Finite Group Databasebeta

← C3×#70719 #11785546

Subgroups — C3×#70719 #11785546

Order 1152.

565 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#901C211yesmaximal, characteristic
2C2×C32×#108C211yesmaximal, characteristic, complemented
3C3×#5139C212yesmaximal
4C3×#5106C211yesmaximal, characteristic, complemented
5C3×#4556C212yesmaximal, complemented
6C3×#1814—31nomaximal
7#70719C311yesmaximal, characteristic, direct product factor
8C2×C3×#235C2211yescharacteristic
9C2×C32×M16C2212yes—
10C22×C32×C8C2211yescharacteristic
11C3×#926C2212yes—
12C2×C3×#208C2211yescharacteristic
13C3×#901—21no—
14C3×#901—21no—
15C3×#901C412yescomplemented
16C3×#901C412yescomplemented
17C2×C3×ΓL(1,𝔽25)—21no—
18C2×C3×D3×C8—21no—
19C32×#108—21no—
20C32×#108—21no—
21C3×#825—24no—
22C3×#818—21no—
23C3×#817—21no—
24C2×C3×#131—31no—
25C2×#901C2×C311yescharacteristic
26C2×C3×#108—21no—
27C2×C3×#108D311yescharacteristic, complemented
28C2×C3×#108C2×C311yescharacteristic, complemented
29C3×#427—31no—
30C3×#421—32no—
31#5139C2×C312yes—
32#5106C2×C311yescharacteristic, complemented
33C3×#344—32no—
34#4556C2×C312yescomplemented
35C22×C32×C4C2311yescharacteristic
36C2×C3×#67—21no—
37C3×#235—21no—
38C3×#235—21no—
39C3×#235C2×C412yes—
40C3×#235D412yes—
41C2×C3×D12Q811yescharacteristic
42C2×C3×C4×D3—21no—
43C2×C3×Dic6Q811yescharacteristic
44C32×M16—22no—
45C32×M16—22no—
46C2×C32×C8—22no—
47C2×C32×C8C2×C411yescharacteristic
48C2×C32×C8C2×C411yescharacteristic
49C2×C32×C8D411yescharacteristic
50C2×C32×C8D411yescharacteristic
51C2×C32×C8D411yescharacteristic
52C2×C32×C8D411yescharacteristic
53C3×#217—22no—
54C3×C4⋊Dic3—22no—
55C3×C4×Dic3—22no—
56C3×#208—21no—
57C3×#208C2×C412yes—
58C2×C3×C3⋊C8—21no—
59C3×ΓL(1,𝔽25)—22no—
60C3×ΓL(1,𝔽25)—22no—
61C3×D3×C8—22no—
62C3×D3×C8—22no—
63#1814—31noSylow
64C2×C3×#41—31no—
65C2×#235C22×C311yescharacteristic
66C3×#131—32no—
67C3×#131—32no—
68C3×#131—61no—
69C3×#131—61no—
70C2×C3×M16—22no—
71C2×C3×M16—31no—
72C2×C3×M16—61no—
73C2×C3×M16C2×D312yes—
74C2×C3×M16C22×C312yes—
75C22×C3×C8—21no—
76C22×C3×C8—61no—
77C22×C3×C8C2×D311yescharacteristic
78C22×C3×C8C22×C311yescharacteristic
79C3×#117—32no—
80#926C22×C312yes—
81C2×#208C22×C311yescharacteristic
82#901—21no—
83#901—21no—
84#901C3×C412yescomplemented
85#901C3×C412yescomplemented
86C2×ΓL(1,𝔽25)—21no—
87C2×D3×C8—21no—
88C3×#108—22no—
89C3×#108—22no—
90C3×#108—21no—
91C3×#108—21no—
92C3×#108—21no—
93C3×#108—21no—
94C3×#107—61no—
95C3×#106—61no—
96C3×#104—64no—
97#825—24no—
98#818—21no—
99#817—21no—
100C23×C32#4111yescharacteristic
101C22×C3×D3—21no—
102C2×C32×C4C22×C411yescharacteristic
103C2×C32×C4C2×D411yescharacteristic
104C2×C32×C4C2×D411yescharacteristic
105C2×C32×C4C2×D411yescharacteristic
106C2×C32×C4C2×Q811yescharacteristic
107C2×C32×C4#4111yescharacteristic
108C3×#67—22no—
109C3×#67—22no—
110C2×C3×Dic3—21no—
111C2×C3×Dic3—42no—
112C3×D12—21no—
113C3×D12C4⋊C412yes—
114C3×C4×D3—22no—
115C3×C4×D3—22no—
116C3×Dic6—21no—
117C3×Dic6C4⋊C412yes—
118C32×C8—42no—
119C32×C8C22⋊C411yescharacteristic
120C32×C8C22⋊C411yescharacteristic
121C32×C8C22⋊C411yescharacteristic
122C32×C8C22⋊C411yescharacteristic
123C3×C3⋊C8—22no—
124C2×#131—31no—
125C2×#108C3×D311yescharacteristic, complemented
126#427—31no—
127#421—32no—
128#344—32no—
129C3×#41—32no—
130C3×#41—32no—
131C3×#41—61no—
132C3×#41—61no—
133C2×C3×Q8—31no—
134C2×C3×D4—31no—
135C2×C3×D4—61no—
136C22×C3×C4—21no—
137C22×C3×C4—61no—
138C22×C3×C4C22×D311yescharacteristic
139C22×C3×C4C23×C311yescharacteristic
140C2×#67—21no—
141#235—21no—
142#235—21no—
143#235C2×C3×C412yes—
144#235C3×D412yes—
145C2×D12C3×Q811yescharacteristic
146C2×C4×D3—21no—
147C2×Dic6C3×Q811yescharacteristic
148C3×M16—24no—
149C3×M16—22no—
150C3×M16—22no—
151C3×M16—22no—
152C3×M16—22no—
153C3×M16—32no—
154C3×M16—42no—
155C3×M16—61no—
156C3×M16—62no—
157C3×M16—62no—
158C2×C3×C8—22no—
159C2×C3×C8—22no—
160C2×C3×C8—21no—
161C2×C3×C8—21no—
162C2×C3×C8—21no—
163C2×C3×C8—21no—
164C2×C3×C8—21no—
165C2×C3×C8—21no—
166C2×C3×C8—42no—
167C2×C3×C8—61no—
168C2×C3×C8—62no—
169C2×C3×C8—62no—
170C2×C3×C8C4×D311yescharacteristic
171C2×C3×C8C4×D311yescharacteristic
172C2×C3×C8D1211yescharacteristic
173C2×C3×C8D1211yescharacteristic
174C2×C3×C8#6711yescharacteristic
175C2×C3×C8#6711yescharacteristic
176C2×C3×C8C2×C3×C411yescharacteristic
177C2×C3×C8C2×C3×C411yescharacteristic
178C2×C3×C8C3×D411yescharacteristic
179C2×C3×C8C3×D411yescharacteristic
180C2×C3×C8C3×D411yescharacteristic
181C2×C3×C8C3×D411yescharacteristic
182C3×C4⋊C4—62no—
183C3×C22⋊C4—62no—
184C3×C42—62no—
185#217—22no—
186C4⋊Dic3—22no—
187C4×Dic3—22no—
188#208—21no—
189#208C2×C3×C412yes—
190C2×C3⋊C8—21no—
191ΓL(1,𝔽25)—22no—
192ΓL(1,𝔽25)—22no—
193D3×C8—22no—
194D3×C8—22no—
195C22×C32—41no—
196C22×C32#11711yescharacteristic
197C22×C32#12111yescharacteristic
198C22×C32C22⋊Q811yescharacteristic
199C2×C3×D3—22no—
200C2×C3×D3—42no—
201C32×C4C2×C22⋊C411yescharacteristic
202C32×C4C2×C4⋊C411yescharacteristic
203C32×C4#12111yescharacteristic
204C32×C4C22⋊Q811yescharacteristic
205C3×Dic3—22no—
206C3×Dic3—82no—
207C2×#41—31no—
208#131—32no—
209#131—32no—
210#131—61no—
211#131—61no—
212C2×M16—31no—
213C2×M16—61no—
214C2×M16C2×C3×D312yes—
215C22×C8—61no—
216C22×C8C2×C3×D311yescharacteristic
217#117—32no—
218#108—21no—
219#108—21no—
220#107—61no—
221#106—61no—
222#104—64no—
223C23×C3—21no—
224C23×C3—61no—
225C23×C3#23911yescharacteristic
226C23×C3C3×#4111yescharacteristic
227C22×D3—21no—
228C3×Q8—32no—
229C3×Q8—61no—
230C3×D4—32no—
231C3×D4—62no—
232C3×D4—62no—
233C3×D4—61no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—61no—
241C2×C3×C4—62no—
242C2×C3×C4—62no—
243C2×C3×C4—122no—
244C2×C3×C4C2×C4×D311yescharacteristic
245C2×C3×C4C2×D1211yescharacteristic
246C2×C3×C4D3×D411yescharacteristic
247C2×C3×C4#23711yescharacteristic
248C2×C3×C4D3×Q811yescharacteristic
249C2×C3×C4C2×#6711yescharacteristic
250C2×C3×C4C22×C3×C411yescharacteristic
251C2×C3×C4C2×C3×D411yescharacteristic
252C2×C3×C4C2×C3×D411yescharacteristic
253C2×C3×C4C2×C3×D411yescharacteristic
254C2×C3×C4C2×C3×Q811yescharacteristic
255C2×C3×C4C3×#4111yescharacteristic
256#67—22no—
257#67—22no—
258C2×Dic3—21no—
259C2×Dic3—42no—
260D12—21no—
261D12C3×C4⋊C412yes—
262C4×D3—22no—
263C4×D3—22no—
264Dic6—21no—
265Dic6C3×C4⋊C412yes—
266C3×C8—21no—
267C3×C8—21no—
268C3×C8—21no—
269C3×C8—21no—
270C3×C8—44no—
271C3×C8—42no—
272C3×C8—42no—
273C3×C8—62no—
274C3×C8(C2×D3)⋊C411yescharacteristic
275C3×C8(C2×D3)⋊C411yescharacteristic
276C3×C8(C2×D3)⋊C411yescharacteristic
277C3×C8(C2×D3)⋊C411yescharacteristic
278C3×C8C3×C22⋊C411yescharacteristic
279C3×C8C3×C22⋊C411yescharacteristic
280C3×C8C3×C22⋊C411yescharacteristic
281C3×C8C3×C22⋊C411yescharacteristic
282C3⋊C8—22no—
283C2×C32—21no—
284C2×C32—21no—
285C2×C32—21no—
286C2×C32#38011yescharacteristic
287C3×D3—42no—
288#41—32no—
289#41—32no—
290#41—61no—
291#41—61no—
292C2×Q8—31no—
293C2×D4—31no—
294C2×D4—61no—
295C22×C4—61no—
296C22×C4C22×C3×D311yescharacteristic
297M16—22no—
298M16—22no—
299M16—32no—
300M16—61no—
301M16—62no—
302M16—62no—
303C2×C8—22no—
304C2×C8—61no—
305C2×C8—62no—
306C2×C8—62no—
307C2×C8C3×C4×D311yescharacteristic
308C2×C8C3×C4×D311yescharacteristic
309C2×C8C3×D1211yescharacteristic
310C2×C8C3×D1211yescharacteristic
311C2×C8C3×#6711yescharacteristic
312C2×C8C3×#6711yescharacteristic
313C4⋊C4—62no—
314C22⋊C4—62no—
315C42—62no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—41no—
320C22×C3—41no—
321C22×C3—42no—
322C22×C3—62no—
323C22×C3—122no—
324C22×C3#89211yescharacteristic
325C22×C3#89511yescharacteristic
326C22×C3#94611yescharacteristic
327C22×C3C3×#11711yescharacteristic
328C22×C3C3×#12111yescharacteristic
329C22×C3C3×C22⋊Q811yescharacteristic
330C2×D3—22no—
331C2×D3—42no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—62no—
337C3×C4—242no—
338C3×C4D3×C4⋊C411yescharacteristic
339C3×C4#89711yescharacteristic
340C3×C4C2×(C2×D3)⋊C411yescharacteristic
341C3×C4#93811yescharacteristic
342C3×C4C2×C3×C22⋊C411yescharacteristic
343C3×C4C2×C3×C4⋊C411yescharacteristic
344C3×C4C3×#12111yescharacteristic
345C3×C4C3×C22⋊Q811yescharacteristic
346Dic3—22no—
347Dic3—82no—
348C32#181411yescharacteristic, complemented, Sylow
349C23—61no—
350C23C3×#23911yescharacteristic
351Q8—32no—
352Q8—61no—
353D4—32no—
354D4—62no—
355D4—62no—
356D4—61no—
357C2×C4—61no—
358C2×C4—62no—
359C2×C4—62no—
360C2×C4—122no—
361C2×C4C2×C3×C4×D311yescharacteristic
362C2×C4C2×C3×D1211yescharacteristic
363C2×C4C3×D3×D411yescharacteristic
364C2×C4C3×#23711yescharacteristic
365C2×C4C3×D3×Q811yescharacteristic
366C2×C4C2×C3×#6711yescharacteristic
367C8—42no—
368C8—62no—
369C8C3×(C2×D3)⋊C411yescharacteristic
370C8C3×(C2×D3)⋊C411yescharacteristic
371C8C3×(C2×D3)⋊C411yescharacteristic
372C8C3×(C2×D3)⋊C411yescharacteristic
373C2×C3—22no—
374C2×C3—21no—
375C2×C3—21no—
376C2×C3—21no—
377C2×C3—21no—
378C2×C3—21no—
379C2×C3—21no—
380C2×C3—21no—
381C2×C3—41no—
382C2×C3—41no—
383C2×C3—122no—
384C2×C3#498711yescharacteristic
385C2×C3C3×#38011yescharacteristic
386D3—42no—
387C22—41no—
388C22—62no—
389C22—122no—
390C22C3×#89211yescharacteristic
391C22C3×#89511yescharacteristic
392C22C3×#94611yescharacteristic
393C4—62no—
394C4—242no—
395C4C3×D3×C4⋊C411yescharacteristic
396C4C3×#89711yescharacteristic
397C4C2×C3×(C2×D3)⋊C411yescharacteristic
398C4C3×#93811yescharacteristic
399C3—21no—
400C3#7071911yescharacteristic, direct product factor
401C3C3×#181411yescharacteristic, complemented
402C2—21no—
403C2—21no—
404C2—21no—
405C2—122no—
406C2C3×#498711yescharacteristic
subgroupG/H
centerC3×C4#938
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC2×C32×#108C2
socleC2×C32#380
prime radicalC2×C3×#235C22
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×#108D3
hypocenterC3C3×#1814
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C2×C3×#108

Complements

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

NH ≅ G/Ntype
2. C2×C32×#108C2semidirect
4. C3×#5106C2semidirect
5. C3×#4556C2semidirect
7. #70719C3direct (N × H)
15. C3×#901C4semidirect
16. C3×#901C4semidirect
27. C2×C3×#108D3semidirect
28. C2×C3×#108C2×C3semidirect
32. #5106C2×C3semidirect
34. #4556C2×C3semidirect
84. #901C3×C4semidirect
85. #901C3×C4semidirect
125. C2×#108C3×D3semidirect
348. C32#1814semidirect
400. C3#70719direct (N × H)
401. C3C3×#1814semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9012
C2×C32×#1082
C3×#51392
C3×#51062
C3×#45562
C3×#18143
#707193