Finite Group Databasebeta

← C3×#10055 #10628662

Subgroups — C3×#10055 #10628662

Order 768.

702 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#492C211yesmaximal, characteristic
2C2×C3×#477C211yesmaximal, characteristic, complemented
3C3×#2548C211yesmaximal, characteristic
4C3×#1987C211yesmaximal, characteristic, complemented
5C2×C3×#331C211yesmaximal, characteristic
6C2×C3×Q8⋊C8C211yesmaximal, characteristic, complemented
7C3×#1279C211yesmaximal, characteristic, complemented
8#10055C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×SD16—22no—
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×#127C2211yescharacteristic
12C2×C3×#124—21no—
13C2×C3×C4×Q8C2211yescharacteristic
14C3×#492—22no—
15C3×#492—22no—
16C3×#477—22no—
17C3×#477—22no—
18C2×C3×#107—21no—
19C2×C3×C4⋊C8C2211yescharacteristic
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×#103—21no—
22C2×C3×C4×C8C2211yescharacteristic
23C3×#394—22no—
24C3×#390—21no—
25C3×#390—22no—
26C3×C4×C4⋊C4C2211yescharacteristic
27C3×#340—22no—
28C3×#331—22no—
29C3×Q8⋊C8—22no—
30C2×#492C2×C311yescharacteristic
31C2×#477C2×C311yescharacteristic, complemented
32#2548C2×C311yescharacteristic
33#1987C2×C311yescharacteristic, complemented
34C2×#331C2×C311yescharacteristic
35C2×Q8⋊C8C2×C311yescharacteristic, complemented
36#1279C2×C311yescharacteristic, complemented
37C22×C3×Q8—21no—
38C22×C3×Q8D411yescharacteristic
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4D412yes—
42C2×C3×SD16—28no—
43C2×C3×SD16—48no—
44C22×C3×C8—21no—
45C22×C3×C8—21no—
46C22×C3×C8—21no—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8—21no—
49C3×C4⋊Q8—21no—
50C3×C4⋊Q8D412yes—
51C3×#127—21no—
52C3×#127—21no—
53C3×#127—21no—
54C3×#127—21no—
55C3×#124—42no—
56C3×#124—42no—
57C3×C4×Q8—21no—
58C3×C4×Q8—21no—
59C3×C4×Q8—21no—
60C3×C4×Q8—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4D412yes—
65C2×C3×C4⋊C4D411yescharacteristic
66C2×C3×C22⋊C4—24no—
67C2×C3×C22⋊C4—24no—
68C2×C3×C42—21no—
69C2×C3×C42—21no—
70C2×C3×C42C2311yescharacteristic
71C3×#107—24no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8—21no—
74C3×C4⋊C8—21no—
75C3×C4⋊C8D412yes—
76C3×#103—24no—
77C3×C4×C8—21no—
78C3×C4×C8D412yes—
79C3×#95—21no—
80C22×SD16—22no—
81C2×C4⋊Q8C22×C311yescharacteristic
82C2×#127C22×C311yescharacteristic
83C2×#124—21no—
84C2×C4×Q8C22×C311yescharacteristic
85#492—22no—
86#492—22no—
87#477—22no—
88#477—22no—
89C2×#107—21no—
90C2×C4⋊C8C22×C311yescharacteristic
91C2×C4⋊C8C22×C311yescharacteristic
92C2×#103—21no—
93C2×C4×C8C22×C311yescharacteristic
94#394—22no—
95#390—22no—
96#390—21no—
97C4×C4⋊C4C22×C311yescharacteristic
98#340—22no—
99#331—22no—
100Q8⋊C8—22no—
101C24×C3—22no—
102C2×C3×Q8—21no—
103C2×C3×Q8—21no—
104C2×C3×Q8—22no—
105C2×C3×Q8—21no—
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—44no—
109C2×C3×Q8—41no—
110C2×C3×Q8SD1612yescomplemented
111C2×C3×D4—22no—
112C2×C3×D4—22no—
113C2×C3×D4—44no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—42no—
121C2×C3×D4—41no—
122C2×C3×D4—41no—
123C2×C3×D4—42no—
124C22×C3×C4—21no—
125C22×C3×C4—22no—
126C22×C3×C4—21no—
127C22×C3×C4—41no—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C3×SD16—416no—
132C2×C3×C8—24no—
133C2×C3×C8—22no—
134C2×C3×C8—22no—
135C2×C3×C8—41no—
136C2×C3×C8—41no—
137C2×C3×C8—41no—
138C2×C3×C8—42no—
139C3×C4⋊C4—22no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4—22no—
145C3×C4⋊C4—41no—
146C3×C4⋊C4—41no—
147C3×C4⋊C4—41no—
148C3×C4⋊C4—41no—
149C3×C22⋊C4—48no—
150C3×C22⋊C4—84no—
151C3×C42—41no—
152C3×C42—41no—
153C3×C42—41no—
154C3×C42—41no—
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157C3×C42C2×D411yescharacteristic
158C3×C42#4111yescharacteristic
159C22×Q8—21no—
160C22×Q8C3×D411yescharacteristic
161C22×D4—21no—
162C22×D4—21no—
163C22×D4C3×D412yes—
164C2×SD16—28no—
165C2×SD16—48no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8C3×D412yes—
173#127—21no—
174#127—21no—
175#127—21no—
176#127—21no—
177#124—42no—
178#124—42no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4C3×D411yescharacteristic
187C2×C4⋊C4C3×D412yes—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42C23×C311yescharacteristic
193#107—24no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8C3×D412yes—
198#103—24no—
199C4×C8—21no—
200C4×C8C3×D412yes—
201#95—21no—
202C23×C3—42no—
203C23×C3—42no—
204C23×C3—42no—
205C23×C3—44no—
206C23×C3—44no—
207C23×C3#12011yescharacteristic
208C3×Q8—22no—
209C3×Q8—22no—
210C3×Q8—44no—
211C3×Q8—41no—
212C3×Q8—41no—
213C3×Q8—82no—
214C3×D4—44no—
215C3×D4—82no—
216C3×D4—81no—
217C3×D4—81no—
218C3×D4—82no—
219C3×D4—81no—
220C3×D4—81no—
221C3×D4—81no—
222C3×D4—81no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—42no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—82no—
234C2×C3×C4#12011yescharacteristic
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12711yescharacteristic
240C2×C3×C4C2×SD1611yescharacteristic
241C2×C3×C4#13511yescharacteristic
242C2×C3×C4#13512yes—
243C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
244C2×C3×C4#13711yescharacteristic
245C2×C3×C4#13712yes—
246C3×C8—24no—
247C3×C8—42no—
248C3×C8—81no—
249C24—22no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—22no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—44no—
257C2×Q8—41no—
258C2×Q8C3×SD1612yescomplemented
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—44no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—42no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—42no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—41no—
276C22×C4C2×C3×D411yescharacteristic
277C22×C4C2×C3×D411yescharacteristic
278C22×C4C2×C3×D411yescharacteristic
279SD16—416no—
280C2×C8—24no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—42no—
287C4⋊C4—22no—
288C4⋊C4—21no—
289C4⋊C4—22no—
290C4⋊C4—21no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C22⋊C4—48no—
298C22⋊C4—84no—
299C42—41no—
300C42—41no—
301C42—41no—
302C42—41no—
303C42C2×C3×D411yescharacteristic
304C42C2×C3×D411yescharacteristic
305C42C2×C3×D411yescharacteristic
306C42C3×#4111yescharacteristic
307C22×C3—84no—
308C22×C3—84no—
309C22×C3—82no—
310C22×C3—82no—
311C22×C3—82no—
312C22×C3#39211yescharacteristic
313C22×C3#45011yescharacteristic
314C22×C3#45211yescharacteristic
315C22×C3#45212yes—
316C22×C3#45312yes—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—42no—
320C3×C4—82no—
321C3×C4—81no—
322C3×C4—82no—
323C3×C4#44811yescharacteristic
324C3×C4#44911yescharacteristic
325C3×C4#46412yes—
326C3×C4#46511yescharacteristic
327C3×C4#46811yescharacteristic
328C3×C4#49511yescharacteristic
329C3×C4#49711yescharacteristic
330C23—42no—
331C23—42no—
332C23—42no—
333C23—44no—
334C23—44no—
335C23C3×#12011yescharacteristic
336Q8—22no—
337Q8—22no—
338Q8—44no—
339Q8—41no—
340Q8—41no—
341Q8—82no—
342D4—44no—
343D4—82no—
344D4—81no—
345D4—81no—
346D4—82no—
347D4—81no—
348D4—81no—
349D4—81no—
350D4—81no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—82no—
362C2×C4C3×#12011yescharacteristic
363C2×C4C3×#12011yescharacteristic
364C2×C4C3×#12111yescharacteristic
365C2×C4C3×#12111yescharacteristic
366C2×C4C3×#12111yescharacteristic
367C2×C4C3×#12711yescharacteristic
368C2×C4C2×C3×SD1611yescharacteristic
369C2×C4C3×#13511yescharacteristic
370C2×C4C3×#13512yes—
371C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
372C2×C4C3×#13711yescharacteristic
373C2×C4C3×#13712yes—
374C8—24no—
375C8—42no—
376C8—81no—
377C2×C3—162no—
378C2×C3#163012yes—
379C2×C3#163112yes—
380C2×C3#200011yescharacteristic
381C2×C3#200211yescharacteristic
382C2×C3#201511yescharacteristic
383C22—82no—
384C22—82no—
385C22—82no—
386C22—84no—
387C22—84no—
388C22C3×#39211yescharacteristic
389C22C3×#45011yescharacteristic
390C22C3×#45211yescharacteristic
391C22C3×#45212yes—
392C22C3×#45312yes—
393C4—21no—
394C4—21no—
395C4—42no—
396C4—81no—
397C4—82no—
398C4—82no—
399C4C3×#44811yescharacteristic
400C4C3×#44911yescharacteristic
401C4C3×#46412yes—
402C4C3×#46511yescharacteristic
403C4C3×#46811yescharacteristic
404C4C3×#49511yescharacteristic
405C4C3×#49711yescharacteristic
406C3#1005511yescharacteristic, direct product factor, Sylow
407C2—162no—
408C2C3×#163012yes—
409C2C3×#163112yes—
410C2C3×#200011yescharacteristic
411C2C3×#200211yescharacteristic
412C2C3×#201511yescharacteristic
subgroupG/H
center, socleC23×C3#120
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC2×C3×#127C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#477C2semidirect
4. C3×#1987C2semidirect
6. C2×C3×Q8⋊C8C2semidirect
7. C3×#1279C2semidirect
8. #10055C3direct (N × H)
31. C2×#477C2×C3semidirect
33. #1987C2×C3semidirect
35. C2×Q8⋊C8C2×C3semidirect
36. #1279C2×C3semidirect
110. C2×C3×Q8SD16semidirect
258. C2×Q8C3×SD16semidirect
406. C3#10055direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4922
C2×C3×#4772
C3×#25482
C3×#19872
C2×C3×#3312
C2×C3×Q8⋊C82
C3×#12792
#100553