Finite Group Databasebeta

← C2×C3×#1697 #10637745

Subgroups — C2×C3×#1697 #10637745

Order 768.

778 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#486C211yesmaximal, characteristic, complemented
2C2×C3×#476C211yesmaximal, characteristic, complemented
3C2×C3×#459C211yesmaximal, characteristic
4C2×C3×#436C211yesmaximal, characteristic, complemented
5C3×#1697C218yesmaximal, direct product factor
6C2×C3×#335C211yesmaximal, characteristic, complemented
7C2×C3×#333C211yesmaximal, characteristic, complemented
8C2×C3×#331C211yesmaximal, characteristic, complemented
9C2×#1697C311yesmaximal, characteristic, direct product factor, Sylow
10C22×C3×D8—21no—
11C22×C3×M16—21no—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×#127C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C3×#486—22no—
16C3×#486C2214yescomplemented
17C3×#476—22no—
18C3×#476C2214yescomplemented
19C3×#459—22no—
20C3×#459C2214yes—
21C3×#436—22no—
22C3×#436C2214yescomplemented
23C2×C3×#106—21no—
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#102—21no—
28C2×C3×#102—21no—
29C2×C3×#102—21no—
30C2×C3×#102—21no—
31C2×C3×C22⋊C8—21no—
32C2×C3×C4×C8C2211yescharacteristic
33C3×#335C2214yescomplemented
34C3×#333C2214yescomplemented
35C3×#331C2214yescomplemented
36C2×#486C2×C311yescharacteristic, complemented
37C2×#476C2×C311yescharacteristic, complemented
38C2×#459C2×C311yescharacteristic
39C2×#436C2×C311yescharacteristic, complemented
40#1697C2×C318yesdirect product factor
41C2×#335C2×C311yescharacteristic, complemented
42C2×#333C2×C311yescharacteristic, complemented
43C2×#331C2×C311yescharacteristic, complemented
44C22×C3×Q8—21no—
45C22×C3×D4—21no—
46C22×C3×D4—21no—
47C22×C3×D4—21no—
48C22×C3×D4D411yescharacteristic
49C23×C3×C4—21no—
50C2×C3×D8—24no—
51C2×C3×D8—44no—
52C2×C3×M16—24no—
53C2×C3×M16—44no—
54C22×C3×C8—21no—
55C22×C3×C8—21no—
56C22×C3×C8—21no—
57C22×C3×C8D411yescharacteristic
58C22×C3×C8D411yescharacteristic
59C3×C4⋊Q8—21no—
60C3×C4⋊Q8—21no—
61C3×C4⋊Q8—21no—
62C3×C4⋊Q8C2312yes—
63C3×#127—21no—
64C3×#127—21no—
65C3×#127—21no—
66C3×#127C2312yes—
67C3×C4×D4—21no—
68C3×C4×D4—21no—
69C3×C4×D4D412yes—
70C3×C4×D4C2312yes—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4D411yescharacteristic
74C2×C3×C22⋊C4—21no—
75C2×C3×C42C2311yescharacteristic
76C3×#106—24no—
77C3×C4⋊C8—21no—
78C3×C4⋊C8—21no—
79C3×C4⋊C8—21no—
80C3×C4⋊C8C2312yes—
81C3×C4⋊C8C2312yes—
82C3×C4⋊C8C2312yes—
83C3×#102—24no—
84C3×#102—24no—
85C3×#102—24no—
86C3×#102—24no—
87C3×C22⋊C8—24no—
88C3×C4×C8D412yes—
89C3×C4×C8C2312yes—
90C22×D8—21no—
91C22×M16—21no—
92C2×C4⋊Q8C22×C311yescharacteristic
93C2×#127C22×C311yescharacteristic
94C2×C4×D4C22×C311yescharacteristic
95#486—22no—
96#486C22×C314yescomplemented
97#476—22no—
98#476C22×C314yescomplemented
99#459—22no—
100#459C22×C314yes—
101#436—22no—
102#436C22×C314yescomplemented
103C2×#106—21no—
104C2×C4⋊C8C22×C311yescharacteristic
105C2×C4⋊C8C22×C311yescharacteristic
106C2×C4⋊C8C22×C311yescharacteristic
107C2×#102—21no—
108C2×#102—21no—
109C2×#102—21no—
110C2×#102—21no—
111C2×C22⋊C8—21no—
112C2×C4×C8C22×C311yescharacteristic
113#335C22×C314yescomplemented
114#333C22×C314yescomplemented
115#331C22×C314yescomplemented
116C24×C3—21no—
117C24×C3—41no—
118C2×C3×Q8—22no—
119C2×C3×Q8—41no—
120C2×C3×Q8—82no—
121C2×C3×D4—21no—
122C2×C3×D4—22no—
123C2×C3×D4—22no—
124C2×C3×D4—22no—
125C2×C3×D4—22no—
126C2×C3×D4—22no—
127C2×C3×D4—41no—
128C2×C3×D4—41no—
129C2×C3×D4—41no—
130C2×C3×D4—42no—
131C2×C3×D4—42no—
132C2×C3×D4—42no—
133C2×C3×D4—42no—
134C2×C3×D4—82no—
135C2×C3×D4C2×D412yes—
136C22×C3×C4—21no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—42no—
140C22×C3×C4—42no—
141C22×C3×C4—41no—
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4#4111yescharacteristic
145C3×D8—48no—
146C3×M16—48no—
147C2×C3×C8—22no—
148C2×C3×C8—22no—
149C2×C3×C8—22no—
150C2×C3×C8—22no—
151C2×C3×C8—22no—
152C2×C3×C8—42no—
153C2×C3×C8—42no—
154C2×C3×C8—42no—
155C2×C3×C8C2×D412yes—
156C2×C3×C8C2×D412yes—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—41no—
161C3×C4⋊C4—41no—
162C3×C4⋊C4C2×D412yes—
163C3×C22⋊C4—22no—
164C3×C22⋊C4—22no—
165C3×C42C2×D411yescharacteristic
166C3×C42C2×D411yescharacteristic
167C3×C42#4111yescharacteristic
168C3×C42C2411yescharacteristic
169C22×Q8—21no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4C3×D411yescharacteristic
174C23×C4—21no—
175C2×D8—24no—
176C2×D8—44no—
177C2×M16—24no—
178C2×M16—44no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8C3×D411yescharacteristic
183C22×C8C3×D411yescharacteristic
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8C23×C312yes—
188#127—21no—
189#127—21no—
190#127—21no—
191#127C23×C312yes—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4C3×D412yes—
195C4×D4C23×C312yes—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C3×D411yescharacteristic
199C2×C22⋊C4—21no—
200C2×C42C23×C311yescharacteristic
201#106—24no—
202C4⋊C8—21no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8C23×C312yes—
206C4⋊C8C23×C312yes—
207C4⋊C8C23×C312yes—
208#102—24no—
209#102—24no—
210#102—24no—
211#102—24no—
212C22⋊C8—24no—
213C4×C8C3×D412yes—
214C4×C8C23×C312yes—
215C23×C3—22no—
216C23×C3—42no—
217C23×C3—42no—
218C23×C3—42no—
219C23×C3—42no—
220C23×C3—82no—
221C23×C3—82no—
222C23×C3—82no—
223C23×C3#12111yescharacteristic
224C3×Q8—82no—
225C3×Q8—82no—
226C3×D4—22no—
227C3×D4—22no—
228C3×D4—42no—
229C3×D4—42no—
230C3×D4—42no—
231C3×D4—42no—
232C3×D4—42no—
233C3×D4—82no—
234C3×D4—82no—
235C3×D4—82no—
236C3×D4—82no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—44no—
243C2×C3×C4—42no—
244C2×C3×C4—82no—
245C2×C3×C4#12111yescharacteristic
246C2×C3×C4#12111yescharacteristic
247C2×C3×C4#12111yescharacteristic
248C2×C3×C4#13512yes—
249C2×C3×C4AGL(1,ℤ/8)12yes—
250C2×C3×C4AGL(1,ℤ/8)12yes—
251C2×C3×C4#13712yes—
252C2×C3×C4C22×D411yescharacteristic
253C2×C3×C4C22×D411yescharacteristic
254C2×C3×C4C2×#4111yescharacteristic
255C3×C8—22no—
256C3×C8—22no—
257C3×C8—42no—
258C3×C8—42no—
259C3×C8—42no—
260C24—21no—
261C24—41no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—82no—
265C2×D4—21no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—82no—
279C2×D4C2×C3×D412yes—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4C2×C3×D411yescharacteristic
287C22×C4C2×C3×D411yescharacteristic
288C22×C4C3×#4111yescharacteristic
289D8—48no—
290M16—48no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8C2×C3×D412yes—
300C2×C8C2×C3×D412yes—
301C4⋊C4—21no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4C2×C3×D412yes—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C42C2×C3×D411yescharacteristic
310C42C2×C3×D411yescharacteristic
311C42C3×#4111yescharacteristic
312C42C24×C311yescharacteristic
313C22×C3—42no—
314C22×C3—42no—
315C22×C3—42no—
316C22×C3—84no—
317C22×C3—82no—
318C22×C3—82no—
319C22×C3—82no—
320C22×C3—164no—
321C22×C3#46312yes—
322C22×C3#46812yes—
323C22×C3#47212yes—
324C22×C3C2×#12111yescharacteristic
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—42no—
328C3×C4—82no—
329C3×C4#46312yes—
330C3×C4#46812yes—
331C3×C4C2×#13511yescharacteristic
332C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
333C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
334C3×C4C2×#13711yescharacteristic
335C23—22no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—82no—
341C23—82no—
342C23—82no—
343C23C3×#12111yescharacteristic
344Q8—82no—
345Q8—82no—
346D4—22no—
347D4—22no—
348D4—42no—
349D4—42no—
350D4—42no—
351D4—42no—
352D4—42no—
353D4—82no—
354D4—82no—
355D4—82no—
356D4—82no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—82no—
365C2×C4C3×#12111yescharacteristic
366C2×C4C3×#12111yescharacteristic
367C2×C4C3×#12111yescharacteristic
368C2×C4C3×#13512yes—
369C2×C4C3×AGL(1,ℤ/8)12yes—
370C2×C4C3×AGL(1,ℤ/8)12yes—
371C2×C4C3×#13712yes—
372C2×C4C22×C3×D411yescharacteristic
373C2×C4C22×C3×D411yescharacteristic
374C2×C4C2×C3×#4111yescharacteristic
375C8—22no—
376C8—22no—
377C8—42no—
378C8—42no—
379C8—42no—
380C2×C3—82no—
381C2×C3—162no—
382C2×C3#169714yesdirect product factor
383C2×C3C2×#46311yescharacteristic
384C2×C3C2×#46811yescharacteristic
385C2×C3C2×#47211yescharacteristic
386C22—42no—
387C22—42no—
388C22—42no—
389C22—84no—
390C22—82no—
391C22—82no—
392C22—82no—
393C22—164no—
394C22C3×#46312yes—
395C22C3×#46812yes—
396C22C3×#47212yes—
397C22C2×C3×#12111yescharacteristic
398C4—21no—
399C4—21no—
400C4—42no—
401C4—82no—
402C4C3×#46312yes—
403C4C3×#46812yes—
404C4C2×C3×#13511yescharacteristic
405C4C2×C3×AGL(1,ℤ/8)11yescharacteristic
406C4C2×C3×AGL(1,ℤ/8)11yescharacteristic
407C4C2×C3×#13711yescharacteristic
408C3C2×#169711yescharacteristic, direct product factor, Sylow
409C2—82no—
410C2—162no—
411C2C3×#169714yesdirect product factor
412C2C2×C3×#46311yescharacteristic
413C2C2×C3×#46811yescharacteristic
414C2C2×C3×#47211yescharacteristic
subgroupG/H
center, socleC23×C3#121
commutator, Frattini, cosocleC42C24×C3
prime radicalC2×C3×#459C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#486C2semidirect
2. C2×C3×#476C2semidirect
4. C2×C3×#436C2semidirect
5. C3×#1697C2direct (N × H)
6. C2×C3×#335C2semidirect
7. C2×C3×#333C2semidirect
8. C2×C3×#331C2semidirect
9. C2×#1697C3direct (N × H)
16. C3×#486C22semidirect
18. C3×#476C22semidirect
22. C3×#436C22semidirect
33. C3×#335C22semidirect
34. C3×#333C22semidirect
35. C3×#331C22semidirect
36. C2×#486C2×C3semidirect
37. C2×#476C2×C3semidirect
39. C2×#436C2×C3semidirect
40. #1697C2×C3direct (N × H)
41. C2×#335C2×C3semidirect
42. C2×#333C2×C3semidirect
43. C2×#331C2×C3semidirect
96. #486C22×C3semidirect
98. #476C22×C3semidirect
102. #436C22×C3semidirect
113. #335C22×C3semidirect
114. #333C22×C3semidirect
115. #331C22×C3semidirect
382. C2×C3#1697direct (N × H)
408. C3C2×#1697direct (N × H)
411. C2C3×#1697direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4862
C2×C3×#4762
C2×C3×#4592
C2×C3×#4362
C3×#16972
C2×C3×#3352
C2×C3×#3332
C2×C3×#3312
C2×#16973