Finite Group Databasebeta

← C3×#10003 #10628610

Subgroups — C3×#10003 #10628610

Order 768.

500 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3104C211yesmaximal, characteristic
2C3×#3078C211yesmaximal, characteristic
3C3×#2581C211yesmaximal, characteristic
4C3×#1518C211yesmaximal, characteristic
5C3×#1487C211yesmaximal, characteristic
6C3×#1479C211yesmaximal, characteristic
7C3×#1286C211yesmaximal, characteristic
8#10003C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C2×C3×C4×Q8C2211yescharacteristic
12C3×#484—21no—
13C3×#479—22no—
14C3×#477—21no—
15C3×#477—21no—
16C3×#475—21no—
17C3×#462—21no—
18C3×#451—21no—
19C3×C4×Q16—22no—
20C3×#431C2211yescharacteristic
21C3×#431C2211yescharacteristic
22C3×#428—21no—
23C3×#423C2211yescharacteristic
24C3×#419—21no—
25C3×#400—21no—
26C3×#395—21no—
27C3×#391—21no—
28C3×#389—21no—
29C3×#387—22no—
30C3×#382C2211yescharacteristic
31C3×#328—22no—
32C3×Q8⋊C8—22no—
33C3×Q8⋊C8—22no—
34#3104C2×C311yescharacteristic
35#3078C2×C311yescharacteristic
36#2581C2×C311yescharacteristic
37#1518C2×C311yescharacteristic
38#1487C2×C311yescharacteristic
39#1479C2×C311yescharacteristic
40#1286C2×C311yescharacteristic
41C22×C3×Q8D411yescharacteristic
42C22×C3×Q8D411yescharacteristic
43C2×C3×Q16—22no—
44C2×C3×M16—21no—
45C3×C4⋊Q8D411yescharacteristic
46C3×C4⋊Q8D411yescharacteristic
47C3×#125—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
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×Q8—21no—
60C3×#117—21no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4D411yescharacteristic
66C2×C3×C4⋊C4D411yescharacteristic
67C2×C3×C42—21no—
68C2×C3×C42—21no—
69C2×C3×C42C2311yescharacteristic
70C3×#107—22no—
71C3×#107—22no—
72C3×#106—22no—
73C3×C4⋊C8—21no—
74C3×C4⋊C8—21no—
75C3×C4⋊C8—21no—
76C3×C4⋊C8D412yes—
77C3×#103—22no—
78C3×#103—22no—
79C3×#103—22no—
80C3×#103—22no—
81C3×C22⋊C8—21no—
82C3×C22⋊C8—21no—
83C3×C4×C8—21no—
84C3×C4×C8D412yes—
85C3×#95—22no—
86C3×#95—22no—
87C3×#95—22no—
88C3×#95—21no—
89C3×#95D412yes—
90#533C22×C311yescharacteristic
91C2×C4×Q8C22×C311yescharacteristic
92C2×C4×Q8C22×C311yescharacteristic
93#484—21no—
94#479—22no—
95#477—21no—
96#477—21no—
97#475—21no—
98#462—21no—
99#451—21no—
100C4×Q16—22no—
101#431C22×C311yescharacteristic
102#431C22×C311yescharacteristic
103#428—21no—
104#423C22×C311yescharacteristic
105#419—21no—
106#400—21no—
107#395—21no—
108#391—21no—
109#389—21no—
110#387—22no—
111#382C22×C311yescharacteristic
112#328—22no—
113Q8⋊C8—22no—
114Q8⋊C8—22no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—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—42no—
125C2×C3×Q8Q1612yescomplemented
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C3×Q16—44no—
135C3×M16—41no—
136C3×M16—41no—
137C2×C3×C8—22no—
138C2×C3×C8—22no—
139C2×C3×C8—41no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—44no—
149C3×C4⋊C4—41no—
150C3×C4⋊C4—41no—
151C3×C4⋊C4—41no—
152C3×C4⋊C4—41no—
153C3×C4⋊C4—41no—
154C3×C4⋊C4—41no—
155C3×C4⋊C4—41no—
156C3×C4⋊C4—41no—
157C3×C22⋊C4—41no—
158C3×C42—41no—
159C3×C42—41no—
160C3×C42—41no—
161C3×C42—41no—
162C3×C42—41no—
163C3×C42C2×D411yescharacteristic
164C3×C42C2×D411yescharacteristic
165C3×C42C2×D411yescharacteristic
166C3×C42#4111yescharacteristic
167C22×Q8C3×D411yescharacteristic
168C22×Q8C3×D411yescharacteristic
169C2×Q16—22no—
170C2×M16—21no—
171C4⋊Q8C3×D411yescharacteristic
172C4⋊Q8C3×D411yescharacteristic
173#125—21no—
174C22⋊Q8—21no—
175C22⋊Q8—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×Q8—21no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186#117—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4C3×D411yescharacteristic
192C2×C4⋊C4C3×D411yescharacteristic
193C2×C42—21no—
194C2×C42—21no—
195C2×C42C23×C311yescharacteristic
196#107—22no—
197#107—22no—
198#106—22no—
199C4⋊C8—21no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8C3×D412yes—
203#103—22no—
204#103—22no—
205#103—22no—
206#103—22no—
207C22⋊C8—21no—
208C22⋊C8—21no—
209C4×C8—21no—
210C4×C8C3×D412yes—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—21no—
215#95C3×D412yes—
216C23×C3#12011yescharacteristic
217C3×Q8—22no—
218C3×Q8—22no—
219C3×Q8—22no—
220C3×Q8—22no—
221C3×Q8—41no—
222C3×Q8—41no—
223C3×Q8—42no—
224C3×Q8—41no—
225C3×Q8—41no—
226C3×Q8—81no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—42no—
244C2×C3×C4#12011yescharacteristic
245C2×C3×C4#12011yescharacteristic
246C2×C3×C4#12111yescharacteristic
247C2×C3×C4#12111yescharacteristic
248C2×C3×C4#12111yescharacteristic
249C2×C3×C4#12711yescharacteristic
250C2×C3×C4C2×Q1611yescharacteristic
251C2×C3×C4#13511yescharacteristic
252C2×C3×C4#13711yescharacteristic
253C2×C3×C4#13711yescharacteristic
254C3×C8—42no—
255C3×C8—42no—
256C3×C8—81no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8—42no—
267C2×Q8C3×Q1612yescomplemented
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4C2×C3×D411yescharacteristic
274C22×C4C2×C3×D411yescharacteristic
275C22×C4C2×C3×D411yescharacteristic
276Q16—44no—
277M16—41no—
278M16—41no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—41no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—44no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C22⋊C4—41no—
300C42—41no—
301C42—41no—
302C42—41no—
303C42—41no—
304C42—41no—
305C42C2×C3×D411yescharacteristic
306C42C2×C3×D411yescharacteristic
307C42C2×C3×D411yescharacteristic
308C42C3×#4111yescharacteristic
309C22×C3—21no—
310C22×C3—21no—
311C22×C3#39211yescharacteristic
312C22×C3#45111yescharacteristic
313C22×C3#45211yescharacteristic
314C3×C4—21no—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—42no—
319C3×C4—42no—
320C3×C4—82no—
321C3×C4—81no—
322C3×C4—81no—
323C3×C4—81no—
324C3×C4—81no—
325C3×C4#45111yescharacteristic
326C3×C4#45211yescharacteristic
327C3×C4#46711yescharacteristic
328C3×C4#47011yescharacteristic
329C23C3×#12011yescharacteristic
330Q8—22no—
331Q8—22no—
332Q8—22no—
333Q8—22no—
334Q8—41no—
335Q8—41no—
336Q8—42no—
337Q8—41no—
338Q8—41no—
339Q8—81no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4C3×#12011yescharacteristic
358C2×C4C3×#12011yescharacteristic
359C2×C4C3×#12111yescharacteristic
360C2×C4C3×#12111yescharacteristic
361C2×C4C3×#12111yescharacteristic
362C2×C4C3×#12711yescharacteristic
363C2×C4C2×C3×Q1611yescharacteristic
364C2×C4C3×#13511yescharacteristic
365C2×C4C3×#13711yescharacteristic
366C2×C4C3×#13711yescharacteristic
367C8—42no—
368C8—42no—
369C8—81no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3#200111yescharacteristic
373C2×C3#200911yescharacteristic
374C2×C3#201011yescharacteristic
375C22—21no—
376C22—21no—
377C22C3×#39211yescharacteristic
378C22C3×#45111yescharacteristic
379C22C3×#45211yescharacteristic
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—42no—
385C4—42no—
386C4—82no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4—81no—
391C4C3×#45111yescharacteristic
392C4C3×#45211yescharacteristic
393C4C3×#46711yescharacteristic
394C4C3×#47011yescharacteristic
395C3#1000311yescharacteristic, direct product factor, Sylow
396C2—21no—
397C2—21no—
398C2C3×#200111yescharacteristic
399C2C3×#200911yescharacteristic
400C2C3×#201011yescharacteristic
subgroupG/H
center, socleC22×C3#392
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC23×C3#120
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
8. #10003C3direct (N × H)
125. C2×C3×Q8Q16semidirect
267. C2×Q8C3×Q16semidirect
395. C3#10003direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#31042
C3×#30782
C3×#25812
C3×#15182
C3×#14872
C3×#14792
C3×#12862
#100033