Finite Group Databasebeta

← C3×#13391 #10631998

Subgroups — C3×#13391 #10631998

Order 768.

390 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3067C211yesmaximal, characteristic
2C3×#2161C211yesmaximal, characteristic, complemented
3C3×#2148C211yesmaximal, characteristic, complemented
4C2×C3×#360C211yesmaximal, characteristic, complemented
5C3×#2115C211yesmaximal, characteristic, complemented
6C3×#1815C211yesmaximal, characteristic, complemented
7C3×#1383C211yesmaximal, characteristic, complemented
8#13391C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#575—21no—
10C2×C3×#135C2211yescharacteristic
11C2×C3×#131C2211yescharacteristic
12C2×C3×M32C2211yescharacteristic
13C2×C3×M32C2211yescharacteristic
14C3×#473—21no—
15C3×#472—21no—
16C3×#471—22no—
17C2×C3×#108C2211yescharacteristic
18C2×C3×#108C2211yescharacteristic
19C2×C3×#106C2211yescharacteristic
20C3×#417—21no—
21C3×#413—21no—
22C3×#368—22no—
23C3×#362—21no—
24C3×#361—21no—
25C3×#360—22no—
26C3×#349—22no—
27C3×#343—21no—
28#3067C2×C311yescharacteristic
29#2161C2×C311yescharacteristic, complemented
30#2148C2×C311yescharacteristic, complemented
31C2×#360C2×C311yescharacteristic, complemented
32#2115C2×C311yescharacteristic, complemented
33#1815C2×C311yescharacteristic, complemented
34#1383C2×C311yescharacteristic, complemented
35C2×C3×#41—21no—
36C2×C3×#41D411yescharacteristic
37C3×#137—22no—
38C3×#137—41no—
39C3×AGL(1,ℤ/8)—22no—
40C3×AGL(1,ℤ/8)—41no—
41C3×#135—21no—
42C3×#135—21no—
43C3×#135—21no—
44C3×#135—21no—
45C3×#135—42no—
46C2×C3×Q16D411yescharacteristic
47C2×C3×SD16—21no—
48C2×C3×D8D411yescharacteristic
49C3×#131—21no—
50C3×#131—21no—
51C3×#131—21no—
52C3×#131—21no—
53C2×C3×M16—21no—
54C2×C3×M16—21no—
55C2×C3×M16—21no—
56C2×C3×M16D411yescharacteristic
57C22×C3×C8—21no—
58C22×C3×C8C2311yescharacteristic
59C2×C3×C4⋊C4—21no—
60C3×M32—21no—
61C3×M32—21no—
62C3×M32—21no—
63C3×M32D412yes—
64C2×C3×C16—21no—
65C2×C3×C16—21no—
66C3×#108—21no—
67C3×#108—21no—
68C3×#108—21no—
69C3×#108D412yes—
70C3×#106—21no—
71C3×#106—21no—
72C3×#103—41no—
73C3×#102—41no—
74C3×#101—22no—
75C3×#100—22no—
76#575—21no—
77C2×#135C22×C311yescharacteristic
78C2×#131C22×C311yescharacteristic
79C2×M32C22×C311yescharacteristic
80C2×M32C22×C311yescharacteristic
81#473—21no—
82#472—21no—
83#471—22no—
84C2×#108C22×C311yescharacteristic
85C2×#108C22×C311yescharacteristic
86C2×#106C22×C311yescharacteristic
87#417—21no—
88#413—21no—
89#368—22no—
90#362—21no—
91#361—21no—
92#360—22no—
93#349—22no—
94#343—21no—
95C3×#41—21no—
96C3×#41—21no—
97C3×#41—21no—
98C3×#41—21no—
99C3×#41—41no—
100C3×#41—41no—
101C3×#41—41no—
102C3×#41—41no—
103C2×C3×Q8—21no—
104C2×C3×Q8D811yescharacteristic
105C2×C3×D4—21no—
106C2×C3×D4—21no—
107C2×C3×D4—41no—
108C2×C3×D4D811yescharacteristic
109C22×C3×C4—21no—
110C22×C3×C4—41no—
111C22×C3×C4—41no—
112C22×C3×C4C2×D411yescharacteristic
113C3×Q16—21no—
114C3×Q16—21no—
115C3×Q16—42no—
116C3×SD16—42no—
117C3×SD16—42no—
118C3×SD16—41no—
119C3×SD16—41no—
120C3×D8—21no—
121C3×D8—21no—
122C3×D8—42no—
123C3×M16—21no—
124C3×M16—22no—
125C3×M16—21no—
126C3×M16—41no—
127C3×M16—41no—
128C3×M16—41no—
129C3×M16—41no—
130C3×M16—41no—
131C2×C3×C8—21no—
132C2×C3×C8—41no—
133C2×C3×C8—41no—
134C2×C3×C8—41no—
135C2×C3×C8—41no—
136C2×C3×C8C2×D411yescharacteristic
137C2×C3×C8C2×D411yescharacteristic
138C2×C3×C8C2×D411yescharacteristic
139C2×C3×C8#4111yescharacteristic
140C2×C3×C8#4111yescharacteristic
141C2×C3×C8#4111yescharacteristic
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C16—22no—
145C3×C16—41no—
146C2×#41—21no—
147C2×#41C3×D411yescharacteristic
148#137—22no—
149#137—41no—
150AGL(1,ℤ/8)—22no—
151AGL(1,ℤ/8)—41no—
152#135—21no—
153#135—21no—
154#135—21no—
155#135—21no—
156#135—42no—
157C2×Q16C3×D411yescharacteristic
158C2×SD16—21no—
159C2×D8C3×D411yescharacteristic
160#131—21no—
161#131—21no—
162#131—21no—
163#131—21no—
164C2×M16—21no—
165C2×M16—21no—
166C2×M16—21no—
167C2×M16C3×D411yescharacteristic
168C22×C8—21no—
169C22×C8C23×C311yescharacteristic
170C2×C4⋊C4—21no—
171M32—21no—
172M32—21no—
173M32—21no—
174M32C3×D412yes—
175C2×C16—21no—
176C2×C16—21no—
177#108—21no—
178#108—21no—
179#108—21no—
180#108C3×D412yes—
181#106—21no—
182#106—21no—
183#103—41no—
184#102—41no—
185#101—22no—
186#100—22no—
187C23×C3—21no—
188C23×C3—41no—
189C23×C3#13511yescharacteristic
190C3×Q8—21no—
191C3×Q8—21no—
192C3×Q8—41no—
193C3×Q8—41no—
194C3×D4—21no—
195C3×D4—21no—
196C3×D4—41no—
197C3×D4—41no—
198C3×D4—41no—
199C3×D4—41no—
200C3×D4—81no—
201C3×D4—81no—
202C2×C3×C4—21no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—81no—
207C2×C3×C4—81no—
208C2×C3×C4—81no—
209C2×C3×C4—161no—
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12311yescharacteristic
213C2×C3×C4C2×D811yescharacteristic
214C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
215C2×C3×C4#13711yescharacteristic
216C3×C8—41no—
217C3×C8—42no—
218C3×C8—81no—
219C3×C8#12111yescharacteristic
220C3×C8#12111yescharacteristic
221C3×C8#12311yescharacteristic
222C3×C8#12411yescharacteristic
223#41—21no—
224#41—21no—
225#41—21no—
226#41—21no—
227#41—41no—
228#41—41no—
229#41—41no—
230#41—41no—
231C2×Q8—21no—
232C2×Q8C3×D811yescharacteristic
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—41no—
236C2×D4C3×D811yescharacteristic
237C22×C4—21no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4C2×C3×D411yescharacteristic
241Q16—21no—
242Q16—21no—
243Q16—42no—
244SD16—42no—
245SD16—42no—
246SD16—41no—
247SD16—41no—
248D8—21no—
249D8—21no—
250D8—42no—
251M16—21no—
252M16—22no—
253M16—21no—
254M16—41no—
255M16—41no—
256M16—41no—
257M16—41no—
258M16—41no—
259C2×C8—21no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8C2×C3×D411yescharacteristic
265C2×C8C2×C3×D411yescharacteristic
266C2×C8C2×C3×D411yescharacteristic
267C2×C8C3×#4111yescharacteristic
268C2×C8C3×#4111yescharacteristic
269C2×C8C3×#4111yescharacteristic
270C4⋊C4—41no—
271C4⋊C4—41no—
272C16—22no—
273C16—41no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—81no—
277C22×C3—81no—
278C22×C3—161no—
279C22×C3#45211yescharacteristic
280C22×C3#46611yescharacteristic
281C22×C3#48211yescharacteristic
282C3×C4—41no—
283C3×C4—81no—
284C3×C4—161no—
285C3×C4#39411yescharacteristic
286C3×C4#44711yescharacteristic
287C3×C4#46811yescharacteristic
288C3×C4#48011yescharacteristic
289C23—21no—
290C23—41no—
291C23C3×#13511yescharacteristic
292Q8—21no—
293Q8—21no—
294Q8—41no—
295Q8—41no—
296D4—21no—
297D4—21no—
298D4—41no—
299D4—41no—
300D4—41no—
301D4—41no—
302D4—81no—
303D4—81no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—161no—
312C2×C4C3×#12011yescharacteristic
313C2×C4C3×#12111yescharacteristic
314C2×C4C3×#12311yescharacteristic
315C2×C4C2×C3×D811yescharacteristic
316C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
317C2×C4C3×#13711yescharacteristic
318C8—41no—
319C8—42no—
320C8—81no—
321C8C3×#12111yescharacteristic
322C8C3×#12111yescharacteristic
323C8C3×#12311yescharacteristic
324C8C3×#12411yescharacteristic
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—81no—
329C2×C3—161no—
330C2×C3#203311yescharacteristic
331C22—41no—
332C22—41no—
333C22—81no—
334C22—81no—
335C22—161no—
336C22C3×#45211yescharacteristic
337C22C3×#46611yescharacteristic
338C22C3×#48211yescharacteristic
339C4—41no—
340C4—81no—
341C4—161no—
342C4C3×#39411yescharacteristic
343C4C3×#44711yescharacteristic
344C4C3×#46811yescharacteristic
345C4C3×#48011yescharacteristic
346C3#1339111yescharacteristic, direct product factor, Sylow
347C2—21no—
348C2—21no—
349C2—21no—
350C2—81no—
351C2—161no—
352C2C3×#203311yescharacteristic
subgroupG/H
center, socleC2×C3#2033
commutator, Frattini, cosocleC22×C8C23×C3
prime radicalC3×#3067C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2161C2semidirect
3. C3×#2148C2semidirect
4. C2×C3×#360C2semidirect
5. C3×#2115C2semidirect
6. C3×#1815C2semidirect
7. C3×#1383C2semidirect
8. #13391C3direct (N × H)
29. #2161C2×C3semidirect
30. #2148C2×C3semidirect
31. C2×#360C2×C3semidirect
32. #2115C2×C3semidirect
33. #1815C2×C3semidirect
34. #1383C2×C3semidirect
346. C3#13391direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#30672
C3×#21612
C3×#21482
C2×C3×#3602
C3×#21152
C3×#18152
C3×#13832
#133913