Finite Group Databasebeta

← C3×#85378 #11922665

Subgroups — C3×#85378 #11922665

Order 1152.

425 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 425 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#548⋊C3C211yesmaximal, characteristic
2C3×#5426C211yesmaximal, characteristic, complemented
3C3×#5425C211yesmaximal, characteristic, complemented
4C3×#5420C211yesmaximal, characteristic, complemented
5C3×#5419C211yesmaximal, characteristic, complemented
6C3×#5389C211yesmaximal, characteristic, complemented
7C3×C4×#226C211yesmaximal, characteristic, complemented
8#85378C311yesmaximal, characteristic, direct product factor
9C3×#3327—31nomaximal
10C2×C3×#41⋊C3C2211yescharacteristic
11C22×C3×SL(2,𝔽3)C2211yescharacteristic
12C2×C3×#226C2211yescharacteristic
13C3×#879—41nomaximal
14C3×C4×SL(2,𝔽3)C2211yescharacteristic
15C3×#859C2211yescharacteristic
16C3×#859C2211yescharacteristic
17C3×#859C2211yescharacteristic
18C3×#552—31no—
19C3×#548D311yescharacteristic
20#548⋊C3—21no—
21#548⋊C3C2×C311yescharacteristic
22#5426C2×C311yescharacteristic, complemented
23#5425C2×C311yescharacteristic, complemented
24#5420C2×C311yescharacteristic, complemented
25#5419C2×C311yescharacteristic, complemented
26#5389C2×C311yescharacteristic, complemented
27C4×#226C2×C311yescharacteristic, complemented
28C3×#488—31no—
29C3×#484—31no—
30C3×#482—31no—
31C3×#479—31no—
32C3×#470—31no—
33C3×#468—31no—
34C3×#464—31no—
35C3×#452—31no—
36C3×#451—31no—
37C3×C4×Q16—31no—
38C3×#436—31no—
39C3×#419—31no—
40C3×#417—31no—
41C3×#41⋊C3—21no—
42C2×C3×SL(2,𝔽3)—21no—
43C2×C3×SL(2,𝔽3)C2311yescharacteristic
44C3×#226—21no—
45C32×C22⋊C4—41no—
46C3×#217—41no—
47C3×#217—41no—
48C3×C4⋊Dic3—41no—
49C3×Dic3⋊C4—41no—
50C3×Dic3⋊C4—41no—
51C3×C4×Dic3—41no—
52#3327—31noSylow
53C2×C3×#41C2×D311yescharacteristic
54C22×C3×Q8C2×D311yescharacteristic
55C2×#41⋊C3—21no—
56C2×#41⋊C3C22×C311yescharacteristic
57C22×SL(2,𝔽3)—21no—
58C22×SL(2,𝔽3)C22×C311yescharacteristic
59C2×#226C22×C311yescharacteristic
60C2×C3×Q16—31no—
61C2×C3×Q16—31no—
62C2×C3×SD16—31no—
63C2×C3×M16—31no—
64C2×C3×M16—31no—
65C3×#126—61no—
66C3×#125—31no—
67C3×#125—61no—
68C3×#124—31no—
69C3×#123—61no—
70C3×C22⋊Q8—31no—
71C3×C22⋊Q8—31no—
72C3×C22⋊Q8—31no—
73C3×#121—31no—
74C3×C4×Q8—31no—
75C3×C4×Q8C2×D311yescharacteristic
76C3×C4×D4—31no—
77C3×#117—31no—
78C3×#117—61no—
79C2×C3×C4⋊C4—31no—
80#879—41no—
81C4×SL(2,𝔽3)—21no—
82C4×SL(2,𝔽3)C22×C311yescharacteristic
83#859C22×C311yescharacteristic
84#859C22×C311yescharacteristic
85#859C22×C311yescharacteristic
86C3×#107—31no—
87C3×#107—31no—
88C3×#106—31no—
89C3×C4⋊C8—31no—
90C3×#103—31no—
91C3×#103—31no—
92C3×#103—31no—
93C3×#103—31no—
94C3×#103—31no—
95C3×#103—31no—
96C3×#103—31no—
97C3×#102—31no—
98C3×#102—31no—
99C3×#102—31no—
100C3×C22⋊C8—31no—
101C3×C22⋊C8—31no—
102C3×C4×C8—31no—
103C23×C32—41no—
104C2×C32×C4—41no—
105C2×C32×C4—41no—
106C2×C3×Dic3—41no—
107C2×C3×Dic3—41no—
108C2×C3×Dic3—41no—
109C2×C3×Dic3—41no—
110C3×SL(2,𝔽3)#4111yescharacteristic
111#552—31no—
112#548C3×D311yescharacteristic
113#488—31no—
114#484—31no—
115#482—31no—
116#479—31no—
117#470—31no—
118#468—31no—
119#464—31no—
120#452—31no—
121#451—31no—
122C4×Q16—31no—
123#436—31no—
124#419—31no—
125#417—31no—
126C3×#41—21no—
127C3×#41—61no—
128C2×C3×Q8—21no—
129C2×C3×Q8—31no—
130C2×C3×Q8—31no—
131C2×C3×Q8—61no—
132C2×C3×Q8C22×D311yescharacteristic
133C2×C3×D4—31no—
134C2×C3×D4—31no—
135C22×C3×C4—31no—
136C22×C3×C4—31no—
137C22×C3×C4—61no—
138#41⋊C3—21no—
139#41⋊C3—41no—
140C2×SL(2,𝔽3)—21no—
141C2×SL(2,𝔽3)—21no—
142C2×SL(2,𝔽3)—41no—
143C2×SL(2,𝔽3)C23×C311yescharacteristic
144#226—21no—
145C3×Q16—61no—
146C3×Q16—61no—
147C3×Q16—121no—
148C3×SD16—121no—
149C3×M16—121no—
150C3×M16—121no—
151C2×C3×C8—31no—
152C2×C3×C8—31no—
153C2×C3×C8—31no—
154C2×C3×C8—31no—
155C3×C4⋊C4—31no—
156C3×C4⋊C4—31no—
157C3×C4⋊C4—31no—
158C3×C4⋊C4—31no—
159C3×C4⋊C4—31no—
160C3×C4⋊C4—61no—
161C3×C4⋊C4—61no—
162C3×C4⋊C4—61no—
163C3×C4⋊C4—61no—
164C3×C4⋊C4—61no—
165C3×C4⋊C4—61no—
166C3×C22⋊C4—31no—
167C3×C22⋊C4—41no—
168C3×C22⋊C4—61no—
169C3×C22⋊C4—61no—
170C3×C22⋊C4—61no—
171C3×C22⋊C4—61no—
172C3×C22⋊C4—81no—
173C3×C22⋊C4S411yescharacteristic
174C3×C42—31no—
175C3×C42—61no—
176C3×C42—61no—
177#217—41no—
178#217—41no—
179C4⋊Dic3—41no—
180Dic3⋊C4—41no—
181Dic3⋊C4—41no—
182C4×Dic3—41no—
183C22×C32—41no—
184C22×C32—81no—
185C22×C32—81no—
186C22×C32—81no—
187C32×C4—81no—
188C32×C4—81no—
189C3×Dic3—81no—
190C3×Dic3—81no—
191C3×Dic3—81no—
192C3×Dic3—81no—
193C2×#41C2×C3×D311yescharacteristic
194C22×Q8C2×C3×D311yescharacteristic
195C2×Q16—31no—
196C2×Q16—31no—
197C2×SD16—31no—
198C2×M16—31no—
199C2×M16—31no—
200#126—61no—
201#125—31no—
202#125—61no—
203#124—31no—
204#123—61no—
205C22⋊Q8—31no—
206C22⋊Q8—31no—
207C22⋊Q8—31no—
208#121—31no—
209C4×Q8—31no—
210C4×Q8C2×C3×D311yescharacteristic
211C4×D4—31no—
212#117—31no—
213#117—61no—
214C2×C4⋊C4—31no—
215#107—31no—
216#107—31no—
217#106—31no—
218C4⋊C8—31no—
219#103—31no—
220#103—31no—
221#103—31no—
222#103—31no—
223#103—31no—
224#103—31no—
225#103—31no—
226#102—31no—
227#102—31no—
228#102—31no—
229C22⋊C8—31no—
230C22⋊C8—31no—
231C4×C8—31no—
232C23×C3—31no—
233C23×C3—41no—
234C23×C3—81no—
235C23×C3C2×S411yescharacteristic
236C3×Q8—31no—
237C3×Q8—61no—
238C3×Q8—61no—
239C3×Q8—61no—
240C3×Q8—61no—
241C3×Q8#23711yescharacteristic
242C3×D4—61no—
243C3×D4—61no—
244C3×D4—121no—
245C2×C3×C4—31no—
246C2×C3×C4—31no—
247C2×C3×C4—31no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—61no—
251C2×C3×C4—61no—
252C2×C3×C4—61no—
253C2×C3×C4—61no—
254C2×C3×C4—61no—
255C2×C3×C4—61no—
256C2×C3×C4—61no—
257C2×C3×C4—61no—
258C2×C3×C4—81no—
259C2×C3×C4—81no—
260C2×C3×C4—122no—
261C2×C3×C4C2×S411yescharacteristic
262C2×C3×C4C2×S411yescharacteristic
263C2×Dic3—41no—
264C2×Dic3—41no—
265C2×Dic3—41no—
266C2×Dic3—41no—
267SL(2,𝔽3)—21no—
268SL(2,𝔽3)C3×#4111yescharacteristic
269C3×C8—61no—
270C3×C8—61no—
271C3×C8—61no—
272C3×C8—61no—
273C2×C32—41no—
274C2×C32—41no—
275C2×C32—41no—
276C2×C32—161no—
277#41—21no—
278#41—61no—
279C2×Q8—21no—
280C2×Q8—31no—
281C2×Q8—31no—
282C2×Q8—61no—
283C2×Q8C22×C3×D311yescharacteristic
284C2×D4—31no—
285C2×D4—31no—
286C22×C4—31no—
287C22×C4—31no—
288C22×C4—61no—
289Q16—61no—
290Q16—61no—
291Q16—121no—
292SD16—121no—
293M16—121no—
294M16—121no—
295C2×C8—31no—
296C2×C8—31no—
297C2×C8—31no—
298C2×C8—31no—
299C4⋊C4—31no—
300C4⋊C4—31no—
301C4⋊C4—31no—
302C4⋊C4—31no—
303C4⋊C4—31no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C4⋊C4—61no—
309C4⋊C4—61no—
310C22⋊C4—31no—
311C22⋊C4—61no—
312C22⋊C4—61no—
313C22⋊C4—61no—
314C22⋊C4—61no—
315C22⋊C4C3×S411yescharacteristic
316C42—31no—
317C42—61no—
318C42—61no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—41no—
323C22×C3—61no—
324C22×C3—61no—
325C22×C3—61no—
326C22×C3—81no—
327C22×C3—81no—
328C22×C3—81no—
329C22×C3—82no—
330C22×C3—81no—
331C22×C3—161no—
332C22×C3—161no—
333C22×C3C22×S411yescharacteristic
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—31no—
337C3×C4—31no—
338C3×C4—61no—
339C3×C4—61no—
340C3×C4—81no—
341C3×C4—81no—
342C3×C4—121no—
343C3×C4—121no—
344C3×C4—121no—
345C3×C4—121no—
346C3×C4—161no—
347C3×C4—161no—
348Dic3—81no—
349Dic3—81no—
350Dic3—81no—
351Dic3—81no—
352C32—41noSylow
353C23—31no—
354C23C2×C3×S411yescharacteristic
355Q8—31no—
356Q8—61no—
357Q8—61no—
358Q8—61no—
359Q8—61no—
360Q8C3×#23711yescharacteristic
361D4—61no—
362D4—61no—
363D4—121no—
364C2×C4—31no—
365C2×C4—31no—
366C2×C4—31no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—61no—
372C2×C4—61no—
373C2×C4—61no—
374C2×C4—61no—
375C2×C4—122no—
376C2×C4C2×C3×S411yescharacteristic
377C2×C4C2×C3×S411yescharacteristic
378C8—61no—
379C8—61no—
380C8—61no—
381C8—61no—
382C2×C3—41no—
383C2×C3—41no—
384C2×C3—41no—
385C2×C3—41no—
386C2×C3—81no—
387C2×C3—81no—
388C2×C3—81no—
389C2×C3—121no—
390C2×C3—161no—
391C2×C3—162no—
392C2×C3#591411yescharacteristic
393C2×C3#592311yescharacteristic
394C2×C3#592711yescharacteristic
395C22—21no—
396C22—21no—
397C22—21no—
398C22—61no—
399C22—61no—
400C22—61no—
401C22C22×C3×S411yescharacteristic
402C4—21no—
403C4—21no—
404C4—31no—
405C4—31no—
406C4—61no—
407C4—61no—
408C4—121no—
409C4—121no—
410C4—121no—
411C4—121no—
412C3—41no—
413C3—81no—
414C3#8537811yescharacteristic, direct product factor
415C2—41no—
416C2—121no—
417C2C3×#591411yescharacteristic
418C2C3×#592311yescharacteristic
419C2C3×#592711yescharacteristic
subgroupG/H
center, socleC22×C3C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C3
FrattiniC22C22×C3×S4
FittingC3×#548D3
prime radicalC3×#548⋊C3C2
hypercenterC3×C22⋊C4S4
hypocenterSL(2,𝔽3)C3×#41
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#5426C2semidirect
3. C3×#5425C2semidirect
4. C3×#5420C2semidirect
5. C3×#5419C2semidirect
6. C3×#5389C2semidirect
7. C3×C4×#226C2semidirect
8. #85378C3direct (N × H)
22. #5426C2×C3semidirect
23. #5425C2×C3semidirect
24. #5420C2×C3semidirect
25. #5419C2×C3semidirect
26. #5389C2×C3semidirect
27. C4×#226C2×C3semidirect
414. C3#85378direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#548⋊C32
C3×#54262
C3×#54252
C3×#54202
C3×#54192
C3×#53892
C3×C4×#2262
#853783
C3×#33273
C3×#8794