Finite Group Databasebeta

← C3×#10985 #10629592

Subgroups — C3×#10985 #10629592

Order 768.

1270 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#536C211yesmaximal, characteristic
2C2×C3×#419C212yesmaximal, complemented
3C2×C3×#413C211yesmaximal, characteristic
4C2×C3×#410C211yesmaximal, characteristic
5C3×#1733C212yesmaximal, complemented
6#10985C311yesmaximal, characteristic, direct product factor, Sylow
7C22×C3×#41C2211yescharacteristic
8C23×C3×Q8C2211yescharacteristic
9C22×C3×M16C2212yes—
10C3×#536—24no—
11C3×#536—24no—
12C2×C3×C4⋊Q8—21no—
13C2×C3×#124—21no—
14C2×C3×#123—22no—
15C2×C3×C22⋊Q8C414yescomplemented
16C2×C3×#117C2212yes—
17C2×C3×#117C2211yescharacteristic
18C3×#419—24no—
19C3×#419—24no—
20C2×C3×#103—24no—
21C3×#413—24no—
22C2×C3×#101—21no—
23C2×C3×#100—21no—
24C3×#410—24no—
25C2×C3×#99—22no—
26C3×#339—24no—
27C2×#536C2×C311yescharacteristic
28C2×#419C2×C312yescomplemented
29C2×#413C2×C311yescharacteristic
30C2×#410C2×C311yescharacteristic
31#1733C2×C312yescomplemented
32C2×C3×#41—22no—
33C2×C3×#41—22no—
34C2×C3×#41—22no—
35C2×C3×#41—22no—
36C2×C3×#41—22no—
37C2×C3×#41—22no—
38C22×C3×Q8—21no—
39C22×C3×Q8—22no—
40C22×C3×Q8—22no—
41C22×C3×Q8—22no—
42C22×C3×Q8—22no—
43C22×C3×Q8—22no—
44C22×C3×Q8C2×C412yes—
45C22×C3×Q8D414yes—
46C22×C3×Q8D411yescharacteristic
47C22×C3×D4D412yes—
48C22×C3×D4D411yescharacteristic
49C23×C3×C4—21no—
50C23×C3×C4C2×C412yes—
51C23×C3×C4C2311yescharacteristic
52C2×C3×M16—22no—
53C2×C3×M16—22no—
54C2×C3×M16—22no—
55C2×C3×M16—22no—
56C2×C3×M16—22no—
57C2×C3×M16D414yes—
58C22×C3×C8—22no—
59C3×C4⋊Q8—28no—
60C3×#124—28no—
61C3×#123—216no—
62C3×C22⋊Q8—28no—
63C3×C22⋊Q8—28no—
64C3×#117—22no—
65C3×#117—22no—
66C3×#117—22no—
67C3×#117—22no—
68C3×#117D414yes—
69C3×#117D414yes—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4C2×C412yes—
73C2×C3×C22⋊C4—24no—
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C42—21no—
77C2×C3×C42—22no—
78C3×#103—216no—
79C3×#101—24no—
80C3×#100—24no—
81C3×#99—28no—
82C22×#41C22×C311yescharacteristic
83C23×Q8C22×C311yescharacteristic
84C22×M16C22×C312yes—
85#536—24no—
86#536—24no—
87C2×C4⋊Q8—21no—
88C2×#124—21no—
89C2×#123—22no—
90C2×C22⋊Q8C3×C414yescomplemented
91C2×#117C22×C312yes—
92C2×#117C22×C311yescharacteristic
93#419—24no—
94#419—24no—
95C2×#103—24no—
96#413—24no—
97C2×#101—21no—
98C2×#100—21no—
99#410—24no—
100C2×#99—22no—
101#339—24no—
102C24×C3C22⋊C412yes—
103C24×C3C2×D411yescharacteristic
104C3×#41—42no—
105C3×#41—42no—
106C3×#41—42no—
107C3×#41—42no—
108C3×#41—42no—
109C3×#41—42no—
110C3×#41—42no—
111C3×#41—42no—
112C2×C3×Q8—22no—
113C2×C3×Q8—21no—
114C2×C3×Q8—24no—
115C2×C3×Q8—22no—
116C2×C3×Q8—22no—
117C2×C3×Q8—21no—
118C2×C3×Q8—24no—
119C2×C3×Q8—24no—
120C2×C3×Q8—24no—
121C2×C3×Q8—24no—
122C2×C3×Q8—24no—
123C2×C3×Q8—24no—
124C2×C3×Q8—41no—
125C2×C3×Q8—42no—
126C2×C3×Q8—42no—
127C2×C3×Q8—42no—
128C2×C3×Q8—41no—
129C2×C3×Q8—42no—
130C2×C3×Q8—42no—
131C2×C3×Q8C22⋊C414yes—
132C2×C3×D4—22no—
133C2×C3×D4—24no—
134C2×C3×D4—24no—
135C2×C3×D4—22no—
136C2×C3×D4—21no—
137C2×C3×D4—22no—
138C2×C3×D4—22no—
139C2×C3×D4—21no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—24no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—24no—
146C22×C3×C4—24no—
147C22×C3×C4—41no—
148C22×C3×C4—42no—
149C22×C3×C4—41no—
150C22×C3×C4—42no—
151C22×C3×C4C22⋊C412yes—
152C22×C3×C4C22×C411yescharacteristic
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×D412yes—
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4#4112yes—
159C22×C3×C4#4112yes—
160C3×M16—24no—
161C3×M16—24no—
162C3×M16—42no—
163C3×M16—42no—
164C2×C3×C8—24no—
165C2×C3×C8—42no—
166C2×C3×C8—42no—
167C3×C4⋊C4—216no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—24no—
170C3×C4⋊C4—42no—
171C3×C4⋊C4C22⋊C414yes—
172C3×C22⋊C4—216no—
173C3×C22⋊C4—24no—
174C3×C22⋊C4—24no—
175C3×C22⋊C4—42no—
176C3×C42—24no—
177C3×C42—24no—
178C3×C42—42no—
179C2×#41—22no—
180C2×#41—22no—
181C2×#41—22no—
182C2×#41—22no—
183C2×#41—22no—
184C2×#41—22no—
185C22×Q8—21no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8C2×C3×C412yes—
192C22×Q8C3×D414yes—
193C22×Q8C3×D411yescharacteristic
194C22×D4C3×D412yes—
195C22×D4C3×D411yescharacteristic
196C23×C4—21no—
197C23×C4C2×C3×C412yes—
198C23×C4C23×C311yescharacteristic
199C2×M16—22no—
200C2×M16—22no—
201C2×M16—22no—
202C2×M16—22no—
203C2×M16—22no—
204C2×M16C3×D414yes—
205C22×C8—22no—
206C4⋊Q8—28no—
207#124—28no—
208#123—216no—
209C22⋊Q8—28no—
210C22⋊Q8—28no—
211#117—22no—
212#117—22no—
213#117—22no—
214#117—22no—
215#117C3×D414yes—
216#117C3×D414yes—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4C2×C3×C412yes—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—22no—
223C2×C42—21no—
224C2×C42—22no—
225#103—216no—
226#101—24no—
227#100—24no—
228#99—28no—
229C23×C3—22no—
230C23×C3—22no—
231C23×C3—22no—
232C23×C3—42no—
233C23×C3—44no—
234C23×C3C2×C22⋊C411yescharacteristic
235C23×C3#12012yes—
236C23×C3#12011yescharacteristic
237C23×C3#12112yes—
238C23×C3#12311yescharacteristic
239C3×Q8—28no—
240C3×Q8—44no—
241C3×Q8—44no—
242C3×Q8—42no—
243C3×Q8—44no—
244C3×Q8—42no—
245C3×D4—44no—
246C3×D4—44no—
247C3×D4—42no—
248C3×D4—42no—
249C2×C3×C4—24no—
250C2×C3×C4—24no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—44no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—44no—
258C2×C3×C4—44no—
259C2×C3×C4—44no—
260C2×C3×C4—44no—
261C2×C3×C4—44no—
262C2×C3×C4—84no—
263C2×C3×C4C2×C22⋊C411yescharacteristic
264C2×C3×C4C2×C22⋊C411yescharacteristic
265C2×C3×C4C4×D412yes—
266C2×C3×C4C4×D412yes—
267C2×C3×C4#12012yes—
268C2×C3×C4#12011yescharacteristic
269C2×C3×C4#12012yes—
270C2×C3×C4#12112yes—
271C2×C3×C4#12112yes—
272C2×C3×C4#12112yes—
273C2×C3×C4#12112yes—
274C2×C3×C4#12112yes—
275C2×C3×C4#12312yes—
276C2×C3×C4#12311yescharacteristic
277C2×C3×C4#12412yes—
278C2×C3×C4#12712yes—
279C3×C8—44no—
280C24C3×C22⋊C412yes—
281C24C2×C3×D411yescharacteristic
282#41—42no—
283#41—42no—
284#41—42no—
285#41—42no—
286#41—42no—
287#41—42no—
288#41—42no—
289#41—42no—
290C2×Q8—22no—
291C2×Q8—21no—
292C2×Q8—24no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—21no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—41no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—41no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8C3×C22⋊C414yes—
310C2×D4—22no—
311C2×D4—24no—
312C2×D4—24no—
313C2×D4—22no—
314C2×D4—21no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4C3×C22⋊C412yes—
330C22×C4C22×C3×C411yescharacteristic
331C22×C4C2×C3×D412yes—
332C22×C4C2×C3×D412yes—
333C22×C4C2×C3×D412yes—
334C22×C4C2×C3×D412yes—
335C22×C4C2×C3×D411yescharacteristic
336C22×C4C3×#4112yes—
337C22×C4C3×#4112yes—
338M16—24no—
339M16—24no—
340M16—42no—
341M16—42no—
342C2×C8—24no—
343C2×C8—42no—
344C2×C8—42no—
345C4⋊C4—216no—
346C4⋊C4—22no—
347C4⋊C4—24no—
348C4⋊C4—42no—
349C4⋊C4C3×C22⋊C414yes—
350C22⋊C4—216no—
351C22⋊C4—24no—
352C22⋊C4—24no—
353C22⋊C4—42no—
354C42—24no—
355C42—24no—
356C42—42no—
357C22×C3—22no—
358C22×C3—24no—
359C22×C3—41no—
360C22×C3—42no—
361C22×C3—41no—
362C22×C3—44no—
363C22×C3—42no—
364C22×C3—82no—
365C22×C3—82no—
366C22×C3#37911yescharacteristic
367C22×C3#38611yescharacteristic
368C22×C3#38611yescharacteristic
369C22×C3#39212yes—
370C22×C3#39412yes—
371C3×C4—44no—
372C3×C4—42no—
373C3×C4—44no—
374C3×C4—84no—
375C3×C4#38612yes—
376C3×C4#39012yes—
377C3×C4#39212yes—
378C3×C4#39412yes—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—42no—
383C23—44no—
384C23C2×C3×C22⋊C411yescharacteristic
385C23C3×#12012yes—
386C23C3×#12011yescharacteristic
387C23C3×#12112yes—
388C23C3×#12311yescharacteristic
389Q8—28no—
390Q8—44no—
391Q8—44no—
392Q8—42no—
393Q8—44no—
394Q8—42no—
395D4—44no—
396D4—44no—
397D4—42no—
398D4—42no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—84no—
413C2×C4C2×C3×C22⋊C411yescharacteristic
414C2×C4C2×C3×C22⋊C411yescharacteristic
415C2×C4C3×C4×D412yes—
416C2×C4C3×C4×D412yes—
417C2×C4C3×#12012yes—
418C2×C4C3×#12011yescharacteristic
419C2×C4C3×#12012yes—
420C2×C4C3×#12112yes—
421C2×C4C3×#12112yes—
422C2×C4C3×#12112yes—
423C2×C4C3×#12112yes—
424C2×C4C3×#12112yes—
425C2×C4C3×#12312yes—
426C2×C4C3×#12311yescharacteristic
427C2×C4C3×#12412yes—
428C2×C4C3×#12712yes—
429C8—44no—
430C2×C3—22no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—22no—
434C2×C3—82no—
435C2×C3#143811yescharacteristic
436C2×C3#179612yes—
437C22—22no—
438C22—24no—
439C22—41no—
440C22—42no—
441C22—41no—
442C22—44no—
443C22—42no—
444C22—82no—
445C22—82no—
446C22C3×#37911yescharacteristic
447C22C3×#38611yescharacteristic
448C22C3×#38611yescharacteristic
449C22C3×#39212yes—
450C22C3×#39412yes—
451C4—44no—
452C4—42no—
453C4—44no—
454C4—84no—
455C4C3×#38612yes—
456C4C3×#39012yes—
457C4C3×#39212yes—
458C4C3×#39412yes—
459C3#1098511yescharacteristic, direct product factor, Sylow
460C2—22no—
461C2—21no—
462C2—21no—
463C2—22no—
464C2—82no—
465C2C3×#143811yescharacteristic
466C2C3×#179612yes—
subgroupG/H
center, socleC22×C3#379
commutatorC22×C4C22×C3×C4
Frattini, cosocleC23×C4C23×C3
prime radicalC22×C3×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#419C2semidirect
5. C3×#1733C2semidirect
6. #10985C3direct (N × H)
15. C2×C3×C22⋊Q8C4semidirect
28. C2×#419C2×C3semidirect
31. #1733C2×C3semidirect
90. C2×C22⋊Q8C3×C4semidirect
459. C3#10985direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5362
C2×C3×#4192
C2×C3×#4132
C2×C3×#4102
C3×#17332
#109853