Finite Group Databasebeta

← #13138875

Subgroups — #13138875

Order 1728.

528 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C2×#3745⋊C3C211yesmaximal, characteristic, complemented
2#11721144C211yesmaximal, characteristic, complemented
3#11721136C211yesmaximal, characteristic, complemented
4C2×#88432C211yesmaximal, characteristic, complemented
5#11720547C211yesmaximal, characteristic, complemented
6#63648⋊C3C211yesmaximal, characteristic, complemented
7#11720318C211yesmaximal, characteristic, complemented
8#10590765—31nomaximal
9#10590753—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×#6198—21no—
14C2×#6198—21no—
15C2×((C3⋊D3)⋊C3)⋊C4—21no—
16C2×C8×H(3,𝔽3)C2211yescharacteristic
17C4×#1084C2211yescharacteristic
18C2×#6180C2211yescharacteristic
19C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
20#3745⋊C3—22no—
21#88450—21no—
22#88432—22no—
23C2×#3745D311yescharacteristic, complemented
24C2×C3×D24—31no—
25#63911—31no—
26#63911—31no—
27#63894—31no—
28Dic3×D12—31no—
29C2×#3714—31no—
30C2×#3714—31no—
31#63648D311yescharacteristic, complemented
32C3×#821—31no—
33C3×Dic3⋊C8—31no—
34C3×Dic3⋊C8—31no—
35C3×Dic3⋊C8—61no—
36#63564—31no—
37#63563—31no—
38C22×(C3⋊D3)⋊C3—21no—
39C22×(C3⋊D3)⋊C3—21no—
40C22×(C3⋊D3)⋊C3—21no—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084D411yescharacteristic
43C2×#1084D411yescharacteristic
44C2×(C3⋊Dic3)⋊C3—21no—
45C4⋊((C3⋊D3)⋊C3)—21no—
46C4⋊((C3⋊D3)⋊C3)—21no—
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)—21no—
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)D412yescomplemented
51#6198—42no—
52#6198—42no—
53((C3⋊D3)⋊C3)⋊C4—42no—
54C8×H(3,𝔽3)—21no—
55#6180—21no—
56#5150—91no—
57#4843—91no—
58#4786—91no—
59C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
60C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
61C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
62C2×C3×D12—31no—
63C2×C3×D12—31no—
64C2×C3×D12—31no—
65C2×#626—61no—
66C2×#626—61no—
67C2×#626—61no—
68C2×#626—61no—
69C2×#624—61no—
70C2×D3×Dic3—61no—
71C2×C32×C8—21no—
72C2×C32×C8C2×D311yescharacteristic
73C2×C32×C8C2×D311yescharacteristic
74C4⋊(C3⋊Dic3)C2×D311yescharacteristic
75#3745—22no—
76C3×C4⋊Dic3—31no—
77C3×C4×Dic3—31no—
78C3×C4×Dic3—31no—
79C3×C4×Dic3—61no—
80C2×C3×C3⋊C8—31no—
81C2×C3×C3⋊C8—31no—
82C2×C3×C3⋊C8—61no—
83C3×D24—62no—
84#3722—61no—
85(C2×D3)⋊Dic3—61no—
86#3714—62no—
87#3714—62no—
88C22×H(3,𝔽3)C2×D411yescharacteristic
89C2×(C3⋊D3)⋊C3—22no—
90C2×(C3⋊D3)⋊C3—41no—
91C2×(C3⋊D3)⋊C3—41no—
92C2×(C3⋊D3)⋊C3—41no—
93C2×(C3⋊D3)⋊C3—41no—
94C2×(C3⋊D3)⋊C3—41no—
95C2×(C3⋊D3)⋊C3—41no—
96C2×(C3⋊D3)⋊C3—41no—
97C2×(C3⋊D3)⋊C3—41no—
98C4×H(3,𝔽3)C2×D411yescharacteristic
99C4×H(3,𝔽3)#4111yescharacteristic
100#1084—21no—
101#1084D812yescomplemented
102(C3⋊Dic3)⋊C3—41no—
103C2×C3×D8—91no—
104#940—91no—
105#940—91no—
106D4×Dic3—91no—
107C2×#213—91no—
108C2×#213—91no—
109C2×D24—31no—
110C2×D24—91no—
111C2×D24—91no—
112#893—91no—
113#874—91no—
114C4×D12—91no—
115C3×C4⋊C8—91no—
116C3×#102—91no—
117#832—91no—
118#821—31no—
119#821—91no—
120#821—91no—
121Dic3⋊C8—31no—
122Dic3⋊C8—31no—
123Dic3⋊C8—61no—
124#809—91no—
125C22×C3⋊D3—21no—
126C22×C3⋊D3—21no—
127C22×C3⋊D3—21no—
128C22×C3×D3—61no—
129C22×C3×D3—61no—
130C22×C3×D3—61no—
131C2×C32×C4—21no—
132C2×C32×C4C22×D311yescharacteristic
133C2×C32×C4C22×D311yescharacteristic
134C2×C3⋊Dic3—21no—
135C4⋊(C3⋊D3)—21no—
136C4⋊(C3⋊D3)—21no—
137C4⋊(C3⋊D3)—21no—
138C4⋊(C3⋊D3)—21no—
139C4⋊(C3⋊D3)—21no—
140C4⋊(C3⋊D3)#6712yescomplemented
141C2×C3×Dic3—31no—
142C2×C3×Dic3—31no—
143C2×C3×Dic3—31no—
144C2×C3×Dic3—31no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—61no—
147C2×C3×Dic3—61no—
148C3×D12—32no—
149C3×D12—61no—
150C3×D12—61no—
151C3×D12—61no—
152C3×D12—61no—
153C3×D12—61no—
154#626—122no—
155#626—122no—
156#626—122no—
157#626—122no—
158#624—122no—
159D3×Dic3—122no—
160C32×C8—21no—
161C32×C8—21no—
162C32×C8—41no—
163C3×C3⋊C8—61no—
164C3×C3⋊C8—61no—
165C3×C3⋊C8—121no—
166#459—271noSylow
167C2×H(3,𝔽3)#12111yescharacteristic
168C2×H(3,𝔽3)C2×D811yescharacteristic
169C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
170(C3⋊D3)⋊C3—42no—
171(C3⋊D3)⋊C3—81no—
172(C3⋊D3)⋊C3—81no—
173C2×C3×D4—91no—
174C2×C3×D4—91no—
175C2×C3×D4—91no—
176C2×#67—181no—
177C2×#67—181no—
178C2×#67—181no—
179C2×#67—181no—
180C22×Dic3—181no—
181C2×D12—31no—
182C2×D12—31no—
183C2×D12—31no—
184C2×D12—91no—
185C2×D12—91no—
186C2×D12—91no—
187C2×D12—181no—
188C2×D12—181no—
189C2×C4×D3—181no—
190C2×C4×D3—181no—
191C3×D8—182no—
192C2×C3×C8—31no—
193C2×C3×C8—31no—
194C2×C3×C8—61no—
195C2×C3×C8—91no—
196C2×C3×C8D3211yescharacteristic
197C3×C4⋊C4—91no—
198C3×C42—91no—
199#217—181no—
200#213—182no—
201#213—182no—
202(C2×D3)⋊C4—181no—
203(C2×D3)⋊C4—181no—
204C4⋊Dic3—31no—
205C4⋊Dic3—91no—
206C4×Dic3—31no—
207C4×Dic3—31no—
208C4×Dic3—61no—
209C2×C3⋊C8—31no—
210C2×C3⋊C8—31no—
211C2×C3⋊C8—61no—
212D24—62no—
213D24—182no—
214D24—182no—
215C22×C32—21no—
216C22×C32D3×D411yescharacteristic
217C22×C32D3×D411yescharacteristic
218C2×C3⋊D3—22no—
219C2×C3⋊D3—41no—
220C2×C3⋊D3—41no—
221C2×C3⋊D3—41no—
222C2×C3⋊D3—41no—
223C2×C3⋊D3—41no—
224C2×C3⋊D3—41no—
225C2×C3⋊D3—41no—
226C2×C3⋊D3—41no—
227C2×C3×D3—62no—
228C2×C3×D3—121no—
229C2×C3×D3—121no—
230C2×C3×D3—121no—
231C2×C3×D3—121no—
232C2×C3×D3—121no—
233C2×C3×D3—121no—
234C2×C3×D3—121no—
235C2×C3×D3—121no—
236C32×C4—21no—
237C32×C4—21no—
238C32×C4#23511yescharacteristic
239C32×C4D3×D411yescharacteristic
240C32×C4#23711yescharacteristic
241C32×C4C2×#6711yescharacteristic
242C3⋊Dic3—41no—
243C3×Dic3—32no—
244C3×Dic3—32no—
245C3×Dic3—62no—
246C3×Dic3—61no—
247C3×Dic3—61no—
248C3×Dic3—121no—
249C3×Dic3—121no—
250C2×D8—91no—
251C2×D8—271no—
252#127—271no—
253C4×D4—271no—
254C4⋊C8—91no—
255#102—91no—
256#102—271no—
257H(3,𝔽3)#45911yescharacteristic, complemented, Sylow
258C23×C3—181no—
259C23×C3—181no—
260C23×C3—181no—
261C22×D3—61no—
262C22×D3—61no—
263C22×D3—61no—
264C22×D3—181no—
265C22×D3—181no—
266C22×D3—181no—
267C3×D4—92no—
268C3×D4—181no—
269C3×D4—181no—
270C3×D4—181no—
271C3×D4—181no—
272C3×D4—181no—
273C2×C3×C4—31no—
274C2×C3×C4—31no—
275C2×C3×C4—61no—
276C2×C3×C4—91no—
277C2×C3×C4—91no—
278C2×C3×C4—181no—
279C2×C3×C4C2×D3211yescharacteristic
280#67—362no—
281#67—362no—
282#67—362no—
283#67—362no—
284C2×Dic3—31no—
285C2×Dic3—31no—
286C2×Dic3—31no—
287C2×Dic3—31no—
288C2×Dic3—61no—
289C2×Dic3—61no—
290C2×Dic3—61no—
291C2×Dic3—181no—
292C2×Dic3—362no—
293D12—32no—
294D12—61no—
295D12—61no—
296D12—61no—
297D12—61no—
298D12—61no—
299D12—92no—
300D12—181no—
301D12—181no—
302D12—181no—
303D12—181no—
304D12—181no—
305D12—362no—
306D12—362no—
307C4×D3—362no—
308C4×D3—362no—
309C3×C8—21no—
310C3×C8—61no—
311C3×C8—61no—
312C3×C8—121no—
313C3×C8—181no—
314C3⋊C8—61no—
315C3⋊C8—61no—
316C3⋊C8—121no—
317C2×C32—21no—
318C2×C32—21no—
319C2×C32—21no—
320C2×C32#88411yescharacteristic
321C2×C32D3×D811yescharacteristic
322C2×C32D3×D811yescharacteristic
323C2×C32#91111yescharacteristic
324C2×C32#91411yescharacteristic
325C2×C32#93911yescharacteristic
326C3⋊D3—42no—
327C3⋊D3—81no—
328C3⋊D3—81no—
329C3×D3—122no—
330C3×D3—241no—
331C3×D3—241no—
332C2×D4—91no—
333C2×D4—91no—
334C2×D4—91no—
335C2×D4—541no—
336C2×D4—541no—
337C22×C4—541no—
338D8—182no—
339D8—542no—
340C2×C8—91no—
341C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
342C4⋊C4—91no—
343C22⋊C4—541no—
344C42—91no—
345C22×C3—31no—
346C22×C3—31no—
347C22×C3—61no—
348C22×C3—182no—
349C22×C3—361no—
350C22×C3—361no—
351C22×C3—361no—
352C22×C3—361no—
353C22×C3—361no—
354C22×C3—361no—
355C22×C3—361no—
356C22×C3—361no—
357C22×C3#380211yescharacteristic
358C2×D3—62no—
359C2×D3—121no—
360C2×D3—121no—
361C2×D3—121no—
362C2×D3—121no—
363C2×D3—121no—
364C2×D3—121no—
365C2×D3—121no—
366C2×D3—121no—
367C2×D3—182no—
368C2×D3—361no—
369C2×D3—361no—
370C2×D3—361no—
371C2×D3—361no—
372C2×D3—361no—
373C2×D3—361no—
374C2×D3—361no—
375C2×D3—361no—
376C3×C4—31no—
377C3×C4—31no—
378C3×C4—31no—
379C3×C4—31no—
380C3×C4—61no—
381C3×C4—61no—
382C3×C4—92no—
383C3×C4—181no—
384C3×C4—361no—
385C3×C4#379511yescharacteristic
386C3×C4D3×#6711yescharacteristic
387Dic3—32no—
388Dic3—32no—
389Dic3—62no—
390Dic3—61no—
391Dic3—61no—
392Dic3—121no—
393Dic3—121no—
394Dic3—361no—
395C32—21no—
396C32#478611yescharacteristic, complemented
397C32#515011yescharacteristic, complemented
398C23—181no—
399C23—181no—
400C23—181no—
401D4—92no—
402D4—181no—
403D4—181no—
404D4—181no—
405D4—181no—
406D4—181no—
407D4—1082no—
408D4—1082no—
409C2×C4—91no—
410C2×C4—91no—
411C2×C4—181no—
412C2×C4—1082no—
413C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
414C8—21no—
415C8—181no—
416C2×C3—31no—
417C2×C3—31no—
418C2×C3—31no—
419C2×C3—31no—
420C2×C3—31no—
421C2×C3—31no—
422C2×C3—61no—
423C2×C3—61no—
424C2×C3—61no—
425C2×C3—362no—
426C2×C3—721no—
427C2×C3—721no—
428C2×C3#6379711yescharacteristic
429C2×C3#6379811yescharacteristic
430C2×C3#6388511yescharacteristic
431D3—122no—
432D3—241no—
433D3—241no—
434D3—362no—
435D3—721no—
436D3—721no—
437C22—182no—
438C22—361no—
439C22—361no—
440C22—361no—
441C22—361no—
442C22—361no—
443C22—361no—
444C22—361no—
445C22—361no—
446C22#8865511yescharacteristic
447C4—92no—
448C4—181no—
449C4—361no—
450C4#8864811yescharacteristic
451C4#8867611yescharacteristic
452C3—31no—
453C3—31no—
454C3—61no—
455C3#1059074911yescharacteristic
456C2—362no—
457C2—721no—
458C2—721no—
459C2#1172083811yescharacteristic
460C2#1172083911yescharacteristic
461C2#1172112611yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#459
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×#3745⋊C3C2semidirect
2. #11721144C2semidirect
3. #11721136C2semidirect
4. C2×#88432C2semidirect
5. #11720547C2semidirect
6. #63648⋊C3C2semidirect
7. #11720318C2semidirect
23. C2×#3745D3semidirect
31. #63648D3semidirect
50. C4⋊((C3⋊D3)⋊C3)D4semidirect
101. #1084D8semidirect
140. C4⋊(C3⋊D3)#67semidirect
257. H(3,𝔽3)#459semidirect
396. C32#4786semidirect
397. C32#5150semidirect

Maximal subgroups

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

subgroupindex
C2×#3745⋊C32
#117211442
#117211362
C2×#884322
#117205472
#63648⋊C32
#117203182
#105907653
#105907533