Finite Group Databasebeta

← #10698600

Subgroups — #10698600

Order 768.

520 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83240C211yesmaximal, characteristic
2C2×C3×#332C211yesmaximal, characteristic, complemented
3#71787C211yesmaximal, characteristic, complemented
4#70479C211yesmaximal, characteristic
5#70453C211yesmaximal, characteristic, complemented
6C2×#4483C211yesmaximal, characteristic
7#67704C211yesmaximal, characteristic, complemented
8#10416—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#947—21no—
11C2×C4×D12C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14#5229—21no—
15C2×#821—21no—
16C2×#818—21no—
17#4992—22no—
18#4987—21no—
19#4987—21no—
20#4935C2211yescharacteristic
21C4×C4⋊Dic3C2211yescharacteristic
22C2×C4×C3⋊C8C2211yescharacteristic
23C3×#332—22no—
24#4551—21no—
25#4551—21no—
26#4529—22no—
27#4483—22no—
28#2547—31no—
29#1990—31no—
30#1947—31no—
31#1802—31no—
32C2×#332D311yescharacteristic, complemented
33C2×D4⋊C8—31no—
34#1279—31no—
35C22×C3×Q8—21no—
36C22×D12D411yescharacteristic
37C22×C4×D3—21no—
38C22×C3×C8—21no—
39C22×C3×C8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4D412yes—
46C2×C3×C42C2311yescharacteristic
47#947—42no—
48#947—42no—
49C2×(C2×D3)⋊C4—22no—
50C2×(C2×D3)⋊C4—22no—
51C2×(C2×D3)⋊C4—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3D411yescharacteristic
56C2×C4⋊Dic3D411yescharacteristic
57C2×C4⋊Dic3D411yescharacteristic
58C2×C4×Dic3—21no—
59C22×C3⋊C8—21no—
60C4×D12—21no—
61C4×D12—21no—
62C4×D12—21no—
63C4×D12—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8D412yes—
68C3⋊#95—22no—
69C3⋊#95—22no—
70C3⋊#95—21no—
71#821—42no—
72#818—42no—
73C4×C3⋊C8—21no—
74C4×C3⋊C8—21no—
75C2×C4⋊Q8C2×D311yescharacteristic
76C2×#124—61no—
77C2×C4×D4—31no—
78C2×#107—61no—
79C2×C4⋊C8C2×D311yescharacteristic
80C2×C4⋊C8C2×D311yescharacteristic
81C2×#102—61no—
82C2×C4×C8—31no—
83#397—62no—
84#391—61no—
85#384—31no—
86#380—61no—
87#380—61no—
88C4×C4⋊C4—31no—
89#340—61no—
90#340—61no—
91#340—62no—
92#332—22no—
93D4⋊C8—62no—
94C23×D3—21no—
95C2×C3×Q8—41no—
96C2×C3×Q8—41no—
97C2×C3×Q8—41no—
98C2×C3×Q8—41no—
99C2×C3×Q8—41no—
100C2×C3×Q8—41no—
101C22×C3×C4—22no—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—41no—
109C2×D12—21no—
110C2×D12—21no—
111C2×D12—21no—
112C2×D12—21no—
113C2×D12—21no—
114C2×D12D812yes—
115C2×C4×D3—41no—
116C2×C4×D3—41no—
117C2×C4×D3—41no—
118C2×C4×D3—41no—
119C2×C4×D3—41no—
120C2×C4×D3—41no—
121C2×C3×C8—22no—
122C2×C3×C8—41no—
123C2×C3×C8—41no—
124C2×C3×C8—41no—
125C2×C3×C8—41no—
126C2×C3×C8—41no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C42C2×D411yescharacteristic
132C3×C42#4111yescharacteristic
133C3×C42#4111yescharacteristic
134C3×C42#4111yescharacteristic
135(C2×D3)⋊C4—44no—
136(C2×D3)⋊C4—44no—
137(C2×D3)⋊C4—41no—
138(C2×D3)⋊C4—41no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—42no—
146C4⋊Dic3—41no—
147C4⋊Dic3—41no—
148C4⋊Dic3—41no—
149C4⋊Dic3—41no—
150C4×Dic3—41no—
151C4×Dic3—41no—
152C2×C3⋊C8—22no—
153C2×C3⋊C8—41no—
154C2×C3⋊C8—41no—
155C22×Q8—21no—
156C22×D4—31no—
157C23×C4—61no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—61no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165#124—122no—
166#124—122no—
167C4×D4—61no—
168C4×D4—61no—
169C4×D4—61no—
170C4×D4—61no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—31no—
173C2×C4⋊C4—31no—
174C2×C4⋊C4—31no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4D1212yes—
179C2×C22⋊C4—62no—
180C2×C22⋊C4—62no—
181C2×C22⋊C4—61no—
182C2×C42—61no—
183C2×C42C22×D311yescharacteristic
184#107—122no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8#6712yes—
189#102—122no—
190C4×C8—61no—
191C4×C8—61no—
192#95—62no—
193#95—62no—
194#95—61no—
195C23×C3#12011yescharacteristic
196C22×D3—22no—
197C22×D3—41no—
198C22×D3—41no—
199C22×D3—41no—
200C22×D3—41no—
201C22×D3—41no—
202C22×D3—41no—
203C3×Q8—81no—
204C3×Q8—81no—
205C3×Q8—81no—
206C3×Q8—81no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12311yescharacteristic
216C2×C3×C4#12311yescharacteristic
217C2×C3×C4#12411yescharacteristic
218C2×C3×C4C2×D811yescharacteristic
219C2×C3×C4#13511yescharacteristic
220C2×C3×C4#13511yescharacteristic
221C2×C3×C4#13511yescharacteristic
222C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
223C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
224C2×C3×C4#13711yescharacteristic
225C2×C3×C4#13711yescharacteristic
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—81no—
236C2×Dic3—81no—
237C2×Dic3—81no—
238D12—22no—
239D12—22no—
240D12—41no—
241D12—41no—
242C4×D3—81no—
243C4×D3—81no—
244C4×D3—81no—
245C4×D3—81no—
246C3×C8—42no—
247C3×C8—81no—
248C3⋊C8—42no—
249C24—61no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×D4—32no—
257C2×D4—61no—
258C2×D4—61no—
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—61no—
262C22×C4—22no—
263C22×C4—61no—
264C22×C4—61no—
265C22×C4—61no—
266C22×C4—121no—
267C22×C4—121no—
268C22×C4—121no—
269C22×C4—121no—
270C22×C4—121no—
271C22×C4—121no—
272C22×C4—121no—
273C22×C4C2×D1211yescharacteristic
274C22×C4D3×D411yescharacteristic
275C22×C4D3×D411yescharacteristic
276C2×C8—22no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—62no—
283C2×C8—121no—
284C2×C8—121no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—122no—
296C4⋊C4—121no—
297C4⋊C4—121no—
298C4⋊C4—121no—
299C4⋊C4—121no—
300C22⋊C4—124no—
301C22⋊C4—124no—
302C22⋊C4—121no—
303C22⋊C4—121no—
304C42—121no—
305C42—121no—
306C42#23511yescharacteristic
307C42#23711yescharacteristic
308C42#23711yescharacteristic
309C42C2×#6711yescharacteristic
310C22×C3#39411yescharacteristic
311C22×C3#44711yescharacteristic
312C22×C3#44911yescharacteristic
313C22×C3#45211yescharacteristic
314C22×C3#45211yescharacteristic
315C22×C3#45411yescharacteristic
316C22×C3#45611yescharacteristic
317C2×D3—42no—
318C2×D3—42no—
319C2×D3—42no—
320C2×D3—81no—
321C2×D3—81no—
322C2×D3—81no—
323C2×D3—81no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—82no—
327C3×C4#46311yescharacteristic
328C3×C4#46411yescharacteristic
329C3×C4#46611yescharacteristic
330C3×C4#46811yescharacteristic
331C3×C4#48011yescharacteristic
332C3×C4#48211yescharacteristic
333C3×C4#48811yescharacteristic
334C3×C4#48911yescharacteristic
335Dic3—81no—
336Dic3—81no—
337Dic3—81no—
338Dic3—161no—
339C23—62no—
340C23—121no—
341C23—121no—
342C23—121no—
343C23—121no—
344C23—121no—
345C23—121no—
346C23#88211yescharacteristic
347Q8—81no—
348Q8—81no—
349Q8—81no—
350Q8—81no—
351D4—62no—
352D4—62no—
353D4—121no—
354D4—121no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4—241no—
370C2×C4—241no—
371C2×C4—241no—
372C2×C4—241no—
373C2×C4—241no—
374C2×C4—241no—
375C2×C4—241no—
376C2×C4#88311yescharacteristic
377C2×C4#88511yescharacteristic
378C2×C4#88611yescharacteristic
379C2×C4D3×D811yescharacteristic
380C2×C4#91211yescharacteristic
381C2×C4#91411yescharacteristic
382C2×C4#91411yescharacteristic
383C2×C4#91511yescharacteristic
384C2×C4#91611yescharacteristic
385C2×C4#91811yescharacteristic
386C2×C4#91911yescharacteristic
387C2×C4#93011yescharacteristic
388C2×C4#93811yescharacteristic
389C2×C4#93911yescharacteristic
390C8—42no—
391C8—81no—
392C8—122no—
393C2×C3#162511yescharacteristic
394C2×C3#163811yescharacteristic
395C2×C3#164711yescharacteristic
396C2×C3#165211yescharacteristic
397C2×C3#203311yescharacteristic
398C2×C3#203811yescharacteristic
399C2×C3#204211yescharacteristic
400D3—82no—
401C22—122no—
402C22—122no—
403C22—122no—
404C22—241no—
405C22—241no—
406C22—241no—
407C22—241no—
408C22#475211yescharacteristic
409C22#475311yescharacteristic
410C22#477311yescharacteristic
411C22#478311yescharacteristic
412C22#480811yescharacteristic
413C22#481011yescharacteristic
414C22#496111yescharacteristic
415C4—21no—
416C4—21no—
417C4—82no—
418C4—241no—
419C4—241no—
420C4—241no—
421C4—481no—
422C4#477411yescharacteristic
423C4#478411yescharacteristic
424C4#479411yescharacteristic
425C4#480011yescharacteristic
426C4#515711yescharacteristic
427C4#516511yescharacteristic
428C4#517411yescharacteristic
429C4#518511yescharacteristic
430C3#1041611yescharacteristic, complemented, Sylow
431C2—242no—
432C2#6843211yescharacteristic
433C2#6843511yescharacteristic
434C2#6843611yescharacteristic
435C2#6846211yescharacteristic
436C2#7010011yescharacteristic
437C2#7017511yescharacteristic
438C2#7024511yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#332C2
socleC23×C3#120
prime radicalC22×D12D4
hypercenterC2×#332D3
hypocenterC3#10416
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
2. C2×C3×#332C2semidirect
3. #71787C2semidirect
5. #70453C2semidirect
7. #67704C2semidirect
32. C2×#332D3semidirect
430. C3#10416semidirect

Maximal subgroups

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

subgroupindex
#832402
C2×C3×#3322
#717872
#704792
#704532
C2×#44832
#677042
#104163