Finite Group Databasebeta

← C3×#25489 #10644096

Subgroups — C3×#25489 #10644096

Order 768.

664 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
1C3×#3165C211yesmaximal, characteristic, complemented
2C3×#3077C211yesmaximal, characteristic
3C2×C3×#474C211yesmaximal, characteristic
4C3×#2962C211yesmaximal, characteristic, complemented
5C3×#2089C212yesmaximal, complemented
6C3×#2063C212yesmaximal
7C3×#1997C211yesmaximal, characteristic, complemented
8C3×#1952C212yesmaximal, complemented
9C3×#1950C211yesmaximal, characteristic, complemented
10C3×#1931C212yesmaximal
11C3×C4×#100C211yesmaximal, characteristic
12#25489C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#533C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C3×#501—22no—
17C3×#500—21no—
18C3×#498—21no—
19C3×#478—22no—
20C3×#476—21no—
21C3×#474—21no—
22C3×#474—21no—
23C3×#474—22no—
24C3×#474—21no—
25C3×#446—22no—
26C3×#446—21no—
27C3×C8×Q8—21no—
28C2×C3×#108C2212yes—
29C3×#428C2212yes—
30C3×#428C2212yes—
31C3×#428C2212yes—
32C2×C3×#107C2212yes—
33C3×#424C2212yes—
34C3×#423C2211yescharacteristic
35C2×C3×C4⋊C8C2211yescharacteristic
36C3×#418C2212yes—
37C2×C3×#102C2212yes—
38C2×C3×#100C2211yescharacteristic
39C2×C3×#100C2211yescharacteristic
40C3×#405C2212yes—
41C3×C4×M16C2211yescharacteristic
42C3×C4×M16C2211yescharacteristic
43C3×#343C2212yes—
44C3×#343C2212yes—
45C3×#343C2211yescharacteristic
46C3×#343C2211yescharacteristic
47C3×#341C2212yes—
48#3165C2×C311yescharacteristic, complemented
49#3077C2×C311yescharacteristic
50C2×#474C2×C311yescharacteristic
51#2962C2×C311yescharacteristic, complemented
52#2089C2×C312yescomplemented
53#2063C2×C312yes—
54#1997C2×C311yescharacteristic, complemented
55#1952C2×C312yescomplemented
56#1950C2×C311yescharacteristic, complemented
57#1931C2×C312yes—
58C4×#100C2×C311yescharacteristic
59C22×C3×Q8—21no—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C2×C3×M16C2311yescharacteristic
63C2×C3×M16C2312yes—
64C2×C3×M16C2311yescharacteristic
65C2×C3×M16C2311yescharacteristic
66C2×C3×M16C2311yescharacteristic
67C22×C3×C8C2312yes—
68C3×C4⋊Q8—21no—
69C3×C4⋊Q8—21no—
70C3×C4⋊Q8—21no—
71C3×C4⋊Q8—21no—
72C3×C4⋊Q8D411yescharacteristic
73C3×C4⋊Q8D411yescharacteristic
74C3×#125D412yes—
75C3×C22⋊Q8—21no—
76C3×C22⋊Q8—21no—
77C3×C4×Q8—21no—
78C3×C4×Q8—21no—
79C3×C4×D4—21no—
80C3×C4×D4—21no—
81C3×C4×D4—21no—
82C3×C4×D4—21no—
83C3×#117C2312yes—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2312yes—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C22⋊C4—21no—
88C2×C3×C42C2311yescharacteristic
89C3×#108—24no—
90C3×#107—22no—
91C3×#107—22no—
92C3×#107—22no—
93C3×#107—22no—
94C3×#107—22no—
95C3×#106—22no—
96C3×#106—22no—
97C3×#106—22no—
98C3×C4⋊C8—22no—
99C3×C4⋊C8—22no—
100C3×C4⋊C8—21no—
101C3×C4⋊C8—21no—
102C3×C4⋊C8D411yescharacteristic
103C3×C4⋊C8D411yescharacteristic
104C3×C4⋊C8D412yes—
105C3×#102—22no—
106C3×#102—22no—
107C3×#102—22no—
108C3×#102—22no—
109C3×#100—22no—
110C3×#100Q814yescomplemented
111C3×#97—22no—
112C3×#97—21no—
113C3×#97—21no—
114C3×C4×C8—22no—
115C3×C4×C8—21no—
116C3×C4×C8—21no—
117#533C22×C311yescharacteristic
118C2×C4⋊Q8C22×C311yescharacteristic
119C2×C4×D4C22×C311yescharacteristic
120#501—22no—
121#500—21no—
122#498—21no—
123#478—22no—
124#476—21no—
125#474—21no—
126#474—21no—
127#474—22no—
128#474—21no—
129#446—22no—
130#446—21no—
131C8×Q8—21no—
132C2×#108C22×C312yes—
133#428C22×C312yes—
134#428C22×C312yes—
135#428C22×C312yes—
136C2×#107C22×C312yes—
137#424C22×C312yes—
138#423C22×C311yescharacteristic
139C2×C4⋊C8C22×C311yescharacteristic
140#418C22×C312yes—
141C2×#102C22×C312yes—
142C2×#100C22×C311yescharacteristic
143C2×#100C22×C311yescharacteristic
144#405C22×C312yes—
145C4×M16C22×C311yescharacteristic
146C4×M16C22×C311yescharacteristic
147#343C22×C312yes—
148#343C22×C311yescharacteristic
149#343C22×C312yes—
150#343C22×C311yescharacteristic
151#341C22×C312yes—
152C24×C3—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—41no—
156C2×C3×Q8—41no—
157C2×C3×Q8—82no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4C2×Q812yes—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—41no—
167C22×C3×C4—41no—
168C22×C3×C4—82no—
169C22×C3×C4#4111yescharacteristic
170C22×C3×C4#4111yescharacteristic
171C22×C3×C4C2411yescharacteristic
172C3×M16—21no—
173C3×M16—21no—
174C3×M16—21no—
175C3×M16—21no—
176C3×M16—22no—
177C3×M16—21no—
178C3×M16—22no—
179C3×M16C2×Q812yes—
180C3×M16C2×Q812yes—
181C3×M16#4112yes—
182C2×C3×C8—21no—
183C2×C3×C8—21no—
184C2×C3×C8—21no—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C2×C3×C8—21no—
188C2×C3×C8—22no—
189C2×C3×C8C2×D412yes—
190C2×C3×C8C2×D412yes—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—41no—
200C3×C4⋊C4—41no—
201C3×C4⋊C4C2×D412yes—
202C3×C4⋊C4C2×D412yes—
203C3×C22⋊C4—22no—
204C3×C22⋊C4—41no—
205C3×C22⋊C4—41no—
206C3×C42—22no—
207C3×C42C2×D411yescharacteristic
208C3×C42C2×D411yescharacteristic
209C3×C42C2×D411yescharacteristic
210C3×C42C2×D411yescharacteristic
211C22×Q8—21no—
212C22×D4C23×C311yescharacteristic
213C23×C4—21no—
214C2×M16C23×C312yes—
215C2×M16C23×C311yescharacteristic
216C2×M16C23×C311yescharacteristic
217C2×M16C23×C311yescharacteristic
218C2×M16C23×C311yescharacteristic
219C22×C8C23×C312yes—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224C4⋊Q8C3×D411yescharacteristic
225C4⋊Q8C3×D411yescharacteristic
226#125C3×D412yes—
227C22⋊Q8—21no—
228C22⋊Q8—21no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×D4—21no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—21no—
235#117C23×C312yes—
236C2×C4⋊C4—21no—
237C2×C4⋊C4C23×C311yescharacteristic
238C2×C4⋊C4C23×C312yes—
239C2×C22⋊C4—21no—
240C2×C42C23×C311yescharacteristic
241#108—24no—
242#107—22no—
243#107—22no—
244#107—22no—
245#107—22no—
246#107—22no—
247#106—22no—
248#106—22no—
249#106—22no—
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8C3×D411yescharacteristic
255C4⋊C8C3×D411yescharacteristic
256C4⋊C8C3×D412yes—
257#102—22no—
258#102—22no—
259#102—22no—
260#102—22no—
261#100—22no—
262#100C3×Q814yescomplemented
263#97—22no—
264#97—21no—
265#97—21no—
266C4×C8—22no—
267C4×C8—21no—
268C4×C8—21no—
269C23×C3—22no—
270C23×C3—41no—
271C23×C3—41no—
272C23×C3—81no—
273C23×C3—81no—
274C23×C3C2×#4111yescharacteristic
275C3×Q8—42no—
276C3×Q8—81no—
277C3×Q8—82no—
278C3×Q8—82no—
279C3×D4—22no—
280C3×D4—22no—
281C3×D4—41no—
282C3×D4—41no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—42no—
289C2×C3×C4—82no—
290C2×C3×C4—82no—
291C2×C3×C4—81no—
292C2×C3×C4—82no—
293C2×C3×C4#12111yescharacteristic
294C2×C3×C4#12111yescharacteristic
295C2×C3×C4#12111yescharacteristic
296C2×C3×C4#12111yescharacteristic
297C2×C3×C4C22×D411yescharacteristic
298C2×C3×C4C22×D411yescharacteristic
299C2×C3×C4C22×Q811yescharacteristic
300C2×C3×C4C2×#4111yescharacteristic
301C2×C3×C42−1+411yescharacteristic
302C2×C3×C42−1+411yescharacteristic
303C3×C8—22no—
304C3×C8—22no—
305C3×C8—22no—
306C3×C8—22no—
307C3×C8—22no—
308C3×C8—41no—
309C3×C8—42no—
310C24—21no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—82no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4C2×C3×Q812yes—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—82no—
327C22×C4C3×#4111yescharacteristic
328C22×C4C3×#4111yescharacteristic
329C22×C4C24×C311yescharacteristic
330M16—21no—
331M16—21no—
332M16—21no—
333M16—21no—
334M16—22no—
335M16—21no—
336M16—22no—
337M16C2×C3×Q812yes—
338M16C2×C3×Q812yes—
339M16C3×#4112yes—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—21no—
346C2×C8—22no—
347C2×C8C2×C3×D412yes—
348C2×C8C2×C3×D412yes—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—22no—
354C4⋊C4—21no—
355C4⋊C4—22no—
356C4⋊C4—21no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4C2×C3×D412yes—
360C4⋊C4C2×C3×D412yes—
361C22⋊C4—22no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C42—22no—
365C42C2×C3×D411yescharacteristic
366C42C2×C3×D411yescharacteristic
367C42C2×C3×D411yescharacteristic
368C42C2×C3×D411yescharacteristic
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—42no—
372C22×C3—82no—
373C22×C3—82no—
374C22×C3—81no—
375C22×C3—81no—
376C22×C3C2×#12111yescharacteristic
377C22×C3#53311yescharacteristic
378C22×C3#54111yescharacteristic
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—42no—
383C3×C4—42no—
384C3×C4—81no—
385C3×C4—82no—
386C3×C4#47212yes—
387C3×C4#54711yescharacteristic
388C3×C4#54811yescharacteristic
389C3×C4D4×Q811yescharacteristic
390C3×C4D4×Q811yescharacteristic
391C23—22no—
392C23—41no—
393C23—41no—
394C23—81no—
395C23—81no—
396C23C2×C3×#4111yescharacteristic
397Q8—42no—
398Q8—81no—
399Q8—82no—
400Q8—82no—
401D4—22no—
402D4—22no—
403D4—41no—
404D4—41no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—81no—
414C2×C4—82no—
415C2×C4C3×#12111yescharacteristic
416C2×C4C3×#12111yescharacteristic
417C2×C4C3×#12111yescharacteristic
418C2×C4C3×#12111yescharacteristic
419C2×C4C22×C3×D411yescharacteristic
420C2×C4C22×C3×D411yescharacteristic
421C2×C4C22×C3×Q811yescharacteristic
422C2×C4C2×C3×#4111yescharacteristic
423C2×C4C3×2−1+411yescharacteristic
424C2×C4C3×2−1+411yescharacteristic
425C8—22no—
426C8—22no—
427C8—22no—
428C8—22no—
429C8—22no—
430C8—41no—
431C8—42no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—82no—
435C2×C3#242911yescharacteristic
436C2×C3C2×#47211yescharacteristic
437C2×C3#306911yescharacteristic
438C22—21no—
439C22—21no—
440C22—42no—
441C22—82no—
442C22—82no—
443C22—81no—
444C22—81no—
445C22C2×C3×#12111yescharacteristic
446C22C3×#53311yescharacteristic
447C22C3×#54111yescharacteristic
448C4—21no—
449C4—21no—
450C4—21no—
451C4—42no—
452C4—42no—
453C4—81no—
454C4—82no—
455C4C3×#47212yes—
456C4C3×#54711yescharacteristic
457C4C3×#54811yescharacteristic
458C4C3×D4×Q811yescharacteristic
459C4C3×D4×Q811yescharacteristic
460C3#2548911yescharacteristic, direct product factor, Sylow
461C2—21no—
462C2—21no—
463C2—82no—
464C2C3×#242911yescharacteristic
465C2C2×C3×#47211yescharacteristic
466C2C3×#306911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#121
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3165C2semidirect
4. C3×#2962C2semidirect
5. C3×#2089C2semidirect
7. C3×#1997C2semidirect
8. C3×#1952C2semidirect
9. C3×#1950C2semidirect
12. #25489C3direct (N × H)
48. #3165C2×C3semidirect
51. #2962C2×C3semidirect
52. #2089C2×C3semidirect
54. #1997C2×C3semidirect
55. #1952C2×C3semidirect
56. #1950C2×C3semidirect
110. C3×#100Q8semidirect
262. #100C3×Q8semidirect
460. C3#25489direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#31652
C3×#30772
C2×C3×#4742
C3×#29622
C3×#20892
C3×#20632
C3×#19972
C3×#19522
C3×#19502
C3×#19312
C3×C4×#1002
#254893