Finite Group Databasebeta

← C3×#10978 #10629585

Subgroups — C3×#10978 #10629585

Order 768.

1334 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊Q8C211yesmaximal, characteristic
2C2×C3×#419C212yesmaximal
3C22×C3×#101C211yesmaximal, characteristic
4C3×#2310C211yesmaximal, characteristic
5C2×C3×#339C212yesmaximal
6#10978C311yesmaximal, characteristic, direct product factor, Sylow
7C23×C3×Q8C2211yescharacteristic
8C23×C3×Q8C2211yescharacteristic
9C22×C3×M16C2212yes—
10C2×C3×C4⋊Q8—24no—
11C2×C3×C4⋊Q8—22no—
12C2×C3×C4⋊Q8—24no—
13C2×C3×C4⋊Q8C414yescomplemented
14C2×C3×#117C2212yes—
15C22×C3×C4⋊C4—22no—
16C22×C3×C42C2211yescharacteristic
17C3×#419—24no—
18C3×#419—24no—
19C2×C3×#103—24no—
20C2×C3×#101—22no—
21C2×C3×#101—24no—
22C3×#391—24no—
23C3×#381—22no—
24C3×#339—24no—
25C22×C4⋊Q8C2×C311yescharacteristic
26C2×#419C2×C312yes—
27C22×#101C2×C311yescharacteristic
28#2310C2×C311yescharacteristic
29C2×#339C2×C312yes—
30C22×C3×Q8—22no—
31C22×C3×Q8—22no—
32C22×C3×Q8—21no—
33C22×C3×Q8—22no—
34C22×C3×Q8—22no—
35C22×C3×Q8—22no—
36C22×C3×Q8—22no—
37C22×C3×Q8—22no—
38C22×C3×Q8—22no—
39C22×C3×Q8—22no—
40C22×C3×Q8—22no—
41C22×C3×Q8—22no—
42C22×C3×Q8C2×C412yes—
43C22×C3×Q8C2×C412yes—
44C22×C3×Q8D414yes—
45C22×C3×Q8D412yes—
46C23×C3×C4—21no—
47C23×C3×C4—21no—
48C23×C3×C4D412yes—
49C23×C3×C4C2311yescharacteristic
50C2×C3×M16—22no—
51C2×C3×M16—22no—
52C2×C3×M16—22no—
53C2×C3×M16—22no—
54C2×C3×M16—22no—
55C2×C3×M16D414yes—
56C22×C3×C8—22no—
57C3×C4⋊Q8—216no—
58C3×C4⋊Q8—28no—
59C3×C4⋊Q8—28no—
60C3×#117—22no—
61C3×#117—22no—
62C3×#117—22no—
63C3×#117—22no—
64C2×C3×C4⋊C4—216no—
65C2×C3×C4⋊C4—28no—
66C2×C3×C4⋊C4D414yes—
67C2×C3×C22⋊C4—22no—
68C2×C3×C42—22no—
69C2×C3×C42—21no—
70C2×C3×C42—22no—
71C2×C3×C42C2×C412yes—
72C2×C3×C42D414yes—
73C3×#103—216no—
74C3×#101—28no—
75C3×#95—28no—
76C23×Q8C22×C311yescharacteristic
77C23×Q8C22×C311yescharacteristic
78C22×M16C22×C312yes—
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8—22no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8C3×C414yescomplemented
83C2×#117C22×C312yes—
84C22×C4⋊C4—22no—
85C22×C42C22×C311yescharacteristic
86#419—24no—
87#419—24no—
88C2×#103—24no—
89C2×#101—22no—
90C2×#101—24no—
91#391—24no—
92#381—22no—
93#339—24no—
94C24×C3C2×D411yescharacteristic
95C2×C3×Q8—22no—
96C2×C3×Q8—22no—
97C2×C3×Q8—22no—
98C2×C3×Q8—24no—
99C2×C3×Q8—24no—
100C2×C3×Q8—22no—
101C2×C3×Q8—24no—
102C2×C3×Q8—24no—
103C2×C3×Q8—24no—
104C2×C3×Q8—22no—
105C2×C3×Q8—24no—
106C2×C3×Q8—24no—
107C2×C3×Q8—24no—
108C2×C3×Q8—24no—
109C2×C3×Q8—24no—
110C2×C3×Q8—24no—
111C2×C3×Q8—41no—
112C2×C3×Q8—42no—
113C2×C3×Q8—42no—
114C2×C3×Q8—42no—
115C2×C3×Q8—42no—
116C2×C3×Q8—42no—
117C2×C3×Q8—42no—
118C2×C3×Q8—42no—
119C2×C3×Q8—42no—
120C2×C3×Q8—41no—
121C2×C3×Q8—42no—
122C2×C3×Q8—42no—
123C2×C3×Q8—42no—
124C2×C3×Q8—42no—
125C2×C3×Q8—42no—
126C2×C3×Q8C22⋊C414yes—
127C22×C3×C4—28no—
128C22×C3×C4—22no—
129C22×C3×C4—24no—
130C22×C3×C4—22no—
131C22×C3×C4—24no—
132C22×C3×C4—24no—
133C22×C3×C4—22no—
134C22×C3×C4—24no—
135C22×C3×C4—22no—
136C22×C3×C4—24no—
137C22×C3×C4—41no—
138C22×C3×C4—42no—
139C22×C3×C4—41no—
140C22×C3×C4C22⋊C414yes—
141C22×C3×C4C22×C411yescharacteristic
142C22×C3×C4C2×D412yes—
143C22×C3×C4C2×D412yes—
144C22×C3×C4C2×D412yes—
145C22×C3×C4C2×D412yes—
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4#4112yes—
148C22×C3×C4#4112yes—
149C3×M16—24no—
150C3×M16—24no—
151C3×M16—42no—
152C3×M16—42no—
153C2×C3×C8—24no—
154C2×C3×C8—42no—
155C2×C3×C8—42no—
156C3×C4⋊C4—232no—
157C3×C4⋊C4—24no—
158C3×C4⋊C4—24no—
159C3×C22⋊C4—42no—
160C3×C22⋊C4—42no—
161C3×C42—22no—
162C3×C42—24no—
163C3×C42—42no—
164C3×C42—42no—
165C3×C42C22⋊C414yes—
166C22×Q8—21no—
167C22×Q8—22no—
168C22×Q8—22no—
169C22×Q8—22no—
170C22×Q8—22no—
171C22×Q8—22no—
172C22×Q8—22no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8C2×C3×C412yes—
179C22×Q8C2×C3×C412yes—
180C22×Q8C3×D414yes—
181C22×Q8C3×D412yes—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4C3×D412yes—
185C23×C4C23×C311yescharacteristic
186C2×M16—22no—
187C2×M16—22no—
188C2×M16—22no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16C3×D414yes—
192C22×C8—22no—
193C4⋊Q8—216no—
194C4⋊Q8—28no—
195C4⋊Q8—28no—
196#117—22no—
197#117—22no—
198#117—22no—
199#117—22no—
200C2×C4⋊C4—216no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4C3×D414yes—
203C2×C22⋊C4—22no—
204C2×C42—22no—
205C2×C42—21no—
206C2×C42—22no—
207C2×C42C2×C3×C412yes—
208C2×C42C3×D414yes—
209#103—216no—
210#101—28no—
211#95—28no—
212C23×C3—24no—
213C23×C3C2×C22⋊C411yescharacteristic
214C23×C3#12012yes—
215C23×C3#12011yescharacteristic
216C23×C3#12112yes—
217C23×C3#12311yescharacteristic
218C3×Q8—28no—
219C3×Q8—44no—
220C3×Q8—44no—
221C3×Q8—44no—
222C3×Q8—44no—
223C3×Q8—44no—
224C3×Q8—44no—
225C3×Q8—44no—
226C2×C3×C4—28no—
227C2×C3×C4—216no—
228C2×C3×C4—24no—
229C2×C3×C4—24no—
230C2×C3×C4—24no—
231C2×C3×C4—24no—
232C2×C3×C4—24no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4—44no—
236C2×C3×C4—44no—
237C2×C3×C4—44no—
238C2×C3×C4—44no—
239C2×C3×C4—44no—
240C2×C3×C4—48no—
241C2×C3×C4C2×C22⋊C411yescharacteristic
242C2×C3×C4C2×C22⋊C411yescharacteristic
243C2×C3×C4C4×D412yes—
244C2×C3×C4C4×D412yes—
245C2×C3×C4#12012yes—
246C2×C3×C4#12011yescharacteristic
247C2×C3×C4#12012yes—
248C2×C3×C4#12112yes—
249C2×C3×C4#12112yes—
250C2×C3×C4#12112yes—
251C2×C3×C4#12112yes—
252C2×C3×C4#12112yes—
253C2×C3×C4#12312yes—
254C2×C3×C4#12311yescharacteristic
255C2×C3×C4#12412yes—
256C2×C3×C4#12712yes—
257C3×C8—44no—
258C24C2×C3×D411yescharacteristic
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—22no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—22no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—41no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—41no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8C3×C22⋊C414yes—
291C22×C4—28no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4C3×C22⋊C414yes—
305C22×C4C22×C3×C411yescharacteristic
306C22×C4C2×C3×D412yes—
307C22×C4C2×C3×D412yes—
308C22×C4C2×C3×D412yes—
309C22×C4C2×C3×D412yes—
310C22×C4C2×C3×D411yescharacteristic
311C22×C4C3×#4112yes—
312C22×C4C3×#4112yes—
313M16—24no—
314M16—24no—
315M16—42no—
316M16—42no—
317C2×C8—24no—
318C2×C8—42no—
319C2×C8—42no—
320C4⋊C4—232no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C42—22no—
326C42—24no—
327C42—42no—
328C42—42no—
329C42C3×C22⋊C414yes—
330C22×C3—22no—
331C22×C3—22no—
332C22×C3—24no—
333C22×C3—24no—
334C22×C3#37911yescharacteristic
335C22×C3#38611yescharacteristic
336C22×C3#38611yescharacteristic
337C22×C3#39212yes—
338C22×C3#39412yes—
339C22×C3#45614yes—
340C3×C4—28no—
341C3×C4—44no—
342C3×C4—44no—
343C3×C4—84no—
344C3×C4#38612yes—
345C3×C4#39012yes—
346C3×C4#39212yes—
347C3×C4#39412yes—
348C23—24no—
349C23C2×C3×C22⋊C411yescharacteristic
350C23C3×#12012yes—
351C23C3×#12011yescharacteristic
352C23C3×#12112yes—
353C23C3×#12311yescharacteristic
354Q8—28no—
355Q8—44no—
356Q8—44no—
357Q8—44no—
358Q8—44no—
359Q8—44no—
360Q8—44no—
361Q8—44no—
362C2×C4—216no—
363C2×C4—28no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—48no—
377C2×C4C2×C3×C22⋊C411yescharacteristic
378C2×C4C2×C3×C22⋊C411yescharacteristic
379C2×C4C3×C4×D412yes—
380C2×C4C3×C4×D412yes—
381C2×C4C3×#12012yes—
382C2×C4C3×#12011yescharacteristic
383C2×C4C3×#12012yes—
384C2×C4C3×#12112yes—
385C2×C4C3×#12112yes—
386C2×C4C3×#12112yes—
387C2×C4C3×#12112yes—
388C2×C4C3×#12112yes—
389C2×C4C3×#12312yes—
390C2×C4C3×#12311yescharacteristic
391C2×C4C3×#12412yes—
392C2×C4C3×#12712yes—
393C8—44no—
394C2×C3—21no—
395C2×C3—21no—
396C2×C3—22no—
397C2×C3#143811yescharacteristic
398C2×C3#188612yes—
399C2×C3#200514yes—
400C22—24no—
401C22—24no—
402C22—22no—
403C22—22no—
404C22C3×#37911yescharacteristic
405C22C3×#38611yescharacteristic
406C22C3×#38611yescharacteristic
407C22C3×#39212yes—
408C22C3×#39412yes—
409C22C3×#45614yes—
410C4—28no—
411C4—44no—
412C4—44no—
413C4—84no—
414C4C3×#38612yes—
415C4C3×#39012yes—
416C4C3×#39212yes—
417C4C3×#39412yes—
418C3#1097811yescharacteristic, direct product factor, Sylow
419C2—21no—
420C2—21no—
421C2—22no—
422C2C3×#143811yescharacteristic
423C2C3×#188612yes—
424C2C3×#200514yes—
subgroupG/H
center, socleC23×C3#120
commutatorC22×C4C22×C3×C4
Frattini, cosocleC23×C4C23×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
6. #10978C3direct (N × H)
13. C2×C3×C4⋊Q8C4semidirect
82. C2×C4⋊Q8C3×C4semidirect
418. C3#10978direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊Q82
C2×C3×#4192
C22×C3×#1012
C3×#23102
C2×C3×#3392
#109783