—
411C32—69no—
412C32—62no—
413C32—66no—
414C32—66no—
415C32—618no—
416C32—66no—
417C32—66no—
418C32—612no—
419C32—61no—
420C32—63no—
421C32—63no—
422C32—63no—
423C32—69no—
424C32—63no—
425C32—66no—
426C32—66no—
427C32—63no—
428C32—927no—
429C32—93no—
430C32—96no—
431C32—186no—
432C32—186no—
433C32—1812no—
434C32—1827no—
435C32—183no—
436C32—183no—
437C32—189no—
438C32—1827no—
439C32—186no—
440C32—1818no—
441C32—1854no—
442C32—183no—
443C32—186no—
444C32C32×(C3⋊D3)⋊C313yes—
445C32C32×(C3⋊D3)⋊C311yescharacteristic
446C32C3×D3×3−1+213yes—
447C32C3×D3×3−1+219yescomplemented
448C32C2×C32×3−1+211yescharacteristic, complemented
449C32C34×D311yescharacteristic
450C9—29no—
451C9—318no—
452C9—69no—
453C9—618no—
454C9C32×(C3⋊D3)⋊C319yescomplemented
455C2×C3—99no—
456C2×C3—91no—
457C2×C3—93no—
458C2×C3—273no—
459C2×C3—276no—
460D3—31no—
461D3—91no—
462C3—21no—
463C3—23no—
464C3—31no—
465C3—39no—
466C3—33no—
467C3—63no—
468C3—61no—
469C3—63no—
470C3—69no—
471C3—96no—
472C3—183no—
473C3—186no—
474C33−1+2×(C3⋊D3)⋊C313yesdirect product factor
475C3C33×(C3⋊D3)⋊C311yescharacteristic
476C3C32×D3×3−1+211yescharacteristic
477C2—91noSylow
subgroupG/H
centerC32C32×(C3⋊D3)⋊C3
commutator, cosocleC33C2×C34
FrattiniC32C34×D3
FittingC3×H(3,𝔽3)×3−1+2C2
socleC33C33×D3
prime radicalC33×(C3⋊D3)⋊C3C3
hypercenterC3×3−1+2(C3⋊D3)⋊C3
hypocenterC32C2×C32×3−1+2
LCSG ▷ C33 ▷ C32
UCSC1 ◁ C32 ◁ C3×3−1+2

Complements

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

NH ≅ G/Ntype
1. C3×H(3,𝔽3)×3−1+2C2semidirect
2. C3×C3⋊D3×3−1+2C3semidirect
5. 3−1+2×(C3⋊D3)⋊C3C3direct (N × H)
6. C3×#92262C3semidirect
7. C3×#92259C3semidirect
8. C3×C9×(C3⋊D3)⋊C3C3semidirect
10. C33×3−1+2D3semidirect
11. C33×3−1+2C2×C3semidirect
14. H(3,𝔽3)×3−1+2C2×C3semidirect
17. C3×C9⋊H(3,𝔽3)C2×C3semidirect
20. C3×#6891C2×C3semidirect
21. C3×C9×H(3,𝔽3)C2×C3semidirect
25. C3⋊D3×3−1+2C32semidirect
26. C3×C9×C3⋊D3C32semidirect
37. #92262C32semidirect
38. #92259C32semidirect
39. C9×(C3⋊D3)⋊C3C32semidirect
40. C3×(C3⋊D3)⋊C9C32semidirect
64. C32×3−1+2C3×D3semidirect
65. C32×3−1+2C3×D3semidirect
66. C32×3−1+2C2×C32semidirect
76. C33×C9C3×D3semidirect
77. C33×C9C2×C32semidirect
81. C9⋊H(3,𝔽3)C2×C32semidirect
85. #6891C2×C32semidirect
87. C9×H(3,𝔽3)C2×C32semidirect
93. C3×C32⋊C9C2×C32semidirect
109. C9×C3⋊D3C33semidirect
116. C3×(C3⋊D3)⋊C33−1+2direct (N × H)
123. (C3⋊D3)⋊C9C33semidirect
188. C3×3−1+2(C3⋊D3)⋊C3direct (N × H)
191. C3×3−1+2C32×D3semidirect
204. C3×H(3,𝔽3)C2×3−1+2semidirect
222. C32×C9C32×D3semidirect
223. C32×C9C2×C33semidirect
230. C32⋊C9C2×C33semidirect
237. C3×C3⋊D3C3×3−1+2semidirect
257. (C3⋊D3)⋊C3C3×3−1+2direct (N × H)
322. C33D3×3−1+2semidirect
323. C33C2×C3×3−1+2semidirect
343. 3−1+2C3×(C3⋊D3)⋊C3direct (N × H)
351. H(3,𝔽3)C2×C3×3−1+2semidirect
370. C3×C9C3×(C3⋊D3)⋊C3semidirect
381. C3⋊D3C32×3−1+2semidirect
447. C32C3×D3×3−1+2semidirect
448. C32C2×C32×3−1+2semidirect
454. C9C32×(C3⋊D3)⋊C3semidirect
474. C33−1+2×(C3⋊D3)⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C3×H(3,𝔽3)×3−1+22
C3×C3⋊D3×3−1+23
C32×D3×3−1+23
C33×(C3⋊D3)⋊C33
3−1+2×(C3⋊D3)⋊C33
C3×#922623
C3×#922593
C3×C9×(C3⋊D3)⋊C33