Finite Group Databasebeta

← C3×#29837 #10648444

Subgroups — C3×#29837 #10648444

Order 768.

712 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#479C211yesmaximal, characteristic
2C2×C3×#462C211yesmaximal, characteristic
3C3×#2642C211yesmaximal, characteristic
4C3×#2320C211yesmaximal, characteristic
5C3×#2068C212yesmaximal
6C3×#2066C212yesmaximal
7C3×#2056C212yesmaximal
8C3×#2056C212yesmaximal
9C3×#1989C211yesmaximal, characteristic
10C3×#1981C211yesmaximal, characteristic
11C3×#1843C211yesmaximal, characteristic
12#29837C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2212yes—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×#125C2211yescharacteristic
16C2×C3×C4×Q8C2211yescharacteristic
17C3×#479—24no—
18C3×#462—24no—
19C2×C3×#107C2211yescharacteristic
20C2×C3×#106C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C4⋊C8C2212yes—
24C2×C3×C4⋊C8C2212yes—
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#103C2212yes—
28C2×C3×#103C2212yes—
29C2×C3×#103C2212yes—
30C3×#400—21no—
31C3×#400—21no—
32C3×#400—22no—
33C3×#391C2212yes—
34C3×#391C2212yes—
35C3×#391C2212yes—
36C3×#389C2212yes—
37C3×#387—22no—
38C3×#383—21no—
39C3×#382—21no—
40C3×C4×C4⋊C4C2211yescharacteristic
41C3×#376C2211yescharacteristic
42C3×#340C2212yes—
43C3×#340C2212yes—
44C2×#479C2×C311yescharacteristic
45C2×#462C2×C311yescharacteristic
46#2642C2×C311yescharacteristic
47#2320C2×C311yescharacteristic
48#2068C2×C312yes—
49#2066C2×C312yes—
50#2056C2×C312yes—
51#2056C2×C312yes—
52#1989C2×C311yescharacteristic
53#1981C2×C311yescharacteristic
54#1843C2×C311yescharacteristic
55C22×C3×Q8C2312yes—
56C22×C3×Q8C2311yescharacteristic
57C2×C3×Q16—24no—
58C2×C3×Q16—42no—
59C2×C3×Q16—42no—
60C22×C3×C8C2312yes—
61C22×C3×C8C2311yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C3×C4⋊Q8—22no—
64C3×C4⋊Q8—22no—
65C3×#125—22no—
66C3×#125—22no—
67C3×C4×Q8—22no—
68C3×C4×Q8D414yes—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C42—21no—
80C2×C3×C42—21no—
81C2×C3×C42C2312yes—
82C2×C3×C42C2311yescharacteristic
83C3×#107—22no—
84C3×#106—22no—
85C3×C4⋊C8—22no—
86C3×C4⋊C8—22no—
87C3×C4⋊C8—42no—
88C3×C4⋊C8—42no—
89C3×#103—22no—
90C3×#103—22no—
91C3×#103—24no—
92C3×#103—42no—
93C3×#103—42no—
94C3×#95—21no—
95C3×#95—22no—
96C3×#95—22no—
97C3×#95—22no—
98C3×#95—22no—
99C3×#95—22no—
100C3×#95—22no—
101C3×#95—22no—
102C3×#95D411yescharacteristic
103C3×#95D411yescharacteristic
104C3×#95D412yes—
105C22×Q16C22×C312yes—
106C2×C4⋊Q8C22×C311yescharacteristic
107C2×#125C22×C311yescharacteristic
108C2×C4×Q8C22×C311yescharacteristic
109#479—24no—
110#462—24no—
111C2×#107C22×C311yescharacteristic
112C2×#106C22×C311yescharacteristic
113C2×C4⋊C8C22×C311yescharacteristic
114C2×C4⋊C8C22×C311yescharacteristic
115C2×C4⋊C8C22×C312yes—
116C2×C4⋊C8C22×C312yes—
117C2×#103C22×C311yescharacteristic
118C2×#103C22×C311yescharacteristic
119C2×#103C22×C312yes—
120C2×#103C22×C312yes—
121C2×#103C22×C312yes—
122#400—21no—
123#400—21no—
124#400—22no—
125#391C22×C312yes—
126#391C22×C312yes—
127#391C22×C312yes—
128#389C22×C312yes—
129#387—22no—
130#383—21no—
131#382—21no—
132C4×C4⋊C4C22×C311yescharacteristic
133#376C22×C311yescharacteristic
134#340C22×C312yes—
135#340C22×C312yes—
136C2×C3×Q8—21no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8—22no—
143C2×C3×Q8—22no—
144C2×C3×Q8—22no—
145C2×C3×Q8C2×D412yes—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2411yescharacteristic
157C3×Q16—48no—
158C2×C3×C8—21no—
159C2×C3×C8—21no—
160C2×C3×C8—22no—
161C2×C3×C8—22no—
162C2×C3×C8—22no—
163C2×C3×C8—22no—
164C2×C3×C8—22no—
165C3×C4⋊C4—24no—
166C3×C4⋊C4—21no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—42no—
174C3×C4⋊C4—42no—
175C3×C4⋊C4—42no—
176C3×C4⋊C4—42no—
177C3×C4⋊C4—82no—
178C3×C4⋊C4C2×D412yes—
179C3×C42—24no—
180C3×C42—22no—
181C3×C42—22no—
182C3×C42—81no—
183C3×C42C2×D412yes—
184C3×C42#4112yes—
185C22×Q8C23×C312yes—
186C22×Q8C23×C311yescharacteristic
187C2×Q16—24no—
188C2×Q16—42no—
189C2×Q16—42no—
190C22×C8C23×C312yes—
191C22×C8C23×C311yescharacteristic
192C22×C8C23×C311yescharacteristic
193C4⋊Q8—22no—
194C4⋊Q8—22no—
195#125—22no—
196#125—22no—
197C4×Q8—22no—
198C4×Q8C3×D414yes—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4C23×C311yescharacteristic
206C2×C4⋊C4C23×C311yescharacteristic
207C2×C4⋊C4C23×C312yes—
208C2×C4⋊C4C23×C311yescharacteristic
209C2×C42—21no—
210C2×C42—21no—
211C2×C42C23×C312yes—
212C2×C42C23×C311yescharacteristic
213#107—22no—
214#106—22no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—42no—
218C4⋊C8—42no—
219#103—22no—
220#103—22no—
221#103—24no—
222#103—42no—
223#103—42no—
224#95—21no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95C3×D411yescharacteristic
233#95C3×D411yescharacteristic
234#95C3×D412yes—
235C23×C3C22×D411yescharacteristic
236C3×Q8—22no—
237C3×Q8—22no—
238C3×Q8—42no—
239C3×Q8—42no—
240C3×Q8—42no—
241C3×Q8—42no—
242C3×Q8—42no—
243C2×C3×C4—21no—
244C2×C3×C4—22no—
245C2×C3×C4—21no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—42no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—41no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—41no—
268C2×C3×C4#12112yes—
269C2×C3×C4#12112yes—
270C2×C3×C4C22×D411yescharacteristic
271C2×C3×C4C2×#4111yescharacteristic
272C2×C3×C4H(5,𝔽2)12yes—
273C2×C3×C42−1+412yes—
274C3×C8—41no—
275C3×C8—41no—
276C3×C8—42no—
277C2×Q8—21no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8C2×C3×D412yes—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4C2×C3×D412yes—
294C22×C4C2×C3×D412yes—
295C22×C4C2×C3×D411yescharacteristic
296C22×C4C2×C3×D411yescharacteristic
297C22×C4C24×C311yescharacteristic
298Q16—48no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C4⋊C4—24no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—82no—
319C4⋊C4C2×C3×D412yes—
320C42—24no—
321C42—22no—
322C42—22no—
323C42—81no—
324C42C2×C3×D412yes—
325C42C3×#4112yes—
326C22×C3C2×#12111yescharacteristic
327C22×C3#53712yes—
328C22×C3#57711yescharacteristic
329C22×C3#57812yes—
330C22×C3#57811yescharacteristic
331C3×C4—22no—
332C3×C4—22no—
333C3×C4—42no—
334C3×C4—42no—
335C3×C4—42no—
336C3×C4—42no—
337C3×C4—82no—
338C3×C4—82no—
339C3×C4—81no—
340C3×C4—81no—
341C3×C4#53511yescharacteristic
342C3×C4#53611yescharacteristic
343C3×C4#54212yes—
344C23C22×C3×D411yescharacteristic
345Q8—22no—
346Q8—22no—
347Q8—42no—
348Q8—42no—
349Q8—42no—
350Q8—42no—
351Q8—42no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4C3×#12112yes—
378C2×C4C3×#12112yes—
379C2×C4C22×C3×D411yescharacteristic
380C2×C4C2×C3×#4111yescharacteristic
381C2×C4C3×H(5,𝔽2)12yes—
382C2×C4C3×2−1+412yes—
383C8—42no—
384C8—41no—
385C8—41no—
386C2×C3#262011yescharacteristic
387C2×C3#304411yescharacteristic
388C2×C3#304611yescharacteristic
389C2×C3#325812yes—
390C2×C3#326112yes—
391C22C2×C3×#12111yescharacteristic
392C22C3×#53712yes—
393C22C3×#57711yescharacteristic
394C22C3×#57812yes—
395C22C3×#57811yescharacteristic
396C4—22no—
397C4—22no—
398C4—42no—
399C4—42no—
400C4—42no—
401C4—42no—
402C4—82no—
403C4—82no—
404C4—81no—
405C4—81no—
406C4C3×#53511yescharacteristic
407C4C3×#53611yescharacteristic
408C4C3×#54212yes—
409C3#2983711yescharacteristic, direct product factor, Sylow
410C2C3×#262011yescharacteristic
411C2C3×#304411yescharacteristic
412C2C3×#304611yescharacteristic
413C2C3×#325812yes—
414C2C3×#326112yes—
subgroupG/H
center, socle, prime radicalC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×#376 ◁ G

Complements

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

NH ≅ G/Ntype
12. #29837C3direct (N × H)
409. C3#29837direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4792
C2×C3×#4622
C3×#26422
C3×#23202
C3×#20682
C3×#20662
C3×#20562
C3×#19892
C3×#19812
C3×#18432
#298373