Finite Group Databasebeta

← C3×#10006 #10628613

Subgroups — C3×#10006 #10628613

Order 768.

572 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 102 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×#3213C211yesmaximal, characteristic
2C3×#3112C211yesmaximal, characteristic, complemented
3C3×#2592C211yesmaximal, characteristic, complemented
4C3×#1980C211yesmaximal, characteristic, complemented
5C3×#1527C211yesmaximal, characteristic, complemented
6C3×#1491C211yesmaximal, characteristic, complemented
7C3×#1285C211yesmaximal, characteristic, complemented
8#10006C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×#124C2211yescharacteristic
11C2×C3×C4×Q8C2211yescharacteristic
12C3×#496—21no—
13C3×#489—21no—
14C3×#480—21no—
15C3×#477—21no—
16C3×#463—22no—
17C3×#460—21no—
18C3×#448—21no—
19C3×#448—21no—
20C3×#447—21no—
21C3×#440—22no—
22C3×#432C2211yescharacteristic
23C3×#432C2211yescharacteristic
24C2×C3×#104—22no—
25C3×C4×M16C2211yescharacteristic
26C3×#397—21no—
27C3×#393—21no—
28C3×#391—21no—
29C3×#390—21no—
30C3×#388—22no—
31C3×C4×C4⋊C4C2211yescharacteristic
32C3×#329—22no—
33C3×Q8⋊C8—22no—
34#3213C2×C311yescharacteristic
35#3112C2×C311yescharacteristic, complemented
36#2592C2×C311yescharacteristic, complemented
37#1980C2×C311yescharacteristic, complemented
38#1527C2×C311yescharacteristic, complemented
39#1491C2×C311yescharacteristic, complemented
40#1285C2×C311yescharacteristic, complemented
41C2×C3×#41—21no—
42C22×C3×Q8D411yescharacteristic
43C22×C3×Q8D411yescharacteristic
44C22×C3×D4D411yescharacteristic
45C2×C3×SD16—22no—
46C2×C3×SD16—22no—
47C2×C3×D8—22no—
48C2×C3×M16—21no—
49C3×C4⋊Q8D411yescharacteristic
50C3×#127D411yescharacteristic
51C3×#124—21no—
52C3×#124—21no—
53C3×#124—21no—
54C3×#124—21no—
55C3×#124—21no—
56C3×#121—21no—
57C3×#121—21no—
58C3×C4×Q8—21no—
59C3×C4×Q8—21no—
60C3×C4×Q8—21no—
61C3×C4×Q8—21no—
62C3×C4×D4—21no—
63C3×C4×D4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4D411yescharacteristic
67C2×C3×C22⋊C4—22no—
68C2×C3×C22⋊C4—21no—
69C2×C3×C22⋊C4—21no—
70C2×C3×C22⋊C4—22no—
71C2×C3×C42—21no—
72C2×C3×C42D412yes—
73C2×C3×C42C2311yescharacteristic
74C3×#107—22no—
75C3×C4⋊C8—21no—
76C3×C4⋊C8D412yes—
77C3×#104—44no—
78C3×#103—22no—
79C3×#103—22no—
80C3×#102—22no—
81C3×#102—22no—
82C3×C22⋊C8—21no—
83C3×C22⋊C8—21no—
84C3×#97—21no—
85C3×#97—21no—
86C3×#97D412yes—
87C3×C4×C8—21no—
88C3×#95—22no—
89C3×#95—22no—
90C3×#95—21no—
91#532C22×C311yescharacteristic
92C2×#124C22×C311yescharacteristic
93C2×C4×Q8C22×C311yescharacteristic
94#496—21no—
95#489—21no—
96#480—21no—
97#477—21no—
98#463—22no—
99#460—21no—
100#448—21no—
101#448—21no—
102#447—21no—
103#440—22no—
104#432C22×C311yescharacteristic
105#432C22×C311yescharacteristic
106C2×#104—22no—
107C4×M16C22×C311yescharacteristic
108#397—21no—
109#393—21no—
110#391—21no—
111#390—21no—
112#388—22no—
113C4×C4⋊C4C22×C311yescharacteristic
114#329—22no—
115Q8⋊C8—22no—
116C24×C3—21no—
117C3×#41—42no—
118C3×#41—42no—
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×Q8—42no—
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—21no—
129C2×C3×D4—21no—
130C2×C3×D4—21no—
131C2×C3×D4—42no—
132C2×C3×D4—41no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—22no—
136C22×C3×C4—21no—
137C22×C3×C4—41no—
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C3×SD16—44no—
142C3×SD16—82no—
143C3×D8—82no—
144C3×M16—22no—
145C3×M16—41no—
146C2×C3×C8—22no—
147C2×C3×C8—22no—
148C2×C3×C8—41no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—42no—
154C3×C4⋊C4—41no—
155C3×C4⋊C4—41no—
156C3×C22⋊C4—44no—
157C3×C22⋊C4—41no—
158C3×C22⋊C4—41no—
159C3×C22⋊C4—41no—
160C3×C22⋊C4—41no—
161C3×C22⋊C4—41no—
162C3×C22⋊C4—41no—
163C3×C22⋊C4—82no—
164C3×C42—42no—
165C3×C42—41no—
166C3×C42—41no—
167C3×C42C2×D411yescharacteristic
168C3×C42C2×D411yescharacteristic
169C3×C42C2×D411yescharacteristic
170C3×C42#4111yescharacteristic
171C2×#41—21no—
172C22×Q8C3×D411yescharacteristic
173C22×Q8C3×D411yescharacteristic
174C22×D4C3×D411yescharacteristic
175C2×SD16—22no—
176C2×SD16—22no—
177C2×D8—22no—
178C2×M16—21no—
179C4⋊Q8C3×D411yescharacteristic
180#127C3×D411yescharacteristic
181#124—21no—
182#124—21no—
183#124—21no—
184#124—21no—
185#124—21no—
186#121—21no—
187#121—21no—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×D4—21no—
193C4×D4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4C3×D411yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C42—21no—
202C2×C42C3×D412yes—
203C2×C42C23×C311yescharacteristic
204#107—22no—
205C4⋊C8—21no—
206C4⋊C8C3×D412yes—
207#104—44no—
208#103—22no—
209#103—22no—
210#102—22no—
211#102—22no—
212C22⋊C8—21no—
213C22⋊C8—21no—
214#97—21no—
215#97—21no—
216#97C3×D412yes—
217C4×C8—21no—
218#95—22no—
219#95—22no—
220#95—21no—
221C23×C3—42no—
222C23×C3—42no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3#12011yescharacteristic
228C3×Q8—22no—
229C3×Q8—22no—
230C3×Q8—41no—
231C3×Q8—41no—
232C3×Q8—42no—
233C3×Q8—42no—
234C3×Q8—81no—
235C3×D4—42no—
236C3×D4—42no—
237C3×D4—42no—
238C3×D4—42no—
239C3×D4—81no—
240C3×D4—81no—
241C3×D4—81no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—42no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—82no—
259C2×C3×C4#12011yescharacteristic
260C2×C3×C4#12011yescharacteristic
261C2×C3×C4#12111yescharacteristic
262C2×C3×C4#12111yescharacteristic
263C2×C3×C4#12111yescharacteristic
264C2×C3×C4#12711yescharacteristic
265C3×C8—42no—
266C3×C8—42no—
267C3×C8—42no—
268C24—21no—
269#41—42no—
270#41—42no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—42no—
284C2×D4—41no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—41no—
290C22×C4C2×C3×D411yescharacteristic
291C22×C4C2×C3×D411yescharacteristic
292C22×C4C2×C3×D411yescharacteristic
293SD16—44no—
294SD16—82no—
295D8—82no—
296M16—22no—
297M16—41no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—41no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—42no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C22⋊C4—44no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—82no—
316C42—42no—
317C42—41no—
318C42—41no—
319C42C2×C3×D411yescharacteristic
320C42C2×C3×D411yescharacteristic
321C42C2×C3×D411yescharacteristic
322C42C3×#4111yescharacteristic
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—82no—
326C22×C3—82no—
327C22×C3—81no—
328C22×C3—81no—
329C22×C3—81no—
330C22×C3—81no—
331C22×C3—81no—
332C22×C3—81no—
333C22×C3#39211yescharacteristic
334C22×C3#45311yescharacteristic
335C22×C3#45411yescharacteristic
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—42no—
342C3×C4—82no—
343C3×C4—81no—
344C3×C4—81no—
345C3×C4—81no—
346C3×C4#45512yes—
347C23—42no—
348C23—42no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23C3×#12011yescharacteristic
354Q8—22no—
355Q8—22no—
356Q8—41no—
357Q8—41no—
358Q8—42no—
359Q8—42no—
360Q8—81no—
361D4—42no—
362D4—42no—
363D4—42no—
364D4—42no—
365D4—81no—
366D4—81no—
367D4—81no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—82no—
385C2×C4C3×#12011yescharacteristic
386C2×C4C3×#12011yescharacteristic
387C2×C4C3×#12111yescharacteristic
388C2×C4C3×#12111yescharacteristic
389C2×C4C3×#12111yescharacteristic
390C2×C4C3×#12711yescharacteristic
391C8—42no—
392C8—42no—
393C8—42no—
394C2×C3—21no—
395C2×C3—21no—
396C2×C3—161no—
397C2×C3—161no—
398C2×C3#200411yescharacteristic
399C2×C3#200611yescharacteristic
400C2×C3#201011yescharacteristic
401C22—21no—
402C22—21no—
403C22—82no—
404C22—82no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22C3×#39211yescharacteristic
412C22C3×#45311yescharacteristic
413C22C3×#45411yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—42no—
420C4—82no—
421C4—81no—
422C4—81no—
423C4—81no—
424C4C3×#45512yes—
425C3#1000611yescharacteristic, direct product factor, Sylow
426C2—21no—
427C2—21no—
428C2—161no—
429C2—161no—
430C2C3×#200411yescharacteristic
431C2C3×#200611yescharacteristic
432C2C3×#201011yescharacteristic
subgroupG/H
center, socleC22×C3#392
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC3×#3213C2
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
2. C3×#3112C2semidirect
3. C3×#2592C2semidirect
4. C3×#1980C2semidirect
5. C3×#1527C2semidirect
6. C3×#1491C2semidirect
7. C3×#1285C2semidirect
8. #10006C3direct (N × H)
35. #3112C2×C3semidirect
36. #2592C2×C3semidirect
37. #1980C2×C3semidirect
38. #1527C2×C3semidirect
39. #1491C2×C3semidirect
40. #1285C2×C3semidirect
425. C3#10006direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#32132
C3×#31122
C3×#25922
C3×#19802
C3×#15272
C3×#14912
C3×#12852
#100063