Finite Group Databasebeta

← C23×#8473 #19156099

Subgroups — C23×#8473 #19156099

Order 5888.

732 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3503C211yesmaximal, characteristic
2C23×#1997C211yesmaximal, characteristic, complemented
3C23×#1966C211yesmaximal, characteristic, complemented
4C23×#1561C211yesmaximal, characteristic, complemented
5C23×#1526C211yesmaximal, characteristic, complemented
6C23×#1489C211yesmaximal, characteristic
7C23×#1489C211yesmaximal, characteristic
8Q82×C23—21no—
9C23×#557—21no—
10C23×#554—21no—
11C23×#554—22no—
12D4×Q8×C23—22no—
13D4×Q8×C23—21no—
14C23×#544—22no—
15C23×#544C412yes—
16C23×#533—21no—
17C23×#533C2211yescharacteristic
18C2×C23×C4⋊Q8—21no—
19C2×C23×C4⋊Q8C2211yescharacteristic
20C23×#520C412yes—
21C4×#41×C23C2211yescharacteristic
22C23×#423C2211yescharacteristic
23C23×#423C2211yescharacteristic
24C2×C23×#101—22no—
25C4×M16×C23C2211yescharacteristic
26C4×M16×C23C2211yescharacteristic
27C23×#343—22no—
28C23×#332—21no—
29C23×#332—21no—
30C23×#330—22no—
31C23×Q8⋊C8—21no—
32C23×Q8⋊C8—21no—
33C23×D4⋊C8—21no—
34C23×D4⋊C8—21no—
35C2×#41×C23C2×C411yescharacteristic
36C22×Q8×C23—22no—
37C22×Q8×C23D412yes—
38C2×M16×C23—21no—
39C2×M16×C23—21no—
40C2×M16×C23—21no—
41C2×M16×C23—21no—
42C23×C4⋊Q8—22no—
43C23×C4⋊Q8—24no—
44C23×C4⋊Q8—22no—
45C23×C4⋊Q8—22no—
46C23×C4⋊Q8—22no—
47C23×C4⋊Q8—21no—
48C23×C4⋊Q8—21no—
49C23×C4⋊Q8—21no—
50C23×C4⋊Q8—21no—
51C23×C4⋊Q8D411yescharacteristic
52C23×C4⋊Q8D411yescharacteristic
53C23×#125—24no—
54C23×#125—22no—
55C23×#125—22no—
56C23×#125D412yes—
57C23×C22⋊Q8—22no—
58C23×C22⋊Q8—24no—
59C23×C22⋊Q8—24no—
60C23×C22⋊Q8—22no—
61C23×C22⋊Q8—24no—
62C23×C22⋊Q8—21no—
63C23×C22⋊Q8—21no—
64C4×Q8×C23—22no—
65C4×Q8×C23—22no—
66C4×Q8×C23—22no—
67C4×Q8×C23—21no—
68C4×Q8×C23—21no—
69C4×Q8×C23D411yescharacteristic
70C4×Q8×C23D411yescharacteristic
71C4×D4×C23—22no—
72C4×D4×C23—24no—
73C4×D4×C23—21no—
74C4×D4×C23—21no—
75C4×D4×C23D411yescharacteristic
76C4×D4×C23D411yescharacteristic
77C23×#117—22no—
78C23×#117—21no—
79C23×#117C2×C411yescharacteristic
80C23×#117C2×C412yes—
81C2×C4⋊C4×C23—22no—
82C2×C4⋊C4×C23—22no—
83C2×C4⋊C4×C23C2×C412yes—
84C2×C4⋊C4×C23D412yes—
85C2×C42×C23—21no—
86C2×C42×C23C2311yescharacteristic
87C23×C4⋊C8—21no—
88C23×C4⋊C8—21no—
89C23×C4⋊C8—21no—
90C23×C4⋊C8—21no—
91C23×#101—44no—
92C23×#97—21no—
93C23×#97—21no—
94C4×C8×C23—21no—
95C4×C8×C23—21no—
96#41×C23—22no—
97#41×C23—22no—
98C2×Q8×C23—24no—
99C2×Q8×C23—21no—
100C2×Q8×C23—22no—
101C2×Q8×C23—21no—
102C2×Q8×C23—22no—
103C2×Q8×C23—42no—
104C2×Q8×C23—42no—
105C2×Q8×C23—82no—
106C2×Q8×C23—82no—
107C2×Q8×C23C22⋊C411yescharacteristic
108C2×D4×C23—21no—
109C2×D4×C23C22⋊C411yescharacteristic
110C22×C4×C23—21no—
111C22×C4×C23—24no—
112C22×C4×C23—21no—
113C22×C4×C23—21no—
114C22×C4×C23C22⋊C414yes—
115C22×C4×C23C22×C411yescharacteristic
116C22×C4×C23C2×D411yescharacteristic
117C22×C4×C23C2×D411yescharacteristic
118M16×C23—22no—
119M16×C23—22no—
120M16×C23—41no—
121M16×C23—41no—
122M16×C23—42no—
123M16×C23—42no—
124C2×C8×C23—41no—
125C2×C8×C23—41no—
126C2×C8×C23—41no—
127C2×C8×C23—41no—
128C4⋊C4×C23—24no—
129C4⋊C4×C23—24no—
130C4⋊C4×C23—24no—
131C4⋊C4×C23—24no—
132C4⋊C4×C23—22no—
133C4⋊C4×C23—22no—
134C4⋊C4×C23—21no—
135C4⋊C4×C23—22no—
136C4⋊C4×C23—22no—
137C4⋊C4×C23—21no—
138C4⋊C4×C23—21no—
139C4⋊C4×C23—21no—
140C4⋊C4×C23—22no—
141C4⋊C4×C23—22no—
142C4⋊C4×C23C22⋊C412yes—
143C4⋊C4×C23C22⋊C412yes—
144C4⋊C4×C23C22⋊C411yescharacteristic
145C4⋊C4×C23C22⋊C411yescharacteristic
146C22⋊C4×C23—24no—
147C22⋊C4×C23—22no—
148C22⋊C4×C23—21no—
149C22⋊C4×C23—21no—
150C22⋊C4×C23—21no—
151C42×C23—24no—
152C42×C23—21no—
153C42×C23—22no—
154C42×C23—21no—
155C42×C23—21no—
156C42×C23C2×D411yescharacteristic
157C42×C23C2×D411yescharacteristic
158C42×C23C2×D411yescharacteristic
159C42×C23C2×D411yescharacteristic
160#8473C2311yesmaximal, characteristic, direct product factor, Sylow
161C23×C23—21no—
162C23×C23C2×C22⋊C411yescharacteristic
163Q8×C23—21no—
164Q8×C23—42no—
165Q8×C23—41no—
166Q8×C23—41no—
167Q8×C23—42no—
168Q8×C23—41no—
169Q8×C23—41no—
170Q8×C23—82no—
171Q8×C23—84no—
172Q8×C23#10112yescomplemented
173D4×C23—24no—
174D4×C23—21no—
175D4×C23#10112yescomplemented
176C2×C4×C23—24no—
177C2×C4×C23—21no—
178C2×C4×C23—21no—
179C2×C4×C23—21no—
180C2×C4×C23—21no—
181C2×C4×C23—21no—
182C2×C4×C23—21no—
183C2×C4×C23—21no—
184C2×C4×C23—22no—
185C2×C4×C23—21no—
186C2×C4×C23—21no—
187C2×C4×C23—24no—
188C2×C4×C23—22no—
189C2×C4×C23—41no—
190C2×C4×C23—41no—
191C2×C4×C23—41no—
192C2×C4×C23—41no—
193C2×C4×C23—44no—
194C2×C4×C23—84no—
195C2×C4×C23C2×C22⋊C411yescharacteristic
196C2×C4×C23C2×C22⋊C411yescharacteristic
197C2×C4×C23#12011yescharacteristic
198C2×C4×C23#12011yescharacteristic
199C2×C4×C23#12011yescharacteristic
200C2×C4×C23#12011yescharacteristic
201C8×C23—42no—
202C8×C23—42no—
203C8×C23—81no—
204C8×C23—81no—
205#3503C2×C2311yescharacteristic
206#1997C2×C2311yescharacteristic, complemented
207#1966C2×C2311yescharacteristic, complemented
208#1561C2×C2311yescharacteristic, complemented
209#1526C2×C2311yescharacteristic, complemented
210#1489C2×C2311yescharacteristic
211#1489C2×C2311yescharacteristic
212C22×C23—21no—
213C22×C23—21no—
214C22×C23—21no—
215C22×C23—22no—
216C22×C23—41no—
217C22×C23—41no—
218C22×C23#37911yescharacteristic
219C4×C23—21no—
220C4×C23—21no—
221C4×C23—21no—
222C4×C23—22no—
223C4×C23—41no—
224C4×C23—44no—
225C4×C23—42no—
226C4×C23—41no—
227C4×C23—42no—
228C4×C23C2×#10111yescharacteristic
229C4×C23C2×#10111yescharacteristic
230C4×C23#45311yescharacteristic
231C4×C23#45411yescharacteristic
232C4×C23#45511yescharacteristic
233C4×C23#45611yescharacteristic
234Q82—21no—
235#557—21no—
236#554—21no—
237#554—22no—
238D4×Q8—22no—
239D4×Q8—21no—
240#544—22no—
241#544C4×C2312yes—
242#533—21no—
243#533C22×C2311yescharacteristic
244C2×C4⋊Q8—21no—
245C2×C4⋊Q8C22×C2311yescharacteristic
246#520C4×C2312yes—
247C4×#41C22×C2311yescharacteristic
248#423C22×C2311yescharacteristic
249#423C22×C2311yescharacteristic
250C2×#101—22no—
251C4×M16C22×C2311yescharacteristic
252C4×M16C22×C2311yescharacteristic
253#343—22no—
254#332—21no—
255#332—21no—
256#330—22no—
257Q8⋊C8—21no—
258Q8⋊C8—21no—
259D4⋊C8—21no—
260D4⋊C8—21no—
261C2×C23—41no—
262C2×C23—42no—
263C2×C23#178411yescharacteristic
264C2×C23#179211yescharacteristic
265C2×C23#179311yescharacteristic
266C2×#41C2×C4×C2311yescharacteristic
267C22×Q8—22no—
268C22×Q8D4×C2312yes—
269C2×M16—21no—
270C2×M16—21no—
271C2×M16—21no—
272C2×M16—21no—
273C4⋊Q8—22no—
274C4⋊Q8—22no—
275C4⋊Q8—24no—
276C4⋊Q8—22no—
277C4⋊Q8—22no—
278C4⋊Q8—21no—
279C4⋊Q8—21no—
280C4⋊Q8—21no—
281C4⋊Q8—21no—
282C4⋊Q8D4×C2311yescharacteristic
283C4⋊Q8D4×C2311yescharacteristic
284#125—24no—
285#125—22no—
286#125—22no—
287#125D4×C2312yes—
288C22⋊Q8—22no—
289C22⋊Q8—22no—
290C22⋊Q8—24no—
291C22⋊Q8—24no—
292C22⋊Q8—24no—
293C22⋊Q8—21no—
294C22⋊Q8—21no—
295C4×Q8—22no—
296C4×Q8—22no—
297C4×Q8—22no—
298C4×Q8—21no—
299C4×Q8—21no—
300C4×Q8D4×C2311yescharacteristic
301C4×Q8D4×C2311yescharacteristic
302C4×D4—22no—
303C4×D4—24no—
304C4×D4—21no—
305C4×D4—21no—
306C4×D4D4×C2311yescharacteristic
307C4×D4D4×C2311yescharacteristic
308#117—22no—
309#117—21no—
310#117C2×C4×C2311yescharacteristic
311#117C2×C4×C2312yes—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4C2×C4×C2312yes—
315C2×C4⋊C4D4×C2312yes—
316C2×C42—21no—
317C2×C42C23×C2311yescharacteristic
318C4⋊C8—21no—
319C4⋊C8—21no—
320C4⋊C8—21no—
321C4⋊C8—21no—
322#101—44no—
323#97—21no—
324#97—21no—
325C4×C8—21no—
326C4×C8—21no—
327C23#847311yescharacteristic, direct product factor, Sylow
328#41—22no—
329#41—22no—
330C2×Q8—24no—
331C2×Q8—21no—
332C2×Q8—22no—
333C2×Q8—21no—
334C2×Q8—22no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—82no—
338C2×Q8—82no—
339C2×Q8C22⋊C4×C2311yescharacteristic
340C2×D4—21no—
341C2×D4C22⋊C4×C2311yescharacteristic
342C22×C4—21no—
343C22×C4—24no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4C22⋊C4×C2314yes—
347C22×C4C22×C4×C2311yescharacteristic
348C22×C4C2×D4×C2311yescharacteristic
349C22×C4C2×D4×C2311yescharacteristic
350M16—22no—
351M16—22no—
352M16—41no—
353M16—41no—
354M16—42no—
355M16—42no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—21no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4C22⋊C4×C2312yes—
375C4⋊C4C22⋊C4×C2312yes—
376C4⋊C4C22⋊C4×C2311yescharacteristic
377C4⋊C4C22⋊C4×C2311yescharacteristic
378C22⋊C4—24no—
379C22⋊C4—22no—
380C22⋊C4—21no—
381C22⋊C4—21no—
382C22⋊C4—21no—
383C42—24no—
384C42—21no—
385C42—22no—
386C42—21no—
387C42—21no—
388C42C2×D4×C2311yescharacteristic
389C42C2×D4×C2311yescharacteristic
390C42C2×D4×C2311yescharacteristic
391C42C2×D4×C2311yescharacteristic
392C23—21no—
393C23C2×C22⋊C4×C2311yescharacteristic
394Q8—21no—
395Q8—42no—
396Q8—41no—
397Q8—41no—
398Q8—42no—
399Q8—41no—
400Q8—41no—
401Q8—82no—
402Q8—84no—
403Q8C23×#10112yescomplemented
404D4—24no—
405D4—21no—
406D4C23×#10112yescomplemented
407C2×C4—24no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—44no—
425C2×C4—84no—
426C2×C4C2×C22⋊C4×C2311yescharacteristic
427C2×C4C2×C22⋊C4×C2311yescharacteristic
428C2×C4C23×#12011yescharacteristic
429C2×C4C23×#12011yescharacteristic
430C2×C4C23×#12011yescharacteristic
431C2×C4C23×#12011yescharacteristic
432C8—42no—
433C8—42no—
434C8—81no—
435C8—81no—
436C22—21no—
437C22—22no—
438C22—21no—
439C22—21no—
440C22—41no—
441C22—41no—
442C22C23×#37911yescharacteristic
443C4—21no—
444C4—22no—
445C4—21no—
446C4—21no—
447C4—41no—
448C4—44no—
449C4—42no—
450C4—41no—
451C4—42no—
452C4C2×C23×#10111yescharacteristic
453C4C2×C23×#10111yescharacteristic
454C4C23×#45311yescharacteristic
455C4C23×#45411yescharacteristic
456C4C23×#45511yescharacteristic
457C4C23×#45611yescharacteristic
458C2—41no—
459C2—42no—
460C2C23×#178411yescharacteristic
461C2C23×#179211yescharacteristic
462C2C23×#179311yescharacteristic
subgroupG/H
center, socleC22×C23#379
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC2×#41×C23C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1997C2semidirect
3. C23×#1966C2semidirect
4. C23×#1561C2semidirect
5. C23×#1526C2semidirect
160. #8473C23direct (N × H)
172. Q8×C23#101semidirect
175. D4×C23#101semidirect
206. #1997C2×C23semidirect
207. #1966C2×C23semidirect
208. #1561C2×C23semidirect
209. #1526C2×C23semidirect
327. C23#8473direct (N × H)
403. Q8C23×#101semidirect
406. D4C23×#101semidirect

Maximal subgroups

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

subgroupindex
C23×#35032
C23×#19972
C23×#19662
C23×#15612
C23×#15262
C23×#14892
#847323