Finite Group Databasebeta

← C3×#33061 #10651668

Subgroups — C3×#33061 #10651668

Order 768.

770 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#487C211yesmaximal, characteristic, complemented
2C3×#3079C211yesmaximal, characteristic
3C3×#3039C211yesmaximal, characteristic
4C3×C4×#131C211yesmaximal, characteristic
5C3×#1997C211yesmaximal, characteristic, complemented
6C3×#1966C211yesmaximal, characteristic, complemented
7C3×#1930C214yesmaximal, complemented
8C3×#1908C214yesmaximal
9C3×#1837C211yesmaximal, characteristic, complemented
10#33061C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×#137C2212yes—
12C2×C3×#131C2212yes—
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C3×C4×#41C2211yescharacteristic
16C3×#487—21no—
17C3×#487—21no—
18C3×#487—22no—
19C3×#477—21no—
20C3×#476—21no—
21C3×#475—21no—
22C3×#474—21no—
23C3×#462—21no—
24C3×#462—21no—
25C3×#461—21no—
26C3×#461—21no—
27C3×#428C2212yes—
28C3×#423C2211yescharacteristic
29C3×#423C2211yescharacteristic
30C3×#419C2212yes—
31C3×#417C2212yes—
32C2×C3×#103C2214yes—
33C2×C3×#101C2214yes—
34C3×#405—22no—
35C3×#405C2212yes—
36C3×C4×M16—21no—
37C3×C4×M16C2211yescharacteristic
38C2×C3×C4×C8—21no—
39C2×C3×C4×C8C2211yescharacteristic
40C3×#343C2214yes—
41C3×#341C2214yes—
42C2×#487C2×C311yescharacteristic, complemented
43#3079C2×C311yescharacteristic
44#3039C2×C311yescharacteristic
45C4×#131C2×C311yescharacteristic
46#1997C2×C311yescharacteristic, complemented
47#1966C2×C311yescharacteristic, complemented
48#1930C2×C314yescomplemented
49#1908C2×C314yes—
50#1837C2×C311yescharacteristic, complemented
51C2×C3×#41C2311yescharacteristic
52C22×C3×Q8—21no—
53C22×C3×Q8C2312yes—
54C3×#137—28no—
55C2×C3×Q16—22no—
56C2×C3×SD16—22no—
57C3×#131—24no—
58C3×#131—24no—
59C2×C3×M16—22no—
60C2×C3×M16C2312yes—
61C2×C3×M16C2312yes—
62C2×C3×M16C2312yes—
63C22×C3×C8—22no—
64C22×C3×C8C2312yes—
65C3×C4⋊Q8—21no—
66C3×C4⋊Q8—21no—
67C3×C4⋊Q8—21no—
68C3×C4⋊Q8—21no—
69C3×C4⋊Q8—21no—
70C3×C4⋊Q8—21no—
71C3×C4⋊Q8—21no—
72C3×C4⋊Q8—21no—
73C3×C4×Q8D411yescharacteristic
74C3×C4×Q8D411yescharacteristic
75C3×C4×D4—21no—
76C3×C4×D4—21no—
77C3×C4×D4D411yescharacteristic
78C3×C4×D4D411yescharacteristic
79C3×#117—21no—
80C3×#117C2311yescharacteristic
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C42—21no—
86C2×C3×C42C2311yescharacteristic
87C3×#107—22no—
88C3×#106—22no—
89C3×C4⋊C8—21no—
90C3×C4⋊C8—21no—
91C3×C4⋊C8—21no—
92C3×C4⋊C8—21no—
93C3×#103—22no—
94C3×#103—24no—
95C3×#103—24no—
96C3×#103—22no—
97C3×#103—22no—
98C3×#102—22no—
99C3×#101—28no—
100C3×#97—22no—
101C3×#97—21no—
102C3×C4×C8—22no—
103C3×C4×C8—21no—
104C3×C4×C8D411yescharacteristic
105C3×C4×C8D411yescharacteristic
106C3×C4×C8D412yes—
107C2×#137C22×C312yes—
108C2×#131C22×C312yes—
109C2×C4⋊Q8C22×C311yescharacteristic
110C2×C4⋊Q8C22×C311yescharacteristic
111C4×#41C22×C311yescharacteristic
112#487—21no—
113#487—21no—
114#487—22no—
115#477—21no—
116#476—21no—
117#475—21no—
118#474—21no—
119#462—21no—
120#462—21no—
121#461—21no—
122#461—21no—
123#428C22×C312yes—
124#423C22×C311yescharacteristic
125#423C22×C311yescharacteristic
126#419C22×C312yes—
127#417C22×C312yes—
128C2×#103C22×C314yes—
129C2×#101C22×C314yes—
130#405—22no—
131#405C22×C312yes—
132C4×M16—21no—
133C4×M16C22×C311yescharacteristic
134C2×C4×C8—21no—
135C2×C4×C8C22×C311yescharacteristic
136#343C22×C314yes—
137#341C22×C314yes—
138C3×#41—22no—
139C3×#41#4114yes—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8—24no—
143C2×C3×Q8—24no—
144C2×C3×Q8—41no—
145C2×C3×Q8—41no—
146C2×C3×Q8—82no—
147C2×C3×Q8C2×D411yescharacteristic
148C2×C3×D4—21no—
149C2×C3×D4C2×D411yescharacteristic
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4#4111yescharacteristic
156C22×C3×C4#4111yescharacteristic
157C22×C3×C4C2411yescharacteristic
158C3×Q16—44no—
159C3×SD16—44no—
160C3×M16—28no—
161C3×M16—22no—
162C3×M16—24no—
163C3×M16—24no—
164C3×M16#4114yes—
165C2×C3×C8—28no—
166C2×C3×C8—22no—
167C2×C3×C8—22no—
168C2×C3×C8—22no—
169C2×C3×C8—22no—
170C2×C3×C8—22no—
171C2×C3×C8C2×D412yes—
172C2×C3×C8C2×D412yes—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—41no—
178C3×C4⋊C4—41no—
179C3×C4⋊C4—41no—
180C3×C4⋊C4—41no—
181C3×C4⋊C4—41no—
182C3×C4⋊C4—41no—
183C3×C4⋊C4C2×D411yescharacteristic
184C3×C4⋊C4C2×D411yescharacteristic
185C3×C22⋊C4—21no—
186C3×C22⋊C4—21no—
187C3×C22⋊C4—21no—
188C3×C42—21no—
189C3×C42—21no—
190C3×C42—21no—
191C3×C42C2×D411yescharacteristic
192C3×C42C2×D411yescharacteristic
193C3×C42C2×D411yescharacteristic
194C3×C42C2×D411yescharacteristic
195C2×#41C23×C311yescharacteristic
196C22×Q8—21no—
197C22×Q8C23×C312yes—
198#137—28no—
199C2×Q16—22no—
200C2×SD16—22no—
201#131—24no—
202#131—24no—
203C2×M16—22no—
204C2×M16C23×C312yes—
205C2×M16C23×C312yes—
206C2×M16C23×C312yes—
207C22×C8—22no—
208C22×C8C23×C312yes—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4×Q8C3×D411yescharacteristic
218C4×Q8C3×D411yescharacteristic
219C4×D4—21no—
220C4×D4—21no—
221C4×D4C3×D411yescharacteristic
222C4×D4C3×D411yescharacteristic
223#117—21no—
224#117C23×C311yescharacteristic
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4C23×C312yes—
229C2×C42—21no—
230C2×C42C23×C311yescharacteristic
231#107—22no—
232#106—22no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237#103—22no—
238#103—24no—
239#103—24no—
240#103—22no—
241#103—22no—
242#102—22no—
243#101—28no—
244#97—22no—
245#97—21no—
246C4×C8—22no—
247C4×C8—21no—
248C4×C8C3×D411yescharacteristic
249C4×C8C3×D411yescharacteristic
250C4×C8C3×D412yes—
251C23×C3—21no—
252C23×C3C2×#4111yescharacteristic
253C3×Q8—21no—
254C3×Q8—44no—
255C3×Q8—42no—
256C3×Q8—42no—
257C3×Q8—82no—
258C3×Q8—82no—
259C3×Q8#12412yes—
260C3×D4—24no—
261C3×D4—21no—
262C3×D4#12412yes—
263C2×C3×C4—21no—
264C2×C3×C4—24no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—44no—
274C2×C3×C4—82no—
275C2×C3×C4#12111yescharacteristic
276C2×C3×C4#12111yescharacteristic
277C2×C3×C4#12111yescharacteristic
278C2×C3×C4#12111yescharacteristic
279C2×C3×C4C22×D411yescharacteristic
280C2×C3×C4C22×D411yescharacteristic
281C2×C3×C4C2×#4112yes—
282C2×C3×C4C2×#4112yes—
283C3×C8—22no—
284C3×C8—24no—
285C3×C8—22no—
286C3×C8—42no—
287C3×C8—42no—
288#41—22no—
289#41C3×#4114yes—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—24no—
293C2×Q8—24no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—82no—
297C2×Q8C2×C3×D411yescharacteristic
298C2×D4—21no—
299C2×D4C2×C3×D411yescharacteristic
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4C3×#4111yescharacteristic
306C22×C4C3×#4111yescharacteristic
307C22×C4C24×C311yescharacteristic
308Q16—44no—
309SD16—44no—
310M16—28no—
311M16—22no—
312M16—24no—
313M16—24no—
314M16C3×#4114yes—
315C2×C8—28no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8C2×C3×D412yes—
322C2×C8C2×C3×D412yes—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4C2×C3×D411yescharacteristic
334C4⋊C4C2×C3×D411yescharacteristic
335C22⋊C4—21no—
336C22⋊C4—21no—
337C22⋊C4—21no—
338C42—21no—
339C42—21no—
340C42—21no—
341C42C2×C3×D411yescharacteristic
342C42C2×C3×D411yescharacteristic
343C42C2×C3×D411yescharacteristic
344C42C2×C3×D411yescharacteristic
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—22no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3C2×#12111yescharacteristic
351C22×C3#52912yes—
352C3×C4—22no—
353C3×C4—21no—
354C3×C4—21no—
355C3×C4—41no—
356C3×C4—41no—
357C3×C4—82no—
358C3×C4—82no—
359C3×C4#47112yes—
360C3×C4#47312yes—
361C3×C4#52512yes—
362C3×C4C2×#12411yescharacteristic
363C3×C4#53211yescharacteristic
364C23—21no—
365C23C2×C3×#4111yescharacteristic
366Q8—21no—
367Q8—44no—
368Q8—42no—
369Q8—42no—
370Q8—82no—
371Q8—82no—
372Q8C3×#12412yes—
373D4—24no—
374D4—21no—
375D4C3×#12412yes—
376C2×C4—21no—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—82no—
388C2×C4C3×#12111yescharacteristic
389C2×C4C3×#12111yescharacteristic
390C2×C4C3×#12111yescharacteristic
391C2×C4C3×#12111yescharacteristic
392C2×C4C22×C3×D411yescharacteristic
393C2×C4C22×C3×D411yescharacteristic
394C2×C4C2×C3×#4112yes—
395C2×C4C2×C3×#4112yes—
396C8—22no—
397C8—24no—
398C8—22no—
399C8—42no—
400C8—42no—
401C2×C3—22no—
402C2×C3—42no—
403C2×C3#285011yescharacteristic
404C2×C3C2×#47111yescharacteristic
405C2×C3C2×#47311yescharacteristic
406C22—21no—
407C22—21no—
408C22—22no—
409C22—41no—
410C22—41no—
411C22C2×C3×#12111yescharacteristic
412C22C3×#52912yes—
413C4—21no—
414C4—21no—
415C4—22no—
416C4—41no—
417C4—41no—
418C4—82no—
419C4—82no—
420C4C3×#47112yes—
421C4C3×#47312yes—
422C4C3×#52512yes—
423C4C2×C3×#12411yescharacteristic
424C4C3×#53211yescharacteristic
425C3#3306111yescharacteristic, direct product factor, Sylow
426C2—22no—
427C2—42no—
428C2C3×#285011yescharacteristic
429C2C2×C3×#47111yescharacteristic
430C2C2×C3×#47311yescharacteristic
subgroupG/H
center, socleC22×C3C2×#121
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#41C23
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. C2×C3×#487C2semidirect
5. C3×#1997C2semidirect
6. C3×#1966C2semidirect
7. C3×#1930C2semidirect
9. C3×#1837C2semidirect
10. #33061C3direct (N × H)
42. C2×#487C2×C3semidirect
46. #1997C2×C3semidirect
47. #1966C2×C3semidirect
48. #1930C2×C3semidirect
50. #1837C2×C3semidirect
425. C3#33061direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4872
C3×#30792
C3×#30392
C3×C4×#1312
C3×#19972
C3×#19662
C3×#19302
C3×#19082
C3×#18372
#330613