Finite Group Databasebeta

← C3×#35845 #10654452

Subgroups — C3×#35845 #10654452

Order 768.

2174 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#557C211yesmaximal, characteristic, complemented
2C2×C3×#519C211yesmaximal, characteristic
3C3×#2628C213yesmaximal
4C3×#2581C213yesmaximal, complemented
5C3×#2567C213yesmaximal, complemented
6C3×#2453C216yesmaximal, complemented
7C3×#2441C216yesmaximal
8C3×#2396C213yesmaximal
9C3×#2351C211yesmaximal, characteristic, complemented
10C3×Q8×C4⋊C4C212yesmaximal, complemented
11C3×Q8×C22⋊C4C212yesmaximal
12#35845C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C3×#557—28no—
15C2×C3×C4⋊Q8C2213yes—
16C2×C3×#125C2216yes—
17C3×#519—28no—
18C2×C3×C4×Q8C2213yes—
19C2×C3×C4×Q8C2213yes—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×Q8C2213yes—
22C2×C3×C4×Q8C2212yes—
23C2×C3×C4×Q8C2212yes—
24C2×C3×C4×Q8C2211yescharacteristic
25C2×C3×#117C2213yes—
26C2×C3×#117C2213yes—
27C3×#400C2216yes—
28C3×#400C2213yes—
29C3×#399C2216yes—
30C3×#399C2211yescharacteristic
31C3×#398C2216yes—
32C3×#396C2216yes—
33C3×#395C2216yes—
34C3×#395C2213yes—
35C3×#393C2213yes—
36C3×#391C2216yes—
37C3×#391C2216yes—
38C3×#389C2216yes—
39C3×#389C22112yes—
40C3×#389C2216yes—
41C3×#389C2216yes—
42C3×#387C22112yes—
43C3×#384C2216yes—
44C3×#382C2213yes—
45C3×#380C2216yes—
46C3×C4×C4⋊C4C2216yes—
47C3×C4×C4⋊C4C2213yes—
48C3×C4×C22⋊C4C2216yes—
49C2×#557C2×C311yescharacteristic, complemented
50C2×#519C2×C311yescharacteristic
51#2628C2×C313yes—
52#2581C2×C313yescomplemented
53#2567C2×C313yescomplemented
54#2453C2×C316yescomplemented
55#2441C2×C316yes—
56#2396C2×C313yes—
57#2351C2×C311yescharacteristic, complemented
58Q8×C4⋊C4C2×C312yescomplemented
59Q8×C22⋊C4C2×C312yes—
60C22×C3×Q8—24no—
61C22×C3×Q8—24no—
62C22×C3×Q8Q818yescomplemented
63C22×C3×Q8C2313yes—
64C22×C3×Q8C2313yes—
65C22×C3×Q8C2311yescharacteristic
66C23×C3×C4C2313yes—
67C3×C4⋊Q8—212no—
68C3×#125—224no—
69C3×C4×Q8—24no—
70C3×C4×Q8—28no—
71C3×C4×Q8—28no—
72C3×C4×Q8—212no—
73C3×C4×Q8—212no—
74C3×C4×Q8—212no—
75C3×C4×Q8D418yes—
76C3×#117—212no—
77C3×#117—212no—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C4⋊C4C2316yes—
80C2×C3×C4⋊C4C2316yes—
81C2×C3×C4⋊C4C2316yes—
82C2×C3×C4⋊C4C2313yes—
83C2×C3×C4⋊C4C2316yes—
84C2×C3×C4⋊C4C2316yes—
85C2×C3×C4⋊C4C2313yes—
86C2×C3×C4⋊C4C2313yes—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2316yes—
89C2×C3×C4⋊C4C23112yes—
90C2×C3×C4⋊C4C2313yes—
91C2×C3×C22⋊C4C2312yes—
92C2×C3×C22⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4C2313yes—
94C2×C3×C22⋊C4C2316yes—
95C2×C3×C42C2316yes—
96C2×C3×C42C2316yes—
97C2×C3×C42C2313yes—
98C2×C3×C42C2316yes—
99C2×C3×C42C2313yes—
100C3×#95C2316yes—
101C3×#95C2316yes—
102C3×#95C2311yescharacteristic
103C3×#95C2313yes—
104C3×#95C2316yes—
105C3×#95C2316yes—
106C3×#95C23112yes—
107C3×#95C2316yes—
108C23×Q8C22×C311yescharacteristic
109#557—28no—
110C2×C4⋊Q8C22×C313yes—
111C2×#125C22×C316yes—
112#519—28no—
113C2×C4×Q8C22×C313yes—
114C2×C4×Q8C22×C313yes—
115C2×C4×Q8C22×C312yes—
116C2×C4×Q8C22×C312yes—
117C2×C4×Q8C22×C311yescharacteristic
118C2×C4×Q8C22×C313yes—
119C2×C4×Q8C22×C311yescharacteristic
120C2×#117C22×C313yes—
121C2×#117C22×C313yes—
122#400C22×C313yes—
123#400C22×C316yes—
124#399C22×C316yes—
125#399C22×C311yescharacteristic
126#398C22×C316yes—
127#396C22×C316yes—
128#395C22×C313yes—
129#395C22×C316yes—
130#393C22×C313yes—
131#391C22×C316yes—
132#391C22×C316yes—
133#389C22×C3112yes—
134#389C22×C316yes—
135#389C22×C316yes—
136#389C22×C316yes—
137#387C22×C3112yes—
138#384C22×C316yes—
139#382C22×C313yes—
140#380C22×C316yes—
141C4×C4⋊C4C22×C313yes—
142C4×C4⋊C4C22×C316yes—
143C4×C22⋊C4C22×C316yes—
144C24×C3C2411yescharacteristic
145C2×C3×Q8—232no—
146C2×C3×Q8—26no—
147C2×C3×Q8—26no—
148C2×C3×Q8—26no—
149C2×C3×Q8—46no—
150C2×C3×Q8C2×D414yes—
151C2×C3×Q8C2×Q8112yes—
152C2×C3×Q8C2×Q814yes—
153C2×C3×Q8#4114yes—
154C22×C3×C4—26no—
155C22×C3×C4—26no—
156C22×C3×C4C2×Q8112yes—
157C22×C3×C4C2416yes—
158C22×C3×C4C2411yescharacteristic
159C22×C3×C4C2412yes—
160C22×C3×C4C2416yes—
161C22×C3×C4C2413yes—
162C22×C3×C4C2411yescharacteristic
163C22×C3×C4C2413yes—
164C22×C3×C4C2413yes—
165C22×C3×C4C2412yes—
166C22×C3×C4C2413yes—
167C3×C4⋊C4—26no—
168C3×C4⋊C4—26no—
169C3×C4⋊C4—26no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—212no—
172C3×C4⋊C4—212no—
173C3×C4⋊C4—212no—
174C3×C4⋊C4—212no—
175C3×C4⋊C4—212no—
176C3×C4⋊C4—24no—
177C3×C4⋊C4—46no—
178C3×C4⋊C4—412no—
179C3×C4⋊C4C2×D4112yes—
180C3×C22⋊C4—26no—
181C3×C22⋊C4—22no—
182C3×C22⋊C4—24no—
183C3×C22⋊C4—46no—
184C3×C42—212no—
185C3×C42—26no—
186C3×C42—212no—
187C3×C42—212no—
188C3×C42C2×D4112yes—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8C3×Q818yescomplemented
192C22×Q8C23×C313yes—
193C22×Q8C23×C313yes—
194C22×Q8C23×C311yescharacteristic
195C23×C4C23×C313yes—
196C4⋊Q8—212no—
197#125—224no—
198C4×Q8—24no—
199C4×Q8—28no—
200C4×Q8—28no—
201C4×Q8—212no—
202C4×Q8—212no—
203C4×Q8—212no—
204C4×Q8C3×D418yes—
205#117—212no—
206#117—212no—
207C2×C4⋊C4C23×C313yes—
208C2×C4⋊C4C23×C313yes—
209C2×C4⋊C4C23×C313yes—
210C2×C4⋊C4C23×C311yescharacteristic
211C2×C4⋊C4C23×C316yes—
212C2×C4⋊C4C23×C316yes—
213C2×C4⋊C4C23×C316yes—
214C2×C4⋊C4C23×C316yes—
215C2×C4⋊C4C23×C312yes—
216C2×C4⋊C4C23×C316yes—
217C2×C4⋊C4C23×C316yes—
218C2×C4⋊C4C23×C3112yes—
219C2×C4⋊C4C23×C313yes—
220C2×C22⋊C4C23×C311yescharacteristic
221C2×C22⋊C4C23×C313yes—
222C2×C22⋊C4C23×C312yes—
223C2×C22⋊C4C23×C316yes—
224C2×C42C23×C313yes—
225C2×C42C23×C316yes—
226C2×C42C23×C316yes—
227C2×C42C23×C316yes—
228C2×C42C23×C313yes—
229#95C23×C316yes—
230#95C23×C316yes—
231#95C23×C311yescharacteristic
232#95C23×C313yes—
233#95C23×C316yes—
234#95C23×C316yes—
235#95C23×C3112yes—
236#95C23×C316yes—
237C23×C3—21no—
238C23×C3—21no—
239C23×C3—22no—
240C23×C3—22no—
241C23×C3C22×Q812yes—
242C23×C3C2511yescharacteristic
243C3×Q8—224no—
244C3×Q8—412no—
245C3×Q8C22⋊Q8116yes—
246C2×C3×C4—23no—
247C2×C3×C4—22no—
248C2×C3×C4—23no—
249C2×C3×C4—26no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—224no—
253C2×C3×C4—23no—
254C2×C3×C4—21no—
255C2×C3×C4—22no—
256C2×C3×C4—26no—
257C2×C3×C4—26no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—26no—
262C2×C3×C4—26no—
263C2×C3×C4—26no—
264C2×C3×C4—26no—
265C2×C3×C4—26no—
266C2×C3×C4C22×D416yes—
267C2×C3×C4C22×D412yes—
268C2×C3×C4C22×D416yes—
269C2×C3×C4C22×Q816yes—
270C2×C3×C4C22×Q816yes—
271C2×C3×C4C2×#4116yes—
272C2×C3×C42−1+414yes—
273C24C24×C311yescharacteristic
274C2×Q8—232no—
275C2×Q8—26no—
276C2×Q8—26no—
277C2×Q8—26no—
278C2×Q8—46no—
279C2×Q8C2×C3×D414yes—
280C2×Q8C2×C3×Q8112yes—
281C2×Q8C2×C3×Q814yes—
282C2×Q8C3×#4114yes—
283C22×C4—26no—
284C22×C4—26no—
285C22×C4C2×C3×Q8112yes—
286C22×C4C24×C316yes—
287C22×C4C24×C311yescharacteristic
288C22×C4C24×C312yes—
289C22×C4C24×C316yes—
290C22×C4C24×C313yes—
291C22×C4C24×C311yescharacteristic
292C22×C4C24×C313yes—
293C22×C4C24×C313yes—
294C22×C4C24×C312yes—
295C22×C4C24×C313yes—
296C4⋊C4—26no—
297C4⋊C4—26no—
298C4⋊C4—26no—
299C4⋊C4—22no—
300C4⋊C4—212no—
301C4⋊C4—212no—
302C4⋊C4—212no—
303C4⋊C4—212no—
304C4⋊C4—212no—
305C4⋊C4—24no—
306C4⋊C4—46no—
307C4⋊C4—412no—
308C4⋊C4C2×C3×D4112yes—
309C22⋊C4—26no—
310C22⋊C4—22no—
311C22⋊C4—24no—
312C22⋊C4—46no—
313C42—212no—
314C42—26no—
315C42—212no—
316C42—212no—
317C42C2×C3×D4112yes—
318C22×C3—22no—
319C22×C3—24no—
320C22×C3—41no—
321C22×C3—41no—
322C22×C3—42no—
323C22×C3C23×D411yescharacteristic
324C22×C3C23×Q811yescharacteristic
325C22×C3C22×#4111yescharacteristic
326C22×C3C2×2−1+412yes—
327C22×C3#58512yes—
328C3×C4—26no—
329C3×C4—22no—
330C3×C4—26no—
331C3×C4—24no—
332C3×C4—43no—
333C3×C4—46no—
334C3×C4—41no—
335C3×C4—42no—
336C3×C4—46no—
337C3×C4C2×C22⋊Q8112yes—
338C23—21no—
339C23—21no—
340C23—22no—
341C23—22no—
342C23C22×C3×Q812yes—
343C23C25×C311yescharacteristic
344Q8—224no—
345Q8—412no—
346Q8C3×C22⋊Q8116yes—
347C2×C4—23no—
348C2×C4—22no—
349C2×C4—23no—
350C2×C4—26no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—224no—
354C2×C4—23no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—26no—
358C2×C4—26no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—26no—
363C2×C4—26no—
364C2×C4—26no—
365C2×C4—26no—
366C2×C4—26no—
367C2×C4C22×C3×D416yes—
368C2×C4C22×C3×D412yes—
369C2×C4C22×C3×D416yes—
370C2×C4C22×C3×Q816yes—
371C2×C4C22×C3×Q816yes—
372C2×C4C2×C3×#4116yes—
373C2×C4C3×2−1+414yes—
374C2×C3—42no—
375C2×C3C22×C22⋊Q811yescharacteristic
376C2×C3#348711yescharacteristic
377C2×C3#349911yescharacteristic
378C2×C3#350212yes—
379C2×C3#351612yes—
380C22—22no—
381C22—24no—
382C22—41no—
383C22—41no—
384C22—42no—
385C22C23×C3×D411yescharacteristic
386C22C23×C3×Q811yescharacteristic
387C22C22×C3×#4111yescharacteristic
388C22C2×C3×2−1+412yes—
389C22C3×#58512yes—
390C4—26no—
391C4—22no—
392C4—26no—
393C4—24no—
394C4—46no—
395C4—43no—
396C4—41no—
397C4—42no—
398C4—46no—
399C4C2×C3×C22⋊Q8112yes—
400C3#3584511yescharacteristic, direct product factor, Sylow
401C2—42no—
402C2C22×C3×C22⋊Q811yescharacteristic
403C2C3×#348711yescharacteristic
404C2C3×#349911yescharacteristic
405C2C3×#350212yes—
406C2C3×#351612yes—
subgroupG/H
center, socleC23×C3C25
commutator, Frattini, cosocleC23C25×C3
prime radicalC24×C3C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#557C2semidirect
4. C3×#2581C2semidirect
5. C3×#2567C2semidirect
6. C3×#2453C2semidirect
9. C3×#2351C2semidirect
10. C3×Q8×C4⋊C4C2semidirect
12. #35845C3direct (N × H)
49. C2×#557C2×C3semidirect
52. #2581C2×C3semidirect
53. #2567C2×C3semidirect
54. #2453C2×C3semidirect
57. #2351C2×C3semidirect
58. Q8×C4⋊C4C2×C3semidirect
62. C22×C3×Q8Q8semidirect
191. C22×Q8C3×Q8semidirect
400. C3#35845direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5572
C2×C3×#5192
C3×#26282
C3×#25812
C3×#25672
C3×#24532
C3×#24412
C3×#23962
C3×#23512
C3×Q8×C4⋊C42
C3×Q8×C22⋊C42
#358453