Finite Group Databasebeta

← C3×#10001 #10628608

Subgroups — C3×#10001 #10628608

Order 768.

568 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3101C211yesmaximal, characteristic
2C3×#3078C211yesmaximal, characteristic, complemented
3C3×#2536C211yesmaximal, characteristic
4C3×#1518C211yesmaximal, characteristic, complemented
5C3×#1486C211yesmaximal, characteristic
6C3×#1479C211yesmaximal, characteristic, complemented
7C3×#1286C211yesmaximal, characteristic, complemented
8#10001C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C22⋊Q8—21no—
11C2×C3×C4×Q8C2211yescharacteristic
12C2×C3×C4×D4C2211yescharacteristic
13C3×#483—21no—
14C3×#479—22no—
15C3×#477—21no—
16C3×#476—21no—
17C3×#475—21no—
18C3×#461—21no—
19C3×#450—21no—
20C3×C4×SD16—22no—
21C3×#431C2211yescharacteristic
22C3×#431C2211yescharacteristic
23C3×#428—21no—
24C3×#423C2211yescharacteristic
25C3×#419—21no—
26C3×#398—21no—
27C3×#394—21no—
28C3×#388—22no—
29C3×#382C2211yescharacteristic
30C3×#380—21no—
31C3×#328—22no—
32C3×Q8⋊C8—22no—
33C3×D4⋊C8—22no—
34#3101C2×C311yescharacteristic
35#3078C2×C311yescharacteristic, complemented
36#2536C2×C311yescharacteristic
37#1518C2×C311yescharacteristic, complemented
38#1486C2×C311yescharacteristic
39#1479C2×C311yescharacteristic, complemented
40#1286C2×C311yescharacteristic, complemented
41C22×C3×Q8D411yescharacteristic
42C22×C3×D4D411yescharacteristic
43C23×C3×C4—21no—
44C2×C3×SD16—22no—
45C2×C3×M16—21no—
46C3×C4⋊Q8D411yescharacteristic
47C3×C4⋊Q8D411yescharacteristic
48C3×#125—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C3×C22⋊Q8—42no—
52C3×C22⋊Q8—42no—
53C3×C4×Q8—21no—
54C3×C4×Q8—21no—
55C3×C4×Q8—21no—
56C3×C4×Q8—21no—
57C3×C4×Q8—21no—
58C3×C4×Q8—21no—
59C3×C4×D4—21no—
60C3×C4×D4—21no—
61C3×C4×D4—21no—
62C3×C4×D4—21no—
63C3×#117—21no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4D411yescharacteristic
68C2×C3×C4⋊C4D411yescharacteristic
69C2×C3×C22⋊C4—22no—
70C2×C3×C22⋊C4—22no—
71C2×C3×C22⋊C4—21no—
72C2×C3×C42—21no—
73C2×C3×C42C2311yescharacteristic
74C3×#107—22no—
75C3×#106—22no—
76C3×#106—22no—
77C3×C4⋊C8—21no—
78C3×C4⋊C8—21no—
79C3×C4⋊C8—21no—
80C3×C4⋊C8D412yes—
81C3×#103—22no—
82C3×#103—22no—
83C3×#103—22no—
84C3×#102—22no—
85C3×C22⋊C8—21no—
86C3×C22⋊C8—21no—
87C3×C4×C8—21no—
88C3×C4×C8D412yes—
89C3×#95—22no—
90C3×#95—21no—
91C3×#95D412yes—
92#533C22×C311yescharacteristic
93C2×C22⋊Q8—21no—
94C2×C4×Q8C22×C311yescharacteristic
95C2×C4×D4C22×C311yescharacteristic
96#483—21no—
97#479—22no—
98#477—21no—
99#476—21no—
100#475—21no—
101#461—21no—
102#450—21no—
103C4×SD16—22no—
104#431C22×C311yescharacteristic
105#431C22×C311yescharacteristic
106#428—21no—
107#423C22×C311yescharacteristic
108#419—21no—
109#398—21no—
110#394—21no—
111#388—22no—
112#382C22×C311yescharacteristic
113#380—21no—
114#328—22no—
115Q8⋊C8—22no—
116D4⋊C8—22no—
117C24×C3—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—21no—
121C2×C3×Q8—21no—
122C2×C3×Q8—21no—
123C2×C3×Q8—21no—
124C2×C3×Q8—21no—
125C2×C3×Q8SD1612yescomplemented
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—42no—
129C22×C3×C4—21no—
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—41no—
134C22×C3×C4—41no—
135C22×C3×C4—41no—
136C22×C3×C4—41no—
137C22×C3×C4—41no—
138C22×C3×C4—41no—
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C3×SD16—44no—
143C3×M16—41no—
144C3×M16—41no—
145C2×C3×C8—22no—
146C2×C3×C8—22no—
147C2×C3×C8—41no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—44no—
157C3×C4⋊C4—41no—
158C3×C4⋊C4—41no—
159C3×C4⋊C4—41no—
160C3×C4⋊C4—41no—
161C3×C4⋊C4—41no—
162C3×C4⋊C4—41no—
163C3×C22⋊C4—44no—
164C3×C22⋊C4—44no—
165C3×C22⋊C4—41no—
166C3×C22⋊C4—41no—
167C3×C22⋊C4—41no—
168C3×C42—41no—
169C3×C42—41no—
170C3×C42—41no—
171C3×C42C2×D411yescharacteristic
172C3×C42C2×D411yescharacteristic
173C3×C42C2×D411yescharacteristic
174C3×C42#4111yescharacteristic
175C22×Q8C3×D411yescharacteristic
176C22×D4C3×D411yescharacteristic
177C23×C4—21no—
178C2×SD16—22no—
179C2×M16—21no—
180C4⋊Q8C3×D411yescharacteristic
181C4⋊Q8C3×D411yescharacteristic
182#125—21no—
183C22⋊Q8—21no—
184C22⋊Q8—21no—
185C22⋊Q8—42no—
186C22⋊Q8—42no—
187C4×Q8—21no—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×Q8—21no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4—21no—
196C4×D4—21no—
197#117—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4C3×D411yescharacteristic
202C2×C4⋊C4C3×D411yescharacteristic
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—21no—
206C2×C42—21no—
207C2×C42C23×C311yescharacteristic
208#107—22no—
209#106—22no—
210#106—22no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8C3×D412yes—
215#103—22no—
216#103—22no—
217#103—22no—
218#102—22no—
219C22⋊C8—21no—
220C22⋊C8—21no—
221C4×C8—21no—
222C4×C8C3×D412yes—
223#95—22no—
224#95—21no—
225#95C3×D412yes—
226C23×C3—22no—
227C23×C3—41no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3#12011yescharacteristic
234C3×Q8—22no—
235C3×Q8—22no—
236C3×Q8—42no—
237C3×Q8—41no—
238C3×Q8—41no—
239C3×Q8—81no—
240C3×D4—22no—
241C3×D4—22no—
242C3×D4—41no—
243C3×D4—41no—
244C2×C3×C4—22no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—42no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—81no—
260C2×C3×C4—81no—
261C2×C3×C4—81no—
262C2×C3×C4—81no—
263C2×C3×C4#12011yescharacteristic
264C2×C3×C4#12011yescharacteristic
265C2×C3×C4#12111yescharacteristic
266C2×C3×C4#12111yescharacteristic
267C2×C3×C4#12111yescharacteristic
268C2×C3×C4#12711yescharacteristic
269C2×C3×C4C2×SD1611yescharacteristic
270C2×C3×C4#13511yescharacteristic
271C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
272C2×C3×C4#13711yescharacteristic
273C3×C8—42no—
274C3×C8—42no—
275C3×C8—81no—
276C24—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8C3×SD1612yescomplemented
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—42no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4C2×C3×D411yescharacteristic
299C22×C4C2×C3×D411yescharacteristic
300C22×C4C2×C3×D411yescharacteristic
301SD16—44no—
302M16—41no—
303M16—41no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—41no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—44no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C22⋊C4—44no—
323C22⋊C4—44no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C42—41no—
328C42—41no—
329C42—41no—
330C42C2×C3×D411yescharacteristic
331C42C2×C3×D411yescharacteristic
332C42C2×C3×D411yescharacteristic
333C42C3×#4111yescharacteristic
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—42no—
337C22×C3—42no—
338C22×C3—42no—
339C22×C3—81no—
340C22×C3—81no—
341C22×C3—81no—
342C22×C3—81no—
343C22×C3#39211yescharacteristic
344C22×C3#45011yescharacteristic
345C22×C3#45211yescharacteristic
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—42no—
351C3×C4—82no—
352C3×C4—81no—
353C3×C4—81no—
354C3×C4—81no—
355C3×C4—81no—
356C3×C4#44811yescharacteristic
357C3×C4#44911yescharacteristic
358C3×C4#46511yescharacteristic
359C3×C4#46811yescharacteristic
360C23—22no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23C3×#12011yescharacteristic
368Q8—22no—
369Q8—22no—
370Q8—42no—
371Q8—41no—
372Q8—41no—
373Q8—81no—
374D4—22no—
375D4—22no—
376D4—41no—
377D4—41no—
378C2×C4—22no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4C3×#12011yescharacteristic
398C2×C4C3×#12011yescharacteristic
399C2×C4C3×#12111yescharacteristic
400C2×C4C3×#12111yescharacteristic
401C2×C4C3×#12111yescharacteristic
402C2×C4C3×#12711yescharacteristic
403C2×C4C2×C3×SD1611yescharacteristic
404C2×C4C3×#13511yescharacteristic
405C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
406C2×C4C3×#13711yescharacteristic
407C8—42no—
408C8—42no—
409C8—81no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—82no—
413C2×C3#200011yescharacteristic
414C2×C3#200911yescharacteristic
415C2×C3#201011yescharacteristic
416C22—21no—
417C22—21no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22C3×#39211yescharacteristic
426C22C3×#45011yescharacteristic
427C22C3×#45211yescharacteristic
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—42no—
433C4—82no—
434C4—81no—
435C4—81no—
436C4—81no—
437C4—81no—
438C4C3×#44811yescharacteristic
439C4C3×#44911yescharacteristic
440C4C3×#46511yescharacteristic
441C4C3×#46811yescharacteristic
442C3#1000111yescharacteristic, direct product factor, Sylow
443C2—21no—
444C2—21no—
445C2—82no—
446C2C3×#200011yescharacteristic
447C2C3×#200911yescharacteristic
448C2C3×#201011yescharacteristic
subgroupG/H
center, socleC22×C3#392
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC22×C3×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3078C2semidirect
4. C3×#1518C2semidirect
6. C3×#1479C2semidirect
7. C3×#1286C2semidirect
8. #10001C3direct (N × H)
35. #3078C2×C3semidirect
37. #1518C2×C3semidirect
39. #1479C2×C3semidirect
40. #1286C2×C3semidirect
125. C2×C3×Q8SD16semidirect
284. C2×Q8C3×SD16semidirect
442. C3#10001direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#31012
C3×#30782
C3×#25362
C3×#15182
C3×#14862
C3×#14792
C3×#12862
#100013