Finite Group Databasebeta

← C3×#13317 #10631924

Subgroups — C3×#13317 #10631924

Order 768.

552 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#510C211yesmaximal, characteristic, complemented
2C3×#3054C211yesmaximal, characteristic
3C2×C3×#470C211yesmaximal, characteristic, complemented
4C2×C3×#360C211yesmaximal, characteristic, complemented
5C3×#2141C211yesmaximal, characteristic, complemented
6C2×C3×#349C211yesmaximal, characteristic, complemented
7C3×#1386C211yesmaximal, characteristic, complemented
8#13317C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×#135C2211yescharacteristic
10C22×C3×Q16C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12C3×#525—21no—
13C2×C3×C22⋊Q8—21no—
14C3×#510—22no—
15C3×#510—22no—
16C2×C3×Q32—21no—
17C2×C3×SD32—21no—
18C2×C3×M32C2211yescharacteristic
19C2×C3×M32C2211yescharacteristic
20C3×#470—21no—
21C3×#470—22no—
22C3×#470—22no—
23C3×#469—21no—
24C3×#468—22no—
25C3×#427C2211yescharacteristic
26C2×C3×#106C2211yescharacteristic
27C3×#416—21no—
28C2×C3×#103—21no—
29C3×#360—22no—
30C3×#358—22no—
31C3×#349—22no—
32C2×#510C2×C311yescharacteristic, complemented
33#3054C2×C311yescharacteristic
34C2×#470C2×C311yescharacteristic, complemented
35C2×#360C2×C311yescharacteristic, complemented
36#2141C2×C311yescharacteristic, complemented
37C2×#349C2×C311yescharacteristic, complemented
38#1386C2×C311yescharacteristic, complemented
39C2×C3×#41—21no—
40C22×C3×Q8—21no—
41C23×C3×C4D411yescharacteristic
42C3×#135—21no—
43C3×#135—21no—
44C3×#135—21no—
45C3×#135—21no—
46C2×C3×Q16—21no—
47C2×C3×Q16—21no—
48C2×C3×Q16—21no—
49C2×C3×Q16—21no—
50C2×C3×Q16—21no—
51C2×C3×Q16D411yescharacteristic
52C2×C3×Q16D412yescomplemented
53C2×C3×SD16—21no—
54C2×C3×D8D411yescharacteristic
55C2×C3×M16—21no—
56C2×C3×M16—21no—
57C2×C3×M16—21no—
58C2×C3×M16—21no—
59C2×C3×M16—21no—
60C2×C3×M16—21no—
61C22×C3×C8D411yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C3×#123—41no—
64C3×C22⋊Q8—24no—
65C3×C22⋊Q8—22no—
66C3×C22⋊Q8—42no—
67C3×#121—22no—
68C3×#120—41no—
69C3×C4×D4—42no—
70C3×#117—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—41no—
73C2×C3×C22⋊C4—41no—
74C3×Q32—24no—
75C3×SD32—24no—
76C3×M32—21no—
77C3×M32—21no—
78C3×M32—21no—
79C3×M32D412yescomplemented
80C2×C3×C16—21no—
81C2×C3×C16—21no—
82C3×#107D412yes—
83C3×#106—21no—
84C3×#106—21no—
85C3×#106D412yescomplemented
86C3×#103—24no—
87C3×#103—22no—
88C3×#102—22no—
89C2×#135C22×C311yescharacteristic
90C22×Q16C22×C311yescharacteristic
91C22×M16C22×C311yescharacteristic
92#525—21no—
93C2×C22⋊Q8—21no—
94#510—22no—
95#510—22no—
96C2×Q32—21no—
97C2×SD32—21no—
98C2×M32C22×C311yescharacteristic
99C2×M32C22×C311yescharacteristic
100#470—21no—
101#470—22no—
102#470—22no—
103#469—21no—
104#468—22no—
105#427C22×C311yescharacteristic
106C2×#106C22×C311yescharacteristic
107#416—21no—
108C2×#103—21no—
109#360—22no—
110#358—22no—
111#349—22no—
112C24×C3D811yescharacteristic
113C3×#41—41no—
114C3×#41—41no—
115C3×#41—41no—
116C3×#41—41no—
117C2×C3×Q8—21no—
118C2×C3×Q8—22no—
119C2×C3×Q8—41no—
120C2×C3×Q8—41no—
121C2×C3×Q8—41no—
122C2×C3×Q8—41no—
123C2×C3×Q8—41no—
124C2×C3×D4—21no—
125C2×C3×D4—42no—
126C2×C3×D4—41no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—41no—
134C22×C3×C4—41no—
135C22×C3×C4—41no—
136C22×C3×C4D811yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C3×Q16—22no—
139C3×Q16—21no—
140C3×Q16—22no—
141C3×Q16—21no—
142C3×Q16—41no—
143C3×Q16—41no—
144C3×SD16—41no—
145C3×SD16—41no—
146C3×D8—21no—
147C3×D8—21no—
148C3×M16—41no—
149C3×M16—41no—
150C3×M16—41no—
151C3×M16—41no—
152C2×C3×C8—21no—
153C2×C3×C8—21no—
154C2×C3×C8—21no—
155C2×C3×C8C2×D411yescharacteristic
156C2×C3×C8C2×D411yescharacteristic
157C2×C3×C8C2×D411yescharacteristic
158C2×C3×C8C2×D411yescharacteristic
159C2×C3×C8C2×D411yescharacteristic
160C2×C3×C8#4111yescharacteristic
161C3×C4⋊C4—22no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—44no—
164C3×C4⋊C4—42no—
165C3×C4⋊C4—41no—
166C3×C22⋊C4—42no—
167C3×C22⋊C4—44no—
168C3×C22⋊C4—42no—
169C3×C22⋊C4—41no—
170C3×C42—41no—
171C3×C16—22no—
172C3×C16—41no—
173C2×#41—21no—
174C22×Q8—21no—
175C23×C4C3×D411yescharacteristic
176#135—21no—
177#135—21no—
178#135—21no—
179#135—21no—
180C2×Q16—21no—
181C2×Q16—21no—
182C2×Q16—21no—
183C2×Q16—21no—
184C2×Q16—21no—
185C2×Q16C3×D411yescharacteristic
186C2×Q16C3×D412yescomplemented
187C2×SD16—21no—
188C2×D8C3×D411yescharacteristic
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C22×C8C3×D411yescharacteristic
196C22×C8C23×C311yescharacteristic
197#123—41no—
198C22⋊Q8—24no—
199C22⋊Q8—22no—
200C22⋊Q8—42no—
201#121—22no—
202#120—41no—
203C4×D4—42no—
204#117—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—41no—
207C2×C22⋊C4—41no—
208Q32—24no—
209SD32—24no—
210M32—21no—
211M32—21no—
212M32—21no—
213M32C3×D412yescomplemented
214C2×C16—21no—
215C2×C16—21no—
216#107C3×D412yes—
217#106—21no—
218#106—21no—
219#106C3×D412yescomplemented
220#103—24no—
221#103—22no—
222#102—22no—
223C23×C3—21no—
224C23×C3—21no—
225C23×C3—21no—
226C23×C3—42no—
227C23×C3—41no—
228C23×C3C2×D811yescharacteristic
229C3×Q8—41no—
230C3×Q8—42no—
231C3×Q8—42no—
232C3×Q8—41no—
233C3×Q8—81no—
234C3×Q8—81no—
235C3×D4—41no—
236C3×D4—41no—
237C3×D4—84no—
238C3×D4—81no—
239C3×D4—81no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4—41no—
251C2×C3×C4—82no—
252C2×C3×C4—81no—
253C2×C3×C4—81no—
254C2×C3×C4—81no—
255C2×C3×C4—81no—
256C2×C3×C4#12011yescharacteristic
257C2×C3×C4#12011yescharacteristic
258C2×C3×C4#12111yescharacteristic
259C2×C3×C4#13511yescharacteristic
260C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
261C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
262C3×C8—41no—
263C3×C8#12011yescharacteristic
264C3×C8#12111yescharacteristic
265C3×C8#12111yescharacteristic
266C3×C8#12711yescharacteristic
267C24C3×D811yescharacteristic
268#41—41no—
269#41—41no—
270#41—41no—
271#41—41no—
272C2×Q8—21no—
273C2×Q8—22no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×D4—21no—
280C2×D4—42no—
281C2×D4—41no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4C3×D811yescharacteristic
292C22×C4C2×C3×D411yescharacteristic
293Q16—22no—
294Q16—21no—
295Q16—22no—
296Q16—21no—
297Q16—41no—
298Q16—41no—
299SD16—41no—
300SD16—41no—
301D8—21no—
302D8—21no—
303M16—41no—
304M16—41no—
305M16—41no—
306M16—41no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8C2×C3×D411yescharacteristic
311C2×C8C2×C3×D411yescharacteristic
312C2×C8C2×C3×D411yescharacteristic
313C2×C8C2×C3×D411yescharacteristic
314C2×C8C2×C3×D411yescharacteristic
315C2×C8C3×#4111yescharacteristic
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—44no—
319C4⋊C4—42no—
320C4⋊C4—41no—
321C22⋊C4—42no—
322C22⋊C4—44no—
323C22⋊C4—42no—
324C22⋊C4—41no—
325C42—41no—
326C16—22no—
327C16—41no—
328C22×C3—22no—
329C22×C3—42no—
330C22×C3—42no—
331C22×C3—41no—
332C22×C3—81no—
333C22×C3—82no—
334C22×C3—81no—
335C22×C3#44711yescharacteristic
336C22×C3#44711yescharacteristic
337C22×C3#46611yescharacteristic
338C3×C4—41no—
339C3×C4—82no—
340C3×C4—82no—
341C3×C4—81no—
342C3×C4—82no—
343C3×C4#39211yescharacteristic
344C3×C4#44911yescharacteristic
345C3×C4#44911yescharacteristic
346C3×C4#46911yescharacteristic
347C23—21no—
348C23—21no—
349C23—21no—
350C23—42no—
351C23—41no—
352C23C2×C3×D811yescharacteristic
353Q8—41no—
354Q8—42no—
355Q8—42no—
356Q8—41no—
357Q8—81no—
358Q8—81no—
359D4—41no—
360D4—41no—
361D4—84no—
362D4—81no—
363D4—81no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—82no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4C3×#12011yescharacteristic
381C2×C4C3×#12011yescharacteristic
382C2×C4C3×#12111yescharacteristic
383C2×C4C3×#13511yescharacteristic
384C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
385C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
386C8—41no—
387C8C3×#12011yescharacteristic
388C8C3×#12111yescharacteristic
389C8C3×#12111yescharacteristic
390C8C3×#12711yescharacteristic
391C2×C3—21no—
392C2×C3—21no—
393C2×C3—81no—
394C2×C3—161no—
395C2×C3#199911yescharacteristic
396C2×C3#219112yes—
397C22—22no—
398C22—42no—
399C22—41no—
400C22—42no—
401C22—81no—
402C22—81no—
403C22—82no—
404C22C3×#44711yescharacteristic
405C22C3×#44711yescharacteristic
406C22C3×#46611yescharacteristic
407C4—41no—
408C4—82no—
409C4—81no—
410C4—82no—
411C4—82no—
412C4C3×#39211yescharacteristic
413C4C3×#44911yescharacteristic
414C4C3×#44911yescharacteristic
415C4C3×#46911yescharacteristic
416C3#1331711yescharacteristic, direct product factor, Sylow
417C2—21no—
418C2—21no—
419C2—81no—
420C2—161no—
421C2C3×#199911yescharacteristic
422C2C3×#219112yes—
subgroupG/H
center, socleC22×C3#447
commutator, Frattini, cosocleC22×C8C23×C3
prime radicalC3×#3054C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×M16 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#510C2semidirect
3. C2×C3×#470C2semidirect
4. C2×C3×#360C2semidirect
5. C3×#2141C2semidirect
6. C2×C3×#349C2semidirect
7. C3×#1386C2semidirect
8. #13317C3direct (N × H)
32. C2×#510C2×C3semidirect
34. C2×#470C2×C3semidirect
35. C2×#360C2×C3semidirect
36. #2141C2×C3semidirect
37. C2×#349C2×C3semidirect
38. #1386C2×C3semidirect
52. C2×C3×Q16D4semidirect
79. C3×M32D4semidirect
85. C3×#106D4semidirect
186. C2×Q16C3×D4semidirect
213. M32C3×D4semidirect
219. #106C3×D4semidirect
416. C3#13317direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5102
C3×#30542
C2×C3×#4702
C2×C3×#3602
C3×#21412
C2×C3×#3492
C3×#13862
#133173